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 Monday 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, July has the highest number of users with a value of 5792. The month with the lowest number of users is January with an average of 1927.

The largest decrease in month to month users was November to December with an average change of -1161.

The largest increase in month to month users was March to April with an average change of 970.

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, Monday 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=967.17
## cnt ~ season + yr + mnth + weathersit + temp + atemp + hum + 
##     windspeed
## 
##              Df Sum of Sq      RSS     AIC
## - mnth       11   5292354 27963666  960.48
## <none>                    22671312  967.17
## - temp        1    858022 23529335  967.88
## - windspeed   1   1221154 23892467  969.00
## - atemp       1   1784997 24456309  970.70
## - hum         1   2576612 25247924  973.03
## - season      3  10321627 32992940  988.56
## - weathersit  2  14453386 37124699  999.17
## - yr          1  41256750 63928062 1040.85
## 
## Step:  AIC=960.48
## cnt ~ season + yr + weathersit + temp + atemp + hum + windspeed
## 
##              Df Sum of Sq      RSS     AIC
## - temp        1    604910 28568576  960.05
## <none>                    27963666  960.48
## - windspeed   1   1677270 29640936  962.74
## - atemp       1   1751823 29715489  962.92
## - hum         1   4956187 32919853  970.40
## - weathersit  2  16072632 44036298  989.63
## - season      3  23525654 51489320  999.05
## - yr          1  47288538 75252204 1030.75
## 
## Step:  AIC=960.05
## cnt ~ season + yr + weathersit + atemp + hum + windspeed
## 
##              Df Sum of Sq      RSS     AIC
## <none>                    28568576  960.05
## - windspeed   1   2166378 30734954  963.38
## - hum         1   4518507 33087083  968.77
## - weathersit  2  15656221 44224797  987.95
## - atemp       1  16967285 45535861  992.08
## - season      3  24905739 53474315  999.81
## - yr          1  49874459 78443035 1031.78

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

## [1] "cnt"        "season"     "yr"         "weathersit" "atemp"      "hum"       
## [7] "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 
## 
## 73 samples
##  6 predictor
## 
## Pre-processing: centered (9), scaled (9) 
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 65, 65, 66, 65, 66, 66, ... 
## Resampling results:
## 
##   RMSE     Rsquared   MAE    
##   795.377  0.8202952  595.303
## 
## Tuning parameter 'intercept' was held constant at a value of TRUE

Our first linear model has an RMSE of 795.38.

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
##  6 predictor
## 
## Pre-processing: centered (41), scaled (41) 
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 65, 65, 66, 65, 66, 66, ... 
## Resampling results:
## 
##   RMSE      Rsquared   MAE     
##   1054.246  0.7548595  725.5586
## 
## Tuning parameter 'intercept' was held constant at a value of TRUE

The RMSE value of the model changed to 1054.25.

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    1411.1480  0.5677018  1174.7260
##    2    1186.9712  0.6377193   986.3843
##    3    1094.4024  0.6753772   879.8229
##    4    1046.2410  0.6982560   821.1445
##    5    1011.3763  0.7123635   777.2291
##    6     994.8440  0.7167495   748.5663
##    7     980.1268  0.7242386   728.5467
##    8     969.6457  0.7266090   710.0647
##    9     967.1118  0.7252636   698.3667
##   10     959.1956  0.7273668   687.2306
## 
## 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 959.2.

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      1738.0909  0.5099345  1393.9075
##   0.001      1                  100      1697.2178  0.5213435  1365.3508
##   0.001      1                  500      1538.2159  0.5789252  1245.2809
##   0.001      3                   20      1735.8144  0.6076083  1391.0560
##   0.001      3                  100      1684.6202  0.6182622  1351.6036
##   0.001      3                  500      1486.8582  0.6470900  1187.1183
##   0.001      5                   20      1735.9764  0.6187760  1391.1311
##   0.001      5                  100      1684.7865  0.6233656  1351.4338
##   0.001      5                  500      1487.9485  0.6494165  1187.7423
##   0.010      1                   20      1649.1287  0.5327076  1331.3932
##   0.010      1                  100      1398.2924  0.6163837  1128.9400
##   0.010      1                  500      1028.4594  0.7129727   807.9488
##   0.010      3                   20      1628.2342  0.6385100  1307.2839
##   0.010      3                  100      1320.7608  0.6691349  1039.9535
##   0.010      3                  500       962.2232  0.7357640   735.5945
##   0.010      5                   20      1630.8654  0.6228646  1309.6462
##   0.010      5                  100      1332.5015  0.6626448  1050.1001
##   0.010      5                  500       971.9830  0.7323832   744.7821
##   0.100      1                   20      1240.5887  0.6355194  1007.1107
##   0.100      1                  100       983.5621  0.7345245   755.5325
##   0.100      1                  500       952.6332  0.7578123   731.4043
##   0.100      3                   20      1139.2996  0.6719673   890.3783
##   0.100      3                  100       932.5326  0.7459887   698.0666
##   0.100      3                  500       896.4063  0.7807940   659.3739
##   0.100      5                   20      1145.8177  0.6871101   901.0546
##   0.100      5                  100       922.2325  0.7542120   691.7070
##   0.100      5                  500       884.9143  0.7918749   644.3137
## 
## 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 Boosting.Model achieves the lowest test MSE of 5.0929405^{5} for Monday 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')