The goal of this notebookis to review the fundamental concepts of regression modelling. We will also begin to practice using the statistical programming language R.
-> There are two goals. The first is to predict the variable COUNT as a function of the other variables. The second is to determine the effect of bad weather on the number of bikes rented.
Adding a new variable BADWEATHER, which is “YES” if there is light or heavy rain or snow (if WEATHERSIT is 3 or 4), and “NO” otherwise (if WEATHERSIT is 1 or 2).
bikeShare$WEATHERSIT <- as.numeric(bikeShare$WEATHERSIT)
bikeShare$BADWEATHER <- ifelse(bikeShare$WEATHERSIT > 2 , "YES", "NO")
#ScatterPlot for COUNT
ggplot(bikeShare, aes(ATEMP,COUNT, color = BADWEATHER)) + geom_point()
#Scotter Plot for CASUAL users
ggplot(bikeShare, aes(ATEMP,CASUAL, color = BADWEATHER)) + geom_point()
#Scatter Plot for RESGISTERED users
ggplot(bikeShare, aes(ATEMP,REGISTERED, color = BADWEATHER)) + geom_point()
As we observe that the count of bike rides forms a curve. The count of rides is maximum when ATEMP is ~ 0.6 which approximates to 24 degree Celsius, which is pleasant weather.
In case of registered users, we see the trend similar to the count of total rides. But for casual rides the curve is different. Also, we can see that registered users are more active even on bad weather days.
Running a multivariate regression for COUNT, using the factors listed as well as TEMP, ATEMP, HUMIDITY, and WINDSPEED.
bikeShare$BADWEATHER <- as.factor(bikeShare$BADWEATHER)
Regression1 <- lm(COUNT~MONTH + WEEKDAY + HOLIDAY + TEMP + ATEMP +
HUMIDITY + WINDSPEED,
data = bikeShare)
summary(Regression1)##
## Call:
## lm(formula = COUNT ~ MONTH + WEEKDAY + HOLIDAY + TEMP + ATEMP +
## HUMIDITY + WINDSPEED, data = bikeShare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5400.6 -1001.0 -183.8 1096.2 3457.9
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4568.38 455.96 10.019 < 2e-16 ***
## MONTHAUGUST -633.37 298.39 -2.123 0.03413 *
## MONTHDECEMBER 42.59 264.00 0.161 0.87188
## MONTHFEBRUARY -752.73 270.23 -2.786 0.00549 **
## MONTHJANUARY -722.00 287.42 -2.512 0.01223 *
## MONTHJULY -1267.77 313.63 -4.042 5.87e-05 ***
## MONTHJUNE -563.97 287.41 -1.962 0.05012 .
## MONTHMARCH -286.83 245.56 -1.168 0.24317
## MONTHMAY 144.33 256.82 0.562 0.57430
## MONTHNOVEMBER 437.12 253.84 1.722 0.08550 .
## MONTHOCTOBER 771.43 244.51 3.155 0.00167 **
## MONTHSEPTEMBER 444.71 267.86 1.660 0.09731 .
## WEEKDAYYES 71.13 109.18 0.651 0.51494
## HOLIDAYYES -745.85 297.06 -2.511 0.01227 *
## TEMP 5779.00 2377.96 2.430 0.01533 *
## ATEMP 1688.42 2491.75 0.678 0.49824
## HUMIDITY -4244.54 378.61 -11.211 < 2e-16 ***
## WINDSPEED -4692.14 689.88 -6.801 2.19e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1321 on 713 degrees of freedom
## Multiple R-squared: 0.546, Adjusted R-squared: 0.5352
## F-statistic: 50.44 on 17 and 713 DF, p-value: < 2.2e-16
#Predict the COUNT
sum(Regression1$coefficients*c(1,0,0,0,1,0,0,0,0,0,0,0,1,0,0.35,0.30,0.21,0.30,1))## Warning in Regression1$coefficients * c(1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, :
## longer object length is not a multiple of shorter object length
## [1] 8716.079
The p-value of BADWEATHERYES is very low, hence it has a significant effect on the count of bikeshare requests. The coefficient if 1701, means on average if the weather is not bad, the count is 1701 more than if it was bad.
- Running a simple regression to determine the effect of bad weather on COUNT when other variables are not included in the model.
Regression2 <- lm(COUNT~BADWEATHER, data = bikeShare)
summary(Regression2)##
## Call:
## lm(formula = COUNT ~ BADWEATHER, data = bikeShare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4153.2 -1257.7 1.8 1404.8 4129.8
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4584.24 70.63 64.908 < 2e-16 ***
## BADWEATHERYES -2780.95 416.69 -6.674 4.93e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1882 on 729 degrees of freedom
## Multiple R-squared: 0.05758, Adjusted R-squared: 0.05629
## F-statistic: 44.54 on 1 and 729 DF, p-value: 4.934e-11
The difference in the coefficients is due to the inclusion of other variables. In the previous regression we had all the variables hence the effect gets reduced.
- Regression with interaction term BADWEATHER*WEEKDAY
#Regression with interaction term BADWEATHER*WEEKDAY
Regression3 <- lm(COUNT~ BADWEATHER + WEEKDAY + BADWEATHER*WEEKDAY, data = bikeShare)
summary(Regression3)##
## Call:
## lm(formula = COUNT ~ BADWEATHER + WEEKDAY + BADWEATHER * WEEKDAY,
## data = bikeShare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4206.7 -1262.1 -3.7 1405.3 4261.5
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4452.5 131.5 33.861 < 2e-16 ***
## BADWEATHERYES -2637.1 852.2 -3.095 0.00205 **
## WEEKDAYYES 185.3 155.9 1.188 0.23514
## BADWEATHERYES:WEEKDAYYES -201.2 977.1 -0.206 0.83695
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1883 on 727 degrees of freedom
## Multiple R-squared: 0.05941, Adjusted R-squared: 0.05553
## F-statistic: 15.31 on 3 and 727 DF, p-value: 1.15e-09
We start by adding an interaction term BADWEATHERWEEKDAY. Since the coefficient of the interaction term BADWEATHERWEEKDAY is negative, bike share use is more negatively affected if it’s a weekday. But since the p-value of the interaction term is 0.83, the significance is very low.
- Running a regression with TEMP & ATEMP
Regression4 <- lm(COUNT ~ MONTH + HOLIDAY + WEEKDAY + BADWEATHER + TEMP + ATEMP
,data = bikeShare)
summary(Regression4)##
## Call:
## lm(formula = COUNT ~ MONTH + HOLIDAY + WEEKDAY + BADWEATHER +
## TEMP + ATEMP, data = bikeShare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4320.9 -1086.0 -233.3 1193.3 3357.5
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1619.86 380.94 4.252 2.4e-05 ***
## MONTHAUGUST -348.14 306.96 -1.134 0.25711
## MONTHDECEMBER -156.94 269.77 -0.582 0.56092
## MONTHFEBRUARY -831.14 281.71 -2.950 0.00328 **
## MONTHJANUARY -914.49 298.45 -3.064 0.00227 **
## MONTHJULY -679.49 321.59 -2.113 0.03496 *
## MONTHJUNE -105.55 295.80 -0.357 0.72132
## MONTHMARCH -335.19 256.12 -1.309 0.19105
## MONTHMAY 22.22 265.00 0.084 0.93319
## MONTHNOVEMBER 438.14 261.99 1.672 0.09490 .
## MONTHOCTOBER 746.61 250.48 2.981 0.00297 **
## MONTHSEPTEMBER 456.58 273.54 1.669 0.09552 .
## HOLIDAYYES -758.59 310.36 -2.444 0.01476 *
## WEEKDAYYES 98.85 114.03 0.867 0.38630
## BADWEATHERYES -2696.63 310.01 -8.699 < 2e-16 ***
## TEMP 4401.10 2430.46 1.811 0.07059 .
## ATEMP 1843.24 2537.36 0.726 0.46781
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1379 on 714 degrees of freedom
## Multiple R-squared: 0.5042, Adjusted R-squared: 0.4931
## F-statistic: 45.38 on 16 and 714 DF, p-value: < 2.2e-16
- Running a regression without TEMP & ATEMP
Regression5 <- lm(COUNT ~ MONTH + HOLIDAY + WEEKDAY + BADWEATHER,
data = bikeShare)
summary(Regression5) ##
## Call:
## lm(formula = COUNT ~ MONTH + HOLIDAY + WEEKDAY + BADWEATHER,
## data = bikeShare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4410.3 -1114.1 -275.5 1271.0 4528.2
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4468.6 205.0 21.803 < 2e-16 ***
## MONTHAUGUST 1056.7 263.1 4.017 6.53e-05 ***
## MONTHDECEMBER -1038.5 262.6 -3.955 8.42e-05 ***
## MONTHFEBRUARY -1876.2 268.2 -6.995 6.08e-12 ***
## MONTHJANUARY -2345.6 262.6 -8.931 < 2e-16 ***
## MONTHJULY 1031.2 262.6 3.927 9.44e-05 ***
## MONTHJUNE 1169.4 265.2 4.410 1.19e-05 ***
## MONTHMARCH -822.3 262.8 -3.129 0.00183 **
## MONTHMAY 766.2 262.8 2.916 0.00366 **
## MONTHNOVEMBER -175.6 265.0 -0.663 0.50771
## MONTHOCTOBER 844.1 263.0 3.209 0.00139 **
## MONTHSEPTEMBER 1328.2 264.7 5.017 6.63e-07 ***
## HOLIDAYYES -653.0 325.4 -2.007 0.04516 *
## WEEKDAYYES 187.5 119.4 1.571 0.11672
## BADWEATHERYES -2795.2 324.7 -8.608 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1450 on 716 degrees of freedom
## Multiple R-squared: 0.4507, Adjusted R-squared: 0.44
## F-statistic: 41.96 on 14 and 716 DF, p-value: < 2.2e-16
When we are trying to predict the value of COUNT, the model with higher adjusted R-Squared is better (which has both ATEMP and TEMP), But if our use case is just to make an inference TEMP and ATEMP do not have any significance in determining the COUNT.