This analysis explores the relationship between temperature (°C) and hourly bike rental demand in Seoul using simple linear regression. The goal is to identify whether temperature is a statistically significant predictor of demand and to evaluate the strengths and limitations of a single-predictor model.
Data Pre-processing
Data Cleaning:
Manually converted predictors (column names) to snake case for improved readability, easier referencing, and standardization.
Temperatures range from approximately -20 °C to 40 °C.
Most observations fall between 0 °C and 30 °C.
The values seem to closely follow a normal distribution.
# Creating histogram for Temperaturehist( SeoulBikes$temperature_c,main ="Histogram of Temperature",xlab ="Temperature (°C)",col ="pink" )
Hourly Bike Rentals
Distribution is right-skewed.
Most hours have “lower” rental counts, with fewer high-demand hours.
# Creating histogram for Hourly Bike Rentalshist( SeoulBikes$rented_bike_count,main ="Histogram of Hourly Bike Rentals",xlab ="Hourly Bike Rentals",col ="light blue" )
Simple Linear Model
Estimated Regression Equation:
\(\hat{y} = 329.9525 + 29.0811x\)
SeoulBikesModel <-lm(rented_bike_count ~ temperature_c, data=SeoulBikes) #Fitting modelsummary(SeoulBikesModel) #Getting summary stats of model
Call:
lm(formula = rented_bike_count ~ temperature_c, data = SeoulBikes)
Residuals:
Min 1Q Median 3Q Max
-1100.60 -336.57 -49.69 233.81 2525.19
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 329.9525 8.5411 38.63 <2e-16 ***
temperature_c 29.0811 0.4862 59.82 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 543.5 on 8758 degrees of freedom
Multiple R-squared: 0.29, Adjusted R-squared: 0.29
F-statistic: 3578 on 1 and 8758 DF, p-value: < 2.2e-16
Interpretation
Interpretation:
Rented bike count is expected to increase by approximately 29 additional bikes per hour for every 1 °C increase in temperature.
Significant?
Hypothesis Test:
\(H_0:\beta_1 = 0\)
\(H_a: \beta_1 \ne 0\)
\(a = 0.01\)
\(p-value : 2.2 * 10^{-16}\)
At a 99% confidence level \((\alpha = 0.01)\), we have enough statistical evidence \((p-value < a)\) to reject the null hypothesis and further conclude that temperature is a statistically significant predictor of hourly bike rentals.
R-Squared & F-Statistic
\(R^2 = 0.29\)
\(F-Statistic = 3578\)
The model explains 29% of variance, suggesting that additional predictors are needed to better explain hourly bike rental demand. The f-statistic also shows that the model is highly significant.
Residual Analysis
Residual Plot of Bike Rentals vs Temperature
Increase in spread at higher temperatures.
Greater variability in bike demand during warmer conditions.
The limitations of the sole predictor, temperature, start to show when looking at the confidence and prediction interval of our model. The confidence interval (green) captures the uncertainty around the estimated mean bike rental count at each temperature, while the prediction interval (red) reflects the range within which an individual hourly observation is expected to fall. The notably wide prediction interval highlights the considerable variability in the bike rentals that temperature alone cannot account for, reinforcing the need for additional predictors in future modeling.
temp_range <-data.frame(temperature_c =seq(-20, 40, length.out =100))ci <-predict(SeoulBikesModel, temp_range, interval ="confidence")pi <-predict(SeoulBikesModel, temp_range, interval ="prediction")plot(SeoulBikes$temperature_c, SeoulBikes$rented_bike_count,main ="Bike Rentals vs Temperature",xlab ="Temperature (°C)",ylab ="Rented Bike Count",pch =16, col ="gray70")abline(SeoulBikesModel, col ="blue", lwd =2)lines(temp_range$temperature_c, ci[,"lwr"], col ="darkgreen", lty =2)lines(temp_range$temperature_c, ci[,"upr"], col ="darkgreen", lty =2)lines(temp_range$temperature_c, pi[,"lwr"], col ="red", lty =2)lines(temp_range$temperature_c, pi[,"upr"], col ="red", lty =2)legend("topleft",legend =c("Regression Line", "Confidence Interval", "Prediction Interval"),col =c("blue", "darkgreen", "red"),lty =c(1, 2, 2),lwd =c(2, 1.5, 1.5),bty ="n")
At 22 °C, the model predicts ~970 rentals per hour, with a 95% confidence interval of (955.42, 984.06) for the true mean and a 95% prediction interval of (-95.74, 2035.22) for an individual observation.
Summary
Using simple linear regression, we modeled the relationship between temperature and hourly bike rental demand across 8,760 observations in Seoul. The estimated regression equation \(\hat{y} = 329.95 + 29.08x\), indicates that for every 1 °C increase in temperature, hourly bike rentals are expected to increase by ~29 bikes. At a 99% confidence level \((a = 0.01)\), temperature was found to be a statistically significant predictor \((p-value = 2.2 * 10^{-16})\), concluding rejection of the null hypothesis that \(\beta_1 = 0\)
However, with an \(R^2\) of 0.29, temperature alone explains only 29% of the variation in hourly rentals. At 22 °C, the model predicts ~970 rentals per hour, with a 95% confidence interval of (955.42, 984.06) for the true mean and a 95% prediction interval of (-95.74, 2035.22) for an individual observation. The wide prediction interval, combined with the heteroscedasticity observed in the residual plot, suggests that while temperature is a meaningful predictor, additional variables such as hour of day, season, and weather conditions would be needed to build a more reliable model.
