Detecting Anomolous Events in Washington DC Bike Sharing Dataset

We want to detect anomolous events in the Washington DC Bike Sharing dataset.

Data Set Information:

Bike sharing systems are new generation of traditional bike rentals where whole process from membership, rental and return back has become automatic. Through these systems, user is able to easily rent a bike from a particular position and return back at another position. Currently, there are about over 500 bike-sharing programs around the world which is composed of over 500 thousands bicycles. Today, there exists great interest in these systems due to their important role in traffic, environmental and health issues.

Apart from interesting real world applications of bike sharing systems, the characteristics of data being generated by these systems make them attractive for the research. Opposed to other transport services such as bus or subway, the duration of travel, departure and arrival position is explicitly recorded in these systems. This feature turns bike sharing system into a virtual sensor network that can be used for sensing mobility in the city. Hence, it is expected that most of important events in the city could be detected via monitoring these data.

Attribute Information:

Both hour.csv and day.csv have the following fields, except hr which is not available in day.csv

• instant: record index
• dteday : date
• season : season (1:springer, 2:summer, 3:fall, 4:winter)
• yr : year (0: 2011, 1:2012)
• mnth : month ( 1 to 12)
• hr : hour (0 to 23)
• holiday : weather day is holiday or not (extracted from [Web Link])
• weekday : day of the week
• workingday : if day is neither weekend nor holiday is 1, otherwise is 0.
• weathersit :
• 1: Clear, Few clouds, Partly cloudy, Partly cloudy
• 2: Mist + Cloudy, Mist + Broken clouds, Mist + Few clouds, Mist
• 3: Light Snow, Light Rain + Thunderstorm + Scattered clouds, Light Rain + Scattered clouds
• 4: Heavy Rain + Ice Pallets + Thunderstorm + Mist, Snow + Fog
• temp : Normalized temperature in Celsius. The values are derived via (t-t_min)/(t_max-t_min), t_min=-8, t_max=+39 (only in hourly scale)
• atemp: Normalized feeling temperature in Celsius. The values are derived via (t-t_min)/(t_max-t_min), t_min=-16, t_max=+50 (only in hourly scale)
• hum: Normalized humidity. The values are divided to 100 (max)
• windspeed: Normalized wind speed. The values are divided to 67 (max)
• casual: count of casual users
• registered: count of registered users
• cnt: count of total rental bikes including both casual and registered

First we load the required libraries as well as read in the “day” data from the bike sharing dataset.

rm(list=ls())

library('ggplot2')

## Warning: package 'ggplot2' was built under R version 3.2.4

library('dplyr')

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

day <- read.csv("~/NYCDataScience_course/Final_Project/Bike-Sharing-Dataset/day.csv")
#View(day)

Next we set season, yr, mnth, holiday, weekday, workingday and weathersit as factors.

day$season = as.factor(day$season)
day$yr = as.factor(day$yr)
day$mnth = as.factor(day$mnth)
day$holiday = as.factor(day$holiday)
day$weekday = as.factor(day$weekday)
day$workingday = as.factor(day$workingday)
day$weathersit = as.factor(day$weathersit)

We now do some visualizations by plotting various variables against each other with points color coded according the magintude of cnt.

scale_color_gradient(low = "#132B43", high = "#56B1F7", space = "Lab", na.value = "grey50", guide = "colourbar")

## <ScaleContinuous>
##  Range:  
##  Limits:    0 --    1

d1 <- qplot(atemp, hum, data=day, color=cnt)
d1 + scale_colour_gradient(limits=c(22, 8714))

d2 <- qplot(temp, hum, data=day, color=cnt)
d2 + scale_colour_gradient(limits=c(22, 8714))

d3 <- qplot(windspeed, temp, data=day, color=cnt)
d3 + scale_colour_gradient(limits=c(22, 8714))

d3 <- qplot(windspeed, hum, data=day, color=cnt)
d3 + scale_colour_gradient(limits=c(22, 8714))

As you can see from the plots, cnt increases with increasing temp as well as decreasing windspeed. Humdity may or may not have some effect on cnt.

Next we plot hitograms of cnt versus various variables.

qplot(cnt, data=day, binwidth=100) + facet_wrap(~weathersit)

by_weather <-group_by(day, weathersit)
summarize(by_weather, total=sum(cnt))

## Source: local data frame [3 x 2]
## 
##   weathersit   total
##       (fctr)   (int)
## 1          1 2257952
## 2          2  996858
## 3          3   37869

qplot(cnt, data=day, binwidth=100) + facet_wrap(~holiday)

by_holiday <-group_by(day, holiday)
summarize(by_holiday, total=sum(cnt))

## Source: local data frame [2 x 2]
## 
##   holiday   total
##    (fctr)   (int)
## 1       0 3214244
## 2       1   78435

qplot(cnt, data=day, binwidth=100) + facet_wrap(~workingday)       

by_workingday <-group_by(day, workingday)
summarize(by_workingday, total=sum(cnt))

## Source: local data frame [2 x 2]
## 
##   workingday   total
##       (fctr)   (int)
## 1          0 1000269
## 2          1 2292410

qplot(cnt, data=day, binwidth=100) + facet_wrap(~season)

by_season <-group_by(day, season)
summarize(by_season, total=sum(cnt))

## Source: local data frame [4 x 2]
## 
##   season   total
##   (fctr)   (int)
## 1      1  471348
## 2      2  918589
## 3      3 1061129
## 4      4  841613

qplot(cnt, data=day, binwidth=100) + facet_wrap(~weekday)

by_weekday <-group_by(day, weekday)
summarize(by_weekday, total=sum(cnt))

## Source: local data frame [7 x 2]
## 
##   weekday  total
##    (fctr)  (int)
## 1       0 444027
## 2       1 455503
## 3       2 469109
## 4       3 473048
## 5       4 485395
## 6       5 487790
## 7       6 477807

cnt is greatest for weathersit = 1 and then 2 with low values for 3.
cnt is much larger when holiday = 0 than 1
cnt is much larger for workingday = 1 than 0
cnt is largest for season = 3 followed by 2, 4 and 1.
cnt is roughly similar for all weekday.

We now perform multiple linear regressiom on the “day”" data.

model.saturated = lm(cnt ~ . -instant -dteday -casual -registered, data = day)

summary(model.saturated)

## 
## Call:
## lm(formula = cnt ~ . - instant - dteday - casual - registered, 
##     data = day)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3944.7  -348.2    63.8   457.4  2912.7 
## 
## Coefficients: (1 not defined because of singularities)
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1485.84     239.75   6.198 9.77e-10 ***
## season2       884.71     179.49   4.929 1.03e-06 ***
## season3       832.70     213.13   3.907 0.000102 ***
## season4      1575.35     181.00   8.704  < 2e-16 ***
## yr1          2019.74      58.22  34.691  < 2e-16 ***
## mnth2         131.03     143.78   0.911 0.362443    
## mnth3         542.83     165.43   3.281 0.001085 ** 
## mnth4         451.17     247.57   1.822 0.068820 .  
## mnth5         735.51     267.63   2.748 0.006145 ** 
## mnth6         515.40     282.41   1.825 0.068423 .  
## mnth7          30.80     313.82   0.098 0.921854    
## mnth8         444.95     303.17   1.468 0.142639    
## mnth9        1004.17     265.12   3.788 0.000165 ***
## mnth10        519.67     241.55   2.151 0.031787 *  
## mnth11       -116.69     230.78  -0.506 0.613257    
## mnth12        -89.59     182.21  -0.492 0.623098    
## holiday1     -589.70     180.36  -3.270 0.001130 ** 
## weekday1      212.05     109.49   1.937 0.053187 .  
## weekday2      309.53     107.13   2.889 0.003982 ** 
## weekday3      381.36     107.48   3.548 0.000414 ***
## weekday4      386.34     107.53   3.593 0.000350 ***
## weekday5      436.98     107.44   4.067 5.30e-05 ***
## weekday6      440.46     106.56   4.133 4.01e-05 ***
## workingday1       NA         NA      NA       NA    
## weathersit2  -462.54      77.09  -6.000 3.16e-09 ***
## weathersit3 -1965.09     197.05  -9.972  < 2e-16 ***
## temp         2855.01    1398.16   2.042 0.041526 *  
## atemp        1786.16    1462.12   1.222 0.222261    
## hum         -1535.47     292.45  -5.250 2.01e-07 ***
## windspeed   -2823.30     414.55  -6.810 2.09e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 769.2 on 702 degrees of freedom
## Multiple R-squared:  0.8484, Adjusted R-squared:  0.8423 
## F-statistic: 140.3 on 28 and 702 DF,  p-value: < 2.2e-16

plot(model.saturated) #Assessing the assumptions of the model.

library(car) #Companion to applied regression.
influencePlot(model.saturated)

##       StudRes        Hat     CookD
## 595  2.112057 0.64959535 0.5326909
## 669 -5.336784 0.04056466 0.1998979

alias(model.saturated)

## Model :
## cnt ~ (instant + dteday + season + yr + mnth + holiday + weekday + 
##     workingday + weathersit + temp + atemp + hum + windspeed + 
##     casual + registered) - instant - dteday - casual - registered
## 
## Complete :
##             (Intercept) season2 season3 season4 yr1 mnth2 mnth3 mnth4
## workingday1  0           0       0       0       0   0     0     0   
##             mnth5 mnth6 mnth7 mnth8 mnth9 mnth10 mnth11 mnth12 holiday1
## workingday1  0     0     0     0     0     0      0      0     -1      
##             weekday1 weekday2 weekday3 weekday4 weekday5 weekday6
## workingday1  1        1        1        1        1        0      
##             weathersit2 weathersit3 temp atemp hum windspeed
## workingday1  0           0           0    0     0   0

model.saturated seems like a good candidate for linear regression from the above plots.

As you can see from the alias values, workinday1 is perfectly collinear with holiday1.
Hence remove workingday variable in the next model2.

#vif(model.saturated) #Assessing the variance inflation factors for the variables
#in our model.

model2 = lm(cnt ~ . -instant -dteday -workingday -atemp -casual -registered, data = day)

summary(model2) #R^2 adjusted went up, model still significant, etc.

## 
## Call:
## lm(formula = cnt ~ . - instant - dteday - workingday - atemp - 
##     casual - registered, data = day)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3960.9  -350.9    74.1   456.0  2919.9 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1543.552    235.129   6.565 1.01e-10 ***
## season2       889.302    179.516   4.954 9.12e-07 ***
## season3       832.236    213.204   3.903 0.000104 ***
## season4      1578.947    181.040   8.722  < 2e-16 ***
## yr1          2018.063     58.225  34.660  < 2e-16 ***
## mnth2         136.855    143.747   0.952 0.341396    
## mnth3         545.132    165.480   3.294 0.001036 ** 
## mnth4         456.494    247.615   1.844 0.065667 .  
## mnth5         723.520    267.541   2.704 0.007010 ** 
## mnth6         490.552    281.776   1.741 0.082133 .  
## mnth7           8.404    313.395   0.027 0.978613    
## mnth8         404.912    301.494   1.343 0.179700    
## mnth9         983.948    264.698   3.717 0.000217 ***
## mnth10        520.937    241.636   2.156 0.031432 *  
## mnth11       -111.362    230.816  -0.482 0.629621    
## mnth12        -84.389    182.229  -0.463 0.643439    
## holiday1     -603.605    180.066  -3.352 0.000845 ***
## weekday1      214.877    109.508   1.962 0.050133 .  
## weekday2      309.132    107.171   2.884 0.004041 ** 
## weekday3      377.407    107.467   3.512 0.000473 ***
## weekday4      385.206    107.562   3.581 0.000366 ***
## weekday5      428.604    107.258   3.996 7.12e-05 ***
## weekday6      438.699    106.590   4.116 4.32e-05 ***
## weathersit2  -465.202     77.083  -6.035 2.57e-09 ***
## weathersit3 -1981.357    196.670 -10.075  < 2e-16 ***
## temp         4487.305    411.838  10.896  < 2e-16 ***
## hum         -1518.178    292.208  -5.196 2.68e-07 ***
## windspeed   -2925.438    406.175  -7.202 1.53e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 769.5 on 703 degrees of freedom
## Multiple R-squared:  0.848,  Adjusted R-squared:  0.8422 
## F-statistic: 145.3 on 27 and 703 DF,  p-value: < 2.2e-16

plot(model2) #No overt additional violations.

influencePlot(model2) #No overt additional violations; Hawaii actually lowers

##       StudRes        Hat     CookD
## 69  -1.210739 0.16828646 0.1028882
## 668 -4.756498 0.07702696 0.2557740
## 669 -5.356592 0.04026858 0.2033892

#its hat value (leverage).

vif(model2) #VIFs all decrease.

##                  GVIF Df GVIF^(1/(2*Df))
## season     169.713093  3        2.352985
## yr           1.046249  1        1.022863
## mnth       391.715048 11        1.311784
## holiday      1.116829  1        1.056801
## weekday      1.153293  6        1.011956
## weathersit   1.886133  2        1.171907
## temp         7.006223  1        2.646927
## hum          2.135348  1        1.461283
## windspeed    1.221500  1        1.105215

anova(model2, model.saturated)

## Analysis of Variance Table
## 
## Model 1: cnt ~ (instant + dteday + season + yr + mnth + holiday + weekday + 
##     workingday + weathersit + temp + atemp + hum + windspeed + 
##     casual + registered) - instant - dteday - workingday - atemp - 
##     casual - registered
## Model 2: cnt ~ (instant + dteday + season + yr + mnth + holiday + weekday + 
##     workingday + weathersit + temp + atemp + hum + windspeed + 
##     casual + registered) - instant - dteday - casual - registered
##   Res.Df       RSS Df Sum of Sq      F Pr(>F)
## 1    703 416284778                           
## 2    702 415401688  1    883090 1.4924 0.2223

model2 seems like a good candidate for linear regression from the above plots.
The VIF values for all variables are low indicating that collinearity is not a problem.
From analysis of variance, seems like model2 and model.saturated are significantly different.
We now have a reduced model where we remove the hum variable.

model.reduced = lm(cnt ~ . -instant -dteday -workingday -atemp -hum -casual -registered, data = day)

anova(model.reduced, model.saturated)

## Analysis of Variance Table
## 
## Model 1: cnt ~ (instant + dteday + season + yr + mnth + holiday + weekday + 
##     workingday + weathersit + temp + atemp + hum + windspeed + 
##     casual + registered) - instant - dteday - workingday - atemp - 
##     hum - casual - registered
## Model 2: cnt ~ (instant + dteday + season + yr + mnth + holiday + weekday + 
##     workingday + weathersit + temp + atemp + hum + windspeed + 
##     casual + registered) - instant - dteday - casual - registered
##   Res.Df       RSS Df Sum of Sq      F    Pr(>F)    
## 1    704 432269190                                  
## 2    702 415401688  2  16867502 14.252 8.563e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

summary(model.reduced)

## 
## Call:
## lm(formula = cnt ~ . - instant - dteday - workingday - atemp - 
##     hum - casual - registered, data = day)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4189.5  -372.9    67.2   481.8  2893.1 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   715.03     175.95   4.064 5.37e-05 ***
## season2       848.08     182.62   4.644 4.08e-06 ***
## season3       841.83     217.10   3.878 0.000115 ***
## season4      1585.86     184.35   8.603  < 2e-16 ***
## yr1          2065.21      58.57  35.263  < 2e-16 ***
## mnth2         180.92     146.12   1.238 0.216071    
## mnth3         638.19     167.52   3.810 0.000151 ***
## mnth4         606.99     250.41   2.424 0.015603 *  
## mnth5         820.13     271.78   3.018 0.002639 ** 
## mnth6         749.76     282.40   2.655 0.008110 ** 
## mnth7         242.98     315.80   0.769 0.441899    
## mnth8         575.45     305.18   1.886 0.059762 .  
## mnth9        1033.99     269.36   3.839 0.000135 ***
## mnth10        541.10     246.02   2.199 0.028175 *  
## mnth11       -109.31     235.04  -0.465 0.642021    
## mnth12       -119.21     185.44  -0.643 0.520540    
## holiday1     -585.60     183.33  -3.194 0.001464 ** 
## weekday1      222.82     111.50   1.998 0.046061 *  
## weekday2      328.50     109.07   3.012 0.002688 ** 
## weekday3      394.44     109.38   3.606 0.000333 ***
## weekday4      440.09     109.00   4.038 5.99e-05 ***
## weekday5      476.69     108.81   4.381 1.36e-05 ***
## weekday6      467.50     108.39   4.313 1.84e-05 ***
## weathersit2  -702.78      63.19 -11.122  < 2e-16 ***
## weathersit3 -2422.60     180.63 -13.412  < 2e-16 ***
## temp         3923.88     404.57   9.699  < 2e-16 ***
## windspeed   -2328.09     396.69  -5.869 6.76e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 783.6 on 704 degrees of freedom
## Multiple R-squared:  0.8422, Adjusted R-squared:  0.8364 
## F-statistic: 144.5 on 26 and 704 DF,  p-value: < 2.2e-16

plot(model.reduced)

influencePlot(model.reduced)

##       StudRes        Hat     CookD
## 668 -4.825354 0.07620740 0.2625982
## 669 -5.562958 0.03699797 0.2055197
## 725 -1.938558 0.09986872 0.1240252

vif(model.reduced)

##                  GVIF Df GVIF^(1/(2*Df))
## season     169.095113  3        2.351555
## yr           1.020838  1        1.010365
## mnth       354.080512 11        1.305775
## holiday      1.116415  1        1.056606
## weekday      1.135306  6        1.010631
## weathersit   1.115412  2        1.027682
## temp         6.520446  1        2.553516
## windspeed    1.123627  1        1.060013

model.reduced seems like a good candidate for linear regression from the above plots.

We now perform AIC and BIC to see which of these models are most appropriate.

AIC(model.saturated,    #Model with all variables.
    model2,        #Model with all variables EXCEPT Illiteracy.
    model.reduced) #Model with all variables EXCEPT Illiteracy, Area, and Income.

##                 df      AIC
## model.saturated 30 11820.49
## model2          29 11820.04
## model.reduced   28 11845.58

BIC(model.saturated,
    model2,
    model.reduced) #Both the minimum AIC and BIC values appear alongside the

##                 df      BIC
## model.saturated 30 11958.32
## model2          29 11953.28
## model.reduced   28 11974.23

#reduced model that we tested above.

AIC and BIC values are lowest for model2 hence we will go with model2.

We now do a prediction using model2.

#choose model2

p2.conf = predict(model2, se.fit = TRUE, interval = "confidence")
p2.pred = predict(model2, se.fit = TRUE, interval = "prediction")

## Warning in predict.lm(model2, se.fit = TRUE, interval = "prediction"): predictions on current data refer to _future_ responses

We now read in the hour data and do a multiple linear regression on the data.

hour <- read.csv("~/NYCDataScience_course/Final_Project/Bike-Sharing-Dataset/hour.csv")
#View(hour)

hour$season = as.factor(hour$season)
hour$yr = as.factor(hour$yr)
hour$mnth = as.factor(hour$mnth)
hour$hr = as.factor(hour$hr)
hour$holiday = as.factor(hour$holiday)
hour$weekday = as.factor(hour$weekday)
hour$workingday = as.factor(hour$workingday)
hour$weathersit = as.factor(hour$weathersit)

model.hour = lm(cnt ~ . -instant -dteday -workingday -atemp -casual -registered, data = hour)

summary(model.hour)

## 
## Call:
## lm(formula = cnt ~ . - instant - dteday - workingday - atemp - 
##     casual - registered, data = hour)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -392.88  -60.82   -7.68   51.17  440.65 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -79.627      6.567 -12.126  < 2e-16 ***
## season2       38.495      4.858   7.925 2.43e-15 ***
## season3       31.999      5.752   5.563 2.69e-08 ***
## season4       68.257      4.884  13.976  < 2e-16 ***
## yr1           85.337      1.564  54.577  < 2e-16 ***
## mnth2          3.873      3.921   0.988 0.323281    
## mnth3         14.469      4.409   3.282 0.001034 ** 
## mnth4          6.626      6.551   1.012 0.311788    
## mnth5         19.834      7.007   2.830 0.004654 ** 
## mnth6          4.502      7.196   0.626 0.531540    
## mnth7        -14.838      8.077  -1.837 0.066213 .  
## mnth8          5.054      7.853   0.644 0.519865    
## mnth9         30.810      6.996   4.404 1.07e-05 ***
## mnth10        15.854      6.487   2.444 0.014533 *  
## mnth11        -9.485      6.241  -1.520 0.128551    
## mnth12        -5.912      4.956  -1.193 0.232936    
## hr1          -17.393      5.348  -3.253 0.001146 ** 
## hr2          -26.467      5.366  -4.932 8.21e-07 ***
## hr3          -37.167      5.405  -6.876 6.36e-12 ***
## hr4          -40.286      5.411  -7.446 1.01e-13 ***
## hr5          -23.561      5.375  -4.383 1.18e-05 ***
## hr6           35.327      5.361   6.589 4.55e-11 ***
## hr7          170.448      5.351  31.855  < 2e-16 ***
## hr8          310.981      5.344  58.191  < 2e-16 ***
## hr9          163.267      5.350  30.518  < 2e-16 ***
## hr10         108.445      5.372  20.187  < 2e-16 ***
## hr11         133.841      5.412  24.730  < 2e-16 ***
## hr12         173.168      5.458  31.725  < 2e-16 ***
## hr13         168.120      5.496  30.588  < 2e-16 ***
## hr14         152.266      5.527  27.548  < 2e-16 ***
## hr15         161.644      5.538  29.188  < 2e-16 ***
## hr16         223.672      5.526  40.475  < 2e-16 ***
## hr17         377.355      5.494  68.687  < 2e-16 ***
## hr18         345.532      5.458  63.310  < 2e-16 ***
## hr19         236.985      5.407  43.832  < 2e-16 ***
## hr20         157.466      5.377  29.285  < 2e-16 ***
## hr21         108.045      5.355  20.176  < 2e-16 ***
## hr22          71.023      5.345  13.287  < 2e-16 ***
## hr23          32.169      5.341   6.023 1.74e-09 ***
## holiday1     -27.189      4.878  -5.574 2.52e-08 ***
## weekday1       9.456      2.974   3.180 0.001478 ** 
## weekday2      10.812      2.905   3.722 0.000198 ***
## weekday3      13.320      2.900   4.592 4.42e-06 ***
## weekday4      13.048      2.902   4.496 6.98e-06 ***
## weekday5      16.852      2.890   5.831 5.60e-09 ***
## weekday6      15.934      2.879   5.535 3.16e-08 ***
## weathersit2  -10.690      1.920  -5.568 2.61e-08 ***
## weathersit3  -65.927      3.233 -20.394  < 2e-16 ***
## weathersit4  -64.069     58.920  -1.087 0.276874    
## temp         233.297      9.403  24.810  < 2e-16 ***
## hum          -81.333      5.545 -14.667  < 2e-16 ***
## windspeed    -36.309      6.845  -5.304 1.14e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 101.8 on 17327 degrees of freedom
## Multiple R-squared:  0.686,  Adjusted R-squared:  0.6851 
## F-statistic: 742.4 on 51 and 17327 DF,  p-value: < 2.2e-16

plot(model.hour) #Assessing the assumptions of the model.

influencePlot(model.hour)

##         StudRes         Hat      CookD
## 586   -0.253638 0.334718960 0.02494951
## 9124   1.531435 0.334602696 0.15059276
## 14774  4.337326 0.002697825 0.03126733

vif(model.hour) #A

##                  GVIF Df GVIF^(1/(2*Df))
## season     163.930448  3        2.339429
## yr           1.025279  1        1.012561
## mnth       312.113041 11        1.298308
## hr           1.747652 23        1.012210
## holiday      1.115178  1        1.056020
## weekday      1.123621  6        1.009760
## weathersit   1.389828  3        1.056396
## temp         5.499232  1        2.345044
## hum          1.919913  1        1.385609
## windspeed    1.176308  1        1.084577

#Do not use multiple linear regression for "hour" data

As you can see from the above plots, multiple linear regression is not a good candidate for the hour dataset.

Hence we will try Generalized Boosted Regression Models on the hour dataset.

library(gbm)

## Warning: package 'gbm' was built under R version 3.2.3

## Loading required package: survival

## Loading required package: lattice

## Loading required package: splines

## Loading required package: parallel

## Loaded gbm 2.1.1

train = sample(1:nrow(hour), 7*nrow(hour)/10)

#Fitting 10,000 trees with a depth of 4.
set.seed(0)
boost.hour = gbm(cnt ~ . -instant -dteday -workingday -atemp -casual -registered, data = hour[train, ],
                   distribution = "gaussian",
                   n.trees = 10000,
                   interaction.depth = 4)

#Inspecting the relative influence.
par(mfrow = c(1, 1))
summary(boost.hour)

##                   var    rel.inf
## hr                 hr 61.9885912
## temp             temp 11.4149177
## yr                 yr  9.5000267
## weekday       weekday  8.5856425
## season         season  3.5194042
## weathersit weathersit  1.9422920
## hum               hum  1.4143500
## mnth             mnth  1.1357884
## holiday       holiday  0.3748570
## windspeed   windspeed  0.1241303

#Let's make a prediction on the test set. With boosting, the number of trees is
#a tuning parameter; having too many can cause overfitting. In general, we should
#use cross validation to select the number of trees. Instead, we will compute the
#test error as a function of the number of trees and make a plot for illustrative
#purposes.
n.trees = seq(from = 100, to = 10000, by = 100)
predmat = predict(boost.hour, newdata = hour[-train, ], n.trees = n.trees)

#Produces 100 different predictions for each of the 152 observations in our
#test set.
dim(predmat)

## [1] 5214  100

#Calculating the boosted errors.
par(mfrow = c(1, 1))
berr = with(hour[-train, ], apply((predmat - cnt)^2, 2, mean))
plot(n.trees, berr, pch = 16,
     ylab = "Mean Squared Error",
     xlab = "# Trees",
     main = "Boosting Test Error")

berr[which(berr==min(berr))]

##    10000 
## 3567.622

Above we get the number of trees which will give the lowest mean squared error.

We will now increase the shrinkage parameter where a higher proportion of the errors are carried over in order to obtain a lower mean sequared error.

#Increasing the shrinkage parameter; a higher proportion of the errors are
#carried over.
set.seed(0)
boost.hour2 = gbm(cnt ~ . -instant -dteday -workingday -atemp -casual -registered, data = hour[train, ],
                    distribution = "gaussian",
                    n.trees = 10000,
                    interaction.depth = 4,
                    shrinkage = 0.1)
predmat2 = predict(boost.hour2, newdata = hour[-train, ], n.trees = n.trees)

berr2 = with(hour[-train, ], apply((predmat2 - cnt)^2, 2, mean))
plot(n.trees, berr2, pch = 16,
     ylab = "Mean Squared Error",
     xlab = "# Trees",
     main = "Boosting Test Error")

berr2[which(berr2==min(berr2))]

##     1300 
## 1782.969

We get the get the number of trees which will give the lowest mean squared error.
We then use that value to do a prediction.

nt2 = as.numeric(labels(berr2[which(berr2==min(berr2))]))

predgbm = predict(boost.hour2, newdata=hour, n.trees=nt2)

We now compute the residuals which are difference between predicted counts and actual counts.
We calculate the mean of the residuals as well as the standard deviation.
Z scores are then calculated.
We calculate the mean Z-scores for all the hours in a given day.
We choose the highest 15 Z-scores and use that to pick the dates for anomolous events.

res = predgbm - hour$cnt
mres = mean(res)
sdres = sd(res)
n = length(res)
z = abs((res-mres)/(sdres))

hour$z = z
#View(hour)
by_dteday <-group_by(hour, dteday)
s = summarize(by_dteday, avgz=mean(z))

sort(s$avgz, decreasing=TRUE)[1:100]

##   [1] 2.7112381 2.2397319 1.9624200 1.8211334 1.7304263 1.6513585 1.6275855
##   [8] 1.5973282 1.5938915 1.5439886 1.5294643 1.5179849 1.4857667 1.4665028
##  [15] 1.4621685 1.4621259 1.4235389 1.4077666 1.4067821 1.4015984 1.3930406
##  [22] 1.3411182 1.3401727 1.3340643 1.3156862 1.3036272 1.2918769 1.2299950
##  [29] 1.2292920 1.2125380 1.2090872 1.2021152 1.1918650 1.1855863 1.1745246
##  [36] 1.1742758 1.1674217 1.1653828 1.1635320 1.1620328 1.1601677 1.1573951
##  [43] 1.1307613 1.1268530 1.1154102 1.0996260 1.0969936 1.0950148 1.0905806
##  [50] 1.0904985 1.0864845 1.0853500 1.0756282 1.0736106 1.0638337 1.0604942
##  [57] 1.0493767 1.0341047 1.0336842 1.0304345 1.0279820 1.0196014 1.0192730
##  [64] 1.0177070 1.0171453 1.0129120 1.0086488 1.0080085 1.0057368 1.0052259
##  [71] 1.0051839 1.0025880 0.9992134 0.9988848 0.9962330 0.9881547 0.9875984
##  [78] 0.9874134 0.9855418 0.9850833 0.9831720 0.9805352 0.9780101 0.9751244
##  [85] 0.9747139 0.9714812 0.9710251 0.9692384 0.9690795 0.9690553 0.9689953
##  [92] 0.9675672 0.9654109 0.9600272 0.9511359 0.9504113 0.9417479 0.9363075
##  [99] 0.9351280 0.9334449

o = order(s$avgz, decreasing=TRUE)[1:15]

s$avgz[o[1:15]]

##  [1] 2.711238 2.239732 1.962420 1.821133 1.730426 1.651358 1.627585
##  [8] 1.597328 1.593891 1.543989 1.529464 1.517985 1.485767 1.466503
## [15] 1.462169

day$dteday[o[1:15]]

##  [1] 2012-10-30 2012-11-23 2012-11-21 2011-08-28 2012-10-02 2012-07-04
##  [7] 2012-12-31 2012-03-17 2012-11-24 2012-05-14 2012-10-19 2011-10-19
## [13] 2011-11-25 2012-06-01 2011-08-23
## 731 Levels: 2011-01-01 2011-01-02 2011-01-03 2011-01-04 ... 2012-12-31

Events detected:

2012-10-30: Sandy
2012-11-23: Day after Thanksgiving
2012-11-21: Day before Thanksgiving
2011-08-28: Hurricane Irene
2012-07-04: Independence Day/Washington DC Fireworks
2012-12-31: New Year's Eve
2012-11-24: Saturday after Thanksgiving
2012-10-19: Storm
2011-11-25: Day after Thanksgiving
2012-06-01: Tornado
2011-08-23: Virginia Earthquake

We now use the multiple linear regression results from the day dataset to detect anomolous events.
We compute the residuals which are difference between predicted counts and actual counts.
We calculate the mean of the residuals as well as the standard deviation.
Z scores are then calculated. We choose the highest 15 Z-scores and use that to pick the dates for anomolous events.

res.lm = p2.conf$fit[,1] - day$cnt
mres.lm = mean(res.lm)
sdres.lm = sd(res.lm)
n.lm = length(res.lm)
z.lm = abs((res.lm-mres.lm)/(sdres.lm))

day$zlm = z.lm
#View(day)

sort(z.lm, decreasing=TRUE)[1:15]

##      669      668      442      692      500      694      266      554 
## 5.245143 4.586557 3.866616 3.808029 3.751417 3.606062 3.578995 3.498611 
##      695      185      546      239      204      555      448 
## 3.386125 3.256032 3.134793 3.047427 2.920467 2.890135 2.776642

o.lm = order(z.lm, decreasing=TRUE)[1:100]
z.lm[o.lm[1:15]]

##      669      668      442      692      500      694      266      554 
## 5.245143 4.586557 3.866616 3.808029 3.751417 3.606062 3.578995 3.498611 
##      695      185      546      239      204      555      448 
## 3.386125 3.256032 3.134793 3.047427 2.920467 2.890135 2.776642

day$dteday[o.lm[1:15]]

##  [1] 2012-10-30 2012-10-29 2012-03-17 2012-11-22 2012-05-14 2012-11-24
##  [7] 2011-09-23 2012-07-07 2012-11-25 2011-07-04 2012-06-29 2011-08-27
## [13] 2011-07-23 2012-07-08 2012-03-23
## 731 Levels: 2011-01-01 2011-01-02 2011-01-03 2011-01-04 ... 2012-12-31

Events detected:

2012-10-30: Sandy
2012-10-29: Sandy
2012-11-22: Thanksgiving
2012-11-24: Saturday after Thanksgiving
2012-11-25: Sunday after Thanksgiving
2011-07-04: Independence Day/Washington DC fireworks
2011-08-27: Hurricane Irene
2011-07-23: Heatwave
2012-03-23: National Cherry Blossom Festival