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 Sunday 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, September has the highest number of users with a value of 6160. The month with the lowest number of users is January with an average of 1816.

The largest decrease in month to month users was September to October with an average change of -1425.

The largest increase in month to month users was August to September with an average change of 1457.

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, Sunday 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=994.44
## cnt ~ season + yr + mnth + weathersit + temp + atemp + hum + 
##     windspeed
## 
##              Df Sum of Sq      RSS     AIC
## - atemp       1     10965 32950553  992.46
## - temp        1     54078 32993666  992.56
## - hum         1     85940 33025528  992.63
## - windspeed   1    208618 33148206  992.90
## <none>                    32939588  994.44
## - season      3   6217780 39157368 1001.06
## - mnth       11  19762589 52702178 1006.75
## - weathersit  2  12038473 44978061 1013.18
## - yr          1  36833717 69773305 1047.23
## 
## Step:  AIC=992.46
## cnt ~ season + yr + mnth + weathersit + temp + hum + windspeed
## 
##              Df Sum of Sq      RSS     AIC
## - hum         1     82266 33032819  990.65
## - windspeed   1    280404 33230957  991.08
## <none>                    32950553  992.46
## - temp        1   3064297 36014850  996.96
## - season      3   6264429 39214982  999.17
## - mnth       11  19798796 52749349 1004.81
## - weathersit  2  12037751 44988304 1011.20
## - yr          1  37800612 70751165 1046.25
## 
## Step:  AIC=990.65
## cnt ~ season + yr + mnth + weathersit + temp + windspeed
## 
##              Df Sum of Sq      RSS     AIC
## - windspeed   1    225874 33258693  989.14
## <none>                    33032819  990.65
## - temp        1   3197548 36230367  995.39
## - season      3   6227270 39260089  997.25
## - mnth       11  21588771 54621590 1005.36
## - weathersit  2  20104640 53137459 1021.35
## - yr          1  46462583 79495402 1052.75
## 
## Step:  AIC=989.14
## cnt ~ season + yr + mnth + weathersit + temp
## 
##              Df Sum of Sq      RSS     AIC
## <none>                    33258693  989.14
## - temp        1   3243200 36501893  993.94
## - season      3   6617529 39876222  996.39
## - mnth       11  21833413 55092106 1003.99
## - weathersit  2  21330369 54589062 1021.32
## - yr          1  46291253 79549946 1050.80

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

## [1] "cnt"        "season"     "yr"         "mnth"       "weathersit" "temp"

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 
## 
## 73 samples
##  5 predictor
## 
## Pre-processing: centered (18), scaled (18) 
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 65, 65, 66, 65, 66, 66, ... 
## Resampling results:
## 
##   RMSE      Rsquared   MAE     
##   1021.793  0.7225059  834.4907
## 
## Tuning parameter 'intercept' was held constant at a value of TRUE

Our first linear model has an RMSE of 1021.79.

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 
## 
## 73 samples
##  5 predictor
## 
## Pre-processing: centered (112), scaled (112) 
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 65, 65, 66, 65, 66, 66, ... 
## Resampling results:
## 
##   RMSE      Rsquared  MAE     
##   3930.678  0.312887  2379.364
## 
## Tuning parameter 'intercept' was held constant at a value of TRUE

The RMSE value of the model changed to 3930.68.

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 
## 
## 73 samples
## 10 predictors
## 
## Pre-processing: centered (23), scaled (23) 
## Resampling: Bootstrapped (25 reps) 
## Summary of sample sizes: 73, 73, 73, 73, 73, 73, ... 
## Resampling results across tuning parameters:
## 
##   mtry  RMSE       Rsquared   MAE      
##    1    1503.8622  0.6196766  1230.9688
##    2    1222.4006  0.6723380   998.8947
##    3    1123.6962  0.6997624   907.8914
##    4    1074.1155  0.7190466   859.6767
##    5    1050.3281  0.7273632   835.3506
##    6    1030.8000  0.7356933   818.0587
##    7    1014.1238  0.7399714   800.9845
##    8    1008.4858  0.7412673   793.1774
##    9     998.4443  0.7449565   781.9634
##   10     997.8396  0.7441974   781.4082
## 
## 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 997.84.

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 
## 
## 73 samples
## 10 predictors
## 
## No pre-processing
## Resampling: Cross-Validated (10 fold, repeated 3 times) 
## Summary of sample sizes: 66, 65, 65, 65, 65, 65, ... 
## Resampling results across tuning parameters:
## 
##   shrinkage  interaction.depth  n.trees  RMSE       Rsquared   MAE      
##   0.001      1                   20      1840.3221  0.5757905  1493.0212
##   0.001      1                  100      1771.5998  0.5787438  1435.5235
##   0.001      1                  500      1530.9440  0.6054489  1247.2737
##   0.001      3                   20      1838.2792  0.6687314  1491.3589
##   0.001      3                  100      1761.7237  0.6644909  1427.6608
##   0.001      3                  500      1483.1254  0.6826342  1206.2386
##   0.001      5                   20      1838.7181  0.6456000  1491.7626
##   0.001      5                  100      1762.2846  0.6595348  1428.1076
##   0.001      5                  500      1483.0400  0.6828798  1205.6452
##   0.010      1                   20      1698.4968  0.5655502  1377.1134
##   0.010      1                  100      1353.2797  0.6565321  1108.3018
##   0.010      1                  500      1015.0552  0.7470200   858.7482
##   0.010      3                   20      1678.3392  0.6594732  1361.1719
##   0.010      3                  100      1283.5388  0.7091377  1048.8383
##   0.010      3                  500       969.3181  0.7612156   806.1670
##   0.010      5                   20      1679.7214  0.6532403  1361.6328
##   0.010      5                  100      1288.1646  0.7113764  1053.1609
##   0.010      5                  500       963.0617  0.7656589   802.0552
##   0.100      1                   20      1148.6922  0.7103737   969.4285
##   0.100      1                  100       968.4765  0.7572250   796.7408
##   0.100      1                  500       951.9364  0.7695029   773.2968
##   0.100      3                   20      1108.8465  0.7207993   935.1638
##   0.100      3                  100       962.3362  0.7524299   777.2414
##   0.100      3                  500       975.3889  0.7492407   787.4195
##   0.100      5                   20      1094.5970  0.7362154   924.9330
##   0.100      5                  100       946.0414  0.7609759   762.9541
##   0.100      5                  500       948.7343  0.7555612   774.0997
## 
## 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 = 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 Boosting.Model achieves the lowest test MSE of 4.7308134^{5} for Sunday 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')