ST558 Project 2

Crista Gregg and Halid Kopanski 7/2/2021

• Introduction
• Data
• Summarizations
  • Summary statistics of users
  • Rentals by Year
  • Types of weather by season
  • Rentals by Weather
  • Casual vs. Registered bikers
  • Average bikers by month
  • Holiday and Temperature / Humidity data
  • Correlation among numeric predictors
• Modeling
  • Linear Regression
    • Linear Fit 1
    • Linear Fit 2
  • Ensemble Tree
    • Random Forests
    • Boosting Model
• Comparison

Introduction

The following analysis breaks down bicycle sharing usage based on data gathered for every recorded Thursday in the years 2011 and 2012. The data was gathered from users of Capitol Bikeshare based in Washington DC. In total, the dataset contains 731 entries. For each entry, 16 variables were recorded. The following is the list of the 16 variables and a short description of each:

In addition to summary statistics, this report will also model bicycle users by linear regression, random forests, and boosting. The model will help determine anticipated number of users based on readily available data. To achieve this, the response variables are casual, registered, and cnt. The other variables, not including the date and instant columns, will be the predictors for models developed later in this report.

Data

Here, we set up the data for the selected day of week and convert categorical variables to factors, and then split the data into a train and test set.

set.seed(1) #get the same splits every time
bikes <- read_csv('day.csv')

day_function <- function(x){
  x <- x + 1
  switch(x,"Sunday", 
           "Monday", 
           "Tuesday", 
           "Wednesday", 
           "Thursday", 
           "Friday", 
           "Saturday")
}

season_function <- function(x){
    #x <- as.character(x)
    switch(x, "Spring",
              "Summer",
              "Fall",
              "Winter")
}

bikes <- bikes %>% select(everything()) %>% 
  mutate(weekday = sapply(weekday, day_function), 
         season = sapply(season, season_function)) 
  

bikes$season <- as.factor(bikes$season)
bikes$yr <- as.factor(bikes$yr)
levels(bikes$yr) <- c('2011','2012')
bikes$mnth <- as.factor(bikes$mnth)
bikes$holiday <- as.factor(bikes$holiday)
bikes$weekday <- as.factor(bikes$weekday)
bikes$workingday <- as.factor(bikes$workingday)
bikes$weathersit <- as.factor(bikes$weathersit)
levels(bikes$weathersit) <- c('Clear to some clouds', 'Misty', 'Light snow or rain')

day <- params$day_of_week

#filter bikes by day of week
bikes <- filter(bikes, weekday == day)

#split data into train and test sets
train_rows <- sample(nrow(bikes), 0.7*nrow(bikes))
train <- bikes[train_rows,] %>% 
  select(-instant, -weekday, -casual, -registered, -dteday)
test <- bikes[-train_rows,] %>% 
  select(-instant, -weekday, -casual, -registered, -dteday)

Summarizations

Summary statistics of users

Below shows the summary statistics of bike users: casual, registered, and total.

knitr::kable(summary(bikes[,14:16]))

Rentals by Year

The following table tells us the total number of rentals for each of the two years of collected data, as well as the average number of rentals per day.

bikes %>%
  group_by(yr) %>%
  summarise(total_rentals = sum(cnt), avg_rentals = round(mean(cnt))) %>%
  knitr::kable()

Types of weather by season

Now we will look at the number of days with each type of weather by season. 1 represents ‘Clear to some clouds’, 2 represents ‘Misty’, and 3 represents ‘Light snow or rain’.

knitr::kable(table(bikes$season, bikes$weathersit))

Rentals by Weather

The following box plot shows us how many rentals we have for days that are sunny or partly cloudy, misty, or rainy/snowy. We may expect some differences in behavior between weekend days where less people might be inclined to ride their bikes for pleasure, versus weekdays when more people might brave moderately unpleasant weather to get to work.

ggplot(bikes, aes(factor(weathersit), cnt)) +
  geom_boxplot() +
  labs(x = 'Type of Weather', y = 'Number of Rental Bikes', title = 'Rental Bikes by Type of Weather') +
  theme_minimal()

weather_summary <- bikes %>%
  group_by(weathersit) %>%
  summarise(total_rentals = sum(cnt), avg_rentals = round(mean(cnt)))

weather_min <- switch(which.min(weather_summary$avg_rentals),
                               "clear weather",
                                                             "misty weather",
                                                             "weather with light snow or rain")

According to the above box plot, it can be seen that weather with light snow or rain brings out the least amount of total users.

Casual vs. Registered bikers

Below is a chart of the relationship between casual and registered bikers. We might expect a change in the slope if we look at different days of the week. Perhaps we see more registered bikers riding on the weekday but more casual users on the weekend.

ggplot(bikes, aes(casual, registered)) +
  geom_point() +
  geom_smooth(formula = 'y ~ x', method = 'lm') +
  theme_minimal() +
  labs(title = 'Registered versus Casual Renters')

Average bikers by month

Below we see a plot of the average daily number of bikers by month. We should expect to see more bikers in the spring and summer months, and the least in the winter.

plot_mth <- bikes %>%
  group_by(mnth) %>%
  summarize(avg_bikers = mean(cnt))

ggplot(plot_mth, aes(mnth, avg_bikers)) +
  geom_line(group = 1, color = 'darkblue', size = 1.2) +
  geom_point(size = 2) +
  theme_minimal() +
  labs(title='Average daily number of bikers by month', y = 'Average Daily Bikers', x = 'Month') +
  scale_x_discrete(labels = month.abb)

month_max <- month.name[which.max(plot_mth$avg_bikers)]
month_min <- month.name[which.min(plot_mth$avg_bikers)]

user_max <- max(plot_mth$avg_bikers)
user_min <- min(plot_mth$avg_bikers)

changes <- rep(0, 11)
diff_mth <- rep("x", 11)

for (i in 2:12){
  diff_mth[i - 1] <- paste(month.name[i - 1], "to", month.name[i])
  changes[i - 1] <- round(plot_mth$avg_bikers[i] - plot_mth$avg_bikers[i - 1])
}


diff_tab_mth <- as_tibble(cbind(diff_mth, changes))

According to the graph, August has the highest number of users with a value of 6038. The month with the lowest number of users is January with an average of 2513.

The largest decrease in month to month users was October to November with an average change of -1543.

The largest increase in month to month users was April to May with an average change of 1129.

Holiday and Temperature / Humidity data

We would like to see what effect public holidays have on the types of bicycle users on average for a given day. In this case, Thursday data shows the following relationships:

bikes %>% ggplot(aes(x = as.factor(workingday), y = casual)) + geom_boxplot() + 
                labs(title = paste("Casual Users on", params$day_of_week)) + 
                xlab("") + 
                ylab("Casual Users") + 
                scale_x_discrete(labels = c('Public Holiday', 'Workday')) + 
                theme_minimal()

bikes %>% ggplot(aes(x = as.factor(workingday), y = registered)) + geom_boxplot() + 
                labs(title = paste("Registered Users on", params$day_of_week)) + 
                xlab("") + 
                ylab("Registered Users") +
                scale_x_discrete(labels = c('Public Holiday', 'Workday')) +
                theme_minimal()

Temperature and humidity have an effect on the number of users on a given day.

First, normalized temperature data (both actual temperature and perceived):

bike_temp <- bikes %>% select(cnt, temp, atemp) %>% 
                    gather(key = type, value = temp_norm, temp, atemp, factor_key = FALSE)

ggplot(bike_temp, aes(x = temp_norm, y = cnt, col = type, shape = type)) + 
        geom_point() + geom_smooth(formula = 'y ~ x', method = 'loess') +
        scale_color_discrete(name = "Temp Type", labels = c("Perceived", "Actual")) +
        scale_shape_discrete(name = "Temp Type", labels = c("Perceived", "Actual")) +
        labs(title = paste("Temperature on", params$day_of_week, "(Actual and Perceived)")) +
        xlab("Normalized Temperatures") +
        ylab("Total Users") + 
        theme_minimal()

Next the effect of humidity:

bikes%>% ggplot(aes(x = hum, y = cnt)) + geom_point() + geom_smooth(formula = 'y ~ x', method = 'loess') +
                labs(title = paste("Humidity versus Total Users on", params$day_of_week)) +
                xlab("Humidity (normalized)") +
                ylab("Total Number of Users") +
                theme_minimal()

Correlation among numeric predictors

Here we are checking the correlation between the numeric predictors in the data.

knitr::kable(round(cor(bikes[ , c(11:16)]), 3))

corrplot(cor(bikes[ , c(11:16)]), method = "circle")

Modeling

Now, we will fit two linear regression model, a random forest model, and a boosting model. We will use cross-validation to select the best tuning parameters for the ensemble based methods, and then compare all four models using the test MSE.

Linear Regression

Linear regression is one of the most common methods for modeling. It looks at a set of predictors and estimates what will happen to the response if one of the predictors or a combination of predictors change. This model is highly interpretable, as it shows us the effect of each individual predictor as well as interactions. We can see if the change in the response goes up or down and in what quantity. The model is chosen by minimizing the squares of the distances between the estimated value and the actual value in the testing set. Below we fit two different linear regression models.

Linear Fit 1

The first model will have a subset of predictors chosen by stepwise selection. Once we have chosen an interesting set of predictors, we will use cross-validation to determine the RMSE and R².

lm_fit_select <- lm(cnt ~ ., data = train[ , c(1:3, 6:11)])
model <- step(lm_fit_select)

## Start:  AIC=949.71
## cnt ~ season + yr + mnth + weathersit + temp + atemp + hum + 
##     windspeed
## 
##              Df Sum of Sq      RSS     AIC
## - temp        1      2803 20917299  947.72
## - atemp       1     52294 20966790  947.89
## <none>                    20914496  949.71
## - hum         1    880367 21794863  950.68
## - mnth       11   8869335 29783831  953.16
## - season      3   3224091 24138588  954.03
## - windspeed   1   2497331 23411828  955.83
## - weathersit  2   4447899 25362395  959.59
## - yr          1  71470684 92385180 1054.67
## 
## Step:  AIC=947.72
## cnt ~ season + yr + mnth + weathersit + atemp + hum + windspeed
## 
##              Df Sum of Sq      RSS     AIC
## <none>                    20917299  947.72
## - hum         1    891241 21808540  948.72
## - atemp       1   1384732 22302031  950.33
## - mnth       11   9203097 30120396  951.97
## - season      3   3261984 24179284  952.15
## - windspeed   1   2521646 23438945  953.91
## - weathersit  2   4709139 25626438  958.34
## - yr          1  72029922 92947222 1053.10

variables <- names(model$model)
variables #variables we will use for our model

## [1] "cnt"        "season"     "yr"         "mnth"       "weathersit" "atemp"     
## [7] "hum"        "windspeed"

set.seed(10)
lm.fit <- train(cnt ~ ., data = train[variables], method = 'lm',
                preProcess = c('center', 'scale'),
                trControl = trainControl(method = 'cv', number = 10))
lm.fit

## Linear Regression 
## 
## 72 samples
##  7 predictor
## 
## Pre-processing: centered (20), scaled (20) 
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 64, 64, 65, 64, 65, 66, ... 
## Resampling results:
## 
##   RMSE      Rsquared   MAE     
##   777.7939  0.8312946  630.1502
## 
## Tuning parameter 'intercept' was held constant at a value of TRUE

Our first linear model has an RMSE of 777.79.

Linear Fit 2

Adding interactions to the terms included in the first model.

set.seed(10)
lm.fit1 <- train(cnt ~ . + .*., data = train[variables], method = 'lm',
                preProcess = c('center', 'scale'),
                trControl = trainControl(method = 'cv', number = 10))
lm.fit1

## Linear Regression 
## 
## 72 samples
##  7 predictor
## 
## Pre-processing: centered (151), scaled (151) 
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 64, 64, 65, 64, 65, 66, ... 
## Resampling results:
## 
##   RMSE      Rsquared   MAE     
##   22551.26  0.1764429  13936.99
## 
## Tuning parameter 'intercept' was held constant at a value of TRUE

The RMSE value of the model changed to 2.255126^{4}.

Ensemble Tree

Ensemble trees methods come in many types and are very versatile when it comes to regression or classification. For the following, we will be using the two most common and well known methods: Random Forests (a form of bagging) and Boosting. Both these tree based methods involve optimization during the development process. In the case of random forests, the optimization involves varying the number of predictors used. This is done to mitigate the effects of one or more predictors from overshadowing other predictors. Boosting is a method where the final model is developed through an iterative combination of weaker models where each iteration builds upon the last. While both methods are very flexible and tend to process good results, the models themselves are not as interpretable as linear regression. We normally just analyze the output of the models.

Random Forests

Below is the result of training with the random forest method. This method uses a different subset of predictors for each tree and averages the results across many trees, selected by bootstrapping. By reducing the number of predictors considered in each tree, we may be able to reduce the correlation between trees to improve our results. In the training model below, we vary the number of predictors used in each tree.

rf_fit <- train(cnt ~ ., data = train, method = 'rf',
                preProcess = c('center', 'scale'),
                tuneGrid = data.frame(mtry = 1:10))
rf_fit

## Random Forest 
## 
## 72 samples
## 10 predictors
## 
## Pre-processing: centered (23), scaled (23) 
## Resampling: Bootstrapped (25 reps) 
## Summary of sample sizes: 72, 72, 72, 72, 72, 72, ... 
## Resampling results across tuning parameters:
## 
##   mtry  RMSE       Rsquared   MAE      
##    1    1536.5087  0.6236545  1321.5472
##    2    1225.5796  0.7228303  1046.2283
##    3    1065.7195  0.7753845   898.6175
##    4     960.5088  0.8127159   799.8696
##    5     901.4282  0.8277360   744.6310
##    6     860.5639  0.8352925   707.8102
##    7     835.0243  0.8374360   687.9077
##    8     817.0103  0.8391227   676.8263
##    9     807.0021  0.8388095   670.6317
##   10     798.3287  0.8369205   664.9287
## 
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was mtry = 10.

The best model uses 10 predictors. This gives an RMSE of 798.33.

Boosting Model

The following are the results of Boosting model development using the provided bike data.

trctrl <- trainControl(method = "repeatedcv", 
                       number = 10, 
                       repeats = 3)

set.seed(2020)

boost_grid <- expand.grid(n.trees = c(20, 100, 500),
                          interaction.depth = c(1, 3, 5),
                          shrinkage = c(0.1, 0.01, 0.001),
                          n.minobsinnode = 10)

boost_fit <-  train(cnt ~ ., 
                    data = train, 
                    method = "gbm", 
                    verbose = F, #suppresses excessive printing while model is training
                    trControl = trctrl, 
                    tuneGrid = data.frame(boost_grid))

A total of 27 models were evaluated. Each differing by the combination of boosting parameters. The results are show below:

print(boost_fit)

## Stochastic Gradient Boosting 
## 
## 72 samples
## 10 predictors
## 
## No pre-processing
## Resampling: Cross-Validated (10 fold, repeated 3 times) 
## Summary of sample sizes: 66, 64, 64, 64, 64, 64, ... 
## Resampling results across tuning parameters:
## 
##   shrinkage  interaction.depth  n.trees  RMSE       Rsquared   MAE      
##   0.001      1                   20      1803.7727  0.6192964  1534.6698
##   0.001      1                  100      1742.8711  0.6346525  1473.6384
##   0.001      1                  500      1508.6334  0.7106600  1268.8846
##   0.001      3                   20      1802.9551  0.6223451  1534.1533
##   0.001      3                  100      1739.4533  0.6448948  1470.6783
##   0.001      3                  500      1483.5103  0.7215642  1247.9871
##   0.001      5                   20      1803.0211  0.6302004  1534.0942
##   0.001      5                  100      1739.3401  0.6432298  1470.6524
##   0.001      5                  500      1484.1420  0.7167665  1248.0472
##   0.010      1                   20      1674.8002  0.6618092  1410.3764
##   0.010      1                  100      1308.5856  0.7447496  1098.2456
##   0.010      1                  500       890.2355  0.8099954   748.9062
##   0.010      3                   20      1664.0521  0.6612188  1403.4356
##   0.010      3                  100      1253.9133  0.7654264  1051.7556
##   0.010      3                  500       846.6884  0.8229541   706.1216
##   0.010      5                   20      1669.1199  0.6666112  1407.5024
##   0.010      5                  100      1247.1704  0.7716133  1046.4042
##   0.010      5                  500       848.5163  0.8232512   709.0930
##   0.100      1                   20      1079.9195  0.7586377   903.0052
##   0.100      1                  100       851.1249  0.8240164   687.4616
##   0.100      1                  500       825.6938  0.8306110   664.1847
##   0.100      3                   20      1029.5161  0.7643701   875.5980
##   0.100      3                  100       833.2091  0.8202480   676.6808
##   0.100      3                  500       819.8197  0.8300415   661.8549
##   0.100      5                   20      1017.7244  0.7798504   866.8972
##   0.100      5                  100       806.6355  0.8363901   652.2746
##   0.100      5                  500       790.5919  0.8439755   634.7244
## 
## Tuning parameter 'n.minobsinnode' was held constant at a value of 10
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were n.trees = 500, interaction.depth = 5, shrinkage
##  = 0.1 and n.minobsinnode = 10.

results_tab <- as_tibble(boost_fit$results[,c(1,2,4:6)])

The attributes of the best model is shown here.

boost_min <- which.min(results_tab$RMSE)

knitr::kable(results_tab[boost_min,], digits = 2)

Comparison

Here we compare the 4 models developed earlier. Each model was applied to a test set and the results were then used to calculate MSE. Below are the results.

lm_pred <- predict(lm.fit, newdata = test)
lm_pred1 <- predict(lm.fit1, newdata = test)
rf_pred <- predict(rf_fit, newdata = test)
boost_pred <- predict(boost_fit, newdata = test)

prediction_values <- as_tibble(cbind(lm_pred, lm_pred1, rf_pred, boost_pred))

lm_MSE <- mean((lm_pred - test$cnt)^2)
lm_MSE1 <- mean((lm_pred1 - test$cnt)^2)
rf_MSE <- mean((rf_pred - test$cnt)^2)
boost_MSE <- mean((boost_pred - test$cnt)^2)

comp <- data.frame('Linear Model 1' = lm_MSE, 
                   'Linear Model 2' = lm_MSE1, 
                   'Random Forest Model' = rf_MSE, 
                   'Boosting Model' = boost_MSE)

knitr::kable(t(comp), col.names = "MSE")

It was found that Random.Forest.Model achieves the lowest test MSE of 9.0692082^{5} for Thursday data.

Below is a graph of the Actual vs Predicted results:

index_val <- (which.min(t(comp)))

results_plot <- as_tibble(cbind("preds" = prediction_values[[index_val]], "actual" = test$cnt))

ggplot(data = results_plot, aes(preds, actual)) + geom_point() +
     labs(x = paste(names(which.min(comp)), "Predictions"), y = "Actual Values",
     title = paste(names(which.min(comp)), "Actual vs Predicted Values")) +
     geom_abline(slope = 1, intercept = 0, col = 'red')