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 Friday 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 misty weather 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 5958. The month with the lowest number of users is January with an average of 2446.

The largest decrease in month to month users was October to November with an average change of -1100.

The largest increase in month to month users was April to May with an average change of 1364.

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, Friday 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=964.78
## cnt ~ season + yr + mnth + weathersit + temp + atemp + hum + 
##     windspeed
## 
##              Df Sum of Sq       RSS     AIC
## - atemp       1     43275  26554207  962.90
## - temp        1    181543  26692476  963.27
## <none>                     26510932  964.78
## - windspeed   1   1278308  27789241  966.17
## - season      3   5081719  31592651  971.41
## - weathersit  1   3975312  30486244  972.84
## - hum         1   4741056  31251988  974.63
## - mnth       11  16000402  42511334  976.78
## - yr          1  83257126 109768058 1065.08
## 
## Step:  AIC=962.9
## cnt ~ season + yr + mnth + weathersit + temp + hum + windspeed
## 
##              Df Sum of Sq       RSS     AIC
## <none>                     26554207  962.90
## - temp        1    931852  27486058  963.38
## - windspeed   1   1573211  28127418  965.04
## - season      3   5076977  31631184  969.50
## - weathersit  1   3932647  30486854  970.84
## - hum         1   4991816  31546023  973.30
## - mnth       11  15961549  42515756  974.79
## - yr          1  86113899 112668106 1064.96

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

## [1] "cnt"        "season"     "yr"         "mnth"       "weathersit" "temp"      
## [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     
##   840.3348  0.8058005  639.2514
## 
## Tuning parameter 'intercept' was held constant at a value of TRUE

Our first linear model has an RMSE of 840.33.

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     
##   45284.82  0.1125644  24718.89
## 
## Tuning parameter 'intercept' was held constant at a value of TRUE

The RMSE value of the model changed to 4.528482^{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    1541.3024  0.5232559  1287.1656
##    2    1297.4778  0.6029902  1069.1247
##    3    1149.0791  0.6757222   935.9516
##    4    1076.7362  0.7100077   866.6288
##    5    1012.1832  0.7391830   809.7396
##    6     976.9998  0.7505534   780.0984
##    7     957.1692  0.7563229   758.2646
##    8     938.3539  0.7617937   742.0791
##    9     918.6451  0.7682235   726.6247
##   10     915.6496  0.7653673   722.3094
## 
## 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 915.65.

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      1751.7997  0.5677079  1460.4057
##   0.001      1                  100      1702.0125  0.5923391  1413.6224
##   0.001      1                  500      1506.2732  0.6578963  1239.2935
##   0.001      3                   20      1750.0724  0.5703227  1458.4118
##   0.001      3                  100      1696.8749  0.5972390  1409.2492
##   0.001      3                  500      1480.4995  0.6651292  1217.9744
##   0.001      5                   20      1750.5274  0.5767142  1459.0622
##   0.001      5                  100      1698.3299  0.5940318  1410.6581
##   0.001      5                  500      1479.4173  0.6680127  1216.1789
##   0.010      1                   20      1646.7750  0.6042792  1363.4046
##   0.010      1                  100      1327.8664  0.6957145  1086.9363
##   0.010      1                  500       946.0966  0.7600355   809.6001
##   0.010      3                   20      1632.4629  0.6254759  1350.0838
##   0.010      3                  100      1288.7386  0.7021518  1054.7210
##   0.010      3                  500       916.2767  0.7747765   787.5591
##   0.010      5                   20      1634.9489  0.6088769  1353.8759
##   0.010      5                  100      1286.4647  0.7015301  1050.5948
##   0.010      5                  500       919.5385  0.7696854   787.2890
##   0.100      1                   20      1124.8383  0.7129305   944.6193
##   0.100      1                  100       900.5140  0.7750221   746.4735
##   0.100      1                  500       969.9370  0.7369212   790.6134
##   0.100      3                   20      1079.6352  0.7189592   913.8189
##   0.100      3                  100       875.9833  0.7812878   729.9705
##   0.100      3                  500       948.0103  0.7392601   780.5216
##   0.100      5                   20      1104.4144  0.6977804   936.7669
##   0.100      5                  100       890.1437  0.7707885   737.4515
##   0.100      5                  500       971.3040  0.7241388   811.7497
## 
## 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 = 100, interaction.depth = 3, 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 7.2314152^{5} for Friday 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')