Abstract

I evaluate a technical trading strategy that combines the Relative Strength Index (RSI) with a moving‑average (MA) trend filter. Using daily data for the 30 Dow Jones Industrial Average (DJIA) constituents over July 18, 2015–July 18, 2025, I test RSI deviation thresholds of 10, 15, and 20 points from neutral and apply a quarterly walk‑forward selection of RSI and SMA lookbacks. Returns are computed with next‑day execution and later adjusted for transaction and borrow costs. Across all specifications, Sharpe ratios are not statistically different from zero at the 5% level, and realistic costs eliminate small gross gains. Return distributions display high excess kurtosis and low skewness, consistent with exposure to higher‑order risks. I interpret the RSI and SMA combination less as a price‑forecasting tool and more as a risk filter that concentrates tail exposure. The strategy underperforms unconditionally, but the profile suggests it could perform better in specific market regimes in which such risks are compensated. I discuss implications for market efficiency and for using technical rules to target or avoid priced co‑moment exposures.

1. Introduction

Can a double-signal technical trading strategy generate consistent, risk-adjusted returns over the past decade?

To answer this, I run a 10-year backtest on three RSI deviation thresholds (10, 15, and 20 points from neutral), each paired with MA-based trend confirmation. Performance is measured gross of transaction costs and then adjusted using basis-point cost scenarios. I compute risk-adjusted metrics, including Sharpe ratios with Newey-West standard errors, Calmar ratios, and Sortino ratios, for each variation.

Multiple trading influencers promote variations of the RSI and MA strategy as a ‘magic bullet’. These influencers and channels have large followings and often promote their strategy in tandem with a course or funded trading partnership. Some examples are The Moving Average (YouTube, 1.07M+ subscribers), Trading Lab (YouTube, 1.74M+ subscribers), LuxAlgo (TikTok, Instagram, YouTube, 1.2M+ total followers), and many others. They create videos with bright, attention-grabbing colors and enticing thumbnail covers that draw inexperienced audiences in, promising to teach the “right way” to use these indicators and generate unrealistic returns. While these videos focus on anecdotal success and visual appeal, this study tests the strategy in a systematic, data-driven way.

This paper finds that none of the tested variations deliver statistically significant Sharpe ratios, with 95% confidence intervals spanning zero across all strategies. Some configurations yield small positive gross returns, but realistic transaction costs erase these gains. Our results suggest that this RSI–SMA strategy does not provide persistent predictive power and that any observed profitability likely stems from random variation rather than structural inefficiency.

Many investors employ technical trading strategies, such as the RSI and MA, despite decades of academic skepticism. If markets are efficient and past prices contain no useful information, such strategies should not be effective; yet, they persist in practice. We contribute to the literature by testing a simple yet popular combination of RSI and MA to evaluate risk-adjusted performance and exposure to certain risk profiles.

We focus not only on whether this strategy performs well, but also on whether any consistent performance challenges the Efficient Market Hypothesis (EMH), which states that past price data should not offer predictive advantages. McInish & Puglisi (1980) find that markets are efficient through runs tests of utility-preferred stocks. Brock, Lakonishok, and LeBaron (1992), however, find profitability in technical trading rules through nearly a century of DJIA data and contradict the EMH. This conflict shows how empirical results vary based on methodology, asset class, and time horizon, creating a rift between academic theory and real-world trading behavior. Understanding whether this tension reflects outdated testing methods, behavioral inefficiencies, or changing market dynamics is important for both researchers and investors.

The Relative Strength Index identifies overbought and oversold conditions, while the moving average smooths price action to show trend direction. Online trading communities often promote these indicators as part of a ‘signal confluence’ approach before entering trades. As retail trading grows with the rise of zero-commission trading and options contracts, underexperienced traders often rely on signals as primary strategies. These indicators, widely circulated in trading literature and online forums, influence trading decisions despite limited evidence of predictive power, making them a useful case study for examining whether such strategies expose traders to specific market conditions.

In addition to evaluating strategy performance, we reframe technical analysis as a potential filter for hidden risk exposures. This approach considers whether consistent performance could arise from taking on unattractive risks that most investors avoid, and therefore the most rewarding. We account for tail risk and other exposures not well captured in metrics like beta and volatility. In that case, the strategy would not forecast price action but would filter for market conditions where the trade-off between risk and return is most favorable.

The next section reviews the literature on technical analysis, while section three discusses the theoretical context behind risk exposure in asset pricing. Section four describes our methodology, and section five presents the results. Section six discusses the implications, section seven concludes the study, followed by references.

2. Literature Review

2.1 Early Academic Skepticism

Historically, there has been skepticism towards technical analysis in academia. Technical analysis was originally conceptualized around observed patterns rather than formal theory. Some studies, such as Tabell and Tabell’s (1964), emphasize Dow Theory and the Elliott Wave Principle, highlighting cycles and human behavior, themes still relevant today. Levy (1966) offers a more neutral perspective, critiquing both technical and fundamental approaches to trading. While he finds more empirical support for fundamental analysis, his openness suggests that even early researchers saw potential for technical signals to reflect meaningful information. Other studies go further by testing technical analysis and find no edge, concluding that price behaves randomly. Van Horne and Parker (1968) test a moving average and do not find excess returns, and are often cited as one of the earliest academic rejections of technical analysis. Their paper, however, is methodologically limited by a small sample size and lacks out-of-sample testing, limitations that this paper will address with modern tools and risk framing.

2.2 Conditional Effectiveness and Market Context

More modern studies suggest that technical analysis may work in certain market conditions. Han, Yang, and Zhou (2013) and Brown, Crocker, and Foerster (2009) both find that using moving average and buy-and-hold strategies, assets with higher risk components like volatility and turnover earn higher returns. This supports the idea that investor behavior and market dynamics may create price movements that technical analysis is better suited to capture than fundamental analysis. This may imply success in specific market conditions, particularly conditions that are traditionally avoided.

Building on this conditionality, other studies explore when and why technical analysis might work. Bessembinder and Chan (1998) similarly show that technical trading rules may have statistical predictive power, but lack economic significance due to trading costs and nonsynchronous trading effects. This counters the notion of technical analysis as a magic bullet for market context. Likewise, Qi and Wu (2006) test thousands of trading rules and find conditional effectiveness in the earlier half of their sample period, indicating that technical analysis may exploit inefficient markets. Together, these studies suggest that technical analysis can be used as a lens for inefficiency, a concept that will be explored further in this paper.

2.3 Behavioral and Structural Explanations

Many papers support the idea of technical analysis as a filter for behavior and structure. Smith et al. (2016) and Moosa and Li (2011) argue that technical levels correspond with real behavioral structures in different markets. The former shows that hedge funds using technical analysis have higher returns during periods of high sentiment, highlighting the connection to irrational investor behavior. The latter investigates the disproportionate success of technical strategies in Chinese markets, examining the distortion of markets by government intervention and high retail participation. These behavioral patterns may create a unique environment suitable for technical analysis where rational models fail. Other studies, such as Blume et al. (1994) and Kavajecz and Odders-White (2004), find correlations between specific signals and market information. Blume et al. argue that volume reflects trader confidence, while Kavajecz and Odders-White find an alignment between technical indicators and changes in liquidity. These findings imply that, in addition to behavior, structural market components may help inform the conditions where technical analysis could be successful.

2.4 Agreements and Differences

Across the literature, two general camps emerge. Early research, represented by Van Horne and Parker (1968), contends that technical analysis lacks predictive ability, even after accounting for randomness and naïve benchmarks. More recent research, however, finds pockets of conditional efficacy, whether due to market regimes (Han, Yang, & Zhou, 2013; Qi & Wu, 2006) or behavioral/structural patterns (Smith et al., 2016; Moosa & Li, 2011). Where the camps converge is in emphasizing the role of context: even the most ardent supporters recognize that strategy efficacy relies on particular circumstances, whereas skeptics admit anomalies may appear in particular settings. The disagreement is in whether those circumstances are exploitable after costs, and whether findings represent persistent inefficiencies or fleeting artifacts of market design.

This divide directly shapes our methodological choices. First, to address concerns over sample bias and parameter overfitting, we evaluate a multi-signal strategy rather than isolated rules, mirroring how traders filter entries. Second, we incorporate transaction cost adjustments to test whether any gross profitability survives realistic frictions, a point stressed by cost-sensitive studies like Bessembinder and Chan (1998). Finally, inspired by the state-dependent findings in both behavioral and structural work, we adopt a rolling quarterly parameter optimization to account for changing market conditions. This design allows us to test not only whether the strategy performs overall, but also whether it behaves differently across varying environments, bridging the empirical gap between one-size-fits-all tests and purely context-specific studies.

3. Theoretical Context

3.1 Strategy Risk Profile and Testable Claim

We test whether a dual-signal RSI+MA strategy inherently loads on tail risk, marked by high kurtosis and near-zero or negative skewness, because it systematically buys into short-term weakness and sells into short-term strength relative to prevailing trends. In its mean-reversion phases, the strategy takes long positions when RSI signals oversold conditions in uptrends and short positions when RSI signals overbought conditions in downtrends. This pattern can produce many small gains interrupted by rare but significant losses (a “knife-catching” profile), yielding elevated kurtosis and potentially negative skewness. The SMA trend filter mitigates some of this tail risk but does not fully eliminate it. If these return characteristics align with risks that most investors avoid, the strategy may command a risk premium in expected returns.

3.2 Why Higher-Order Risks Matter

In finance, investors are not only concerned with the magnitude of returns, but also with the distribution of gains and losses. Traditional models like the Capital Asset Pricing Model (CAPM) focus on volatility and market sensitivity as the primary sources of risk. However, this framework often overlooks more complex behaviors in return distributions that describe the shape and extremity of gains and losses. For example, two assets might have the same average return and volatility, but one may suddenly crash while the other steadily gains, with investors generally preferring the latter. By intentionally taking on the types of risk that most investors avoid, a strategy may earn higher expected returns as compensation for bearing that discomfort. This paper explores the idea that technical analysis, though often dismissed as unscientific, can function as a filter for the risk exposures that investors implicitly price. This could allow strategies to profit by taking on return profiles that are mispriced or systematically avoided. Central to identifying these patterns are two higher-order risk moments, skewness and kurtosis.

3.3 Definitions and Investor Preferences

A growing body of research suggests that co-skewness and co-kurtosis are crucial in asset pricing. Skewness captures asymmetry in the return distributions of a portfolio. Positive skewness represents large gains with frequent small losses, while negative skewness represents frequent small gains with the potential for large losses. Kurtosis measures the tailedness of the 0 return distribution, where high kurtosis represents a higher probability of extreme values, while low kurtosis represents a lower probability of extreme values.

Co-skewness and co-kurtosis, then, are the relationships between these metrics when comparing one variable to another, generally an asset compared to the market portfolio. Co-skewness describes how the asymmetry of one variable’s distribution is connected to movements in another variable. Co-kurtosis describes how the tendency for extreme values in one variable is connected to movements in another variable. A portfolio with negative co-skewness and high co-kurtosis, for example, tends to suffer its worst losses at the same time as the market, and experiences extreme returns in the same period, amplifying drawdown. However, CAPM suggests that assets performing poorly when investors most value consumption must offer higher expected returns as compensation. Because bad co-skewness and co-kurtosis are so unattractive to investors, the risk may be compensated for.

3.4 Pricing of Higher-Order Risks

Some studies explain this correlation between investor behavior and higher-order risk moments. Conrad, Dittmar, and Ghysels (2013) argue that skewness isn’t only measurable in hindsight, and that investors price ex ante skewness through the options market. They find that stocks with more expected negative skewness earn higher average returns, consistent with the theory that investments with more inherent risk require more compensation. The authors also find that stocks with positive expected skewness earn lower average returns, but the potential for large positive price changes remains attractive to investors. Frugier (2014) adds to this conclusion, arguing that these higher-order moments are embedded signals of trader psychology. He examines past stock data and finds that negative skewness and high kurtosis arise in times of 1 unresolved uncertainty and fear, indicating these psychological factors. These studies show that skewness and kurtosis must be priced in because investors prefer certain shapes of risk, like the frequency and size of gains and losses. These findings suggest that technical analysis could succeed if it accurately filters undesirable risk exposures that demand higher returns.

3.5 Institutional and Practical Evidence

Similar filters seem to be used in institutional trading. A study by Malkiel and Saha (2005) investigates risk exposure in hedge funds. They find that an overwhelming majority of hedge funds in the TASS database exhibit returns with low skewness and high kurtosis, generally avoided by traditional investors. This is supported by Elkamhi and Stefanova (2015), who show that in volatility-based portfolios, even small exposures to skewness and kurtosis can lead to outsized losses. Because of this, portfolios would benefit from explicit identification of these moments to hedge against them. If technical analysis can isolate exposure to higher-order risks, it should be viewed as a way to expand the investor’s span, to broaden the set of priced risks a portfolio can intentionally target or avoid. Technical strategies can be evaluated not just on signal accuracy, but on whether their return profiles reflect exposure to hidden risks. This perspective will be tested in the analysis that follows.

4. Strategy and Methodology

This strategy will test multiple combinations of the Relative Strength Index (RSI) and moving average (MA). The RSI(14) and MA(200) are the most standard values, and these period lengths are commonly referenced in social media posts. Other lengths modify the quantity, and 2 potentially ‘quality’ of each signal as trends become smoother with greater lengths. However, market conditions may affect this ‘quality’, so we perform a quarterly walk-forward selection of RSI and MA combinations to optimize for shifting regimes.

4.1 Universe & Data

We retrieved daily price data for all 30 constituents of the Dow Jones Industrial Average (DJIA) over ten years (July 18, 2015, to July 18, 2025) from the Yahoo Finance Historical Data API (see Appendix for the full list of tickers). Results are non-point-in-time, backfilled from the most recent members. We used close prices adjusted for dividends and splits.

4.2 Signal Construction

The RSI is computed on adjusted closes using Wilder’s smoothing as:

Where AU_t and AD_t are the exponentially smoothed average gains and losses over n days. The SMA is calculated as:

4.3 Position Sizing and Execution Convention

Signals are generated and executed at the close on day t. Each stock may trigger at most one entry per day, and each position is allocated 1/10 of the total portfolio value to avoid concentration risk. Position weight per stock is fixed-weight per trade. Buy and sell signals are processed independently for each stock.

4.4 Walk-Forward Selection

To adapt the strategy to evolving market conditions, we partitioned the 10-year sample into 40 quarterly segments. For each RSI deviation threshold (10, 15, and 20), an initial quarter runs with default parameters of RSI(50) and SMA(200). At the end of each quarter, we evaluate RSI and SMA combinations based on their Sharpe ratios. We apply the combination with the highest Sharpe ratio to the next quarter. This process repeats in a walk-forward fashion across all 40 quarters, producing a series of returns in which each quarter’s parameters reflect the most profitable configuration from the preceding quarter.

4.5 Shorting

This backtest assumes that short sales are permitted for all DJIA constituents at all times, with full borrow availability. A borrowing cost is applied to all open short positions, representing the annualized interest rate charged for borrowing shares. This cost, denoted c_b , is assumed to fall within the range of 30-50 basis points per year (0.30%-0.50%), with the base case using = c_b 0.50%.

This daily borrow charge is calculated as:

Where ShortNotional_t is the dollar value of the short position on day t . Borrow costs t are deducted from portfolio equity each day the short position remains open. Dividend payments owed on short positions are already incorporated in the adjusted price series used for return calculations, so no separate adjustment is required.

4.6 Transaction Costs

Transaction costs were incorporated by first calculating daily portfolio turnover as the average absolute position change across the 30 DJIA stocks.

We multiplied this turnover by a per-dollar transaction cost rate c to estimate daily cost drag.

We then calculated the Compound Annual Growth Rate (CAGR) from the net return series, with results reported for costs of 0, 2, 5, and 10 basis points per side.

All other performance metrics are based on gross returns, with the impact of transaction costs summarized separately in the CAGR table.

4.7 Risk and Return Metrics

We evaluate portfolio performance using a set of risk-adjusted and distributional metrics. The Sharpe ratio is computed using Newey-West adjusted standard errors to account for serial correlation in daily returns.

r o represents the daily average return of our portfolio, while NW represents the standard deviation of returns that has been adjusted for autocorrelation (the similarity between a variable’s values at different points in time) and heteroskedasticity (the spread of standard deviations of a variable).

We use a Calmar ratio to measure the trade-off between annualized return and maximum drawdown.

Where CAGR is the Compound Annual Growth Rate of our portfolio over the testing period, and the denominator is the absolute value of the maximum drawdown of the portfolio.

The Sortino ratio isolates downside volatility as opposed to both sides.

Where T^— is the number of days with negative returns, and r_t the return on day t. Wholly, the denominator represents downside deviation, or the standard deviation of negative returns.

Skewness and kurtosis are computed to assess asymmetry and tail heaviness in the return distribution.

Where T o represents the total number of days in the sample and represents the standard deviation of all returns. Skewness represents asymmetry of the return distribution (whether it is more tilted to gains or losses). Kurtosis represents the “tailedness” of the distribution (the likelihood of extreme gains or losses). All ratios are computed using gross daily returns unless otherwise noted, with transaction-cost-adjusted returns reported separately.

5. Results

This section presents the performance of an RSI and MA strategy under three different threshold configurations: DEV 10, DEV 15, and DEV 20. These values represent the entry condition deviation in RSI. Results are based on a portfolio spanning July 2015 to July 2025.

5.1 Cumulative Returns

The cumulative return chart shows diverging performance across the three RSI deviation tests, all ending with negative returns. Differences in performance begin to emerge in mid-2017, early in the testing period. DEV 10 (blue) showed brief periods of positive returns and was the most volatile, ending with the highest return of the three. DEV 15 and 20, although considerably more stable in return trajectory, finished with even lower returns. Returns tended to move in the same direction for most of the period, suggesting positive covariance between strategies.

5.2 Annualized Metrics

Annual returns were negative across all three configurations. Annual excess returns versus SHY and SPY were also negative, the lowest being DEV 20 (-2.129%) and the highest being DEV 10 (-1.514%). V olatility followed the same pattern, from 2.162% (DEV 20) to 3.154% (DEV 10). The Sharpe ratio also decreased with higher deviation thresholds, ranging from -0.480 (DEV 10) to -0.985 (DEV 20). However, 95% Newey-West adjusted confidence intervals for the Sharpe ratios include 0 in all three strategies, implying that the Sharpe ratios are statistically insignificant. Small, negative Sortino and Calmar ratios suggest poor performance compared to risk-free assets and drawdown, indicating greater downside risk.

5.3 Transaction Costs

To address the effects of transaction costs, we recalculated CAGR for each RSI Dev strategy after applying per-dollar transaction costs of 0, 2, 5, and 10 basis points per side. Across all three strategies, higher transaction costs reduced CAGR approximately linearly with cost magnitude. For further context, we calculated total transaction costs as a percentage of portfolio value. Dev 10 exhibited the highest turnover (5.25% at 10 bps), confirming that smaller deviation thresholds are more sensitive to cost assumptions.

The above graphs show trade logs for each strategy, blue representing long trades and orange representing short trades. This log contextualizes higher transaction costs for the lower threshold strategy, Dev 10, which had the highest trade volume.

5.4 Return Distributions

Returns for all three strategies were density-adjusted and compared to normal distributions. Return distribution plots for each test show a sharper central peak and heavier tails than Gaussian (high excess kurtosis). All three strategies display high kurtosis with values ranging from 18.006 (DEV 15) to 21.788 (DEV 20). Although most returns are clustered around the mean, this shows that these strategies experienced a significantly higher frequency of extreme return days than a normal distribution.

Skewness values ranged from low (DEV 10 and DEV 15) to low-moderate (DEV 20). DEV 10 and DEV 20 had positive skew, implying a higher probability of large positive gains, while DEV 15 had a negative skew and a tendency towards large negative gains.

6. Discussion

This section interprets the performance of the RSI and MA strategy beyond raw returns. While the strategy underperformed, its results reveal insights into market efficiency, hidden risk exposures, and the conditional role of technical analysis.

6.1 Underperformance and Market Efficiency

The RSI and MA strategy tested in this paper underperformed across all annualized metrics in the ten-year testing period, but its implications suggest a broader role of technical analysis in risk exposure. All three strategy variants show negative risk-adjusted values when compared to SHY and display negative Sharpe, Calmar, and Sortino ratios. These outcomes are consistent with the weak form of the Efficient Market Hypothesis, which holds that past price data contains no predictive or actionable information. However, this underperformance does not imply that the signals were entirely random or meaningless. The return paths of DEV 10, 15, and 20 were similar for most of the ten years. Positive covariance in the three strategy variations suggests exposure to similar structural patterns in the market. The strategy may not unconditionally generate alpha, but it might expose investors to certain risks hidden in asset prices.

6.2 Hidden Risk Exposure and Conditional Opportunity

The most compelling pattern in the results is the consistently high kurtosis. This indicates frequent extreme returns, which traditional investors tend to avoid due to financial and psychological discomfort. Relatively low skewness values indicate minor asymmetry in returns, but the heavy-tailed return distributions suggest that the RSI and MA strategy filters for hidden, 2 non-diversifiable risk. This aligns with research by Conrad et al. (2013) and Frugier (2014), who argue that investors price these risks because they reflect the large, abrupt losses during ‘bad times’.

Consistency across all three thresholds suggests that the strategy systematically targets certain classes of risk that are not rewarded over long periods, but may be under unique conditions. This helps explain the underperformance, as strategies with undesirable risk (high kurtosis and low skewness) should not be expected to outperform in all periods, especially in a mostly efficient market. However, it might work over certain regimes where these risks are mispriced, such as volatility clusters and trend reversals.

6.3 Reframing Technical Analysis

Instead of interpreting the negative returns as evidence against technical analysis, these results suggest a more nuanced framing. The value of the RSI and MA strategy is likely not in forecasting price but in allowing investors to bear discomfort in exchange for higher compensation. This is only possible, of course, with research into exactly when and where these strategies conditionally succeed. This can be tested across periods segmented by different macroeconomic factors, and doing so might reveal how technical analysis’s payoffs align with inefficiency-prone conditions.

Ultimately, although this study does not contradict market efficiency, it reveals how technical analysis could account for shortcomings of traditional investment strategies. It might add value by identifying risks that investors expect compensation for. In this way, it can be used as a tool for portfolio construction to access unconventional sources of returns, or simply to hedge against risks in traditional portfolios.

7. Conclusion

In conclusion, this study tested a widely used technical trading strategy, a combination of the Relative Strength Index and moving average. The analysis spanned ten years and evaluated three different RSI deviation thresholds. While the strategy underperformed across all key return metrics, its characteristics offer important insights into the academic and practical discourse around technical analysis. The strategy’s exposure to extreme returns, seen in its high kurtosis, suggests that it filters for hidden, non-diversifiable risk. This finding is relevant in today’s markets, where signal-based strategies are frequently promoted across social media platforms and adopted by traders in search of an edge. Understanding what these strategies are actually capturing, even when they fail to generate alpha, is essential to evaluating their role in constructing portfolios and risk management.

For researchers, these findings support a reframing of technical analysis as a mechanism for identifying priced risk exposures. Further study should test performance in different market conditions where these risks might be mispriced. For traders and portfolio managers, these findings emphasize the risk structure behind technical analysis. While signals like the RSI and MA might not yield consistent profits, they can help in designing portfolios that take advantage of risks that are compensated under certain conditions. This understanding can also help retail traders avoid misusing signals as forecasters of price action. In this way, the lasting contribution of technical analysis is not its forecasts, but its ability to expose the risks that markets reward.

References

Bessembinder, H., & Chan, K. (1998). Market Efficiency and the Returns to Technical Analysis. Financial Management, 27(2), 5–17. https://doi.org/10.2307/3666289

Blume, L., Easley, D., & O’Hara, M. (1994). Market Statistics and Technical Analysis: The Role of V olume. The Journal of Finance, 49(1), 153–181. https://doi.org/10.2307/2329139

Brock, W., Lakonishok, J., & LeBaron, B. (1992). Simple Technical Trading Rules and the Stochastic Properties of Stock Returns. The Journal of Finance, 47(5), 1731–1764. https://doi.org/10.2307/2328994

Brown, J. H., Crocker, D. K., & Foerster, S. R. (2009). Trading V olume and Stock Investments. Financial Analysts Journal, 65(2), 67–84. http://www.jstor.org/stable/40390354

Conrad, J., Dittmar, R. F., & Ghysels, E. (2013). Ex Ante Skewness and Expected Stock Returns. The Journal of Finance, 68(1), 85–124. http://www.jstor.org/stable/23324392

Elkamhi, R., & Stefanova, D. (2015). Dynamic Hedging and Extreme Asset Co-movements. The Review of Financial Studies, 28(3), 743–790. http://www.jstor.org/stable/24465726

Frugier, A. (2014). Higher-order Moments and Investor Sentiment (Alles’ Model Revisited). Quarterly Journal of Finance and Accounting, 51(3–4), 45–70. http://www.jstor.org/stable/qjfinacct.51.3-4.45

Han, Y ., Yang, K., & Zhou, G. (2013). A New Anomaly: The Cross-Sectional Profitability of Technical Analysis. The Journal of Financial and Quantitative Analysis, 48(5), 1433–1461. http://www.jstor.org/stable/43303847

Kavajecz, K. A., & Odders-White, E. R. (2004). Technical Analysis and Liquidity Provision. The Review of Financial Studies, 17(4), 1043–1071. http://www.jstor.org/stable/3598058 5

Levy, R. A. (1966). Conceptual Foundations of Technical Analysis. Financial Analysts Journal, 22(4), 83–89. http://www.jstor.org/stable/4470026

Malkiel, B. G., & Saha, A. (2005). Hedge Funds: Risk and Return. Financial Analysts Journal, 61(6), 80–88. http://www.jstor.org/stable/4480718

McInish, T., & Puglisi, D. J. (1980). Technical Analysis and Utility Preferred Stocks. Nebraska Journal of Economics and Business, 19(3), 55–63. http://www.jstor.org/stable/40472670

Moosa, I., & Li, L. (2011). Technical and Fundamental Trading in the Chinese Stock Market: Evidence Based on Time-Series and Panel Data. Emerging Markets Finance & Trade, 47, 23–31. http://www.jstor.org/stable/27917672

Neely, C. J., Rapach, D. E., Tu, J., & Zhou, G. (2014). Forecasting the Equity Risk Premium: The Role of Technical Indicators. Management Science, 60(7), 1772–1791. http://www.jstor.org/stable/42919633

Qi, M., & Wu, Y . (2006). Technical Trading-Rule Profitability, Data Snooping, and Reality Check: Evidence from the Foreign Exchange Market. Journal of Money, Credit and Banking, 38(8), 2135–2158. http://www.jstor.org/stable/4123046

Smith, D. M., Wang, N., Wang, Y ., & Zychowicz, E. J. (2016). Sentiment and the Effectiveness of Technical Analysis: Evidence from the Hedge Fund Industry. The Journal of Financial and Quantitative Analysis, 51(6), 1991–2013. http://www.jstor.org/stable/44157641

Tabell, Edmund W., & Tabell, Anthony W. (1964). The Case for Technical Analysis. Financial Analysts Journal, 20(2), 67–76. http://www.jstor.org/stable/4469619

Van Horne, J. C., & George G. C. Parker. (1968). Technical Trading Rules: A Comment. Financial Analysts Journal, 24(4), 128–132. http://www.jstor.org/stable/4470382

Yahoo Finance. (2025). Dow Jones Industrial Average historical data [Data set]. Yahoo. Retrieved July 18, 2025, from https://finance.yahoo.com/

About the author

Roshan Shah

Roshan is currently a senior at Glenbrook South High School in Glenview, Illinois. He intends to complete an undergraduate degree in Applied Mathematics. He is particularly interested in the intersection of mathematics, computer science, and finance, and looks forward to pursuing a career in financial technology. In his free time, Roshan enjoys a wide variety of extracurricular activities, including Model United Nations, digital music production, working out at the gym, and spending time with friends. He also enjoys helping his community by providing food and clothing to the homeless in Chicago.

The post Technical Trading and Higher-Order Risks: An Empirical Evaluation of Relative Strength Index and Moving Average Strategies appeared first on Exploratio Journal.

Abstract  

For my research project, I wanted to combine my interest in hockey with my curiosity about data and math. I explored how a Kalman filter—a tool often used in engineering and statistics—could be applied to track NHL player performance during games. Instead of relying on end-of-season stats, which can sometimes be misleading, I built a system that updates each player’s rating shift by shift. It takes into account who was on the ice and the expected goal differential, giving a more accurate and real-time picture of how players are performing. This helps smooth out the effects of one unusually good or bad game and offers more meaningful insights. I hope this project not only helps fans and fantasy players better understand player impact but also shows how math and data science can be used to analyze the sports we love.  

Introduction  

In hockey, stats like goals and assists only tell part of the story. They don’t show what a player does without the puck or during defensive plays, which are just as important. That’s why analysts use a stat called expected goals (or xG), which measures how likely a shot is to become a goal based on things like where it was taken from and what kind of shot it was. While xG is more helpful than just counting goals, it still doesn’t tell you how much each individual player contributed to creating or preventing those chances. For my project, I wanted to build a better way to track player performance during games. I used a tool called a Kalman filter, which is great for situations that change quickly and randomly—just like in hockey. It allowed me to update each player’s rating shift by shift, giving a more accurate and real-time view of how they’re playing. 

Background on Kalman Filters  

The Kalman Filter, introduced by Rudolf E. Kálmán in 1960 [1], is widely used in control systems and aerospace to estimate hidden states over time from noisy measurements. Its ability to extract a “true” underlying value from uncertain data makes it well-suited for tracking player performance shift-by-shift in hockey, where single-game results can be noisy or misleading[3].  

At its core, the Kalman filter is a recursive algorithm that estimates the state of a system over time[2]. The “state” is something we want to track (like a player’s impact), and the filter uses a combination of past estimates and new observations to improve that estimate. 

The Kalman filter works by treating what we’re trying to measure—like a player’s skill— as something that exists but can’t be directly seen. Instead, we get noisy signals (like xG differential from shifts) that give us partial information. The Kalman filter then tries to  “guess” the true skill level over time by combining what it previously thought (the prediction) with the new noisy measurement (the update). 

Think of it like trying to track a car in fog. You can’t see it clearly, but you occasionally catch glimpses of it. Each glimpse is uncertain, but by combining all the glimpses and accounting for how much things could change or how noisy the observations are, you can make a better estimate of where the car is. That’s essentially what the Kalman filter does —but with math. To better understand why this works, we’ll break down the math and where it comes from. 

The Kalman filter is grounded in two core ideas: prediction and correction. It begins with a guess (called the prior) of a hidden variable—like player skill—and as new, imperfect data arrives, it updates that guess. The power of the filter lies in how it mathematically balances how much to trust the old guess versus the new data using probabilities. 

In mathematical terms, it minimizes the mean squared error of the estimate, assuming  Gaussian (bell-curve) noise. This gives it optimality in linear systems, which is why it’s used in everything from radar tracking and robotics to sports analytics.

The filter runs in a loop; every time new data comes in—in this case, each new shift in a game—and updates player ratings by combining what we already know with what just happened. 

The two main steps are called the prediction step and the update step. 

1. Prediction 

In this step, the filter predicts the next state of the system and how uncertain that prediction is. 

    •    State prediction: 

  xₖ₋₁ → x̂ₖ⁻ = x̂ₖ₋₁ 

    •    Uncertainty prediction: 

  Pₖ⁻ = Pₖ₋₁ + Q 

Here: 

    •    x̂ₖ⁻ is the predicted state estimate before seeing new data 

    •    Pₖ⁻ is the predicted uncertainty 

    •    Q is the process noise (random natural changes) 

2. Update 

Now the filter corrects its prediction using the actual observation from the new data.  

   •    Kalman Gain (how much we trust the new data): 

    Kₖ = Pₖ⁻ Hᵀ (H Pₖ⁻ Hᵀ + R)⁻¹ 

    •    Updated estimate:  x̂ₖ = x̂ₖ⁻ + Kₖ (zₖ − H x̂ₖ⁻) 

    •    Updated uncertainty:  Pₖ = (I − Kₖ H) Pₖ⁻

 where:

Zₖ: is the new observation (like xG_diff for a shift) 

   H:  is the matrix that maps the true state to what we can observe     

   R: is the measurement noise (how noisy our observation is) 

   Kₖ: is the Kalman Gain 

   x̂ₖ: is the updated estimate of the state 

   Pₖ: is the updated uncertainty 

These equations come from linear algebra and probability theory. The Kalman filter assumes the state evolves according to a linear model and that both the system and the measurement noise are Gaussian (bell curve-shaped) with known variances. 

The matrices Q and R represent how much randomness or noise we expect in our system and observations. If Q is high, we assume the underlying player skill changes a lot between shifts. If R is high, we assume our xG measurements are very noisy. Tuning these helps the model balance trust between past beliefs and new data. 

Where the Equations Come From  

The Kalman filter is based on the idea of minimizing uncertainty. It assumes that the current state (player rating) follows a simple rule: it either stays the same or changes slightly due to random factors. Observations (like xG_diff) are linked to the state through a matrix (H) and contain noise. The equations are derived by using Bayes’ Theorem to find the most likely state given the prior estimate and the new observation.  

In simple terms, each equation is designed to balance two forces:  

• “The prior (what we thought before this shift)”  
• “The measurement (what the shift data just told us)” 

The Kalman Gain (K) adjusts how much we move toward the new observation. If the data is noisy, we move less. If the prediction was way off, we move more. The math ensures this is done in the optimal way to reduce error. 

The update formula (posterior = prior + gain × error) is essentially a Bayesian update. It  answers: ‘Given what I used to believe and what I just observed, what’s the best new  belief?’ The Kalman gain determines how much we shift our belief, and this is computed based on how uncertain we are about both the prior (P) and the measurement (R).

Why the Kalman Filter Works  

The Kalman Filter works best in situations where you need to track something uncertain over time using noisy information. In hockey, a player’s true impact can’t be directly measured—we only see glimpses through stats like xG_diff. By combining the model’s previous belief with the new evidence from each shift, the Kalman Filter gives a smarter,  smoother estimate of how good each player really is. 

It also adjusts how much to “believe” each new data point based on how noisy it thinks that data is. If one shift has a weird result, the filter won’t overreact. But if lots of shifts show a consistent trend, it will gradually adapt the rating. 

Data and Setup  

I used public 5v5 shift data from Natural Stat Trick [5]. For each shift, I collected:  the list of offensive players and defensive players, and the expected goals differential (xG_diff)  

Each player was given a unique index number. Everyone started with a rating of 0, and I set small initial values for uncertainty and noise because I assumed performance doesn’t change too quickly. This setup allowed the model to slowly adapt as more shifts occurred.  

I used scalar values for the process noise and measurement noise. Specifically, Q=q⋅I and R=r, where I is the identity matrix. I manually tuned q and r to be small (e.g., q=0.01, r=0.1) to ensure stability and avoid overreacting to noisy shifts early in the season.  

Assumptions  

I assumed that the system is linear, noise is Gaussian with a zero mean, and that the covariance is known. 

Method: Using the Kalman Filter  

For every shift in the dataset, I built a vector H, where:  

 Offensive players = +1 

 Defensive players = –1 

This vector H∈R^1xn represents the players on the ice during a shift, where n is the total number of players in the dataset. Each element of H corresponds to a player’s index:  players on offense are set to +1, players on defense are set to –1, and all other values are  0. When we multiply H x X, we get the predicted xG differential for that shift, based on current ratings. 

Then I used the xG_diff from that shift as the observation (z) and ran the Kalman filter to update player ratings. The state vector x∈R^n holds the current rating of every player. Although only a small number of players are involved in any one shift, the Kalman Filter maintains this full vector across all players. The update only affects players present in the shift.  

Importantly, this means we’re not calculating each player’s contribution in isolation. The Kalman filter sees how the group of players on the ice performed together, then uses that to adjust everyone’s ratings just a little. So if a defensive player is consistently on the ice when their team allows fewer expected goals, their rating will slowly increase. This teamwork-based approach is one reason the filter gives more reliable ratings than just counting goals or points.  

How the Kalman Filter is applied to Player Ratings  

In each shift, I record the expected goals for and against (xG differential). Because multiple players are on the ice, the xG differential is influenced by everyone’s performance. I model the team-level xGD as a sum of the ratings of the skaters on the ice.  Using a Kalman Filter, I estimate and update each player’s rating over time. The filter updates a player’s performance estimate based on how the team performed while they were on the ice and how reliable that measurement is.

Let xk be a player’s latent rating at shift k, and zk be the observed xG differential for that shift. 

Prediction Step: 

x̂k | k − 1 = x̂k − 1| k − 1 

P_k|k−1 = P_k−1|k−1 + Q 

Update Step: 

K_{k =} P_k|k−1 

P_k|k−1 + R 

x̂k | k = x̂k | k − 1 + K_k(z_k − x̂_k|k−1) 

P_k|k = (1 − K_k)P_k|k−1

• x^k∣k−1: predicted rating before seeing the new shift 
• P: uncertainty in our estimate 
• Q: process noise (how much a player’s performance might vary) • R: measurement noise (uncertainty in xG) 
• K: Kalman Gain (how much we trust the new shift vs prior rating) 

I chose the Kalman Filter because it balances past performance with new shift data in a mathematically consistent way. Traditional models either rely too much on averages or overreact to single games. By using xG differential and updating shift by shift, this method smooths out variance while still responding to meaningful changes[6]. The assumptions I make include linear player contributions and Gaussian noise, which simplify computation while still producing stable ratings. 

For each shift, we follow this sequence:” 

1. Build matrix H, encoding who is on the ice (offense = +1, defense = –1).
2. Predict the expected xG_diff using current player ratings. 
3. Observe the actual xG_diff for that shift. 
4. Update each involved player’s rating based on the difference between prediction and observation. 
5. Repeat for every shift throughout the season. 

1. Prediction Step  

We first predict the current state before seeing the new data:  

x_prior = x 

This means we start by assuming the ratings from the last shift are still correct.  P_prior = P + Q 

 P is our current uncertainty (a covariance matrix), and Q is the extra uncertainty we add to account for performance possibly changing naturally. 

So this step gives us our best guess before seeing new data and tells us how confident we are in that guess.  

2. Update Step  

Now we use the new shift data to update our estimates:  

K = P_priorH^⊤S⁻¹

This tells us how uncertain we are about our prediction for the xG_diff on this shift.  

 H · P_prior · Hᵀ: how uncertainty in player ratings translates into uncertainty in xG_diff.  

 R: The noise in our observation of xG_diff.  

K = P_priorH^⊤S⁻¹ 

This is the Kalman Gain, which tells us how much to trust the new observation.  

Here, we assume the underlying ratings change slowly (low Q), and that the measurement (xG_diff) has moderate noise (R). The filter adapts depending on these values:  

 High R = less trust in new data, more weight on prior estimates  • High Q = belief that ratings can change quickly, so the model updates faster  

x_updated = x_prior ^{+ K} ₍z − Hx^prior) 

This is the main update:   z is the actual xG_diff observed.  H · Xprior is what we expected the xG_diff to be, based on current ratings. The difference is the “error,” which we scale with K and use to update ratings.  

P_updated = (I − KH) P_prior

Finally, we reduce our uncertainty. After seeing new data, we’re more confident in our updated ratings.

How It All Comes Together  

For every shift, we loop through the following sequence:  

1. Predict ratings and uncertainty 
2. Observe actual xG_diff 
3. Compare prediction to reality 
4. Update ratings and reduce uncertainty 

This repeats for every shift in the season, and over time, the ratings become smarter and more stable. Before applying the Kalman filter, it’s useful to see how shot share (CF%) compares to expected goal share (xGF%) across all players. 

Suppose a shift has three players: Player A, B, and C. A and B are offense, C is defense.

 The vector H=[1,1,−1], and the observed xG differential z=0.3. If we assume that the prior ratings x=[0.1,0.1,−0.1]. The predicted xG is H⋅x=0.1+0.1+0.1=0.3, which matches the observation, so the update is small. However, if z=0.6, the Kalman Gain would increase the offensive ratings and penalize the defensive rating accordingly. This is how the model incrementally adjusts player values. 

This example shows how, over time, each player’s rating becomes a reflection of their repeated impact. A player who always ends up on the ice during good shifts will see their score rise—even if they’re not scoring themselves. 

This process also prevents overreaction. If a player has one really strong shift, their rating only adjusts slightly. But if they consistently influence the xG_diff over many shifts, the filter learns to trust that signal more. This is what gives Kalman ratings their strength— they balance recency with consistency. 

Experiments and Results  

I ran this model on a full NHL season of shift-by-shift data[4]. As the season went on, each player’s rating changed depending on how much they helped or hurt the expected goals during shifts.  This yielded  the following observations:

 Players who weren’t big scorers still rated highly because they consistently created chances or prevented goals.  Players with big goal totals but poor defensive play sometimes ranked lower. Finally, the ratings were more stable than raw xG—less noise, fewer random spikes.  

 For example, Matty Beniers rated higher than expected due to consistent defensive contributions despite modest goal totals, while Patrick Kane had a lower rating due to poor xG_diff when on the ice, even though he scored often. 

As shown in Figure 2 below, some players with strong overall performance (based on xGF%) ranked highly in the Kalman model even if they didn’t lead in goals.. This demonstrates how the model can highlight impactful players who may not lead in traditional scoring stats.  

Implications  

This type of model has real potential:   NHL teams could use it to find undervalued players or make smarter trade decisions.  Scouts could see who consistently impacts the game beyond goals. Fantasy hockey tools or betting models could use it to get an edge.  The method could work in other sports, for example, basketball or soccer.  The filter could be improved by adding more features for example ice time, face-off zones, or goalie performance.  

Conclusion  

Kalman filters offer a powerful, more balanced way to rate players. They combine old data with new information and avoid overreacting to one outlier shift. Compared to just looking at box scores, this model gives a fairer and more complete picture. This is a solid starting point—I didn’t include power play data or goalie stats— providing considerable potential for development. 

Code

The code is publicly  available in the repository below: https://github.com/Dimi Baguette/Kalman-Filter 

References  

[1]Kalman, R. E. (1960). A New Approach to Linear Filtering and Prediction Problems.  Journal of Basic Engineering, 82(1), 35–45.   https://doi.org/10.1115/1.3662552

[2]Welch, G., & Bishop, G. (1995). An Introduction to the Kalman Filter.  University of North Carolina at Chapel Hill.   https://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf

[3]DraftKings Engineering. Kalman Filters for NBA Player Ratings.  https://www.draftkings.com/playbook/nba/kalman-filters-nba-player-ratings  

[4]GitHub Repository – Kalman Filter for NHL Player Ratings:   https://github.com/Dimi-Baguette/Kalman-Filter

[5]Natural Stat Trick – NHL Shift and Player Stats:   https://www.naturalstattrick.com

[6]Simon, D. (2006). Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches.  Wiley Publishing.   https://onlinelibrary.wiley.com/doi/book/10.1002/0470045345

About the author

Dimitri Thivaios

Dimitri is a UK born French citizen living in the US. He is currently studying at Mamorenck High School in NY, expecting to graduate in 2026. Dimitri has a strong interest in computer science, applied mathematics, and data analysis and he’s a passionate ice hockey player and captain on the varsity team.

The post Using a Kalman Filter to Rate NHL Players appeared first on Exploratio Journal.

Introduction

Background Information

Our initial conceptualization

Historically, telescopes have been monumental achievements of stationary precision, built to capture the mysteries of the universe from a fixed vantage point. Their designs prioritized maximizing aperture and achieving optical accuracy to probe the depths of space. While these systems have undoubtedly contributed to groundbreaking discoveries, they remain constrained by their immobility and singular focus. However, as technology advances, so do the opportunities to rethink and redefine these systems. This project explores the integration of telescopes with kinematic mechanisms, envisioning a more dynamic, versatile approach. By mounting a telescope on a mobile, wheeled base with mechanisms enabling 360-degree rotation and tilt, we introduce a new dimension of adaptability to sky observation.

But why move beyond traditional designs? The answer lies in both practicality and potential. Fixed telescopes are excellent for specific observational needs but fall short in scenarios that require broader sky coverage or adaptable positioning. By incorporating mobility and kinematic design, we address this limitation while opening up avenues for new applications, from amateur astronomy to educational outreach and even real-time tracking of celestial events. The incorporation of mechanisms not only enhances the telescope’s functionality but also democratizes its usage, making sophisticated observational tools more accessible and versatile.

This approach also taps into a largely unexplored area of research: the intersection of optical precision and mechanical adaptability. While traditional designs have focused on either optical performance or static stability, integrating kinematic mechanisms introduces a system capable of evolving with future demands. Whether for amateur use, portable scientific observation, or scalable systems for larger projects, this concept represents a shift toward innovation that is not merely about convenience but about laying the foundation for a more dynamic and inclusive future in astronomy.

Specific Fundamentals about Kinematic Mechanism Designs

Kinetic mechanism designs are all about how nodes, linkages, and motors come together to make something move. Each part plays its role — nodes can either be locked in place or free to move, and it’s the interaction between the motors and these nodes that defines how the whole system operates. Take this, for example: if a node is close to a motor, it’ll move almost in sync with the motor. But the further away a node is, the more unpredictable or complex its movement becomes, because it’s now influenced by the other nodes and linkages in between.

Fixed nodes, on the other hand, are crucial for setting limits on movement. The way they interact with motors can change the entire behavior of the system. Sometimes, if a fixed node constrains the motor’s movement too much, you end up with restrictions or even unexpected behaviors, which just goes to show how important node placement really is. It’s like every part is connected in a web of motion, and one adjustment in one area affects everything else.

Now, when you’re dealing with systems that have multiple nodes and linkages, things get more complicated. As you increase the number of nodes, the system’s degrees of freedom expand, meaning there’s a wider range of possible movements. But, at the same time, more linkages also mean more constraints, which limit how much movement you can actually achieve. So, it’s always a balancing act — you want enough nodes to give flexibility but not so many that you start restricting yourself.

Understanding how these nodes move in relation to each other, especially when you’ve got motors and fixed points, is key to making everything work smoothly. In something like a telescope system, for instance, while the motor might be able to rotate a full degrees, a constrained node (like the one holding the lens) might only move along a partial arc. This difference in movement changes how the entire telescope behaves. It’s about finding that sweet spot where everything moves as intended, without unwanted limitations.

Specific Fundamentals about Telescope Designs

The two primary categories of telescopes in astronomy fundamentals are refracting and reflecting.

Refraction is the process of bending light through lenses, which is how reflective telescopes function. Specifically, light comes into the telescope through a convex lens and focuses at a point known as the focal point. These telescopes are straightforward in design making them perfect for getting a clear view of nearby celestial objects like the moon. However, there’s a catch: their size is constrained, and the lens occasionally tinkers with the colors, leading to chromatic aberration.

Conversely, reflective telescopes collect and concentrate light using mirrors. Here, a concave mirror that reflects light back to a focal point is the essential component. Reflectors have a leg up on refractors because they can be built larger without the risk of color distortion. This makes them better suited for looking at distant, faint objects. Interestingly, in 1668, Isaac Newton was the one who created this type of telescope, and even today, they’re still the backbone of most large observatories.

Both of these designs are important for modern telescope technology, giving us the tools to explore everything from the nearby moon to deep space. These will be the main types of telescopes we’ll rely on for the research in this project.

Thesis Statement

The primary goal of this paper is to determine which telescope design and mechanism offer the most reliable, full-sky coverage. Our aim is to create a system that allows the telescope to rotate 360 degrees and move seamlessly back and forth, essentially scanning the entire sky.

We will begin by focusing on the exploration of kinetic designs, simulating different mechanisms to identify which one performs best for this particular setup. This section will delve into the planned designs, highlighting what has been realized so far and identifying areas for further investigation.

Next, we will conduct a case study comparing telescopes, analyzing their strengths and limitations to determine which one is the most suitable. This evaluation will include an in-depth discussion of our decision-making process, addressing areas that warrant additional research.

Once the best mechanism and telescope combination is established, we will explore why this synergy is optimal for the project’s objectives and how it advances our goals. This will lead us into the conclusion, where we summarize key findings and outline potential paths for future research.

Mechanism

Design

Setup of Simple Mechanism Design

The integration of a kinetic mechanism into the telescope system is motivated by the need to achieve precise and comprehensive sky coverage while maintaining flexibility and stability. Telescopes are traditionally stationary or limited in their range of motion, which restricts their ability to scan the entire sky or adjust dynamically to track objects. By combining a robust kinetic mechanism with the telescope, this system can provide both rotational and translational motion, allowing it to explore and capture a wider expanse of the sky with greater accuracy. The pan-tilt mechanism is particularly advantageous for its versatility in achieving precise angles, critical for astronomical observations where even slight misalignments can lead to significant data inaccuracies. Furthermore, incorporating mobility via a wheeled platform addresses the need for repositioning the telescope, making the system adaptable to various observational scenarios.

Mechanisms as a concept are fundamental in robotics and mechanical engineering, enabling controlled motion in machines. They are often used in systems where precise, repeatable

movements are essential, such as robotic arms, cranes, and now, telescopes. A mechanism’s design hinges on the interplay between nodes (points of rotation or translation), linkages (rigid components connecting nodes), and actuators (motors or servos providing motion). These elements dictate the degrees of freedom, constraints, and overall functionality of the system. In the context of telescopes, mechanisms provide a means to combine structural rigidity with dynamic motion, ensuring the telescope can both support heavy components and move fluidly to align with celestial objects.

The kinetic mechanism in this system relies on the interplay between ground nodes, fixed nodes, and motors. Ground nodes allow for limited movement, and their absence results in unrestricted motion dictated solely by the motor’s output. As the tracked node’s position changes relative to the motor, the shape traced by its path also changes. Nodes closer to the motor mimic its motion more closely, while those farther away experience more complex paths influenced by additional nodes and linkages.

Fixed nodes play a crucial role in influencing motor behavior by imposing constraints on motion. If a fixed node limits the motor’s path, the resulting motion may be restricted or altered, leading to unexpected system behavior. For instance, linkages can become tangled, causing disturbances that compromise functionality. Conversely, nodes directly connected to the motor without sufficient constraint might spin randomly without productive output.

The complexity of the mechanism increases with the addition of nodes and linkages. While more nodes introduce additional degrees of freedom, they can also impose constraints that limit the system’s flexibility. Balancing freedom and constraint is pivotal to designing an effective mechanism that maintains both structural integrity and operational versatility.

In this project, rotational and translational motion are key to achieving full-sky coverage. A pan-tilt mechanism has been selected to enable the lens to rotate 360 degrees in both directions and tilt up or down at precise angles. This mechanism uses two servos: one continuous servo for full rotational motion and another for tilt control. The synchronized movement of these servos ensures the lens can scan the sky efficiently and comprehensively.

To enhance its adaptability, the system incorporates translational motion via a wheeled base, similar to a remote-controlled vehicle. Motors or servos control the wheels, and movement is managed through an Arduino or Raspberry Pi with optional Bluetooth control. An ultrasonic sensor is integrated to detect obstacles, allowing the system to avoid collisions autonomously.

The interaction between nodes is optimized to achieve smooth rotational motion. For example, while the motor may execute a full 360-degree rotation, a constrained node carrying the lens will follow a partial arc due to its limitations. This dynamic interaction ensures precise control over the telescope’s movement while maintaining the stability necessary for capturing clear, accurate images.

By blending these mechanisms into a single system, this design aims to redefine how telescopes can function dynamically, addressing the limitations of static models while opening new possibilities for observational astronomy.

Simulation

Present Simulation

As part of understanding the behavior of the mechanism, we decided to dive deeper into the kinematics of bar AB by tracking its midpoint coordinates and orientation. This would allow us to visualize how bar AB moves and rotates over time. The code was adjusted to follow the x– and y-coordinates of the midpoint (P_x and P_y) and to calculate the changing orientation of the bar, represented as the angle θ_AB .

Through this approach, we could gain a clearer perspective on how the system behaves. The focus here was on capturing the essence of bar AB’s motion—how its position shifts and its orientation changes over time. By simulating this and plotting the results, we could make sense of how the mechanism interacts with its environment, something that static analysis just can’t reveal.

The first image (see Figure 1 below) shows a snapshot of the mechanism. Bar AB is the horizontal section, indicated by the black dashed line. The surrounding linkages form part of the larger mechanism, with their positions and movement constrained by the interactions between nodes and motors.

[Simulation from gabemorris12’s code : https://pypi.org/project/mechanism/ ]

Next, we used this data to produce three key plots that describe the motion and rotation of bar AB. These are represented in Figure 2 below, where you can observe how P_x, P_y, and θ_AB evolve over time. These three plots are crucial because they let us visualize the continuous change in position and orientation, offering a dynamic representation of how the mechanism operates in real-time.

[Simulation from gabemorris12’s code : https://pypi.org/project/mechanism/ ]

In these plots, the x-axis represents time, while the y-axes represent P_x, P_y, and θ_AB respectively. We see smooth, sinusoidal curves, which indicate that the motion of bar cyclic, with the midpoint and orientation oscillating in a consistent pattern.

Understanding this movement is key as it not only helps us determine the effectiveness of the mechanism but also opens up the possibility of refining it. By exploring these trajectories, we get a glimpse of how the mechanism could potentially be optimized for full-sky coverage, allowing us to adjust the system parameters for more efficient and smoother motion.

Both of these visualizations—Figure 1 for the mechanism’s structure and Figure 2 for the time-dependent movement—are essential in understanding the practicality and precision of our system.

Parametric Study

Now that we’ve gathered data from the simulation, the real work begins. The goal here is to analyze the information we have and figure out how it connects back to the larger project objectives. By looking closely at how bar AB moves—specifically its midpoint coordinates and orientation—we can start to see patterns that help us understand how well our mechanism is performing.

From the simulation, we gathered time-dependent data on P_x, P_y, and θ_AB These three variables give us a clear view of both the movement and the constraints acting on bar AB. What’s interesting is how the trajectory of P_x and P_y, forms a smooth sinusoidal path, which tells us that the motion is periodic. This periodicity is useful because it confirms that our mechanism is functioning as expected, without erratic or unpredictable movement. The angle θ_AB on the other hand, shows a steady back-and-forth rotation, which is exactly what we need for sky scanning.

[Simulation from gabemorris12’s code : https://pypi.org/project/mechanism/ ]

This information is valuable because it helps us understand the mechanical efficiency of the system. Since bar AB’s motion is smooth and predictable, we know that the node and motor interactions are working well together. But what’s even more important is how this periodic motion allows us to tweak the design for better performance. By adjusting the placement of the nodes and the length of the linkages, we could either amplify or reduce this motion, depending on what’s needed for a specific observation task.

One thing that stands out from the data is the relationship between node placement and the behavior of the motor. As mentioned earlier, the closer the tracked node is to the motor, the more the movement mimics the motor’s motion. If the node is farther away, the movement becomes more constrained, as seen in our plot for θ_AB, where we observe a limited angular range. This is crucial because it helps us pinpoint the best placement for the lens so that it covers the most area in the sky.

[Design made using Decode Links Simulator: https://decode.mit.edu/projects/links/ ]

What we’re also noticing is how the number of linkages and nodes impacts the system’s degree of freedom. More nodes, as expected, increase the degree of freedom, which allows for more complex motions but also introduces a higher level of unpredictability. This is where fine-tuning the number of nodes and linkages comes in. Fewer nodes give us less flexibility but offer more control, whereas more nodes introduce complexity and the possibility of the system overextending or tangling, which could lead to mechanical failure. We saw a bit of this in the simulations, where the nodes started to pull the motor into undesired orientations, slightly reducing the efficiency of the system.

This parametric analysis helps us focus on a key question: What’s the best balance between system flexibility and control? The data suggests that by carefully choosing the placement of the tracked node and adjusting the number of linkages, we can optimize the system for the full -degree sky scanning we need. It’s a balancing act between giving the mechanism enough freedom to cover as much of the sky as possible while also ensuring it remains within functional constraints.

[Design made using Decode Links Simulator: https://decode.mit.edu/projects/links/ ]

Overall, this parametric study has given us crucial insights. The data gathered shows us where improvements can be made, but it also confirms that our base design is solid. By fine-tuning the parameters and making slight adjustments based on node placement and linkage count, we can ensure that our mechanism not only covers the sky efficiently but does so in a controlled and predictable way. This will be especially important when it comes to capturing clear images of the solar system, where precision is everything.

Optics

Refractive Telescope

Sizing and Analysis

Refractive telescopes are known for their simple design compared to more complex systems like reflectors or compound telescopes. The design mainly revolves around a large, curved lens at the front of the telescope and an eyepiece at the back. One of the core features that sets refractors apart is their ability to capture light and form images by bending light through a convex lens. Although effective, refractive telescopes come with inherent limitations, most notably their focal length. Due to this, they are primarily used for viewing closer celestial bodies, with the furthest object typically visible being the moon.

The path of light within the telescope is fairly straightforward. Light enters through the front lens, which bends the rays and converges them into a focal point. This process forms a triangular path from the lens to the focal point, which is critical for creating the image. Once the light reaches the focal point, it then travels through a filter to the eyepiece, allowing the observer to see a clear, magnified image.

Compared to telescopes with longer focal lengths or reflectors, refractors are more limited in their ability to observe distant objects. However, their simplicity and the fact that they don’t require frequent maintenance like mirror alignment make them a popular choice for amateur astronomers.

[Picture from https://personal.math.ubc.ca/~cass/courses/m309-03a/m309-projects/lcheng/project3.html]

Inspiration for this analysis was drawn from a detailed review of the Cambridge paper on telescope design and functionality, which delves deeper into the geometry and practical applications of refractive telescopes. One key takeaway from this work is the emphasis on the refractor’s role in providing stable, low-maintenance viewing for objects within a close range in our solar system.

Case Study

In this case study, we analyze how a refractive telescope captures objects at different distances by calculating the necessary angles to view specific points. We will consider two cases: one involving a nearby object, the Las Vegas Sphere, and another involving a distant object, the Moon. By comparing these two, we can gain insights into how drastically the viewing angle changes depending on the distance.

Case 1: The Las Vegas Sphere

The Sphere, located in Las Vegas, presents an interesting scenario due to its proximity. The telescope we are using has a lens positioned 1 meter (or km) off the ground. The object itself has a size of 0.01 km, and we are analyzing its angular size at a distance of km from the telescope.

Viewing the Bottom of the Sphere:

We can calculate the angle required to view the bottom of the Sphere using basic trigonometry. The distance from the telescope to the object is km, and we are looking downward from a height of km.

Since the result is negative, the telescope must be tilted downward by degrees to view the bottom of the Sphere.

Viewing the Center of the Sphere:

To find the angle to view the center, we first calculate the midpoint of the object, accounting for the height of our telescope lens. The height of the Sphere is 0.157 km, so we subtract the .001km (telescope height) from the total and divide by 2:

Now, using the distance to the Sphere, we can calculate the angle to the center:

The telescope must be tilted upward at an -degree angle to capture the center of the Sphere.

Case 2: The Moon

The Moon, much farther away, presents a different challenge in terms of viewing angle and field

of view. Its constant distance from Earth is , and we are approximating the height relative to Earth’s horizon to be around km. We will use this figure to calculate the angle required to view both the bottom and center of the Moon.

Viewing the Bottom of the Moon:

We assume that the height of the Moon’s bottom is km above the horizon. Using this and the distance to the Moon, the angle can be calculated as:

The telescope would need to be adjusted to an angle of degrees to view the bottom of the Moon.

Viewing the Center of the Moon:

To find the angle to the center, we add half the size of the Moon ( 3.7478 km) to the initial 10 km:

Now, calculating the angle:

Thus, the telescope needs to be angled at degrees to view the center of the Moon.

Angular Size and Field of View Comparison

The comparison between the two cases shows how drastically the required angle changes depending on the distance. For nearby objects like the Sphere, even small changes in position (like moving from the bottom to the center) result in significant angular changes. Conversely, for far-away objects like the Moon, the angular change is minimal, even when viewing different parts of the object.

The table below illustrates how angular size and field of view (FOV) interact when viewing different objects. The angular size is determined by the object’s size and distance, and it must be compared to the telescope’s FOV to assess whether the object can be viewed entirely or not.

Table: Angular Size and Field of View for Different Objects

In this case study, we observe how angular size drastically changes between nearby and distant objects. When an object is close, the required angle changes significantly between points on the object, while for distant objects, these changes become negligible. Furthermore, the telescope’s field of view plays a crucial role in determining how much of an object can be captured, with closer objects often exceeding the FOV, and distant objects appearing smaller within it.

Reflective Telescope

Sizing and Analysis

Reflective telescopes are an essential part of modern astronomy due to their ability to gather and focus light through the use of mirrors. Unlike refractors, which rely on lenses, reflective telescopes employ a concave mirror to form an image. This difference in design offers several advantages, particularly in terms of image quality and structural integrity.

One of the key benefits of using mirrors is that they reflect all wavelengths of light equally, which prevents color distortion. This is crucial when observing celestial objects, as it ensures that the light entering the telescope is not altered in color, providing a more accurate view. Another major advantage is that mirrors can be made much larger than lenses, making reflectors better suited for deep-sky observations. This is because a mirror can be supported across its entire back surface, unlike a lens that can only be supported at the edges. This added support allows for much larger and more stable telescopes, improving their ability to capture faint or distant objects.

Historically, the development of reflective telescopes can be traced back to the work of Isaac Newton, who pioneered their use in the 17th century. His design has since become a foundation for many of the world’s most powerful telescopes, including those used for cutting-edge astronomical research.

In this section, diagrams will be essential to visually demonstrate how light travels through a reflective telescope. These will illustrate the path light takes when it hits the concave mirror, reflects back toward a focal point, and is then viewed through an eyepiece.

In summary, the use of mirrors in reflective telescopes not only allows for larger apertures and better support but also prevents color distortions. This makes them the preferred choice for many astronomers today.

[Picture from https://www.britannica.com/science/optical-telescope/Reflecting-telescopes ]
In the next section, we will conduct a similar case study to the one performed with the refractive telescope, comparing angles, object sizes, and distances.

Case Study

In this mini case study, we focus on analyzing the performance of a reflective telescope under various observational conditions. The parameters for the telescope are:

● Magnification: 500x (using a 10mm eyepiece)
● Aperture: 257mm
● Focal Length: 1200mm
● Telescope Field of View (FOV): .4166 degrees

We’ll investigate how the telescope captures objects of different angular sizes and compare the field of view with these angular dimensions. The following table summarizes the key scenarios we will examine.

In the case of the reflective telescope, the field of view remains the same across all observations due to the fixed parameters, but the ability to capture the object depends heavily on its angular size. As with any high-magnification telescope, the closer the object, the larger its angular size and the more likely the telescope will exceed its FOV, making it impossible to capture the entire image.

Angle of the Telescope When Observing Specific Points

When working with reflective telescopes, knowing the angle at which to position the device is crucial for accurately capturing objects at different heights and distances. Using basic trigonometry, we can calculate these angles based on the object’s distance from the telescope and its size.

Las Vegas Sphere

●  Size: 0.157 km
●  Distance from Telescope: 0.01km
●  Telescope Height: 0.001 km (assumed)

To find the angle needed to observe the bottom and the center of the sphere, we first calculate the angles using:

●  Bottom: This shows the telescope must be tilted downwards by degrees.

●  Center: After adjusting for the telescope height and calculating the center, we find:

Thus, the telescope must be tilted upwards at an angle of degrees to view the center. The Moon

●  Size: 3,747.8 km
●  Distance: 384,400 km
●  Height relative to horizon: 10km

To view the bottom of the moon:

To view the center, adding half the moon’s size:

Despite the moon’s enormous size, the angle difference is minimal due to its great distance, illustrating how viewing angles change significantly for nearby objects but only slightly for distant ones.

We also include a table comparing the angular sizes and resulting angles for reference during the discussion. This reinforces how reflective telescopes behave in various real-world observations, making the advantages of using reflective mirrors—such as the lack of color distortion and larger size capability—apparent.

Telescope System Study

Comparison between both telescopes

In the world of astronomy, both refractive and reflective telescopes have carved their own significant paths, each offering distinct benefits. However, after carefully analyzing both types in the previous sections, it becomes clear that reflective telescopes offer superior performance and versatility, making them the preferred choice for serious observation, particularly when considering larger and more detailed celestial objects.

Refractive Telescopes: A Solid Start, but Limited Potential

Refractive telescopes, as we explored earlier, operate using lenses to bend light and form an image. While they are often easier to use and produce sharp, high-contrast views of objects (especially terrestrial ones), their limitations become apparent when pushing beyond a certain size. For one, the construction of larger lenses presents a significant challenge. Lenses can only be supported at their edges, which not only makes manufacturing difficult but also limits how large the aperture can realistically get.

Moreover, refractors suffer from chromatic aberration, a distortion caused by the bending of different wavelengths of light at slightly different angles. This leads to unwanted color fringing around objects, a problem that reflects a fundamental limitation in how light behaves as it passes through glass. Though advances in lens coatings and multi-lens designs can help mitigate this, the issue persists and remains a challenge for refractors.

The aperture of refractors is inherently restricted, and the field of view (FOV) becomes tighter, particularly when observing larger celestial bodies. As we saw in the earlier analysis, refractors struggled to capture entire objects at times, leading to either incomplete views or challenges in gathering adequate light for faint objects. While refractive telescopes are excellent for smaller,

precise observations, their capabilities are often outshined by their reflective counterparts, especially in the context of astronomical exploration.

Reflective Telescopes: Overcoming Limitations with Mirrors

Reflective telescopes, pioneered by Isaac Newton, offer a much more flexible and efficient system. By utilizing mirrors instead of lenses, they sidestep many of the fundamental issues faced by refractors. Mirrors reflect all wavelengths of light equally, so chromatic aberration is entirely eliminated. This alone is a massive advantage, particularly when observing distant objects where color accuracy is critical.

One of the key benefits of a reflective system is the ability to scale up. Mirrors can be supported across their entire back surface, allowing much larger apertures without the structural issues that plague lenses. This means that reflective telescopes can gather significantly more light, which is essential for observing faint, distant objects such as galaxies, nebulae, or dim stars. In the case studies we explored earlier, the reflective telescope’s large aperture allowed for much clearer and more comprehensive views of celestial bodies compared to its refractive counterpart.

Reflectors also handle magnification more efficiently. With larger apertures and longer focal lengths, they provide a wider range of magnifications without sacrificing clarity. This is essential for viewing both nearby objects, like the moon or planets, and distant deep-sky objects, which require higher magnification levels without compromising image quality.

Conclusion: The Reflective Telescope’s Superiority

When we compare the two systems side by side, the reflective telescope clearly emerges as the superior instrument for serious astronomical observation. Its ability to avoid chromatic aberration, scale up in size, and handle greater magnifications while maintaining clarity make it the clear choice for observing the vast expanse of space. Whether for amateur astronomers or professional observatories, the reflective telescope’s mirror-based design overcomes the physical limitations of lenses, allowing it to capture more light, provide clearer images, and offer more flexibility overall.

While refractors certainly have their place, particularly for beginners or those interested in smaller, more precise observations, the reflective telescope’s advantages are undeniable. For those who want a telescope that can grow with their needs and deliver high-quality observations of a wide variety of celestial phenomena, the reflector is the clear path forward.

Conclusion

Summarizing Key Points

Throughout this paper, we have explored the intricate dynamics behind the optical systems and mechanical designs that drive telescope performance. In doing so, we’ve analyzed the behavior of both refractive and reflective telescopes, as well as the critical role that various mechanisms play in ensuring full sky coverage and optimal image capture.

Optics

In the Optics section, we began by exploring the core differences between refractive and reflective telescopes. Refractors, while providing clear and sharp images, are fundamentally limited by chromatic aberration and the difficulty of scaling up lens size. As light passes through the glass lens, different wavelengths bend at slightly different angles, causing color distortions that become problematic for accurate astronomical observation. This issue is compounded by the fact that larger lenses can only be supported at their edges, which further restricts the size of refractors, limiting the amount of light they can gather and narrowing their field of view.

Reflectors, on the other hand, offer a more versatile system. By employing mirrors instead of lenses, they eliminate chromatic aberration entirely. The use of concave mirrors to focus light allows reflective telescopes to scale much larger without the physical constraints imposed on refractors. Mirrors can be supported across their entire surface, enabling greater apertures to collect more light and produce sharper, more detailed images. This makes them ideal for viewing both nearby and distant celestial objects.

Mechanisms

The mechanism design section of this paper examined the practical considerations necessary for creating a telescope capable of 360-degree movement and full sky coverage. We detailed the implementation of pan-tilt mechanisms, as well as motor and servo systems that allow telescopes to move both horizontally and vertically. This combination of degrees of freedom ensures that the telescope can scan the sky thoroughly, while also maintaining stability and precision in its movement.

We further analyzed the importance of node and linkage systems, specifically focusing on how different mechanical components interact to provide the necessary motion for optimal telescope positioning. Key to this analysis was the exploration of motor-node interactions, which allow telescopes to translate motion on wheels and rotate continuously, ensuring smooth and reliable tracking of celestial objects. The inclusion of an ultrasonic sensor for obstacle avoidance added an extra layer of reliability, ensuring that the system can operate without interference from its environment.

Case Studies

Throughout the paper, our case studies provided practical applications of these optical and mechanical principles. We analyzed both refractive and reflective telescopes in real-world scenarios, focusing on their ability to capture objects of varying angular sizes. These case studies highlighted the superior performance of reflective telescopes, particularly when observing larger, distant objects like the moon or even smaller celestial bodies. Reflective systems consistently demonstrated their ability to gather more light and deliver clearer, more detailed images while overcoming the limitations faced by refractors.

Similarly, the mechanism case study on the positioning of the telescope when observing objects like the Las Vegas Sphere and the moon demonstrated how precise angles and mechanical stability are crucial for ensuring accurate and consistent observations. These examples underscored the value of combining robust optics with finely tuned mechanical systems to achieve the highest level of performance.

Concluding Thoughts

This project, while initially appearing straightforward, has revealed the profound intricacies at the intersection of optical systems and mechanical design. What began as an investigation into the basic differences between refractive and reflective telescopes quickly evolved into a deeper exploration of how these systems function in real-world applications. The mechanical challenges of ensuring full sky coverage, combined with the optical considerations of capturing both nearby and distant objects, highlight just how complex even a seemingly simple project can become.

The truth is, there’s still much more to uncover in this field. Reflective telescopes, while proving superior in this study, are far from perfect. Their reliance on precise mechanical movement and intricate mirror alignments opens the door for further refinement. There are countless factors—from weather conditions to the quality of materials—that impact their performance, and each introduces new challenges for both the optics and mechanisms involved. In addition, sensor technologies and control systems are continually advancing, offering opportunities for greater precision and automation in future telescope designs.

Moreover, the interplay between optical physics and mechanical engineering requires ongoing research. The development of more advanced materials for lenses and mirrors, as well as more efficient servo systems and linkage designs, can transform how we think about observing the cosmos. In this paper, we’ve only scratched the surface of what is possible when these disciplines converge. The potential for innovation is immense, and future work could lead to breakthroughs that allow us to see farther, clearer, and in more detail than ever before.

In essence, this project has demonstrated that even small-scale undertakings are often layered with complexities that call for a deeper understanding and more collaborative research. Whether it’s optimizing the movement of a motor or refining the way light is focused through a mirror, each step presents a new challenge, and with it, new opportunities for discovery. As we

push forward, it’s clear that further exploration is not just beneficial—it’s essential for making meaningful strides in both astronomy and engineering.

References

1. MIT Decode. “Links.” Decode: MIT, decode.mit.edu/projects/links/. Accessed 11 Oct. 2024.

2. PyPi Project. “Mechanism.” PyPi, pypi.org/project/mechanism/. Accessed 11 Oct. 2024.

3. Cambridge Books. Excerpt from Manual. First Light Optics, www.firstlightoptics.com/user/manuals/cambridge_books_9781107619609_excerpt.pdf. Accessed 11 Oct. 2024.

4. Amazon. “Telescopes Astronomy Refractor Telescope Telescopio.” Amazon.ca, www.amazon.ca/Telescopes-Astronomy-Refractor-Telescope-Telescopio/dp/B094CDL8H5/. Accessed 11 Oct. 2024.

5. StackExchange Astronomy. “How Large Must an Object Be to Be Seen through a Telescope?” Astronomy StackExchange, astronomy.stackexchange.com/questions/11409/how-large-must-an-object-be-to-be-seen-throu gh-a-telescope. Accessed 11 Oct. 2024.

6. Essential Pathology. “Helpful Methods.” Essential Pathology, www.essentialpathology.info/gradingschema/index.html?HelpfulMethods. Accessed 11 Oct. 2024.

7. Amazon. “SkyWatcher S11620 Traditional Dobsonian 10-Inch.” Amazon.ca, www.amazon.ca/SkyWatcher-S11620-Traditional-Dobsonian-10-Inch/dp/B00Z4G3CW8/. Accessed 11 Oct. 2024.

8. Ecuip. “Reflective Telescopes.” University of Chicago Multiwavelength Astronomy, ecuip.lib.uchicago.edu/multiwavelength-astronomy/optical/history/04.html. Accessed 11 Oct. 2024.

9. Van Physics Illinois. “Reflective Telescopes.” Van Physics Illinois, van.physics.illinois.edu/ask/listing/2078. Accessed 11 Oct. 2024.

About the author

Gurdit Sekhon

Gurdit is a passionate learner with a keen interest in science, history, and problem-solving. His curiosity drives his involvement in projects blending robotics, physics, and innovative mechanism design—like building a 360-degree telescope with advanced motion control. Alongside his technical pursuits, Gurdit leads his school’s History Club and co-organizes interdisciplinary events, fostering collaboration and creativity.

Beyond academics, he finds joy in playing the harmonium, teaching at community camps, and tackling challenging math contests. Whether it’s organizing a club meeting, solving a tricky physics problem, or brainstorming ideas for his next invention, Gurdit strives to push boundaries and inspire others along the way.

The post Innovations in Telescope Motion Systems appeared first on Exploratio Journal.

Introduction 

Objects and planets in space are much bigger than daily objects we encounter on Earth and, therefore, they experience much larger gravitational forces that cause them to orbit around, collapse on, or escape from another object. The motion of extraterrestrial objects has always intrigued me, especially the NASA DART project, which is a mission to protect the Earth from potential asteroids impacting its surface. I find the collision of objects in space very interesting because the trajectory of the objects after colliding has to take in so many factors like the mass of the objects, their velocities, and any surrounding objects. 

Before I can explain more about the NASA DART project, however, I need to introduce the basics of gravitation and space physics. I will explain the different parts of space physics, like Newton’s universal law of gravitation, the acceleration of objects due to gravitational forces of the Earth and other objects, and escape speed, the speed it takes for an object to escape an object’s orbit. I will also go into the concept of gravitational potential energy, the energy an object has while in orbit, the energy required to place an object in orbit, and the nature of objects orbiting Earth, also known as Earth satellites. Additionally, I will explain Johannes Kepler’s famous 3 laws of planetary motion for a better understanding of how planets move in space. 

Finally, I will introduce the NASA DART (Double Asteroid Redirection Test), a mission where NASA tries to develop technology to protect the Earth in the unlikely event that an asteroid is headed for Earth. Their goal is to make an object, like a satellite, hit the asteroid, thus changing the trajectory of the asteroid and making it miss the Earth. 

This report is also complemented by Python code that simulates planetary motion. It is available for download on my GitHub here: https://github.com/alyang21/solarsystem 

2. Gravitation 

2.1 Universal Law of Gravitation

On Earth, the acceleration at which an object falls toward the Earth is a constant 9.8 m/s². However, this rate is different on other extraterrestrial objects. This is because the force of gravity exerted on an object depends on its mass as well as the mass of the objects around it. Knowing this, famed physicist Sir Issac Newton derived the Universal Law of Gravitation in 1687 [8]. His equation is

where G is the universal gravitational constant, at 6.67 x 10^-11. Furthermore, this equation suggests that the force depends on both objects’ masses and how far apart they are separated.  In vector form, the equation can be written as 

Furthermore, the sum of the forces on an object by the surrounding objects is just the vector sum of all the forces. 

Figure 1.1 The sum of two vectors is found by placing the two vectors tail to tip, and the resulting vector is from the tail of the first vector to the tip of the second. [1]

2.2 Acceleration Due to Gravity of the Earth

The Earth can be visualized as a number of spherical shells centered at the same point. Since the mass of all the shells combined is the mass of the Earth, and the force of gravity by the Earth comes from the center of the Earth. By taking into account the Earth’s density using its volume and mass, we can derive that the force of gravity by the Earth on an object is F_g = (GM_Em)/R_E² [8]. Since F_g = mg where g is the acceleration by the Earth according to Newton’s second Law, 

2.3 Gravitational Potential Energy

The gravitational potential energy of an object on Earth depends on its distance from the center of the Earth. We also know that work equals force multiplied by displacement, so the work done by the Earth to bring a body of mass m from the height h2 to the height h1 is given by:

In other words, the work done on an object is the difference of potential energy from the initial to final positions of the object. If we say that the potential energy W(h) at a height h above the surface of the Earth so that W(h) = mgh + W₀ where W₀ is a constant, then W₁₂ = W(h₂) – W(h₁) [8]. It is also important to note that h = 0 means points on the surface of the Earth.

If we lift the particle along a vertical path where r₁ is the distance from the center of the Earth at its first point and r₂ is the distance from the center at its second point, then we get

And as a result, 

2.4 Escape Speed

Using the law of conservation of energy, we can find the escape speed for an object out of a planet, or the speed it needs to break through the pull of the planet [8]. If we can find the distance where the object has no more potential energy and only kinetic energy, we can set the energies of the object at those two points equal to each other, thus allowing us to find the initial velocity that the object has to leave the planet with. 

As long as the final velocity is greater than or equal to 0, the object can reach infinity. So,

The initial velocity is the minimum velocity for the object to escape the atmosphere, so

If the object is thrown from the surface of the Earth, h = 0, and 

Thus, we come to the equation 

where R_E is the radius of the Earth. This means that the escape speed is independent of the object’s own mass. Additionally, with the knowledge of the Earth’s radius, we can find that the escape speed is 11.2 km/s.

2.5 Earth Satellites

Earth satellites are objects which revolve around the Earth, usually in the shape of an ellipse. The Moon is the only natural satellite of the Earth, and it has a near-circular orbit. Other satellites have been sent up by humans for telecommunication, geophysics, and meteorology. To find the period that these satellites orbit around the Earth once, we can use the equation for centripetal force, where m is the mass of the satellite and V is its speed [8].

This centripetal force is provided by the gravitational force, similar to equation (1.1) but after substituting the variables for the mass and radius of the Earth, we get 

Setting the two equations together, we find that 

A satellite travels a distance 2πR_E with speed V if the satellite is so close to the Earth’s surface that h can be neglected. The time period the satellite takes to orbit the Earth therefore is

and using the relation g = GM/R_E², we arrive at the equation

Substituting the numerical values, we get

Which is about 85 minutes.

2.6 Energy of an Orbiting Satellite

Notice how K is positive and U_g is negative. When added up, the total energy of the satellite is 

It makes sense that the satellite’s total energy is negative because if the total energy is positive, it would leave the orbit and escape to infinity. 

2.7 Energy Required to Orbit a Satellite

The energy required to put a satellite into Earth’s orbit is the difference between the satellite’s total energy in orbit and its energy at Earth’s surface. For example, if we want to lift the 9000-kg Soyuz vehicle from the Earth’s surface up to the ISS, which is 400 km above the Earth’s surface, we would have to find its energy at the Earth’s surface, as well as its total energy in orbit at the ISS. Using Eq 1.19, we get that the total energy of the Soyuz in the same orbit as the ISS is   where m is 9000 kg and h is 0. Plugging the numbers in, we get that E_orbit is -2.65 x 10¹¹ J. The total energy at the surface is just -GmM_e/R_e because E_surface = K_surface + U_surface and K_surface is 0. Plugging the numbers in, we get E_surface = -5.63 x 10¹¹ J. As explained earlier, the energy required is the change in energy, so the energy required is   = -2.65 x 10¹¹ – (-5.63 x 10¹¹) = 2.98 x 10¹¹ J [8].

2.8 Kepler’s Laws of Planetary Motion

After German astronomer Johannes Kepler obtained the data collected by Tycho Brahe, he was able to analyze the positions of all the known planets and our moon. He realized that the orbits of the planets around the sun were elliptical, and was able to come up with three basic laws of planetary motion [8].

Kepler’s first law states that all planets orbit along an ellipse, where the Sun is one of the foci of the ellipse. An ellipse is the set of all points where the sum of the distance from each point to the two foci is a constant. 

In an elliptical orbit, the point where the planet is the closest to the Sun is called the perihelion, which is represented by point A in Figure 1.4. The figure also shows point B, the farthest point from the Sun. This point is called the aphelion. 

The ellipse is a specific example of a conic section, given by the equation

The variables r and θ from Eq. 1.20 are shown in Figure 1.5. The other two variables,   and e, are constants determined by the total energy and angular momentum of the satellite at a point on the ellipse. The constant e is the eccentricity, which determines how close to being a circle the ellipse is. The closer to 0, the more circular the ellipse is, and the closer to 1, the flatter it is.

Kepler’s second law states that over equal periods of time, a planet will sweep out equal areas. In other words, the area it sweeps divided by the time, also known as the areal velocity, is a constant.

This makes sense when you consider that when the planet is closer to the Sun, it is moving faster. Since the energy of the planet-sun system is conserved, when the planet gets closer to the sun, its gravitational potential energy decreases, so its kinetic energy and velocity must increase.

Figure 1.7 The area ∂  swept out during time   as the planet moves through angle  . The angle between the radial direction of r and   is  . [4]

Kepler’s third law states that the square of the period is proportional to the cube of the semi-major axis of the orbit. For this law, we have the equation

In this equation, a is the semi-major axis of the ellipse and T is the period. Interestingly, this law can also be derived from Newtonian principles and the principle of conservation of energy [8]. Additionally, his equation applies to any satellite orbiting any large mass, not just our Sun. If we use this equation for a circular orbit of r about the Earth, we get

3. DART Project NASA

It is widely believed that millions of years ago, the dinosaurs were put into extinction when a meteoroid hit the surface of the Earth. Although no meteor has gotten close enough to Earth since then to cause humans to panic, the scientific community agrees that another meteor will eventually cross paths with the Earth. To combat this, NASA started the Double Asteroid Redirection Test, or DART, to see if it is possible to alter the course of an asteroid by sending an object to impact it. 

I first learned about DART when I visited the Kennedy Space Center in Florida and watched a video about its mission. I was immediately intrigued by DART because I had an interest in object collisions from playing pool and baseball. The DART mission added an interesting element that wasn’t involved when playing on a flat billiards table: the gravitational force of other extraterrestrial objects. This mission pushed me to learn about gravitation, planetary motion, and overall space physics in order to understand the DART mission from a scientific perspective.

DART’s target is the binary asteroid system Didymos. Since Didymos is not on a path that would impact the Earth, it is the ideal candidate for the first planetary defense experiment. The impact would be safe, even if something were to go wrong. The asteroid system consists of two asteroids: the larger asteroid named Didymos, and its moonlet, Dimorphos. DART’s plan was to collide with the moonlet Dimorphos, and then we would examine the changes in Dimorphos’ orbit as a result of the impact. 

The journey to Dimorphos was complicated and required many different state-of-the-art technologies. One was the Small-body Maneuvering Autonomous Real Time Navigation (SMART Nav), developed for guidance, navigation, and control (GNC). The system had to be autonomous because NASA cannot control a satellite when it is 11 million kilometers away from Earth. The system was able to distinguish between Didymos and Dimorphos, and accurately navigate to the moonlet, eventually colliding with the smaller asteroid. DART was also equipped with an ion propulsion system that is solar-powered and incredibly fuel-efficient. Speaking of solar-powered, DART had a Roll-Out Solar Array (ROSA), extending 8.5 meters in length on each side. These solar arrays were used before on the ISS, but DART was the first to use them on a planetary spacecraft. Finally, the LICIACube allowed the DART team back on Earth to see images of the impact and the ejecta cloud, helping them assess the impact and its effects on Dimorphos. These technologies, paired with great antennas to send and receive data from the satellite allowed the DART mission to be incredibly successful.

Figure 1.8 The three images above show the various technologies the DART satellite used throughout its mission. SMART Nav (left) helped the satellite accurately impact Dimorphos. ROSA (center) gave the satellite its power for its ion propulsion system (right). [5]

Figure 1.9 DART would impact Dimorphos from the direction Dimorphos is moving towards, slowing it down. This would cause Dimorphos’ new orbit to be closer to Didymos since its orbiting velocity decreased. At the same time, the LICIA Cube, which DART would eject 15 days before impact, would be able to capture images of the impact and send them back to Earth for NASA to examine. [6]

Overall, the DART project was a massive success, lifting off in November 2021 and colliding with Dimorphos in September 2022. However, the mission is not complete. The DART team is still examining the data from the impact in order to explore all the effects of the impact on Dimorphos. You can watch videos about the mission at this link: https://dart.jhuapl.edu/Gallery/ [7]

References 

[1] https://tikz.net/vector_sum/

[2] https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy

[3] https://openstax.org/books/university-physics-volume-1/pages/13-4-satellite-orbits-and-Energy

[4] https://openstax.org/books/university-physics-volume-1/pages/13-5-keplers-laws-of-Planetary-motion

[5] https://dart.jhuapl.edu/Mission/Impactor-Spacecraft.php

[6] https://dart.jhuapl.edu/Mission/index.php

[7] https://dart.jhuapl.edu/Gallery/

[8] This work is partially based on the content of this book: NCERT Books for Class 11 Physics, https://www.ncertbooks.guru/ncert-books-class-11-physics/amp/

Appendix

The following code computes the planetary motion of the Earth, Mars, and a fictional comet orbiting around the sun according to gravitational physics. The trajectories of these planets are calculated using Newton’s Universal Law of Gravitation, with given initial conditions for the position and velocities of each object. These trajectories are computed over a 5-year period and are visualized using an animation. The code is written in the Python language and is taken from this blog post: https://towardsdatascience.com/simulate-a-tiny-solar-system-with-python-fbbb68d8207b

Available on my Github page here: https://github.com/alyang21/solarsystem

# Ensure the right backend for Spyder

import matplotlib

matplotlib.use(“Qt5Agg”)

import matplotlib.pyplot as plt

from matplotlib import animation

# Constants and initial setup with constants and the objects’ masses, velocities, and gravitational constants.

G = 6.67e-11                  # constant G

Ms = 2.0e30                   # sun

Me = 5.972e24                 # earth        

Mm = 6.39e23                  # mars

Mc = 6.39e20                  # comet

AU = 1.5e11                   # earth sun distance

daysec = 24.0*60*60           # seconds of a day

e_ap_v = 29290                # earth velocity at aphelion

m_ap_v = 21970                # mars velocity at aphelion

commet_v = 7000               # comet velocity

gravconst_e = G*Me*Ms

gravconst_m = G*Mm*Ms

gravconst_c = G*Mc*Ms

# Starting positions

# earth

xe, ye, ze = 1.0167*AU, 0, 0

xve, yve, zve = 0, e_ap_v, 0

# mars

xm, ym, zm = 1.666*AU, 0, 0

xvm, yvm, zvm = 0, m_ap_v, 0

#comet

xc, yc, zc = 2*AU, 0, 0

xvc, yvc, zvc = 0, commet_v, 0

# sun

xs, ys, zs = 0, 0, 0

xvs, yvs, zvs = 0, 0, 0

t = 0.0

dt = 1*daysec

# these lists store the points that the objects are at

xelist, yelist, zelist = [], [], []

xmlist, ymlist, zmlist = [], [], []

xclist, yclist, zclist = [], [], []

xslist, yslist, zslist = [], [], []

# save the initial position in their respective lists

#earth

xelist.append(xe)

yelist.append(ye)

zelist.append(ze)

#mars

xmlist.append(xm)

ymlist.append(ym)

zmlist.append(zm)

#comet

xclist.append(xc)

yclist.append(yc)

zclist.append(zc)

# Simulation

# The new radii, forces, velocities, and positions are calculated at each second for 5 years. The new position is then added to the object’s list. 

while t < 5*365*daysec:

    ################ earth #############

    # compute G force on earth

    rx,ry,rz = xe – xs, ye – ys, ze – zs

    modr3_e = (rx**2+ry**2+rz**2)**1.5

    fx_e = -gravconst_e*rx/modr3_e     

    fy_e = -gravconst_e*ry/modr3_e

    fz_e = -gravconst_e*rz/modr3_e

    # update quantities how is this calculated?  F = ma -> a = F/m

    xve += fx_e*dt/Me

    yve += fy_e*dt/Me

    zve += fz_e*dt/Me

    # update position

    xe += xve*dt

    ye += yve*dt 

    ze += zve*dt

    # save the position in list

    xelist.append(xe)

    yelist.append(ye)

    zelist.append(ze)

    ################ mars #############

    # compute G force on mars

    rx_m,ry_m,rz_m = xm – xs, ym – ys, zm – zs

    modr3_m = (rx_m**2+ry_m**2+rz_m**2)**1.5

    fx_m = -gravconst_m*rx_m/modr3_m

    fy_m = -gravconst_m*ry_m/modr3_m

    fz_m = -gravconst_m*rz_m/modr3_m

    xvm += fx_m*dt/Mm

    yvm += fy_m*dt/Mm

    zvm += fz_m*dt/Mm

    # update position

    xm += xvm*dt

    ym += yvm*dt 

    zm += zvm*dt

    # save the position in list

    xmlist.append(xm)

    ymlist.append(ym)

    zmlist.append(zm)

    ################ comet ##############

    # compute G force on comet

    rx_c,ry_c,rz_c = xc – xs, yc – ys, zc – zs

    modr3_c = (rx_c**2+ry_c**2+rz_c**2)**1.5

    fx_c = -gravconst_c*rx_c/modr3_c

    fy_c = -gravconst_c*ry_c/modr3_c

    fz_c = -gravconst_c*rz_c/modr3_c

    xvc += fx_c*dt/Mc

    yvc += fy_c*dt/Mc

    zvc += fz_c*dt/Mc

    # update position

    xc += xvc*dt

    yc += yvc*dt 

    zc += zvc*dt

    # add to list

    xclist.append(xc)

    yclist.append(yc)

    zclist.append(zc)

    ################ the sun ###########

    # update quantities how is this calculated?  F = ma -> a = F/m

    xvs += -(fx_e+fx_m)*dt/Ms

    yvs += -(fy_e+fy_m)*dt/Ms

    zvs += -(fz_e+fz_m)*dt/Ms

    # # update position

    xs += xvs*dt

    ys += yvs*dt 

    zs += zvs*dt

    xslist.append(xs)

    yslist.append(ys)

    zslist.append(zs)

    # update dt

    t +=dt

print(‘data ready’)

# Animation setup

# grid size

fig, ax = plt.subplots(figsize=(6,6))

ax.set_aspect(‘equal’)

ax.grid()

# earth is blue. The text “Earth” follows point_e as it moves

line_e, = ax.plot([], [], lw=1, c=’blue’)

point_e, = ax.plot([AU], [0], marker=”o”, markersize=4, markeredgecolor=”blue”, markerfacecolor=”blue”)

text_e = ax.text(AU, 0, ‘Earth’)

# mars is red. The text “Mars” follows point_m as it moves

line_m, = ax.plot([], [], lw=1, c=’red’)

point_m, = ax.plot([1.666*AU], [0], marker=”o”, markersize=3, markeredgecolor=”red”, markerfacecolor=”red”)

text_m = ax.text(1.666*AU, 0, ‘Mars’)

# comet is black. The text “Comet” follows point_c as it moves

line_c, = ax.plot([],[], lw=1, c=’black’)

point_c, = ax.plot([2*AU], [0], marker=”o”, markersize=2, markeredgecolor=”black”, markerfacecolor=”black”)

text_c = ax.text(2*AU,0,’Comet’)

# the sun is yellow

point_s, = ax.plot([0], [0], marker=”o”, markersize=7, markeredgecolor=”yellow”, markerfacecolor=”yellow”)

text_s = ax.text(0, 0, ‘Sun’)

ax.axis(‘equal’)

ax.set_xlim(-3*AU, 3*AU)

ax.set_ylim(-3*AU, 3*AU)

exdata, eydata = [], []

mxdata, mydata = [], []

cxdata, cydata = [], []

# The points for each object are put into their respective data sets to be plotted on grid

def update(i):

    exdata.append(xelist[i])

    eydata.append(yelist[i])

    mxdata.append(xmlist[i])

    mydata.append(ymlist[i])

    cxdata.append(xclist[i])

    cydata.append(yclist[i])

    line_e.set_data(exdata,eydata)

    point_e.set_data(xelist[i],yelist[i])

    text_e.set_position((xelist[i],yelist[i]))

    line_m.set_data(mxdata,mydata)

    point_m.set_data(xmlist[i],ymlist[i])

    text_m.set_position((xmlist[i],ymlist[i]))

    line_c.set_data(cxdata,cydata)

    point_c.set_data(xclist[i],yclist[i])

    text_c.set_position((xclist[i],yclist[i]))

    point_s.set_data(xslist[i],yslist[i])

    text_s.set_position((xslist[i],yslist[i]))

    ax.axis(‘equal’)

    ax.set_xlim(-3*AU,3*AU)

    ax.set_ylim(-3*AU,3*AU)

    #print(i)

    return line_e,line_m,line_c,point_s,point_e,point_m,point_c,text_e,text_s,text_m,text_c

anim = animation.FuncAnimation(fig, func=update, frames=len(xelist), interval=1, blit=False)

plt.show(block=True)

About the author

Alexander Yang

Alex is currently a 12th grader at the Livingston High School. He is a dedicated singer-student-athlete with a passion for Math and Physics who is fascinated with data analysis and calculations related to aerospace. He founded his high school’s Rocketry Club, competing in the American Rocketry Challenge and also holding educational community launches to spark interest in rocketry and aerospace. Alex has been a part of his school’s Math Team for all four years of high school, and rising to the Math Honor Society’s Vice President in his Junior year. He was also a camp counselor at the Delaware Aerospace Academy, teaching young students about aviation, space, and rockets. He taught the students to construct and launch model rockets, maglev trains, and solar robots.

In addition to these activities, Alex also plays varsity baseball for his school, being the starting second baseman and starting shortstop in his sophomore and junior years respectively. He has also been an active singer, singing in his school chorus, select chorus, and an outside volunteer chorus. He has auditioned into the NJ All-State Chorus both of the last two years, and he is currently ranked 6th in the state in the Tenor 1 voice part. He is deeply interested in math, data science, physics, and computer science and would like to apply his math and physics knowledge to improve technology. Alex looks to further his knowledge and interest in STEM by studying data science related topics in higher education.

The post Space Physics: The motion of extraterrestrial objects appeared first on Exploratio Journal.

Introduction

In the realm of aviation, there is a constant quest to achieve precision and efficiency. As  modern aircraft technology advances and global air travel demands continue to rise, optimizing flight variables becomes increasingly important. The pursuit of optimal flight variables is driven  by two necessities, safety and economics. The safety of passengers and crew is paramount in aviation, and calculations of aircraft performance are fundamental in ensuring the flight’s safety  and success. The other important factor in the pursuit of optimal flight variables is economic  viability. In a world of finite resources, the management of fuel consumption along with  numerous other variables directly impacts the flight’s economic success.

Over the last couple of months, under the guidance of Dr. Ella Atkins, I have studied various aspects of flight for fixed-wing aircraft systems. I have been able to learn valuable fundamental principles in aircraft systems, manipulating and applying mathematical formulas to the various parts of an aircraft system. I have been able to apply my newfound mathematical skills in an engineering software known as MATLAB, which I learned under the guidance of Professor Atkins. With my knowledge of fixed-wing aircraft systems, I will be able to find steady flight variables for a twin jet engine aircraft and set up the necessary variables for a flight plan from Denver, Colorado to Anchorage, Alaska. First, we will dive into the parameters of the aircraft and how they are used. We will then discuss atmospheric pressure, temperature, and density, and how those variables will play a role in our flight. Once we have all of our parameters and our atmospheric variables, we will begin calculating variables that are required for steady-level flight, steady-level turning flight, climbing flight, and descending flight. All of the calculations will be made through MATLAB, and all the code used in this paper can be found in the appendix. Through this paper, we will come to have a better understanding of the factors that come into play when determining an aircraft’s flight plan.

 Aircraft parameters

 The first few variables we will discuss are weight variables, with max weight and fuel weight being given. This allows us to find the weight of the aircraft at different stages of flight, so for climbing flight, we will use max weight, for steady flight, we will use the difference between max weight and half the fuel weight, and for descending flight, we will use the difference between max weight and fuel weight (assuming that almost all of the fuel is used up by the time of descent). While weight is measured in pound-force (lbf), we must convert it to a force in Newtons (N) using the ratio of one lbf per 4.4482189 N. Another variable that is given in lbf is the max thrust which we will convert to N using the ratio above.

 The next three variables are factors that describe the shape and geometry of the wings of the airplane. The first variable, planform area, is the total area of both wings from a top-down view. Planform area is measured in square meters and generally determines the lift and drag generated by the airplane. The next variable is span, also known as wingspan. Span is the distance from the tip of one wing to the other wing and is measured in meters. Lastly, the aspect ratio is the relationship between span and planform area. It is calculated by the quotient of the span squared and the planform area.

 Dimensionless coefficients are values with no units that can be scaled up or down to accommodate better testing. There are numerous dimensionless coefficients in aviation, with aspect ratio being a great example. The aspect ratio allows a wing to be scaled down on that ratio so that engineers can test its properties in wind tunnels. Along with aspect ratio, other given dimensionless coefficients are maximum lift coefficient, zero-alpha lift coefficient, lift slope, parasitic drag coefficient, and span efficiency factor.

 These next few dimensionless coefficients will deal with the lift coefficient. The lift coefficient is the ratio of the aircraft’s lift over the product of force times planform area. The maximum lift coefficient, which is given as 2.79 for our calculations, is the lift coefficient when the angle of attack (the angle between the aircraft’s nose and the x-axis, also known as alpha) is greatest. The zero-alpha lift coefficient is the lift coefficient when the angle of attack is zero (the aircraft is directed parallel to the x-axis). Lastly, the lift slope represents the change in lift coefficient over the change in angle of attack. This means that it measures how much more lift can be generated by increasing the angle of attack.

 Our final two dimensionless coefficients to discuss are the parasitic drag coefficient and span efficiency factor. The parasitic drag coefficient is a constant due to viscosity in an aircraft which is part of determining the overall drag coefficient. Along with parasitic drag, the other important factors in drag coefficient are lift coefficient, aspect ratio, and span efficiency. As we have already talked about lift coefficient and aspect ratio, we only have span efficiency to discuss. The span efficiency factor is a constant between zero and one that depends on the shape of the wing and evaluates the efficiency of the lift distribution along the wing’s span. The closer the value of span efficiency is to one, the more effective the lift distribution is for the plane.

The values for all of the given variables discussed can be found in Table 1.

Atmospheric temperature, pressure, and density as functions of altitude

 One of the most important factors when doing calculations related to aircraft flight is the atmospheric conditions. There are three ways to measure altitude in aerospace engineering: absolute, geometric, and true altitude. Absolute altitude is the height above the center of the Earth and is mainly used in space travel. True altitude, which is useful when flying very low to the ground, measures the height above a point on the surface of the earth and is only called upon for aircraft above mountain ranges or other tall structures. Geometric altitude is the most straightforward concept of altitude, representing how high an object is above the Earth’s mean sea level (MSL). Generally, geometric altitude is used when describing the altitude and will be used for this paper’s calculations.

When it comes to aircraft, the three key factors that change as elevation increases are temperature, pressure, and density. Temperature is the average kinetic energy of a gas molecule and fluctuates at various intervals in our atmosphere. The levels of the atmosphere can be divided into two groups when discussing temperature. The first group is isothermal layers, which are atmosphere layers that maintain a constant temperature. The other group is constant-gradient layers, which change in temperature linearly with altitude at a specific lapse rate. The temperature and lapse rate at different points in the atmosphere can be seen below (Table 2).

 Atmospheric pressure is the force per unit area exerted by the weight of the air above a certain point. The weight of the air column is influenced by gravity and the density of the air. As one moves higher in the atmosphere, the weight of the air decreases, resulting in lower atmospheric pressure. Pressure is measured in force per unit area (N/m²), also known as Pascals (Pa), and the standard atmospheric pressure at sea level is 101325 Pascals (Pa). Knowing the standard atmospheric pressure at sea level and the temperature at different levels in the atmosphere allows us to calculate the pressure at a specific height using two different equations, one pertaining to isothermal layers, and the other pertaining to constant-gradient layers. In isothermal layers, the equation for pressure change is given.

 In this equation and many others used in physics, g is the gravitational constant (acceleration due to gravity on Earth) measured as 9.81 m/s², R is the gas constant for air measured as 287.053 J/(kg * k), and T is the temperature measured in K. By integrating the equation from the bottom of the layer to the top of the layer using h₁/p₁ as altitude/pressure at the bottom of the layer and h₂/p₂ as altitude/pressure at the top of the layer, we can simplify the equation to solve for p₂.

 The equation for constant gradient layers is given below.

 It is almost the same equation as the equation for isothermal layers, but the value of dh (change in height) is replaced by dT / T (change in temperature over temperature). Another change is the introduction of the symbol ξ, which represents the lapse rate. The integral of this equation from the bottom of the layer to the top of the layer using T₁/p₁ as temperature/pressure at the bottom of the layer and T₂/p₂ as temperature/pressure at the top of the layer gives us the equation for p₂.

With the equations for temperature and pressure at different altitudes, we are able to calculate the final and most important value for aviation, air density. The standard density is used in numerous calculations involving lift and drag, which means pretty much every calculation involving aircraft flight. Standard density, represented by the Greek letter rho (ρ), can be measured using the ideal gas law shown below.

 Manipulating the variables allows us to put density in terms of temperature and pressure, as shown below.

 Thus, all of the calculations for temperature and pressure have culminated in us being able to find density at different points in the standard atmosphere, which, as said before, is extremely important for our calculations of steady flight, turning flight, climbing flight, and descent flight. The values for temperature (K), pressure (Pa), and density (kg / m³) from sea level to 100,000 meters can be seen in Figure 1.

As we can see, standard pressure and density follow a logarithmic pattern as altitude increases. This is due to the fact that at lower altitudes, the force of gravity compresses the air at a much higher rate than it does when altitude rises, so the air pressure and density are much higher at lower altitudes. As we look to optimize values such as airspeed, throttle, and angle of attack, it is important to recognize how temperature, pressure, and density play a significant role in our calculations.

Steady-level flight parameters

 In steady-level flight for a twin-jet engine airplane, the most important variables are airspeed (velocity), thrust, throttle, and altitude. If we were discussing a propeller airplane, we would have to also account for power, which is the force that is created by a propeller instead of thrust, which is created by a jet engine, however, we will not discuss power for now. The first variable to discuss is altitude, which we will set at one km or 10,000 meters for our calculations. 10,000 meters, which equates to about 32,000 feet, is the general cruising altitude for subsonic airplanes, as it is known to be most efficient in terms of drag due to the air density at 10,000 m. With altitude out of the way, we will move on to discuss the basics of steady-level flight.

Steady level flight means that all forces acting on the airplane are balanced, with lift equaling weight and thrust equaling drag, resulting in zero acceleration and a constant velocity. This is due to the equation F = ma, which means that force = mass x acceleration. Also known as Newton’s Second Law, this basic principle of physics will allow us to substitute lift with weight and thrust with drag during our calculations. The relationship between thrust and velocity which will allow us to calculate both stems from the equation for drag, given below.

 In this equation, we already know density (ρ) and planform area ( S ). We still need a constant to replace Cᴅ, which is the coefficient of drag, which is what the three equations below will help us do.

Now, we have parasitic drag ( C ᴅ₀), aspect ratio ( AR ), and span efficiency factor ( e ), all known variables. We can find a value for the coefficient of lift through the next equation.

 Since the angle of attack ( ) is equal to zero, the value for the coefficient of lift equals the value α of the zero-alpha lift coefficient ( C ⳑ₀), which is a known variable. Lastly, we will replace the known value for the coefficient of lift by putting it in terms of lift ( L ) using the equation below.

 The final equation for drag in terms of lift is then shown below.

 Since we are in steady-level flight, we can replace the drag force with thrust, and the lift force with weight, which turns the equation into a thrust equation with velocity unknown, shown below.

The resulting equation can create a graph with thrust as the y-axis and velocity as the x-axis, which I created below using the Desmos Graphing Calculator (Figure 2).

The optimal velocity and thrust are found at the minimum on the velocity vs. thrust graph, which we can find by setting the derivative of the thrust equation with respect to velocity equal to zero. After taking the derivative of the thrust equation with respect to velocity, setting it equal to zero, and simplifying it to isolate velocity, we can find the minimum velocity equation, shown below.

 Thus, we have put velocity into an equation in which we know all of the constants, giving us a steady-level flight velocity of 183.7885 m/s. Now that we know the velocity, we can plug it back into the thrust equation, giving us a thrust force of 15231 N. As seen in Figure 2, these values correspond to the coordinates at the minimum of the thrust vs. velocity graph. Our final variable to calculate for steady-level flight is throttle. The throttle is a dimensionless value between zero and one which can be changed by the pilot to adjust airspeed and thrust. The equation for thrust in relation to the throttle is given below.

The equation can be simplified to isolate throttle (δ_t ). The equation uses thrust ( T ), density ( ρ), and density at sea level ( ρs ), all known values. The other values T^s_max, and m, must be changed for our calculations. The max thrust (T^s_max) must be multiplied by two because the original max thrust is the value for one of the two jet engines. The value of m is a constant that changes based on the specific aircraft. For our aircraft, the value of m is 1/2. Knowing all of these constants, the value for throttle for our aircraft can be calculated as 0.4337, also known as 43.37% throttle.

Steady-turning flight parameters

 In calculating steady-turning flight variables, the basic principles of balancing forces to create a constant velocity and zero acceleration still apply. The main difference comes in calculating all of our variables to adjust for a higher necessary lift. The changes in thrust, drag, lift, and weight forces on the airplane can be seen in Figure 3.

The value for phi ( φ ) is known as the turn/bank angle, and for our calculations will be 25 degrees. Knowing the value for the bank angle, we are able to calculate lift using the equation below.

 The value for lift is important because it allows us to calculate the value for load factor ( n ). Load factor is a dimensionless value that measures the ratio between lift and weight, and is normally one in steady-level flight, as lift and weight are equal in steady-level flight. The load factor is calculated using the equation below.

 Since we are calculating for steady turning flight, the values of lift (L) and weight (W) are not equal, and as a result, the load factor is greater than one. After calculating the value of lift during steady-turning flight, we are able to calculate the load factor during steady-turning flight, which is 1.1034 at a bank angle of 25 degrees. This is slightly more than the load factor at steady-level flight, meaning if you were sitting on the airplane, you would feel more than normal pressure on yourself when the airplane turns.

 The next variable that is to be calculated is the turning radius, which we will use more physics principles to derive an equation for. As said before, F = ma , also known as Newton’s second law, tells us that force = mass x acceleration. In centripetal motion, the F in F = ma signifies the centripetal force, which is the force directed toward the center of the turning arc, which is calculated as Lsin(φ) . Another important aspect of centripetal motion is that the acceleration in centripetal motion is equal to the difference between velocity squared and radius.

 The mass in this situation can be classified as W/g, with g being the force of gravity. By dividing by the weight value (W), which equals Lcos(φ) , we can get tan(φ) as the value on the left of the equation. As a result of all of this, we are left with the following equation.

We know the bank angle (φ), velocity (V) is the value calculated in steady-level flight, and the force of gravity (g) is 9.8 m/sec² . Knowing all of these constants, we can find the turning radius (R) to be 739.16 meters.

 The next couple of values we will calculate are thrust and throttle. All of them will be slightly more than they were for level flight, as we will need to generate more lift in turning flight. The equation for thrust in turning flight is shown below.

The equation is nearly the same as that of thrust for level flight, with the only addition being the value for (cos(φ))² added. Since we are using the velocity from steady-level flight, we can simply plug in all of the known variables, which outputs a thrust value of 16,886 N of force. Using this thrust value, we can the same equation we used for throttle in steady-level flight, finding our throttle in turning flight to be 0.4809, more easily referred to as 48.09% throttle. This value is about five percent greater than our throttle for steady-level flight.

Climbing and descending flight parameters

 Let’s move on to discuss climbing flight variables. In climbing flight, a new angle is added, the flight path angle (γ). The importance of flight path angle is that it affects the thrust required in the flight. The new value for thrust is shown below.

No longer can we just replace the D with its values, derive with respect to V , and set the derivative equal to zero. We must account for the unknown gamma (γ) value. One way to remove the gamma value from the equation is by using the equation below.

As a result of this equation, we can replace the Wγ in the thrust equation resulting in a new equation.

By isolating the equation above so that Vclimb is alone (the rate of climb), we are able to take the derivative of that equation and set it equal to zero to maximize Vclimb . Since the new equation 14 still has a factor of thrust in it, we can make an assumption of operating at full throttle, resulting in the thrust being calculable with the equation.

Knowing all of the values in the thrust equation, and multiplying by two because of the T^s_max two jet engines, we can find the climbing thrust. We can also find the V in equation using equation 12, which is the equation that puts thrust as a function of velocity. Knowing the thrust in this equation, we can calculate V . Now that we know V , we can plug both velocity and thrust into the rate of climb equation shown below.

After finding Vclimb we know almost every variable to complete out climbing flight. We are able to calculate the value for our angle of attack (γ) using equation 20 and we will also be able to do basic estimates for our climbing flight, such as determining the total time for the climb, altitude at certain points during the climb, or distance traveled during the climb. The final calculations we will have to make will be our descending flight calculations. 

The same principles that apply to climbing flight also apply to descending flight, with the only change required being a negative flight path angle (γ). While in climbing flight you can optimize for the flight path angle to be greatest, in descending flight, there is no optimization for the lowest possible flight path angle as it would result in an uncontrollable nosedive. Therefore, we can choose a safe flight path angle for descending flight and use equations 19-21 to calculate all of our descent variables, which we can use for estimates about total descent time, altitude at certain points during descent, or distance traveled during descent, ensuring a safe and efficient descent.

Conclusion

 My experience learning from Dr. Atkins has given me valuable knowledge about aircraft systems that are heavily applicable to ensuring safety and efficiency in aviation. All of the steady flight, turning flight, climbing, and descending parameters calculated will allow for a flight plan to be created using waypoints from Denver to Anchorage. The calculations in this analysis of a twin-jet engine aircraft show how basic principles of physics and mathematical manipulations are used in the world of aviation. These calculations, used for all fixed-wing aircraft, allow for resource-efficient and safe travel. They save time, lower costs, reduce excess fuel usage, and lessen the environmental impact of the industry, all of which are essential resources for Earth and its people. In summary, efficient calculations in aviation are fundamental to the well-ordered functioning of the aviation industry and the well-being of all stakeholders involved.

References

Fidkowski, C., Atkins, E., & Powell, K. (2019). Introduction to Aerospace Engineering . Ann Arbor, MI: University of Michigan.

Appendix

 Aircraft parameters

 Atmospheric temperature, pressure, and density as functions of altitude

 Steady-level flight and steady-turning flight parameters

About the author

Rahul Gupta

Rahul is a senior at Vandegrift High School in Austin, TX. After high school, he is looking to major in aerospace engineering or possibly double major in mechanical engineering. Outside of academics, Rahul loves anything soccer-related and is a varsity soccer player for his high school.

The post Denver to Anchorage: A Detailed Optimization of Flight Variables appeared first on Exploratio Journal.

Abstract

The Surveillance, Epidemiology, and End Results (SEER) program by the National Cancer Institute provides relative survival by time since diagnosis graphs for different cancers at different age groups [9]. This data is invaluable in computing cancer treatment and survivorship statistics, providing critical insights into disease prognosis and patient outcomes [2]. However, a primary limitation in this data lies in the expansive age groups used (<15, <20, 15-39, 40-64, 50-64, 65-74, 65+, and 75+), which fail to accurately represent survival rates for specific ages. As cancer survival rates can differ based on the age of individuals, and considering that the age composition of cancer patients differs across populations, adjusting for age is essential when comparing cancer survival rates [12]. This study proposes a mathematical and computational approach using polynomial regression to create accurate and precise survival curves for exact ages without requiring additional cancer statistics. Polynomial regression is applied to relative survival rates for specific age groups at different times since diagnosis, generating functions that map age to relative survival rates. Then, these functions are used to create age-specific survival curves. Age-specific survival curves can provide patients and healthcare providers with more accurate survival data for developing treatment programs, while public health systems can optimize medical resource allocation.

Introduction

The Surveillance, Epidemiology, and End Results (SEER) program curates and provides cancer statistics procured from different regions in the USA, offering insights into cancer incidence, prevalence, treatment, and survival rates [1]. Cancer survival rates are essential indicators for assessing treatment efficacy, monitoring patient outcomes, and guiding healthcare decisions. Relative survival rates consider both cancer-related deaths and non-cancer-related deaths and compare the survival rates of cancer patients to those of the overall population, taking into consideration factors such as age, race, and gender [1]. This makes them a preferred metric for assessing cancer-specific survival. However, as the curves are for set age ranges, the data fails to effectively capture survival rates for specific ages. For example, the relative survival rate for acute myeloid leukemia (AML) in the age range 15-39 after one year since diagnosis is 78.9 ± 0.4%, while the relative survival rate for AML in the age range 40-64 after one year since diagnosis is 59.4 ± 0.3%. Therefore, it is challenging to determine the precise survival rate for individual ages within these age ranges.

Polynomial regression is a sophisticated statistical technique to elucidate intricate relationships between variables. It involves fitting a polynomial function to observed data points. Unlike simple linear regression, which assumes a linear relationship between variables, polynomial regression accommodates non-linearity by employing higher-degree polynomial equations. This allows for a more flexible relationship between factors.

This study presents an approach that leverages polynomial regression to construct relative survival by time since diagnosis graphs for exact ages without having to collect additional cancer statistics to determine survival rates for individual ages. This is done by, first, plotting relative survival rates against different ages for each year since diagnosis. Then polynomial regression is performed on these graphs to obtain functions that map a certain age to a relative survival rate for a specific period since diagnosis. The outputs of these functions are then used to calculate the points on the relative survival by time since diagnosis graph for exact ages.

Methods

To illustrate the process of constructing age-specific survival curves, data from SEER on acute myeloid leukemia has been utilized. However, the methods outlined in the study can be used on other relative survival by time since diagnosis graphs of other cancers (see Figure 8).

Data Acquisition and Cleaning

Relative survival rates by time since diagnosis data for acute myeloid leukemia was downloaded in a comma-separated values (CSV) file from SEER [9]. The age ranges in the CSV file were: <15, <20, 15-39, 40-64, 50-64, 65-74, 65+ and 75+. Before applying polynomial regression, the raw CSV data was processed using the ‘pandas’ Python library to remove irrelevant or incomplete information (the ‘pandas’ package provides support for working with different datasets [4]). The CSV file was loaded onto a pandas DataFrame object, and rows that did not contain column headings or survival rates were removed, ensuring the data’s integrity and consistency.

Selection of Age Groups for Regression

A diverse range of survival rates is required to optimize the accuracy of the curve fitting while performing polynomial regression. To achieve this, specific age groups are selected from the dataset, ensuring a wide representation of survival rates. Relative survival rates for AML in the ages 15, 39, 64, and 74 were selected. These ages correspond to the age groups <15, 15-39, 50-64, and 65-74respectively, in the cleaned CSV data. This selection can be modified depending on the availability of data for different cancers.

By modifying the age groups used to represent a broader spectrum of survival rates, potential fluctuations in the trend of the survival curve can be minimized. This enhances the robustness of the polynomial regression and improves the accuracy and precision of interpolated survival data.

Performing Polynomial Regression

With the selected age-specific survival rates, we apply polynomial regression to fit a polynomial curve to the data points. The polynomial regression model aims to find the best-fitting curve that maps ages to relative survival rates. In this case, polynomial regression of the second order is employed, yielding a quadratic function that can effectively approximate the underlying survival trend. However, depending on the trend of other cancers, regression to the third or even fourth order might be necessary.

To perform polynomial regression the pandas, Scikit-learn, and NumPy Python libraries were used. The pandas package was used to load the cleaned DataFrame containing relative survival rates of the chosen age ranges: <15, 15-39, 50-64, and 65-74.

NumPy is a fundamental package for scientific computing in Python [5]. It provides support for matrices, arrays, and mathematical functions to operate on these arrays [5]. The ‘array()’ function in NumPy is used to create an array of the chosen ages (15, 39, 64, and 74). The ‘reshape()’ function with the arguments ‘reshape(-1, 1)’ is used to restructure this array into a single column. Survival rates for these ages are taken from the data provided by SEER and stored in the list ‘survival_data.’

Scikit-learn is a Python package that assists with data analysis, statistical modeling, and machine learning. It also provides support for performing different types of regression [6]. ‘LinearRegression’ is a class from the ‘sklearn.linear_model’ module in the Scikit-learn package. It fits a linear model to input data by finding the coefficients that minimize the residual sum of squares between observed and target values.

‘Polynomial Features’ is another class from the ‘sklearn.preprocessing’ module in Scikit-learn. It generates polynomial features based on input data, enabling the model to capture non-linear relationships between the features and target variables.

The equation of the line of best fit from the polynomial regression can be represented as follows, where y is the relative survival rate (%) and x is age (years):

y = —0.0187x² + 0.9386x + 72.7471.

To assess the goodness of fit for this regression, an R-squared value was calculated. The R-squared value measures the proportion of the variation in the dependent variable (relative survival rates) that is predictable from the independent variable (age). A high R-squared value (close to 1) indicates a strong correlation between age and relative survival, supporting the accuracy of the interpolation. The R-squared value for the above regression was 0.9853. This process was repeated for all the other years since diagnosis (2 – 10 years). The results are shown in Table 1.

Results

After calculating the equations of the lines of best fit in Table 1, survival rates for different times since diagnosis for specific ages can be calculated by passing different ages as inputs into the equations of the lines of best fit. Ideally, ages less than 15 and greater than 85 should be ignored as most clinical trials do not account for these ages, and notable changes in physiology occur in geriatric cases and infants [7][8]. Arbitrary ages of 25, 35, 45, and 55 were taken to illustrate age-specific survival curves, but any other age can also be chosen.

These ages were given as inputs to each of the equations calculated in Table 1. The results are shown in Table 2.

The relative survival rates calculated in Table 2 can then be plotted against time since diagnosis for each age to produce age-specific relative survival by time since diagnosis graphs. This is shown in Figure 7.

To assess the accuracy of the constructed survival curves, a Euclidean pairwise distance measure is used. Pairwise distance measures help quantify the similarity or dissimilarity between data points or objects. In Euclidean distance measures, the straight-line distance between two points in a Euclidean space is measured. It is calculated as the square root of the sum of squared differences between corresponding elements of two lists. The Python package ‘SciPy’ provides support for calculating Euclidean distance measures in the ‘scipy.spatial.distance’ module [11]. This was used to calculate distance measures between the constructed age-specific survival curves and survival curves from SEER, which are displayed in Table 3.

Age-specific survival curves for other cancer types can also be constructed using the same methods. Survival curves for the age of 35 for glioblastoma, adenocarcinoma of the esophagus, acute monocytic leukemia and acute lymphocytic leukemia were constructed (see Figure 8) and their pairwise distance measures from SEER’s data for the age-range 15-39 were calculated (see Table 4).

Conclusion

The results obtained from polynomial regression indicate high goodness of fit, as evidenced by the high R-squared values ranging from 0.9850 to 0.9928. This indicates a strong correlation between age and relative survival rates, validating the accuracy of the interpolation method. The pairwise distance measures are also relatively small. Three of the four distance measures for the survival curves for AML are less than 20% and the pairwise distance measures for glioblastoma, adenocarcinoma of the esophagus, acute monocytic leukemia and acute lymphocytic leukemia are all less than 20%, indicating that the age-specific graphs are spatially close to SEER’s data. However, at higher ages (55 years for AML) the pairwise distance is larger than 20%. This is due to interference from low relativesurvival rates, especially from the age group of 65-74 years. The availability of a greater number of narrower age ranges would greatly help offset the effects of low survival rates at higher ages.

The proposed methodology demonstrates a novel and effective way to construct age-specific relative survival curves from SEER’s age-group based data. By utilizing polynomial regression, accurate survival data for exact ages can be interpolated, addressing the limitations of the expansive age groups provided by SEER.

In conclusion, the presented novel polynomial regression technique provides an efficient means of constructing precise and accurate age-specific survival curves without the need for additional cancer statistics, especially for younger ages. As cancer statistics are updated, it is hoped that these graphs will eventually offer significant value to healthcare providers, patients, and public health systems. For healthcare providers, the availability of accurate survival data for specific ages aids in developing tailored treatment plans, optimizing therapy choices, and enhancing patient outcomes. Patients, too, benefit from more personalized information that empowers them to make informed decisions about their treatment journey. Moreover, public health systems can utilize this detailed survival data to allocate medical resources effectively and ensure equitable access to quality cancer care across all age groups.

In the future, it is hoped that more comprehensive survival data with narrower age ranges will be available, further enhancing the accuracy of interpolation [10].

References

1. DeSantis, C.E., Lin, C.C., Mariotto, A.B., Siegel, R.L., Stein, K.D., Kramer, J.L., Alteri, R.,Robbins, A.S. and Jemal, A. (2014), Cancer treatment and survivorship statistics, 2014. CA: A Cancer Journal for Clinicians, 64: 252-271. https://doi.org/10.3322/caac.21235
2. Understanding Statistics Used to Guide Prognosis and Evaluate Treatment. (2010, April 8). Cancer.Net. https://www.cancer.net/navigating-cancer-care/cancer-basics/understanding-statistics-used-guide-prognosis-and-evaluate-treatment
3. J. D. Hunter, “Matplotlib: A 2D Graphics Environment,” in Computing in Science & Engineering, vol. 9, no. 3, pp. 90-95, May-June 2007, doi: 10.1109/MCSE.2007.55.
4. The pandas development team. (2023). pandas-dev/pandas: Pandas (v2.1.0rc0). Zenodo. https://doi.org/10.5281/zenodo.8239932
5. Harris, C.R., Millman, K.J., van der Walt, S.J. et al. Array programming with NumPy. Nature 585, 357–362 (2020). https://doi.org/10.1038/s41586-020-2649-2
6. Scikit-learn: Machine Learning in Python, Pedregosa et al., JMLR 12, pp. 2825-2830, 2011.
7. Zhang E, DuBois SG. Early Termination of Oncology Clinical Trials in the United States. Cancer Med. 2023; 12: 5517-5525. doi: 10.1002/cam4.5385
8. Shenoy, P., & Harugeri, A. (2015). Elderly patients’ participation in clinical trials. Perspectives in clinical research, 6(4), 184–189. https://doi.org/10.4103/2229-3485.167099
9. SEER*Explorer: An interactive website for SEER cancer statistics [Internet]. Surveillance Research Program, National Cancer Institute; 2023 Apr 19. [updated: 2023 Nov 16; cited 2023 Dec 14]. Available from: https://seer.cancer.gov/statistics-network/explorer/. Data source(s): SEER Incidence Data, November 2022 Submission (1975-2020), SEER 22 registries (excluding Illinois and Massachusetts). Expected Survival Life Tables by Socio-Economic Standards.
10. Simon, R., Altman, D. Statistical aspects of prognostic factor studies in oncology. Br J Cancer 69, 979–985 (1994). https://doi.org/10.1038/bjc.1994.192
11. Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, CJ Carey, İlhan Polat, Yu Feng, Eric W. Moore,Jake VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E.A. Quintero, Charles R Harris, Anne M. Archibald, Antônio H. Ribeiro, Fabian Pedregosa, Paul van Mulbregt, and SciPy 1.0 Contributors. (2020) SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, 17(3), 261-272.
12. Brenner, H., Arndt, V., Gefeller, O., & Hakulinen, T. (2004). An alternative approach to age adjustment of cancer survival rates. European Journal of Cancer, 40(15), 2317–2322. doi:10.1016/j.ejca.2004.07.007

About the author

Vedanth Ramji

Vedanth is a junior at APL Global School, Chennai, India. He is passionate about research in computational biology and creating digital healthcare solutions. Vedanth is currently a long-term student researcher at the Big Data Biology Lab at Queensland University of Technology, where he works on bioinformatics tools and conducts research on antimicrobial resistance and metagenomics.

He is also a software developer for Queromatics, a not-for-profit cancer research and consulting organization, where he developed a cancer treatment planning app – Cancerstop. Vedanth is the founder of Thaavaram, a global surveillance system to collect data, detect, monitor, and act on antimicrobial resistance in plants.

The post Refining Cancer Survival Analysis: Constructing Age-specific Relative Survival Graphs Using Polynomial Regression to Interpolate Age-group Based Data from the Surveillance, Epidemiology, and End Results Program appeared first on Exploratio Journal.

Abstract

This paper proceeded a quantitative analysis on customers’ satisfaction on express companies, compared four express companies’ service satisfaction based on different scoring on the survey and provided relevant developing suggestions for the companies. AHP is utilized to determine the weight, and Fuzzy Comprehensive Evaluation (FCE) and TOPSIS method are used to compare the express companies’ performance. It analyzes the results and offers a future development suggestion for the express companies based on single index score and synthesis score.

Key words: AHP, Fuzzy Comprehensive Evaluation, TOPSIS, Express Company, Service Quality Evaluation, Satisfaction

1. Introduction

With the rapid development of the Internet, online shopping has become an important form of shopping for people, and express delivery business has gradually developed into a highly influential emerging industry. According to the 52ND Statistical Report on the Development of China’s Internet[3], as of June 2023, the number of online shopping users in the country reached 884 million, and according to the operation of the postal industry in 2022[4], the volume of express delivery in the year 2022 exceeded 110.5 billion. It can be seen that the development of express delivery business has an inseparable relationship with online shopping.

Nowadays, service quality has become an important indicator related to the strength and development of enterprises. Therefore, the competition between express companies is becoming more and more fierce, and the companies are paying higher attention to their own service quality of express and the customer satisfaction of delivery services in order to further enhance the company’s durability. Because the service key points of different companies are different, the satisfaction of target customers to the service quality of different express companies also differs. Therefore it is essential for every company to notice their strengths and weaknesses in order to enhance their market position.

Previous studies have already proved the evaluation model to be successful[7]. Yet most of these studies merely proceed analysis and give suggestions to the industry as a whole[6], which neglected the difference between enterprises. Based on the customer satisfaction survey results of four listed express companies, this paper conducts a quantitative study on the customer service quality of SF Express, Yunda Express, YTO Express and STO Express. This paper used two methods to compare the enterprises instead of one, which makes the result more robust. It also quantitatively measures the customer satisfaction of each company and puts forward suggestions for improving express service of different companies.

2 Establish the Evaluation System with AHP

2.1 Hierarchy Construction

An evaluation system is constructed by combining SERVQUAL model and service standard of express delivery industry. This paper used SERVQUAL as one of the standards because SERVQUAL scale proposed that the criteria of service quality is constituted of five indexes, and the combination of SERVQUAL model and the service standards of express delivery industry has been proven to be successful[7]. The overall goal of the evaluation system – customer satisfaction on service quality –is analyzed through five indexes: Empathy, Reliability, Timeliness, Convenience, Tangibles. A total of 5 primary indexes and 19 secondary indexes were selected, as shown in table 1.

2.2 Construction of Pairwise Judgement Matrix

According to the 1-9 scale method which effectively reveal the relative importance of one index compared to another, this paper combines the importance evaluation given by experienced customer opinions and expert opinions to each index, and obtains the evaluation matrix of the primary index and the secondary index. The evaluation matrices are as follows

2.3 Index Weight and Consistency Test

After the matrix is normalized, the principal eigenroots of each matrix are calculated, the weights of each index are calculated, and the consistency of each matrix is judged according to the eigenroot method of finding vectors by pairwise judgment matrix[1].

2.3.1 Weight of Primary Indexes

Through the method of finding eigenroot of the vector, the primary indexes’ weight is calculated as follows.

W_Z=(0.0456, 0.5046, 0.2830, 0.1177, 0.0491) (7)

After testing, the consistency ratio is 0.030 < 0.1, matrix Z meets the consistency requirement.

2.3.2 Weight of Secondary Indexes

Through the method of finding eigenroot of the vector, the empathy indexes’ weight is calculated as follows.

W_A=(0.0595, 0.1896, 0.5727, 0.1782) (8)

After testing, the consistency ratio is 0.054 < 0.1, matrix A meets the consistency requirement. Similarly, the weight of reliability indexes, timeliness indexes, convenience indexes, and tangibles indexes is calculated as follows.

W_B=(0.1399, 0.1327, 0.5994, 0.0562, 0.0718) (9)

W_C=(0.0812, 0.6596, 0.1552, 0.1040) (10)

W_D=(0.0680, 0.3800, 0.1420, 0.4100) (11)

W_E=(0.2500, 0.7500) (12)

After testing, the consistency ratio is 0.086 < 0.1, 0.046 < 0.1, 0.029 < 0.1, 0 < 0.1, respectively. Therefore, matrix B, matrix C, matrix D, and matrix E all meet the consistency requirement.

2.4 The Evaluation Set

2.4.1 Evaluation Level

The survey sets the evaluation criteria to 5 levels. M represents the score received. M1, M2, M3, M4, M5 represents very unsatisfied(1pt), unsatisfied(2pts), general(3pts), satisfied(4pts), very satisfied(5pts), respectively.

2.4.2 Data Collected

This paper takes the form of online questionnaire to obtain data. Specifically, the questions are designed to acquire information for the corresponding indexes. For example, For index B3, the question is how frequent is the lost of item occurring. Then the questionnaire is given out both online and offline. Finally, data is collected through the questionnaire backstage. Up to now, 183 questionnaires have been collected, of which 150 are valid. As the questionnaire is a multiple choice question, the amount of valid questionnaire of each company varies.

3 Comparison based on Fuzzy Comprehensive Evaluation

3.1 Data

According to the statistical data obtained from the questionnaire survey, the relative membership degree of each secondary index is obtained, as shown in Table 2.

Similarly, the relative membership degree of the other three companies are collected. With a total number of 42 questionnaires for Yunda Express, 62 for STO Express, and 41 for YTO Express.

3.2 Fuzzy Matrix Construction

According to the collected questionnaire data, the membership degree of each index of four express delivery enterprises is shown, and the fuzzy matrix of each secondary index R_A, R_B, R_C, R_D, and R_E can be obtained.

3.3 Fuzzy Comprehensive Evaluation

B = WR (13)

Through the formula above, where W is the weight of each index and R is the fuzzy matrix, the fuzzy weight vector and fuzzy matrix of each secondary index are fuzzy transformed[2]. And the fuzzy comprehensive evaluation of each secondary index is obtained as shown follows.

By the same token, the fuzzy comprehensive evaluation of each secondary index of the same three companies are obtained. From these evaluations, the fuzzy matrix R_Z of the primary indexes can be obtained through combining vectors B_A, B_B, B_C, B_D, B_E into a matrix as follows.

3.4 Results

As the FCE method only evaluates the degree membership of the companies, it is a qualitative analysis and is unable to determine which company is better if they both belong to the same set. Therefore, a score is added to every level of satisfaction in order to proceed a quantitative analysis. Note vector T as follows.

T =0 30 60 80 100
(15)

Using the formula

S = BT (16)

Where B is the final membership degree vector for every company as shown in table 4.
By multiplying vector B by vector T, we get a precise score of every company. After the calculation, we get S^sf=90.6670, S^yd=81.5652, S^st=81.6914, and S^yt=82.8017.

From the scores, it’s obvious that SF Express is significantly higher than the other three companies, YTO Express is slightly higher than STO Express, and Yunda Express got the lowest score.

4 Comparison based on TOPSIS

4.1 Introduction

As FCE is suitable for problems that concern fuzzy definitions of variables, it can successfully model the problem studied in this paper. On the other hand, the problem studied in this paper also has the feature of optimization of company strategy and relatively large sample size. Therefore, TOPSIS method is adopted in this paper as it performs well in large sample questions and it does not require an objective function to be maximized.

4.2 Data Processing

Since TOPSIS method compares the scores of different indexes, the normalized criteria is needed. Because the data collected is all maximum type index, there is no need of forward transformation of data. As the questionnaire number of each company varies, summing up all the scores is imprecise. Therefore, this paper processed the data by acquiring the mean score of every index directly: The mean of scores indicates the company’s overall satisfaction, and it is normalized itself in order to be compared between different companies. Thus, this paper used score means as the indicator of satisfaction using the formula

Where S_^kj is the mean score for the j th index for company k, S_^kij is the score received from the i th survey for the j th index for company k, N^k_j is the number of survey collected for the j th index for company k.

As the weight is to be considered in this study, it is essential for the weight to be considered in TOPSIS method. For the weight of every secondary index, this paper used the product of primary and secondary weight to determine the weight of secondary indexes directly. Specifically, this paper used the formula

4.3 Solve

First, the scoring of every index among the four companies has to be determined. The formula below indicates how the scoring was determined.

where Y^j_i is the i th index scoring from company j. Then the scoring is calculated using formula (21) and formula (22) .

Where m is the number of indexes, w_j is the weight of the j th index, Z⁺_j and Z^–_j is the highest and lowest score of the j th index, and Zⁱ_j represents the scoring of the j th index of company i. D_i⁺ and D_i^– indicates the distance between the optimal scoring and company i’s scoring and the distance between the pessimal scoring and company i’s scoring, respectively.[5]

4.4 Results

Formula (23) calculates the final scoring of the companies.

Through this formula, the scoring of the four companies are S^sf = .9654, S^yd=0.1179, S^st=0.1173, S^yt=0.2058.

Obviously, SF Express scored much higher than the other three companies, YTO scored higher than Yunda, and STO got the lowest score.

5 Results and Conclusions

5.1 Results Analysis and Further Studies

Both the FCE method and the TOPSIS method indicated that SF Express’s customer satisfaction is significantly higher than the other three companies. The scoring of the other three companies differs, but the difference is subtle: YTO Express scored higher than STO Express and Yunda Express, yet the difference is inconsequential compared to SF Express’s high score.

After the analysis by synthesis, there must exist a single index that caused the difference between the companies. Therefore, a deeper study is conducted. In FCE method, the formula F = BT can calculate every scoring of every index of every company, as shown below.

Where B^k_ij is the i th secondary index from the j th primary index from company k.

Where B^j_i is the i th primary index from company j.

These two equations allows the scoring on every single index to be revealed directly.

After analysis, we found out that index B and C are very important. Among these two indexes, B₃ and C₂ are the main factors.

For reliability, the four companies scored as follows: B_sf=91.33 , B_yd=84.64, B_st=83.97, and B_yt=85.12.

For timeliness, the four companies scored as follows: C^sf=91.06, C^yd=78.30, C^st=78.18, and C^yt=80.44.

For ”Lost of Item”, the four companies scored as follows: B^sf =91.93, B^yd=87.86, B^st=79.84, and B^yt=88.30.

For ”Arrive on Time”, the four companies scored as follows: C^sf =91.66, C^yd=79.05, C^st=78.06, and C^yt=80.24.

The ranking of these four indexes’ scores is exactly the same as the synthesis analysis. Overall, on the specific indexes, SF Express still scored the highest and the other three didn’t reveal significant difference.

Through the weight analysis, we found that reliability and timeliness hold the highest weight, they held 50% and 28% of the total weight respectively. Among the secondary indexes, ”Lost of Item” and ”Arrive on Time” hold 30% and 18% of weight of the total evaluation system. Therefore, it is essential for express companies to guarantee that the package is precise and arrives on time in order to increase customer satisfaction.

From the data provided, SF Express scored the highest, the other three companies’ score ranking is consistent with their final ranking, this further emphasized the importance of these indexes.

5.2 Suggestions

Yunda Express, STO Express and YTO Express should pay more attention to the security of their packages in the future, specifically, make sure it does not get lost. On the other hand, arriving on time should be taken attention on since these indexes constituted a high proportion of customer satisfaction. SF Express is the only company that has a absolute advantage on almost every index except for one – price. Considering SF Express’s overnight delivery service, low satisfaction caused by the high service price is inevitable. Therefore, SF Express may lower the service price a little bit based on the current situation to further increase customer satisfaction. Yet this suggestion is inconclusive since it does not consider the pricing model, further research can be conducted studying the pricing model of express companies to determine whether the prices are able to be optimized.

In fact, the significant difference between SF Express and the other three express companies may be attributed to their operating mode. After interviewing the workers at the express stations, we were informed that SF Express adopts the strategy of district management which requires workers working at SF Express to directly provide service to the assigned districts. It also regulates the company with a direct operation system to the entire corporation. The other companies adopted external contracting strategy. This kind of strategy has a main drawback that some of the contracted people lacks professional ability, this caused a higher difficulty for these companies to regulate the corporation. On the other hand, the service quality and professional level cannot be guaranteed, which caused the lower level of professional skill, leading to a lower customer satisfaction. This may be the key factor that caused the significant difference between Sf Express which adopts the direct operation system and the other three companies that adopts the external contracting strategy. Therefore, another way to improve customer satisfaction for the three companies is to change the operating mode.

6 Reflections

6.1 Advantages

When determining the weights, we took reference from different people to make the weight determination more robust and lowers the risk of wrong weight determination.

This paper used two models to conduct research on express service satisfaction. Compared to using a single model, this paper has a more robust result and avoids error caused by an inappropriate model.

Previous studies are targeted at the entire express industry’s satisfaction. Although their results are significant, they ignored the difference between companies themselves that may have caused the difference between customer satisfaction. Therefore the suggestions offered in those papers also ignore the difference between companies. This paper conducted a comparison study on different express companies, found difference between the satisfaction and the operating mode, and made different suggestions to different companies. This paper creatively adopts two different models to solve the problem. On the other hand, this paper also studies the problem and give suggestions at a more detailed perspective, which foster the improvement of express companies since the problem is more specified.

6.2 Deficiencies

This paper used AHP to determine the weights, yet this method may be influenced by subjective factors. Although this paper collected suggestions from multiple perspectives to alleviate wrong weight determination, manual weight determination is still easily influenced by subjective matters. In the future, other objective methods such as entropy weight method may be combined with AHP to determine the weight in order to mitigate subjective influence.

The two models have their own advantages and disadvantages, and these models have difference on evaluation. In future studies, model averaging methods may be used to make the results more robust.
This paper only gives suggestions for express companies based on customer satisfaction, yet in real life practices, more complicated situations such as enterprise profit, worker satisfaction have to be considered. Future work may focus on other more complicated situations to provide further suggestions and help express company’s development in the future.

References

1. S Cai. Principles and methods of mathematical modeling activities, 2000.
2. X Cao. Principles and methods of mathematical modeling activities, 2014.
3. CNNIC. The 52nd statistical report on china’s internet development. China Internet, 52(10):1, 2023.
4. State Post Bureau of People’s Republic of China. State post bureau of people’s republic of china announced the operation of the postal industry in 2022, 2023.
5. S Si and X Sun. Mathematical modeling algorithms and applications (first edition).
6. Wang. Z. Research on ahp college express service quality – taking a college in shanxi as an example. Logistics Sci-Tech, 44(11):72–75, 2021.
7. M Zhu, S Miao, and J Zhuo. An empirical study on chinese express industry with servqual. Science and Technology Management Research, 31(8):38–45, 2011.

About the author

Weitan

Weitang is an 11th-grader at The Experimental High School Attached to Beijing Normal University. He has participated in many mathematical competitions, including SMC, AMC, and ARML. Throughout his mathematical journey, Weitang found a deep fascination with data analysis, which illuminates real-life scenarios in a vivid manner. His passion for this field motivated his participation in the HiMCM contest, where he and his team conducted extensive research on the carbon emission problem. Inspired by this experience, Weitang has embarked on his intellectual exploration of data science and statistics, as shown in his paper. Beyond academics, Weitang is an electronic music lover who cherishes “patterns” in both the musical and mathematical realms.

The post Comparison Study on Express Companies Based on AHP-FCE and AHP-TOPSIS appeared first on Exploratio Journal.

Introduction

In light of the instability of the economy today, stock prices are a concern. Whether one should liquidate one’s investments or not is the question on many minds. Many economists have used models in order to predict any trends that a stock might have in the future. Two examples of these models are ARIMA and Geometric Brownian Motion. Although they have the same goal as a model, the two have major differences in accuracy, operation, errors, functions, and variables.

The prediction models GBM and ARIMA share numerous similarities and differences. First of all, both of them are stochastic processes (Floyd and Hadjifrangiskou). In the scientific Brownian Motion, stochastic motion is the random motion of particles that is caused by random collisions with molecules (Floyd and Hadjifrangiskou). This is similar in economics. With both GBM and ARIMA(mostly GBM), the models use random fluctuations in data in order to predict the future of the stock. In finance, using Brownian Motion one can predict random movement of data over some time. Due to this random motion, one can determine a solid trend of data that can solidify the data in the future.

Both these models use intervals of time and the data points taken from each interval in order to forecast the price. Although usually, to get the most accuracy, one should use all points of data in order to make a “best fit line” prediction, the GBM model does not use all of the intervals. In stocks, if one takes a mean of data over certain periods of time, it is going to fluctuate throughout different time frames(Kong). Brownian Motion says that the stochasticity would remain 0 if it exceeds Brownian Motion, which is not related to what happens in stocks (Ermogenous). On the other hand, one disadvantage of GBM, similar to ARIMA, is that the randomness it calculates is shown as constant when, in reality, stock prices are always fluctuating and unpredictable due to news or events that change the projection(Kong).

Methodology of ARIMA

ARIMA stands for Autoregressive Integrated Moving average. AR (Autoregression) is the changing of the variable that develops over its own prior values (Bajaj). I (Integrated) is the raw observations that make the time series able to become stationary (Bajaj). Data values are replaced by the difference between data values and previous values (Hayes and Stapleton). MA (Moving average) is the inclusion of dependency between an observation and an error from the moving average model of the ARIMA that is used with the lagged observations (Hayes and Stapleton). Moving average window is the period of time with the average data in that time period (Hayes and Stapleton). The top averages are then used to interpret and predict the next averages for the model.

• p is the parameter which represents the number of lag observations or what the time is that is passing(Hayes and Stapleton).
• d is the number of times that the raw observations are differenced that occur in the Integrated part of the model(Hayes and Stapleton).
• q is the size of the moving average window(Hayes and Stapleton).
• L is the component of the equation that uses another element of the time series in order to put out the previous element. This is also known as the lag operator.
•  θ_i is the representation of the parameter for the moving average (MA) of the equation.
• ε_t is the variable to show the error terms, which are independent variables, that are sampled from the normal distribution of the time series and spread out with an average equal to zero(Notation for ARIMA Models).
• ⍺_i are the parameters of the autoregressive component of the model (A) (Notation for ARIMA Models).
• X_t is a real number where t is an integer Figure 1 (Autoregressive integrated moving average)

Background of GBM

GBM derived from the Brownian Motion (Wiener Process), which was named by the botanist Robert Brown in 1828 when he was studying how pollen particles that are dispersed in water move randomly microscopically(Holton). The key component of Geometric Brownian Motion relates to that of Brownian Motion, randomness/erratic data. This realization of the stochastic process and its relation to pricing was uncovered by a French mathematician by the name of Louis Bachelier(Ermogenous). Bachelier published this idea in his doctoral thesis which was then looked at by Albert Einstein who then implemented this in thermodynamics(Holton). Later on in 1923, Norbert Wiener finalized the existence of the Brownian motion and its application in mathematics using mathematical theories to correlate the two(Holton). This is why Brownian motion is also sometimes called the “Wiener process”(Holton).

Methodology of GBM

An important part of the GBM model is that it will not spit out after doing its operations a negative value as the output(Kong). For example, when calculating the stock predictions, it cannot predict values and output it as a result in the form of a negative number(Kong). The solution of a problem solved by GBM is determined from the time prediction, which is represented by the t parameter. Predictions would be accurate in stock prices when one has a smaller time horizon/factor/period. Two hours in the future prediction would be more accurate than a prediction for five hours or two days in the future because there is data that is being observed over a shorter period so a comparison of data can be more evident. The W is the Brownian Motion (Wiener Process) itself. It is to represent the random motion, which is a key role in prediction using this model. Nu is shown for percent drift(Ross, 612-14). This is the average of how much the data is drifting and has changed from a baseline data point, which can be like an initial point of what to refer to for the change in data(Ross, 612-14).

Note. Random Walk is a model that falls in the ARIMA model category, and Brownian motion model( Random walk model). The Brownian Motion Graph shows the fluctuation of data.

Note.

• S_t is the Stochastic Process that follows the Geometric Brownian Motion Model (Ermogenous).
• nu is the percent drift/error of the process(Herzog).
• σ is the volatility. In this case, GBM in finance, this is the rate at which the price of the stock will increase or decrease(Herzog).

Accuracy of GBM

As compared to ARIMA, GBM is more widely used for stock prediction due to its great accuracy. For stock price prediction, the geometric Brownian Motion Model uses an algorithm that begins by calculating the return value and then estimating the value of the data fluctuating and randomly changing (Farida Agustini et al.). Then the model uses all of that to predict the stock price forecast to help predict the prices for the user. The accuracy of the GBM is very strong and is around 95% (Farida Agustini et al.). This accuracy rate, also known as confidence level, is supported by the MAPE value (Farida Agustini et al.), where MAPE stands for mean absolute percentage error. Along with a high accuracy rate, the GBM process also outputs a MAPE value of ≤20%, which also supports the reliability of this process compared to others (journal of physics conference series) (Farida Agustini et al.). Another benefit of GBM is that the calculations of the model are simple, and it outputs the IV of the significance of the price of the stock (Kong).

Note.

• n is the sample size in the model
• “actual” is the actual data value of the model
• “forecast” is the forecasted data value of the model
• “| |” is representing the absolute value. This is used in the numerator and denominator of the equation.

Note. On the bottom row of the graphs, the ARIMA model of the precipitation prediction is shown. The key also indicates that the darker points are the predictions. As seen in the graph, many of the points plotted are scattered due to the ARIMA showing the percent drift and fluctuation in forecasting data. The points that are located on the higher end of the y axis show that the prediction is stating a major change in precipitation due to some factors around the 2010 time frame.

Conclusion

GBM and ARIMA both have many things in common like their application in finance, stochastic motion, distribution, and their use of previous data points to make data predictions. On the other hand, the function, variables, and other uses differentiate both models evidently. Although the stochastic processes of GBM and ARIMA are used for predictions, GBM is greater in accuracy and reliability (Farida Agustini et al.).

References

“Autoregressive Integrated Moving Average.” Wikipedia, Wikimedia Foundation, 15 Oct. 2022, https://en.wikipedia.org/wiki/Autoregressive_integrated_moving_average.

Bajaj, Aayush. “Arima & Sarima: Real-World Time Series Forecasting.” Neptune.ai, 14 Nov. 2022, https://neptune.ai/blog/arima-sarima-real-world-time-series-forecasting-guide.

Business, Fuqua School of. “ Random Walk Model.” Random Walk Model, https://people.duke.edu/~rnau/Decision411_2007/411rand.htm.

Ermogenous, Angeliki. “Brownian Motion and Its Applications In The Stock Market.” University of Dayton ECommons, 2006, https://ecommons.udayton.edu/cgi/viewcontent.cgi?article=1010&context=mth_epu md.

Farida Agustini, W, et al. “Stock Price Prediction Using Geometric Brownian Motion.” Journal of Physics: Conference Series, vol. 974, 2018, p. 012047., https://doi.org/10.1088/1742-6596/974/1/012047.

Floyd, K A, and M Hadjifrangiskou. “Adhesion of Bacteria to Surfaces and Biofilm Formation on Medical Devices.” ScienceDirect, 2017, https://www.sciencedirect.com/topics/chemistry/brownian-motion. Accessed 30 Nov. 2022.

Hayes, Adam. “Autoregressive Integrated Moving Average (ARIMA).” Edited by Chip Stapleton, Investopedia, Investopedia, 24 June 2022, https://www.investopedia.com/terms/a/autoregressive-integrated-moving-average-ari ma.asp.

Herzog, Florian. “Stochastic Differential Equations – ETH Z.” Stochastic Differential Equations, 2013, https://ethz.ch/content/dam/ethz/special-interest/mavt/dynamic-systems-n-control/idsc-dam/Lectures/Stochastic-Systems/SDE.pdf.

Holton, Glyn. “Brownian Motion (Wiener Process).” GlynHolton.com, 10 Oct. 2016, https://www.glynholton.com/notes/brownian_motion/.

Kong, Yuxuan. “Geometric Brownian Motion – Mi.uni-Koeln.de.” Geometric Brownian Motion, May 2017, http://www.mi.uni-koeln.de/wp-znikolic/wp-content/uploads/2017/05/4_Geometric_Br ownian_Motion_28042017.pdf.

Lai, Yuchuan, and David A. Dzombak. “Use of the Autoregressive Integrated Moving Average (ARIMA) Model to Forecast Near-Term Regional Temperature and Precipitation.” AMS, vol. 35, no. 3, 1 Apr. 2020, pp. 959–976. “Notation for ARIMA Models.” SAS/ETS, July 2011.

Ross, Sheldon M. Introduction to Probability Models (Eleventh Edition). 11th ed., Academic Press, 2014.

Zach. “What Is Considered a Good Value for Mape?” Statology, 10 May 2021, https://www.statology.org/what-is-a-good-mape/.

About the author

Dwij Patel

Dwij is currently an 11th grader at the Mountain Lakes High School.

The post Relation Between Geometric Brownian Motion and ARIMA in the Prediction of Stock Prices appeared first on Exploratio Journal.

1. Introduction

Machine Learning jobs are growing to become one of the most in de- mand jobs in the world. In the 1940’s, the idea of machine learning first started to grow; it was something that would emulate human think- ing and learning. Machine Learning has since grown to become a big part of our daily lives. For example, in speech recognition software, the software will map the different tones and nuances when someone speaks and try to match this to a specific person. Another example is a translator, which tries to understand the accents of people speaking a language and then translates it to another language. Many applications that we use today, such as Alexa, Siri, and Google Translate, use these machine learning algorithms. Furthermore, we are trying to integrate machine learning into our vehicles. Cars like the Tesla use unsupervised learning algorithms to self-drive in traffic and detect any danger. The future holds many possibilities due to machine learning.

In theory, we input great amounts of data into machine-learning programs, which using statistics, will categorize or predict outcomes by finding and applying patterns in the data. We can further categorize the different types of algorithms used in Machine Learning to supervised, unsupervised learning and reinforcement learning. Supervised learning consists of regression and classification while unsupervised learning consists of clustering and association.

In this report, we will first discuss important terminology needed to understand the contents of the report. We will then begin to dis- cuss the theory behind some of the machine learning algorithms. The algorithms implemented in this report are all regression algorithms, however, we will also discuss the theory behind other algorithms. Finally, we will see how to implement the code. There are GitHub links provided with the actual code.

2. Terminology

Before we can get started with all the theory, we must develop an understanding of some key terminology that we will use quite often when working with machine learning programs. These are some basic terms that we should be familiar with:

2.1 Features

  When we are trying to extrapolate from data using a linear model such as a line of best fit, we want the line to have an equation that best fits the data. In general a line has an equation of h = θ₀ + θ₁x₁ + θ₂x₂ θ_nx_n. Here we consider x₁, x₂, , x_n1, x_nthe features. We will go more in depth about this later on in the report.

2.2 Inputs

When we run a python program, we must somehow store the data so that our program knows what we want it to work with. We then take ’input’ of the data in a convenient way for us to work with it. For example, lets say we had a document that contained a few coordinates. We may want our program to take input of this data where the x- coordinates and y-coordinates are stored separately. The program written to complete this process is called ’taking input’. This process is explained in greater deal in the code.

2.3 Outputs

After our code has calculated what we wanted it to, we want to see this information in an organized manner so that we can study it. We then make our program ’output’ this information. Outputs can consist of words, integers, etc.

2.4 Predicted Values

Let us say that we received input of many coordinates and we wanted our program to calculate the line of best fit. When we are testing different equations to see if they best fit the data, we input the same x-coordinates as the ones in our input data. However, our y-coordinates may not always be the exact same as that of the input data. We thus call our y-coordinates predicted values, since they are what our program predicted the coordinate lies at based on the equation that we came up with.

2.5 Expected Values

The values that we get from the inputted data are our expected values since they are the original values that we are comparing the predicted values to.

3. Supervised Learning

Supervised learning is the most commonly used algorithm in Machine Learning and it is also the simplest to implement. When using super- vised learning, we must train the algorithm by pairing labelled inputs with outputs. The program in this stage is trained to look for patterns that correlate the input to the output. When we have provided the algorithm with a good amount of example pairings, the algorithm will be able to apply this to new inputs it receives. We can further split supervised learning into classification and regression.

3.1 Classification

Classification is a type of supervised learning. In classification, our output will always be a category that the algorithm has mapped the input to. An example of this would be our program receiving input of pictures of animals and then outputting what animal they are (their category). We first have to train the program by inputting many pictures of dogs and cats in their respective categories so that the program will be able to establish patterns between the images of the dogs and the images of the cats. After we have inputted a sufficient number of images, the program will get accurate in determining if an animal is a cat or dog when it receives an input that it has not seen before.

3.2 Regression

Regression is another type of supervised learning. In regression, our output is not a category but rather a value such as money or age. We can take for example the price of houses and the total square footage of the house. Using regression, we identify the function that best fits between these values where we have reduced the amount of error as much as we can. We can then use the equation of this line to predict how much a house with a certain square footage will cost.

3.2.1 Linear Regression

When performing linear regression, the program will take input of data and plot it on a graph. It will then find a line of best fit and be able to make predictions based on this line of best fit. For example, we can graph the number of hours a student watches TV rather than studying compared to their test scores.

As we can see, the graph looks fairly linear and it only has one feature; the amount of time spent watching TV rather than studying. This makes it a perfect model for linear regression. We want our program to come up with an approximate equation with which we can estimate a students’ test score based on how long they spent watching TV instead of studying. Really, we are looking for our program to find the line of best fit, since this line would be best for extrapolating the data and providing an as accurate as possible estimate of a test score based on the number of hours that were spent watching TV. Our program would then test many different lines until it reaches one line that fits the data better than any other line.

As we can deduce, when calculating the equation of the line of best fit, our slope and y-intercept variables matter a lot. In fact, we are just making changes to these variables to try to find the line of best fit. Machine learning algorithms rely on these parameters (y-intercept, slope/bias, etc.) to run. When we want to find the best model for our data, we need to keep adjusting these parameters so that the direction of our line better fits the data and our predicted values are closer to the expected values. We must then introduce a function that changes these parameters by determining the amount of error that we are getting with the current parameters. This function is called the cost function.

4. Cost Function

The cost function essentially helps our program minimize the error it produces compared to the actual data set. When we are doing linear regression, it is very rare that we will get a data-set where the data fits precisely on a line. Therefore, when we are computing the line of best fit, we want to find a line such that it has the least possible difference (error) between the actual coordinates and the coordinates our line gives (predicted values). There are multiple ways of defining the cost function, some examples are explained further in the following sections.

4.1 Mean Absolute Error

When we take the mean absolute error, we are taking the absolute value of the difference between the predicted y-value and the expected y- value. The reasoning for this is that, since we are adding up all the error for each data point, we want to keep track of how much error we are accumulating.

In this graph, the blue points are the original points of the data set, while the orange points are the ‘predicted’ points that our program is currently testing for the line of best fit. As we can see, each d_irepresents the amount of ‘error’ our model/line produces for each point in the data set.

However, if we add negative numbers (our predicted point is below the original point), our program actually thinks it’s producing less error. To deal with this we take the absolute value, which is always non-negative, so that our program does not add negative error. Then our program can plug this into the formula which is defined as

Where mis the number of training examples, yˆ(i) is the predicted value, y(i) is the expected value and i is the index of the data point since we want to sum the error of all the data points.

4.2 Mean Squared Error

When we take the mean squared error, instead of taking the absolute value of the difference between the predicted and expected value, we take their square. In this way, we still don’t add up negative error since any real number squared is non-negative. The equation is defined as:

When using mean absolute error, we took the absolute value of the distance between the predicted value and the expected value. We are now taking the square of the area of the square whose side length is the distance between the predicted value and the expected value. All these regions are summed and averaged.

Now that we have discussed how our program will calculate the error that our model/line is producing, we must find a way to minimize the value our cost function is returning. The gradient descent algorithm is one of the most effective ways of doing so.

For linear regression models, we assume that our data has a linear dependence and therefore can be modelled by using a linear equation as follows;

h_θ(x) = θ^Tx= θ₀ + θ₁x,

where θ₀ is our bias (y-intercept) and θ₁ is our slope. Then, we want to change our parameters θ₀ and θ₁ in such a way that our line better fits the data and the cost function produces less error. In batch gradient descent, we update our theta values continuously with the following equation;

Here, θ_jis the value that we are updating. Again, mis the size of the data (how many points there are). Alpha here represents the learning rate of our algorithm. If alpha is too big, our program may be a lot faster, but it will not be nearly as accurate in determining the equation of a line of best fit as a smaller value of alpha may be. However, when we use too small a value for alpha, our program will be incredibly slow. It is best to find a good median between these two values.

6. Multi-Linear Regression

 Now that we have discussed how to optimize our program so that it can calculate the best line of fit with equation h = θ₀ + θ₁ x₁, we think of what we would do when we have multiple features. Currently we have only been working with one feature, which in the example presented, was the number of hours spent watching TV rather than studying. Let’s take another example of the price of a house. When determining the price of a house, we must determine its area, how many rooms it has, how old it is, among other things. In this instance our data when plotted still looks linear however we cannot use the exact same technique as linear regression, since we have more than one feature. We use multi-linear regression in this situation because of its suitability to deal with more than one feature.

Multi-linear regression can be used with as many features as we’d like. Our equation is now

h= θ₀ + θ₁ ·x₁ + θ₂ ·x₂ + ···+ θ_n·x_n,

where all x_irepresent the different features. When we now implement gradient descent, we must use it to update all θ_iso that our line better fits the data. The cost function can be implemented in much the same way.

The interesting thing to note about multi linear regression is that we need an n-D graph to plot all the points, however, if we take a 3-D graph for example, our program is essentially finding the line of best fit in a plane that best suits all the points.

7. Unsupervised Learning

Unlike supervised learning, in unsupervised learning, we do not train the program with inputs and corresponding outputs. Rather, the pro- gram uses its built-in algorithms to try to find patterns in the unlabelled data and produce an output. For example, if we give input of shapes with different sizes, the algorithm can separate these based on how many sides there are in each shape. In general, unsupervised learning requires much less data then supervised learning. We can further split unsupervised learning into clustering and grouping.

7.1 Clustering

As discussed earlier, in unsupervised learning, we input unlabelled data into our program. Graphing our data, it may look like the following:

Once our program has graphed the data, we want our program to try to find patterns in the data. Specifically, clustering algorithms will try to look for clusters of points that seem to be together. The graph could then be divided into the following clusters:

Among the many applications of clustering, we can use the example of social networks. We may want to find which people seem to be very close friends on their social networks so our algorithm would make clusters of people that appear to be close friends.

A more common example in our daily lives would be our spam filter. Our email uses clustering algorithms to try to group spam emails, update emails, advertisement emails, etc. together.

Furthermore, we can classify clustering as hard clustering and soft clustering. In hard clustering, a data point can either belong in a cluster or not. This type of clustering is useful in binary situations such as whether a movie is good or not. On the contrary, when using soft clustering, a data point can belong to many clusters. This is more useful when we may want to determine which books are similar.

7.2 Association

Association algorithms try to see if two items depend on each other. For example, if we take a customer at a supermarket. If this customer has gone to buy bread, then it is very probable that the customer is also looking to buy butter or milk. In this way, we can associate different items based off of their dependency on each other. Many companies use this technique to place associated items away from each other in a store so that the customer see’s many other items on the way and may consider buying additional things. An example of the different associations in a store are given below:

8 Reinforcement Learning

In reinforcement learning, the program learns what to do by trial and error in its current environment. We can think of it as the program receiving a reward if it does something correct and a penalty if it does something incorrect. Take the analogy of a child, when a child is young, they do not know what is good or bad. The only way the child can learn is by trying new things. The child may touch something electric, get a shock, then instinctively not go near the thing again. The child now knows that that object is something that shouldn’t be touched because it will hurt. A reinforcement learning program works in a similar way. The difference here is that the machine can try thousands of operations in one second and even though it may start by making very bad decisions, it will learn over time and will become a lot more sophisticated in its decision. We can simulate giving a program a reward or penalty by giving it a score in which, if it does something incorrect, the score will lower, and conversely, if it does something correct, the score will increase. This type of program is based entirely on trial and error on the programs part, it is also one of the closest things to a machine’s own creativity.

One of the most useful implementations of reinforcement learning are simulations. For example, the program can be used to help create the optimal rocket engine for a rocket launch. If we put our in a rocket launch environment in which the environment responds to the actions of our program, we can ‘reward’ the program if it’s helping the rocket launch with its actions or ‘punish’ the program if it’s not helping the rocket launch.

9. Linear Regression Implementation

For the linear regression code, we took input of the population of a city in 10, 000s and its profit in $10, 000. We then plotted all of the coordinates and got the resulting graph:

As we can see the graph looks fairly linear, thus we can use linear regression on this.

The full code can be found at: https://github.com/ABehniwal/face-recognition/ blob/main/Numpy-Linear-Regression.ipynb

10. Multi-Linear Regression Implementation

For the multi-linear regression code, we took input of the different features of a car (Engine Size, Cylinders, Fuel Consumption (City), Fuel Consumption (Comb)) and the resulting CO2 emission. We then plotted all of these features of the car separately with the CO2 Emissions to get a visual of how the different graphs look. This resulted in the following graphs.

10.1 Engine Size Graph

10.2 Cylinders Graph

10.3 Fuel Consumption (City) Graph

10.4 Fuel Consumption (Comb) Graph

Again, we see that all the graphs look fairly linear, however, since we have multiple different features of the car that we must take into account, we use multi-linear regression. The full code can be found at: https://github.com/ABehniwal/face-recognition/blob/main/Multi-Linear-Regression. ipynb

About the author

Amanbir Behniwal

Amanbir is currently an 11th grader at the Vincent Massey Secondary School in Ontario, Canada. He enjoys challenging myself with difficult math and computer science problems by participating in various contests. Amanbir is an avid fan of Barcelona and has been playing soccer for many years. Amongst other things, he likes to read books, help others with problem-solving, and delve deeper into the field of computer science.

The post The Theory and Implementation of Common Machine Learning Algorithms appeared first on Exploratio Journal.

Background

Dengue is a vector-borne, viral infection commonly found in tropical and sub-tropical areas of Central America, South America, Africa, Asia and Oceania [1]. It is caused by 4 types of viruses (DENV-1, DENV-2, DENV-3 and DENV-4), and is spread by the Aedes albopictus and Aedes aegypti species of mosquito, both of which thrive in areas containing standing water like puddles, pails, water tanks and tires [2]. The first dengue case was picked up at Pulau Pinang in 1901, and since then dengue has been a major problem affecting the Malaysian healthcare system and population as a whole for more than a century. By the 1960s, dengue had become endemic and in 1962 the first confirmed case of dengue-haemorrhagic fever was discovered in Pulau Pinang, eventually leading to the first epidemic outbreak in 1973 [3]. 

The current COVID-19 pandemic and the interventions put forth by the government in response have had their effects on recent dengue incidence rates, particularly in the year 2021. From the epidemiologic weeks 1 to 32 of 2021, a total of 16,565 dengue cases were reported, a drastic decrease of 47,423 (74.1%) from the same period in 2020 [4]. 

There have been a few different Movement Control Orders (MCO) implemented throughout the period of this study. The first MCO which spanned from 18^th March 2020 to 3^rd May 2020 [5] contained sanctions for all Malaysians traveling abroad, 14-day quarantines for those returning from overseas [6], general prohibition of mass movements and gatherings including religious, sports, social and cultural activities [6] and the entry of foreign persons into the country [6]. Some specific locations like Simpang Renggam in Johor were subjected to the Enhanced Movement Control Order (EMCO) for 14 days at a time during the period of the first MCO if a large cluster was detected within the area [7]. All residents living in such areas were forbidden from exiting their homes, outsiders were not allowed into the area and all roads into the area were blocked [7]. The Conditional Movement Control Order (CMCO) was implemented from 4^th May 2020 to 9^th June 2020, and had more relaxed regulations to stimulate the national economy [8]. Most activities and businesses were allowed to operate as long as social distancing was obeyed and interstate travel was not allowed except for work purposes [9]. The Recovery Movement Control Order (RMCO) was implemented from 10^th June 2020 to 31^st August 2020, and allowed interstate travel outside of areas under EMCO and certain religious activities at mosques [10]. Tourism businesses were allowed to open from 1^st July provided that the number of people in crowds was kept to 200-250 people and social distancing measures were obeyed [11]. Private pre-schools, kindergartens and day-care centres were also allowed to operate and many activities such as weddings, seminars, cinemas etc were allowed [12].

This paper examined and analysed the Dengue and COVID-19 incidents more extensively to investigate the reason and the details of this figure, as well as discusses the possible causes of such a drastic decrease.

Objectives

1. To show that the dengue incident rates in Malaysia had behaved unusually and dropped significantly in 2021.

2. To explore the potential contributing factors to the declined dengue incidence rates.

Methodology

This study was conducted during week 44 of 2021, so only data up to that point was used.

In order to illustrate that the dengue incidence rate of 2021 was significant, comparisons with the weekly trend of the Dengue incidence of the previous years was made. The number of reported dengue cases each week starting from week 1 of 2015 to week 44 of 2021 was recorded from public press releases posted on iDengue [13]. Any missing pieces of data were filled by dividing the cumulative number of cases over the number of weeks in the gaps. The epidemiologic weeks were plotted against the dengue incidence rates of each year. 

Correlations between COVID-19 incidence rates and dengue incidence rates were explored. Data on COVID-19 incidences per week from 2020 to week 44 of 2021 was retrieved from the Johns Hopkins University COVID-19 data repository. The maximum value from the COVID-19 case data set was divided by the maximum value from the dengue case data set to end up with a value of 44. All COVID-19 incidences per week were then divided by 44 to normalise the data for graph-plotting. The new COVID-19 data was plotted with the dengue incidence rate from 2020 to week 44 of 2021. The periods of the different Movement Control Orders (MCO) were labelled.

Dengue incidence rates during 2021 may have been low due to underreporting: fear of exposure to the COVID-19 virus and the overwhelmed healthcare system are possible reasons. To investigate this, the ratio of positive dengue tests to total number of dengue tests was analysed. This part of the study only covers the city of Ipoh. Dengue NS1 test data was retrieved from Pantai Hospital Ipoh. The data included the total number of negative and positive tests for dengue from 2015 to 2021 in the population of Ipoh. The total number of positive tests from each year was divided by the total number of tests carried out in the respective years to find the positivity rate. The percentage of positive tests for all years were then analysed.

The method of judgement was as follows: 

• If the percentage of positive tests was significantly lower than previous years, it was a sign of over testing
• If the percentage of positive tests was significantly higher than previous years, it was a sign of undertesting.

Next, the mortality rates of dengue throughout the years from 2018 to 2021 were analysed. The total number of deaths by dengue each studied year in Malaysia was recorded. The data was taken from the World Health Organizations Dengue Situation Update Number 634 [14]. The results between the years were then compared.

The idea of a correlation between vaccination and dengue incidence rates was also explored inside this study. The cumulative number of full vaccinations (2 doses) for each week from week 9 to week 44 of 2021 in Malaysia was recorded. The data was taken from the University of Oxford’s “Our World in Data” website [15]. Each successive value had been deducted by the previous value in order to find the number of complete vaccinations each week. To normalise the data for graph-plotting purposes, the greatest value of complete vaccinations in a week was divided by the greatest number of dengue cases in a week. The value that came out was 666. All values of complete vaccinations each week were then divided by 666 and plotted with the dengue incidence rate of the corresponding weeks.

Results

Graph 1 shows clearly that the dengue incidence rates of 2021 stray far from the trends of the previous years, behaving entirely differently. The most obvious observation is that the number of cases in 2021 appears to be significantly lower than the previous years, with very little deviance from an approximate range of 300-500 cases. Dengue cases usually rise in the months of June to August (week 23 to week 32), and November to February (week 43 to week 60), during the monsoon seasons, as it leaves many areas with standing water which mosquitoes can thrive in [16]. However, dengue incidences in 2021 do not seem to follow this trend which has been consistent for all past years, pointing towards 2021 being highly unusual.  

Graph 2 shows that dengue incidence rates had begun to decrease continuously around the same time as the beginning of the increase in COVID-19 cases. This could indicate a possible correlation between the two, but more investigation and study is required in order to make any conclusions. During March to June, there is a drop in dengue cases compared to earlier years that is not far from the previous years, but the MCOs contribute to the progressive drop of dengue incidence. Besides that, there does not appear to be much correlation between dengue incidence rates and COVID-19. 

There was a general decrease in percentage of positive tests from 2015 to 2018. The rates then increased in the years of 2019 and 2020 then decreased to 1.16% in 2021. It should be noted that the total number of tests done in 2021 was only 614, which is a 79.9% decrease from the average of the past years (2999.5). The low percentage of positive tests shows that undertesting was unlikely. This suggests that the low incidence of Dengue during 2021 is unlikely to cause by under detection.

From Table 2, it can be seen that the total number of cases recorded as well as the total number of recorded deaths were significantly lower than previous years. The total deaths in 2021 were 116 lower than 2020, a decrease of 87.2%. The mortality rate in deaths per 1000 people for 2021 was 0.0751%, drastic decrease from the yearly average of the previous years which was 0.158%. 

Mortality rates are a reliable way to visualise real dengue incidence rates, as deaths will always be reported. This method of analysis and the results decrease the possibility of underreporting being the cause of such low recorded incidence rates. This data points convincingly towards dengue cases dropping in 2021.

Graph 3 reflects no correlation between complete vaccinations and dengue, implying that the COVID-19 vaccinations have no effect on immunity against dengue. Although, majority of COVID-19 vaccinations were only completed near the end of the dengue data recorded, so claiming that there is completely no correlation would not be a reliable conclusion. They may have been a further decrease in dengue cases after week 44. Further study must be done.

Discussion

The data shows unequivocally that there is a significant drop in dengue incidence in the year 2021 compared to previous years, although this decrease seemed to begin around week 30 of 2020. A few speculations were made at an attempt to explain such a decrease. Firstly, one of the possible contributing factors to the decrease was that the cases may have been underreported as the healthcare system may have been overwhelmed, and people who potentially had dengue may have been afraid to seek treatment due to fear of contracting COVID-19 from healthcare facilities. Furthermore, the low dengue mortality rates during 2021 reflects the reduction of dengue incidence, as deaths are less likely to be unreported. The low positive rate of dengue tests and low dengue mortality rate convinced us that the drop of dengue cases in 2021 is unlikely to be caused by under reporting or under detection.

The next hypothesis is that under the movement control orders, people had more time to clean their homes and get rid of potential breeding grounds for vectors. Studies have shown that more frequent cleaning does lead to lower dengue incidence rates [17] [18]. This can only be ascertained with data from a vector (Aedes aegypti and Aedes albopictus) surveillance study.

The restriction on travel might also have affected dengue incidence rates heavily. Previous studies have shown that lesser travel is correlated with lower dengue incidence rates [19]. There have been many travel bans throughout the COVID-19 periods in Malaysia. Further study and analysis of the dengue incidence rates in these time periods is required to draw any conclusions. Unfortunately, month by month data of specific regions was not accessible.

Another potential contributing factor is the closing down of schools and workplaces because of the MCOs and transition to online schooling/working. In 2020, there were 5 million students among the population of 32.37 million in Malaysia [20]. The students were also equally distributed across every household across the nation, making the abstention from schools all the more significant. If this speculation is true, it would prove fruitful to look further into school and working environments as potential reservoirs of Aedes mosquitoes. 

The dengue test data collected had two limitations: patient specific data could not be collected and the data only represented the population of Perak. Different trends and conclusions could be made if all test types and a bigger part of the population was included.

The significant drop of dengue cases during the COVID-19 pandemic has generated a plethora of hypotheses that could be made for further investigation. This pandemic and the MCOs implemented in response have led to a natural experiment in which there was a nationwide change of lifestyle in Malaysia. Hence, a lot of opportunities for further research into the control of dengue and other infectious diseases have opened up. 

Bibliography 

[1] World Health Organization, “Dengue and Severe Dengue,” 10 January 2022. [Online]. Available: https://www.who.int/news-room/fact-sheets/detail/dengue-and-severe-dengue.

[2] Centers for Disease Control and Prevention, “How to Prevent the Spread of the Mosquito that Causes Dengue,” 25 January 2012. [Online]. Available: https://www.cdc.gov/dengue/resources/vectorcontrolsheetdengue.pdf. [Accessed 17 January 2022].

[3] A. Rudnick, “Dengue fever epidemiology in Malaysia, 1901-1980,” ResearchGate, vol. 1, pp. 1269-1272, 1986. 

[4] The Star, “Health Minister: Drastic drop in dengue cases in Malaysia this year,” The Star, 7 August 2021. [Online]. Available: https://www.thestar.com.my/news/nation/2021/08/07/health-minister-drastic-drop-in-dengue-cases-in-malaysia-this-year. [Accessed 17 January 2022].

[5] Bernama, “MCO extended another two weeks to May 12 – Muhyiddin,” Bernama, 23 April 2020. [Online]. Available: https://www.bernama.com/en/general/news.php?id=1835248. [Accessed 17 January 2022].

[6] New Straits Times, “Covid-19: Movement Control Order imposed with only essential sectors operating,” New Straits Times, 16 March 2020. [Online]. Available: https://www.nst.com.my/news/nation/2020/03/575177/covid-19-movement-control-order-imposed-only-essential-sectors-operating. [Accessed 17 January 2022].

[7] N. S. Sham, “COVID-19: PKPD dikuat kuasa di dua kawasan di Simpang Renggam,” Astro Awani, 26 March 2020. [Online]. Available: https://www.astroawani.com/berita-malaysia/covid19-pkpd-dikuat-kuasa-di-dua-kawasan-di-simpang-renggam-235454. [Accessed 17 January 2022].

[8] Bernama, “Perintah Kawalan Pergerakan bersyarat akan dilaksana – Muhyiddin,” Bernama, 1 May 2020. [Online]. Available: https://www.bernama.com/bm/news.php?id=1837419. [Accessed 29 January 2022].

[9] Bernama, “Essence of conditional Movement Control Order,” Bernama, 1 May 2020. [Online]. Available: https://web.archive.org/web/20200502065534/https://www.bernama.com/en/general/news.php?id=1837487. [Accessed 29 January 2022].

[10] The Sun, “Interstate travel among activities allowed from Wednesday – Muhyiddin,” The Sun , 7 June 2020. [Online]. Available: https://web.archive.org/web/20200608041411/https://www.thesundaily.my/home/interstate-travel-among-activities-allowed-from-wednesday-muhyiddin-HN2539119. [Accessed 29 January 2022].

[11] C. Loo, “More sectors to be reopened under RMCO from July 1,” The Sun, 26 June 2020. [Online]. Available: https://web.archive.org/web/20200626081436/https://www.thesundaily.my/covid-19/more-sectors-to-be-reopened-under-rmco-from-july-1-YD2630375. [Accessed 29 January 2022].

[12] The Sun, “Pre-schools, kindergartens to open on Wednesday, preparations underway for the little ones,” The Sun, 29 June 2020. [Online]. Available: https://web.archive.org/web/20200629010541/https://www.thesundaily.my/home/pre-schools-kindergartens-to-open-on-wednesday-preparations-underway-for-the-little-ones-AB2636604. [Accessed 29 January 2022].

[13] Menteri Kesihatan Malaysia, “Demam Denggi Dan Chikungkunya,” 2015-2021. [Online]. Available: https://www.moh.gov.my/index.php/database_stores/store_view/17?items=25&page=6. [Accessed 17 January 2022].

[14] World Health Organization, “Update on the Dengue situation in the Western Pacific Region,” World Health Organization, 2 December 2021. [Online]. Available: https://www.who.int/docs/default-source/wpro—documents/emergency/surveillance/dengue/dengue-20211202.pdf?sfvrsn=fc80101d_106. [Accessed 17 January 2022].

[15] E. M. L. R.-G. C. A. C. G. E. O.-O. J. H. B. M. D. B. a. M. R. Hannah Ritchie, “Our World in Data,” University of Oxford, 2021. [Online]. Available: https://ourworldindata.org/covid-vaccinations?country=MYS. [Accessed 17 January 2022].

[16] H.-Y. Y. et.al, “The effects of seasonal climate variability on dengue annual incidence in Hong Kong: A modelling study,” Scientific Reports, vol. 10, 2020. 

[17] P. R. B. S. Krishna Prasad Bhandari, “Application of GIS Modelling for Dengue Fever Prone Area Based on Socio-Cultural and Environmental Factors,” The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, vol. 37, no. 8, 2008. 

[18] T. Chareonviriyaphap, “Larval habitats and distribution patterns of Aedes aegypti (Linnaeus) and Aedes albopictus (Skuse), in Thailand,” Southeast Asian J Trop Med Public Health, vol. 34, no. 3, pp. 529-535, 2003. 

[19] H. H. e. al, “Dengue viruses circulating in Indonesia: A systematic review and phylogenetic analysis of data from five decades,” Reviews in Medical Virology, vol. 29, no. 4, 2019. 

[20] The Straits Times, “All 5m students in Malaysia back to school for first time since Covid-19 outbreak,” The Straits Times, 5 April 2021. [Online]. Available: https://www.straitstimes.com/asia/se-asia/all-students-in-malaysia-back-to-school-for-first-time-since-covid-19-outbreak. [Accessed 30 January 2022].

About the author

Ze Shan Chan

The post The drastic drop in dengue cases in Malaysia during the COVID-19 pandemic appeared first on Exploratio Journal.