Analysis on Advertisments
Matilda Kadzo
03/06/2022
Defining The Question
A Kenyan entrepreneur has created an online cryptography course and would want to advertise it on her blog. She currently targets audiences originating from various countries. In the past, she ran adverts to advertise a related course on the same blog and collected data in the process. She would now like to employ your services as a Data Science Consultant to help her identify which individuals are most likely to click on her adverts.
Metric of Success
To provide an accurate depiction of the people most likely to view the clients advertisements and provide recommendations to the client based on the results of the univariate and bivariate analysis conducted on the dataset.
Understanding the context
Clicks on adverts can help you understand how appealing your advert is to people who see it. Highly targeted ads are more likely to receive clicks. This can help you gauge how enticing your advert is. In this case, it would help us know how many people would be interested in the online cryptography course through the number of clicks on our client’s blog.
Experimental Design
Steps taken:
- Loading the dataset
- Performing data cleaning
- Exploratory Data Analysis
- Conclusion and recommendation
Data Relevance
- Daily Time Spent on Site - Time spent per day on the blog
- Age - Age of the respondents
- Area Income - Income Distribution of the respondents’ area
- Daily Internet Usage - How much internet is used on a daily
- Ad Topic Line - Topic of the advert
- City - City of respondents
- Male - gender of respondents; 1 if male and 0 if female.
- Country -country of respondents
- Time stamp - the time the data is recorded
- Clicked on Ad - whether the respondents click on the ads; 0 if they don’t and 1 if they do.
Loading the Dataset
advert <- read.csv("/home/binti/Downloads/R/advertising.csv")Previewing the top of our dataset
head(advert)## Daily.Time.Spent.on.Site Age Area.Income Daily.Internet.Usage
## 1 68.95 35 61833.90 256.09
## 2 80.23 31 68441.85 193.77
## 3 69.47 26 59785.94 236.50
## 4 74.15 29 54806.18 245.89
## 5 68.37 35 73889.99 225.58
## 6 59.99 23 59761.56 226.74
## Ad.Topic.Line City Male Country
## 1 Cloned 5thgeneration orchestration Wrightburgh 0 Tunisia
## 2 Monitored national standardization West Jodi 1 Nauru
## 3 Organic bottom-line service-desk Davidton 0 San Marino
## 4 Triple-buffered reciprocal time-frame West Terrifurt 1 Italy
## 5 Robust logistical utilization South Manuel 0 Iceland
## 6 Sharable client-driven software Jamieberg 1 Norway
## Timestamp Clicked.on.Ad
## 1 2016-03-27 00:53:11 0
## 2 2016-04-04 01:39:02 0
## 3 2016-03-13 20:35:42 0
## 4 2016-01-10 02:31:19 0
## 5 2016-06-03 03:36:18 0
## 6 2016-05-19 14:30:17 0Previewing the tail of our dataset
tail(advert)## Daily.Time.Spent.on.Site Age Area.Income Daily.Internet.Usage
## 995 43.70 28 63126.96 173.01
## 996 72.97 30 71384.57 208.58
## 997 51.30 45 67782.17 134.42
## 998 51.63 51 42415.72 120.37
## 999 55.55 19 41920.79 187.95
## 1000 45.01 26 29875.80 178.35
## Ad.Topic.Line City Male
## 995 Front-line bifurcated ability Nicholasland 0
## 996 Fundamental modular algorithm Duffystad 1
## 997 Grass-roots cohesive monitoring New Darlene 1
## 998 Expanded intangible solution South Jessica 1
## 999 Proactive bandwidth-monitored policy West Steven 0
## 1000 Virtual 5thgeneration emulation Ronniemouth 0
## Country Timestamp Clicked.on.Ad
## 995 Mayotte 2016-04-04 03:57:48 1
## 996 Lebanon 2016-02-11 21:49:00 1
## 997 Bosnia and Herzegovina 2016-04-22 02:07:01 1
## 998 Mongolia 2016-02-01 17:24:57 1
## 999 Guatemala 2016-03-24 02:35:54 0
## 1000 Brazil 2016-06-03 21:43:21 1Dataset Columns
names(advert)## [1] "Daily.Time.Spent.on.Site" "Age"
## [3] "Area.Income" "Daily.Internet.Usage"
## [5] "Ad.Topic.Line" "City"
## [7] "Male" "Country"
## [9] "Timestamp" "Clicked.on.Ad"Cleaning Data
Finding the total missing values in our dataset.
colSums(is.na(advert))## Daily.Time.Spent.on.Site Age Area.Income
## 0 0 0
## Daily.Internet.Usage Ad.Topic.Line City
## 0 0 0
## Male Country Timestamp
## 0 0 0
## Clicked.on.Ad
## 0#There are no missing values in our datasetChecking for duplicates across our rows.
sum(advert[duplicated(advert),])## [1] 0#There are no duplicates in this dataset.The dataset had neither missing values or any duplicated values
Exploring the dataset
Checking the descriptive statistics of the dataset
summary(advert)## Daily.Time.Spent.on.Site Age Area.Income Daily.Internet.Usage
## Min. :32.60 Min. :19.00 Min. :13996 Min. :104.8
## 1st Qu.:51.36 1st Qu.:29.00 1st Qu.:47032 1st Qu.:138.8
## Median :68.22 Median :35.00 Median :57012 Median :183.1
## Mean :65.00 Mean :36.01 Mean :55000 Mean :180.0
## 3rd Qu.:78.55 3rd Qu.:42.00 3rd Qu.:65471 3rd Qu.:218.8
## Max. :91.43 Max. :61.00 Max. :79485 Max. :270.0
## Ad.Topic.Line City Male Country
## Length:1000 Length:1000 Min. :0.000 Length:1000
## Class :character Class :character 1st Qu.:0.000 Class :character
## Mode :character Mode :character Median :0.000 Mode :character
## Mean :0.481
## 3rd Qu.:1.000
## Max. :1.000
## Timestamp Clicked.on.Ad
## Length:1000 Min. :0.0
## Class :character 1st Qu.:0.0
## Mode :character Median :0.5
## Mean :0.5
## 3rd Qu.:1.0
## Max. :1.0Checking the structure of the dataframe
str(advert)## 'data.frame': 1000 obs. of 10 variables:
## $ Daily.Time.Spent.on.Site: num 69 80.2 69.5 74.2 68.4 ...
## $ Age : int 35 31 26 29 35 23 33 48 30 20 ...
## $ Area.Income : num 61834 68442 59786 54806 73890 ...
## $ Daily.Internet.Usage : num 256 194 236 246 226 ...
## $ Ad.Topic.Line : chr "Cloned 5thgeneration orchestration" "Monitored national standardization" "Organic bottom-line service-desk" "Triple-buffered reciprocal time-frame" ...
## $ City : chr "Wrightburgh" "West Jodi" "Davidton" "West Terrifurt" ...
## $ Male : int 0 1 0 1 0 1 0 1 1 1 ...
## $ Country : chr "Tunisia" "Nauru" "San Marino" "Italy" ...
## $ Timestamp : chr "2016-03-27 00:53:11" "2016-04-04 01:39:02" "2016-03-13 20:35:42" "2016-01-10 02:31:19" ...
## $ Clicked.on.Ad : int 0 0 0 0 0 0 0 1 0 0 ...Checking for Outliers
Checking for outliers in the dataset. These show a visual shape of our data distribution.
boxplot(advert$Area.Income,
main ="Area Income",
col = "orange",
border = 'brown',
horizontal = TRUE,
notch = TRUE)
#There are a few outliers in the area income column.boxplot(advert$Daily.Time.Spent.on.Site,
main ="Daily Time Spent on Site",
col = "orange",
border = 'brown',
horizontal = TRUE,
notch = TRUE)
#There are no outliers in the daily time spent on site column. boxplot(advert$Age,
main ="Age",
col = "orange",
border = 'brown',
horizontal = TRUE,
notch = TRUE)
#There are no outliers in the age column.boxplot(advert$Daily.Internet.Usage,
main ="Daily Internet Usage",
col = "orange",
border = 'brown',
horizontal = TRUE,
notch = TRUE)
#There are no outliers in the daily internet usage columnExploratory Data Analysis
Univariate Analysis
Measures of Central Tendency
Mean of the numeric columns
colMeans(advert[sapply(advert,is.numeric)])## Daily.Time.Spent.on.Site Age Area.Income
## 65.0002 36.0090 55000.0001
## Daily.Internet.Usage Male Clicked.on.Ad
## 180.0001 0.4810 0.5000Median of our numeric columns
ad_time_median <- median(advert$Daily.Time.Spent.on.Site)
print(ad_time_median)## [1] 68.215ad_age_median <- median(advert$Age)
ad_age_median## [1] 35ad_income_median <- median(advert$Area.Income)
ad_income_median## [1] 57012.3ad_internet_usage_median <- median(advert$Daily.Internet.Usage)
ad_internet_usage_median## [1] 183.13Mode of our numeric columns.
Let’s create the mode function
getmode <- function(v) {
uniqv <- unique(v)
uniqv[which.max(tabulate(match(v, uniqv)))]}Finding the mode in the age column
getmode(advert$Age)## [1] 31getmode(advert$Daily.Time.Spent.on.Site)## [1] 62.26getmode(advert$Area.Income)## [1] 61833.9getmode(advert$Daily.Internet.Usage)## [1] 167.22getmode(advert$City)## [1] "Lisamouth"getmode(advert$Ad.Topic.Line)## [1] "Cloned 5thgeneration orchestration"getmode(advert$Male)## [1] 0getmode(advert$Country)## [1] "Czech Republic"getmode(advert$Timestamp)## [1] "2016-03-27 00:53:11"Minimum values in the numeric columns
min(advert$Age)## [1] 19min(advert$Daily.Time.Spent.on.Site)## [1] 32.6min(advert$Area.Income)## [1] 13996.5min(advert$Daily.Internet.Usage)## [1] 104.78Maximum values in the numeric columns
max(advert$Age)## [1] 61max(advert$Daily.Time.Spent.on.Site)## [1] 91.43max(advert$Area.Income)## [1] 79484.8max(advert$Daily.Internet.Usage)## [1] 269.96Range in the numeric columns
range(advert$Age)## [1] 19 61range(advert$Daily.Time.Spent.on.Site)## [1] 32.60 91.43range(advert$Area.Income)## [1] 13996.5 79484.8range(advert$Daily.Internet.Usage)## [1] 104.78 269.96Summary * The youngest respondent is 19 and the oldest 61 years of age. * The least time spent on her site is 32 minutes and the highest 91 minutes. * The lowest income earner among the respondents earns 13,996 while the highest earns 79,484. * Daily internet usage ranges from 104 - 269.
Quantiles in the columns
quantile(advert$Age)## 0% 25% 50% 75% 100%
## 19 29 35 42 61quantile(advert$Daily.Time.Spent.on.Site)## 0% 25% 50% 75% 100%
## 32.6000 51.3600 68.2150 78.5475 91.4300quantile(advert$Area.Income)## 0% 25% 50% 75% 100%
## 13996.50 47031.80 57012.30 65470.64 79484.80quantile(advert$Daily.Internet.Usage)## 0% 25% 50% 75% 100%
## 104.7800 138.8300 183.1300 218.7925 269.9600Variance of the numeric columns.
This shows how the data values are dispersed around the mean.
var(advert$Age)## [1] 77.18611var(advert$Daily.Time.Spent.on.Site)## [1] 251.3371var(advert$Area.Income)## [1] 179952406var(advert$Daily.Internet.Usage)## [1] 1927.415Finding the standard deviation of the columns.
sd(advert$Age)## [1] 8.785562sd(advert$Daily.Time.Spent.on.Site)## [1] 15.85361sd(advert$Area.Income)## [1] 13414.63sd(advert$Daily.Internet.Usage)## [1] 43.90234Frequency Distribution
requency Distribution in the age column
table(advert$Age)##
## 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
## 6 6 6 13 19 21 27 37 33 48 48 39 60 38 43 39 39 50 36 37 30 36 32 26 23 21
## 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61
## 30 18 13 16 18 20 12 15 10 9 7 2 6 4 2 4 1# Most respondents fall between theage bracket 25-42. The age with the highest number of readers is 31 which has a total of 61 people in total.Histogram
Plotting histograms for the columns
hist(advert$Age, col = "Cyan")
#Most respondents fall in the age bracket 25-40.hist(advert$Area.Income, col = "Purple")
#The respondents mostly earn between 50K - 70Khist(advert$Daily.Time.Spent.on.Site, col = "gold")
hist(advert$Daily.Internet.Usage, col = "pink")
### Plotting count plots for Categorical data
Countplots for categorical data were plotted and it was observed that:
library(ggplot2)
ggplot(advert, aes(x=Male)) + geom_bar(fill=rgb(0.4,0.1,0.5))
There were more male than female users that visited the site and clicked
on the advert
ggplot(advert, aes(x=factor(`Clicked.on.Ad`))) + geom_bar( fill=rgb(0.6,0.4,0.4))
The number of users that clicked the advert are equal to those that did
not click on the advert.
Bivariate Analysis
Ggplots
library(ggplot2)
ggplot(data = advert, aes(x = Area.Income, fill = Clicked.on.Ad))+
geom_histogram(bins =20,col = "orange")+
labs(title = "Income Distribution", x = "Area Income", y= "Frequency", fill = "Clicked on Ad")+ scale_color_brewer(
palette = "Set1"
)
ggplot(data = advert, aes(x = Age, fill = Clicked.on.Ad))+
geom_histogram(bins =20,col = "orange")+
labs(title = "Age Distribution", x = "Age", y= "Frequency", fill = "Clicked on Ad")+ scale_color_brewer(
palette = "Set1"
)
ggplot(data = advert, aes(x =Daily.Time.Spent.on.Site, fill = Clicked.on.Ad))+
geom_histogram(bins =20,col = "orange")+
labs(title = "Daily Time Spent on Site", x = "Time Spent on Site", y= "Frequency", fill = "Clicked on Ad")+ scale_color_brewer(
palette = "Set1"
)
Covariance
Covariance is a statistical representation of the degree to which two variables vary together.
cov(advert$Age, advert$Daily.Time.Spent.on.Site)## [1] -46.17415#There is a negative relationship between the age and the time spent on site which means as the age increases, the daily time spent on the site decreases. The opposite is true.cov(advert$Age, advert$Daily.Internet.Usage)## [1] -141.6348#There is a negative relationship between the age and the daily internet usage as well.cov(advert$Area.Income,advert$Daily.Time.Spent.on.Site)## [1] 66130.81#There is a strong positive relationship between the income and daily time spent on site variables. That goes to say that the higher the income, the more the time spent on site and the lower the income, the less the time spent on site.cov(advert$Age,advert$Area.Income)## [1] -21520.93#There is a negative correlation between the age and income variables.Correlation matrix
cor(advert$Age, advert$Daily.Time.Spent.on.Site)## [1] -0.3315133cor(advert$Age,advert$Daily.Internet.Usage)## [1] -0.3672086cor(advert$Area.Income,advert$Daily.Internet.Usage)## [1] 0.3374955cor(advert$Area.Income,advert$Daily.Time.Spent.on.Site)## [1] 0.3109544cor(advert$Age,advert$Area.Income)## [1] -0.182605cor(advert[, c("Age","Daily.Time.Spent.on.Site","Daily.Internet.Usage")])## Age Daily.Time.Spent.on.Site
## Age 1.0000000 -0.3315133
## Daily.Time.Spent.on.Site -0.3315133 1.0000000
## Daily.Internet.Usage -0.3672086 0.5186585
## Daily.Internet.Usage
## Age -0.3672086
## Daily.Time.Spent.on.Site 0.5186585
## Daily.Internet.Usage 1.0000000cor(advert[,unlist(lapply(advert, is.numeric))])## Daily.Time.Spent.on.Site Age Area.Income
## Daily.Time.Spent.on.Site 1.00000000 -0.33151334 0.310954413
## Age -0.33151334 1.00000000 -0.182604955
## Area.Income 0.31095441 -0.18260496 1.000000000
## Daily.Internet.Usage 0.51865848 -0.36720856 0.337495533
## Male -0.01895085 -0.02104406 0.001322359
## Clicked.on.Ad -0.74811656 0.49253127 -0.476254628
## Daily.Internet.Usage Male Clicked.on.Ad
## Daily.Time.Spent.on.Site 0.51865848 -0.018950855 -0.74811656
## Age -0.36720856 -0.021044064 0.49253127
## Area.Income 0.33749553 0.001322359 -0.47625463
## Daily.Internet.Usage 1.00000000 0.028012326 -0.78653918
## Male 0.02801233 1.000000000 -0.03802747
## Clicked.on.Ad -0.78653918 -0.038027466 1.00000000Plotting a correlation heatmap for the numerical variables
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, unionlibrary(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectlibrary(ggcorrplot)# Selecting the Numerical Variables of the dataset
corr <- dplyr::select(advert,Age,Area.Income,Clicked.on.Ad,Daily.Internet.Usage,Daily.Time.Spent.on.Site,Male )# Plotting the Correlation Heatmap
library(ggcorrplot)
ggcorrplot(cor(corr), hc.order = F,type =
"lower", lab = T,
ggtheme = ggplot2::theme_gray,
colors = c("#00798c", "violet", "#edae49"))
Here, it was noted noted that :
There was a strong negative correlation between the Daily Internet usage and Clicked on Ad variables. This means that the higher ones income the less likely they are to click on the blog ads. The same can also be said for the Daily Time Spent on Site and Click on ad variables.
The Click on Ad variable had a strong positive correlation with the Age Variable, the older users were more likely to click on the ad , as we observed above in our analysis.
The clicked on ad variable was also strongly negatively correlated with the Area Income , where the higher ones income was the less likely they were to click on the ad.
Scatter Plots
Scatter plots are used when we want to see a graphical representation of two different variables. They show how the variables are correlated.
Let’s plot a scatter plot for age and daily time spent on site.
ggplot(advert, aes(Area.Income,Age))+geom_point(aes(colour= factor(`Clicked.on.Ad`)))+
labs(title = "Scatter Plot of Age Distribution vs Area Income",
x = "Area Income",
y = "Age")
The scatter plot for the Area Income against Age showed that , majority
of the users who did not click on the ad were the high income earners
and many of these were aged between 20 and 40 years.
Scatter plot for Income and Daily Internet Usage
ggplot(advert, aes(Area.Income, Daily.Internet.Usage))+
geom_point(aes(colour= factor(`Clicked.on.Ad`)))+
labs(title = "Scatter Plot of Area Income vs Daily Internet Usage",
x = "Area Income",
y = "Daily Internet Usage")
Scatter Plot of Age Distribution vs Time Spent on Site
ggplot(advert, aes(Age, Daily.Time.Spent.on.Site))+
geom_point(aes(colour= factor(`Clicked.on.Ad`)))+
labs(title = "Scatter Plot of Age Distribution vs Time Spent on Site",
x = "Age",
y = "Time Spent on Site")
Plotting the Age against Time spent on the site variable we see that the
younger demographic are less tolerant to ads despite spending
significant amounts of time on the site.
The reason for this may be that younger people , are more tech savvy and therefore are more likely to detect ads and avoid them while using the internet compared to their older counterparts.
Scatter plot for Income Distribution and Daily time spent on site.
ggplot(advert, aes(Daily.Time.Spent.on.Site, Area.Income))+
geom_point(aes(colour= factor(`Clicked.on.Ad`)))+
labs(title = "Time spent on site vs Income",
x = "Daily Time Spent on Site",
y = "Income Distribution")
The people who were least likely to click on the ad were the higher
income earners , this was despite the fact that they seemed to spend a
over an hour a day on the site.
The same sentiments can be echoed for the Usage , total internet usage per day, variable . When plotted against income we see that those who spend over 200 minutes online all day and earn more than 50,000 are the least likely to click on ads on the internet.
Scatter plot for Age and Income Distribution
ggplot(advert, aes(Age, Daily.Internet.Usage))+
geom_point(aes(colour= factor(`Clicked.on.Ad`)))+
labs(title = "Scatter Plot of Age Distribution vs Daily Usage",
x = "Age",
y = "Daily Usage")
Modelling
Multiple Linear Regression
The relationship between the variables has been established by the scatter plots, we’ll therefore begin with modelling.
We’ll begin the variables we’ll use in the model
input <- advert[,c("Clicked.on.Ad","Daily.Time.Spent.on.Site", "Age","Area.Income","Daily.Internet.Usage")]head(input)## Clicked.on.Ad Daily.Time.Spent.on.Site Age Area.Income Daily.Internet.Usage
## 1 0 68.95 35 61833.90 256.09
## 2 0 80.23 31 68441.85 193.77
## 3 0 69.47 26 59785.94 236.50
## 4 0 74.15 29 54806.18 245.89
## 5 0 68.37 35 73889.99 225.58
## 6 0 59.99 23 59761.56 226.74Applying the lm() function
multiple_lm <- lm(Clicked.on.Ad ~ Daily.Time.Spent.on.Site + Age + Area.Income + Daily.Internet.Usage, data = input)
multiple_lm##
## Call:
## lm(formula = Clicked.on.Ad ~ Daily.Time.Spent.on.Site + Age +
## Area.Income + Daily.Internet.Usage, data = input)
##
## Coefficients:
## (Intercept) Daily.Time.Spent.on.Site Age
## 2.293e+00 -1.275e-02 9.017e-03
## Area.Income Daily.Internet.Usage
## -6.170e-06 -5.276e-03Let’s begin with the model assessment
summary(multiple_lm)##
## Call:
## lm(formula = Clicked.on.Ad ~ Daily.Time.Spent.on.Site + Age +
## Area.Income + Daily.Internet.Usage, data = input)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.63848 -0.11736 -0.03329 0.04825 1.02093
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.293e+00 5.722e-02 40.07 <2e-16 ***
## Daily.Time.Spent.on.Site -1.275e-02 5.064e-04 -25.18 <2e-16 ***
## Age 9.017e-03 8.297e-04 10.87 <2e-16 ***
## Area.Income -6.169e-06 5.361e-07 -11.51 <2e-16 ***
## Daily.Internet.Usage -5.276e-03 1.869e-04 -28.22 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2108 on 995 degrees of freedom
## Multiple R-squared: 0.8232, Adjusted R-squared: 0.8225
## F-statistic: 1158 on 4 and 995 DF, p-value: < 2.2e-16To summarize the statistical objects in tidy tibbles, we use the broom package
library(broom)
tidy(multiple_lm)## # A tibble: 5 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 2.29 0.0572 40.1 8.00e-210
## 2 Daily.Time.Spent.on.Site -0.0127 0.000506 -25.2 1.28e-108
## 3 Age 0.00902 0.000830 10.9 4.48e- 26
## 4 Area.Income -0.00000617 0.000000536 -11.5 7.39e- 29
## 5 Daily.Internet.Usage -0.00528 0.000187 -28.2 3.45e-129Let’s check our model’s confidence levels now.
library(MASS)
confint(multiple_lm)## 2.5 % 97.5 %
## (Intercept) 2.180655e+00 2.405221e+00
## Daily.Time.Spent.on.Site -1.374279e-02 -1.175543e-02
## Age 7.388911e-03 1.064533e-02
## Area.Income -7.221617e-06 -5.117457e-06
## Daily.Internet.Usage -5.642504e-03 -4.908785e-03Let’s generate the ANOVA table
anova(multiple_lm)## Analysis of Variance Table
##
## Response: Clicked.on.Ad
## Df Sum Sq Mean Sq F value Pr(>F)
## Daily.Time.Spent.on.Site 1 139.920 139.920 3150.14 < 2.2e-16 ***
## Age 1 16.793 16.793 378.08 < 2.2e-16 ***
## Area.Income 1 13.721 13.721 308.91 < 2.2e-16 ***
## Daily.Internet.Usage 1 35.372 35.372 796.35 < 2.2e-16 ***
## Residuals 995 44.195 0.044
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Predicting the response variable
pred <- predict(multiple_lm, input)
head(pred)## 1 2 3 4 5 6
## -0.003039961 0.105091778 0.025161263 -0.026268696 0.090933941 0.170612160Cross Validating the multiple linear regression model
library(caret)## Loading required package: latticemultiple_lm2 <- train(Clicked.on.Ad ~ Daily.Time.Spent.on.Site + Age + Area.Income + Daily.Internet.Usage, data = input,
method = "lm",
trControl = trainControl(method = "cv",
number = 10,
verboseIter = FALSE))## Warning in train.default(x, y, weights = w, ...): You are trying to do
## regression and your outcome only has two possible values Are you trying to do
## classification? If so, use a 2 level factor as your outcome column.summary(multiple_lm2)##
## Call:
## lm(formula = .outcome ~ ., data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.63848 -0.11736 -0.03329 0.04825 1.02093
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.293e+00 5.722e-02 40.07 <2e-16 ***
## Daily.Time.Spent.on.Site -1.275e-02 5.064e-04 -25.18 <2e-16 ***
## Age 9.017e-03 8.297e-04 10.87 <2e-16 ***
## Area.Income -6.169e-06 5.361e-07 -11.51 <2e-16 ***
## Daily.Internet.Usage -5.276e-03 1.869e-04 -28.22 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2108 on 995 degrees of freedom
## Multiple R-squared: 0.8232, Adjusted R-squared: 0.8225
## F-statistic: 1158 on 4 and 995 DF, p-value: < 2.2e-16multiple_lm2## Linear Regression
##
## 1000 samples
## 4 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 900, 900, 900, 900, 900, 900, ...
## Resampling results:
##
## RMSE Rsquared MAE
## 0.2103976 0.8231955 0.1444346
##
## Tuning parameter 'intercept' was held constant at a value of TRUEFrom the above result, we note that our model has an RMSE value of 0.2103 We’ll use the train object as input to the predict method
pred2 <- predict(multiple_lm2, input)
pred2## 1 2 3 4 5
## -0.0030399606 0.1050917785 0.0251612626 -0.0262686955 0.0909339409
## 6 7 8 9 10
## 0.1706121595 0.0254994749 1.0374704172 0.0198060726 0.2693165738
## 11 12 13 14 15
## 1.2021439565 -0.0360254524 0.9234476809 0.0461290855 1.0944498585
## 16 17 18 19 20
## 0.6274941846 1.0899980252 0.1862988748 1.1037990072 0.8409229118
## 21 22 23 24 25
## -0.0049003846 -0.0353224789 1.1619362991 0.0169481120 0.8272956741
## 26 27 28 29 30
## 0.0611947708 0.9841721204 1.0580825003 0.8738770178 0.1302828474
## 31 32 33 34 35
## 0.0199715636 0.0678829275 1.0034448129 0.4834923881 1.0849082641
## 36 37 38 39 40
## 0.0456002983 0.9650069710 0.1630126102 1.0810072514 1.1234924866
## 41 42 43 44 45
## 0.0461844797 0.0442467204 0.0735508445 0.0143415064 0.0568073546
## 46 47 48 49 50
## 1.0391165016 0.2041424353 0.1298031558 1.0834223345 1.1280128794
## 51 52 53 54 55
## 0.1499916932 -0.0233746955 0.9233760929 1.0128477775 0.9059922821
## 56 57 58 59 60
## 0.1918820744 0.5275875512 1.1562357605 -0.0435118037 1.1512417547
## 61 62 63 64 65
## 0.1828989190 0.0884173867 0.3018106975 0.1603294812 0.9588071457
## 66 67 68 69 70
## 0.2554238662 0.9673749740 0.7569765527 0.0407882698 0.7026867847
## 71 72 73 74 75
## 0.8792769624 0.1934922149 0.6650129282 1.1127418303 1.0522255381
## 76 77 78 79 80
## 0.1618815765 0.9883866088 -0.0615530419 1.1192336485 1.0634452220
## 81 82 83 84 85
## 0.0461265786 0.0286667249 0.9216117369 0.9795876889 0.2388611358
## 86 87 88 89 90
## 1.0762316457 0.1172609608 0.9407618243 1.0250634892 1.0943613643
## 91 92 93 94 95
## 0.6191673992 0.6410743725 0.0367128204 0.9554451957 0.9799556153
## 96 97 98 99 100
## 0.0130218183 1.0834620252 0.9282868033 1.0990107197 0.0381821959
## 101 102 103 104 105
## 1.1522493157 0.1282534105 0.1149975385 0.1636177593 0.2081155784
## 106 107 108 109 110
## 0.0041817405 0.0386686913 0.9241255090 1.0391380409 0.0811725037
## 111 112 113 114 115
## 0.6785797157 0.3182669524 0.1224675618 1.2338812831 0.0863496186
## 116 117 118 119 120
## -0.0481323978 0.5914073003 0.9193166126 0.3907172512 0.4559644081
## 121 122 123 124 125
## 0.0085726791 0.3212707462 0.0334510556 0.7291757915 0.7910507566
## 126 127 128 129 130
## -0.0380281126 0.6476768668 0.1144260972 0.0511052079 0.0845914331
## 131 132 133 134 135
## 1.0861122439 1.0549864050 0.5607684423 0.0354753031 1.1773520010
## 136 137 138 139 140
## 1.0550290989 1.1329893855 1.1575461625 0.2919600141 0.4625410836
## 141 142 143 144 145
## 0.1040320344 0.9038863589 0.9177522240 0.0157398724 0.1929882060
## 146 147 148 149 150
## 1.1813716849 0.9382571281 0.8794097398 0.8274409863 0.7582266919
## 151 152 153 154 155
## 0.4202382368 0.2518391861 1.1924002330 0.0404891399 0.1579902359
## 156 157 158 159 160
## -0.0072789213 1.0341911785 0.5695476061 0.0332810499 0.3624404965
## 161 162 163 164 165
## 0.1475679463 -0.0254908156 -0.0261144198 0.3276541689 1.1624631815
## 166 167 168 169 170
## 0.8140889568 1.0140866996 -0.0156022105 1.0897808911 0.0627848608
## 171 172 173 174 175
## 1.1703668405 0.2779196486 0.0942545225 0.3212357370 0.7607483791
## 176 177 178 179 180
## 0.0465683320 1.0927185639 0.1486650953 1.0650831991 0.0169419364
## 181 182 183 184 185
## 1.1641295235 0.3191266549 0.9761775028 0.0512528771 0.0221707548
## 186 187 188 189 190
## 1.1932857451 1.1311662126 -0.0400241252 0.4262389802 1.1388349582
## 191 192 193 194 195
## 1.0144343807 1.0096775629 1.1123283143 1.1716817861 -0.0383208754
## 196 197 198 199 200
## 0.7320104414 1.0128534594 -0.0418439135 0.1241002921 0.1346520014
## 201 202 203 204 205
## 0.1973207839 0.0071126588 1.0957513695 0.2482315760 0.0761458516
## 206 207 208 209 210
## 0.9439533245 0.0351850331 0.1184753309 1.0728324283 1.1730926351
## 211 212 213 214 215
## 0.0147769569 0.8168573907 -0.0231814346 0.5305604841 -0.0047018895
## 216 217 218 219 220
## 0.6465800409 1.1039675459 1.0524637499 0.7848852185 1.0516078158
## 221 222 223 224 225
## -0.0508546621 0.0948511174 0.8631075426 0.5937434303 0.2630972286
## 226 227 228 229 230
## 0.5235028616 0.9358907877 1.0666257786 0.1618874914 0.1082768448
## 231 232 233 234 235
## 0.0889409906 1.0737833522 0.6211208456 -0.0209295535 0.8667957758
## 236 237 238 239 240
## 1.2290657146 0.6272279248 0.4471550083 0.5839279086 -0.0051576601
## 241 242 243 244 245
## 0.7253477568 1.0776895868 0.0859188983 0.0279490471 0.2655320592
## 246 247 248 249 250
## 0.3467473051 1.0505532079 0.2008930303 1.1010250058 0.7183918942
## 251 252 253 254 255
## 0.0283066985 0.8961146819 0.0359383388 0.6954121808 0.9659419469
## 256 257 258 259 260
## 0.0579922323 -0.0510282371 1.0429133458 -0.0100510918 0.7330724387
## 261 262 263 264 265
## 0.0548188448 0.9227369113 0.6381778162 1.1708992988 -0.0389467444
## 266 267 268 269 270
## 0.7502700994 0.5553280090 0.0265947219 0.8190945981 0.0616845376
## 271 272 273 274 275
## 0.9280308709 0.0706738035 -0.0488573842 0.3190516605 -0.0169553922
## 276 277 278 279 280
## 0.9538522416 -0.0085021936 0.1947670637 0.3371650183 0.0779256731
## 281 282 283 284 285
## 1.2068810034 0.8281972216 1.0856498391 0.0138154997 1.1277967291
## 286 287 288 289 290
## 0.0893091014 0.8560355196 0.0252907465 0.9422025028 0.4428712723
## 291 292 293 294 295
## 0.5770999902 0.0123048254 0.6272574311 -0.0434375465 0.3732462769
## 296 297 298 299 300
## 0.0000365101 -0.0106915254 0.0288894035 0.0058241127 0.0828408064
## 301 302 303 304 305
## -0.0051298706 0.6859772968 0.9555625012 0.5657962861 1.1195858579
## 306 307 308 309 310
## 0.2015992884 -0.0033817940 -0.0435666160 0.1211529730 1.0815419694
## 311 312 313 314 315
## 0.0217623892 0.1512934386 0.3762218727 -0.0286520710 -0.0128913425
## 316 317 318 319 320
## 0.8277072764 0.2295133915 0.0094332366 0.1368163519 1.0720044615
## 321 322 323 324 325
## 0.8154252977 0.1767429496 0.2205077661 0.0509643040 -0.0229211552
## 326 327 328 329 330
## 0.9325675814 0.9194689277 -0.0264119807 0.0944234095 0.5081719766
## 331 332 333 334 335
## 0.0794406315 0.0108643913 0.8875063088 -0.0621670958 0.1497031276
## 336 337 338 339 340
## 1.0582285197 0.1361189279 -0.0173988134 0.0877411659 0.1623605654
## 341 342 343 344 345
## 1.0049307424 0.8709875130 0.0061062027 0.0391482283 0.7124448047
## 346 347 348 349 350
## 0.2303546426 0.1814579145 0.9363142615 0.2503453735 0.3831419809
## 351 352 353 354 355
## -0.0203637425 0.0962020552 -0.0259691459 0.0923183866 0.8678543982
## 356 357 358 359 360
## -0.0200666175 1.1607297074 1.1379133914 1.0720359500 0.0281423910
## 361 362 363 364 365
## 0.9510929748 0.9187257912 0.0992943618 0.5462019414 0.1152267776
## 366 367 368 369 370
## 0.7460658618 0.1538944804 0.0190894285 0.0655792060 0.1002814527
## 371 372 373 374 375
## 1.1167555972 1.1840935280 0.0482547383 1.0460974031 0.1138960739
## 376 377 378 379 380
## 0.1202052138 -0.0019682000 0.9757381684 0.6648339924 -0.0456667305
## 381 382 383 384 385
## -0.0254022457 1.1374412392 0.2633809437 0.0102537924 0.9194736136
## 386 387 388 389 390
## -0.0017893540 0.1090201387 1.2168712499 0.1218668323 0.9758641019
## 391 392 393 394 395
## -0.0328367368 0.0959680974 0.0674776312 0.2307568229 1.0379817918
## 396 397 398 399 400
## 0.1843982512 1.1208963465 0.3294051438 0.1694122222 0.0805246744
## 401 402 403 404 405
## 1.0393501736 -0.0466481375 1.2008351825 0.0612770615 1.1262548311
## 406 407 408 409 410
## -0.0289196622 0.8681576166 0.9548397876 0.3964529080 1.1536412226
## 411 412 413 414 415
## 0.9062148482 0.0451549512 0.2728025320 1.1238171866 0.0407684042
## 416 417 418 419 420
## 1.2154918016 0.6814679869 0.0510841485 0.1785974456 0.4012706022
## 421 422 423 424 425
## 1.0493186491 0.0699762027 0.9194674722 0.9752018146 1.0007245330
## 426 427 428 429 430
## 0.8618207667 0.6524574141 -0.0509509611 0.7821699768 -0.0152009765
## 431 432 433 434 435
## 0.0619735898 0.0097698200 0.2750084032 0.0113346699 0.0064942165
## 436 437 438 439 440
## 0.7653330570 0.2689999241 0.0384996985 1.0714719581 0.2939553901
## 441 442 443 444 445
## 0.8100395182 -0.0480287761 1.0778430589 1.0800608553 1.0746207811
## 446 447 448 449 450
## -0.0327333924 0.6086591483 0.1382421812 1.1312539991 0.3032123075
## 451 452 453 454 455
## 0.7791332120 0.8215625957 -0.0572823073 0.1796222192 0.4767338621
## 456 457 458 459 460
## 0.0578537718 1.0899568594 0.0539237529 0.7278148378 0.0139943104
## 461 462 463 464 465
## 1.1732491412 1.0378471659 0.2427713342 0.9994073760 0.1815817838
## 466 467 468 469 470
## 0.7948835209 0.5583085766 0.7971703937 1.0635155928 0.1006809745
## 471 472 473 474 475
## 0.7398602599 0.1496350032 0.0773010362 -0.0242013826 1.0403489927
## 476 477 478 479 480
## 0.0345889999 0.0582860792 1.1446996159 1.0376776844 0.7820113897
## 481 482 483 484 485
## 0.3020745827 0.1683177690 0.1017050087 1.0222027668 1.0915863103
## 486 487 488 489 490
## 0.7382046202 0.3383238358 0.0399963263 1.1591021500 -0.0481338253
## 491 492 493 494 495
## 1.1852264297 1.0057042766 0.3124953027 0.8466143473 0.8322477443
## 496 497 498 499 500
## -0.0150944064 0.0009543489 1.1785629028 0.2568767561 0.7496476469
## 501 502 503 504 505
## 1.0024061782 -0.0464432182 0.0849743982 1.0320258184 1.0506289878
## 506 507 508 509 510
## 0.0755397140 -0.0417845991 1.0098270220 0.5743569953 -0.0441172386
## 511 512 513 514 515
## 0.8293877313 0.0277820558 -0.0044226122 1.0708587480 -0.0251383958
## 516 517 518 519 520
## 1.0414813512 -0.0010360447 0.5510905556 1.1256654639 1.1254561870
## 521 522 523 524 525
## 0.9526784387 0.7744092320 0.0726701666 0.7160859772 -0.0389198423
## 526 527 528 529 530
## 0.4657109721 0.9627934449 0.3012656774 1.1713040358 0.3415090025
## 531 532 533 534 535
## 0.9149311279 1.1549566037 0.1622024321 0.0817975273 0.0708635834
## 536 537 538 539 540
## 0.0479799363 0.2213670591 0.1727521328 -0.0536569438 0.0260877616
## 541 542 543 544 545
## 0.0188694725 -0.0226797712 0.1242622758 1.0263994367 0.0448332240
## 546 547 548 549 550
## 1.0178545176 0.0318795893 -0.0340971924 -0.0515089744 0.0182348349
## 551 552 553 554 555
## 0.0252704140 0.0271756437 1.0843640319 1.0240284801 1.0413425880
## 556 557 558 559 560
## 0.0671981071 1.0819910762 -0.0281548731 0.3083345558 -0.0432957112
## 561 562 563 564 565
## 1.0200949710 1.1951482902 0.1897388115 0.4748985551 0.7826302242
## 566 567 568 569 570
## 0.1485123874 1.1577322391 -0.0573739713 0.3527096754 0.0009524977
## 571 572 573 574 575
## 1.1766745000 -0.0164541891 -0.0281664050 0.4663629080 1.1775304801
## 576 577 578 579 580
## 1.1194284943 1.1119464028 -0.0383509376 0.1514159993 0.0916321851
## 581 582 583 584 585
## 1.0067291981 0.9715260689 0.8767398152 0.5809819862 0.5260103073
## 586 587 588 589 590
## 0.1048294337 -0.0492472757 1.2391454692 -0.0476787151 0.8708903934
## 591 592 593 594 595
## 0.5594973397 1.0023823320 0.0732301428 0.1291512537 0.8685157366
## 596 597 598 599 600
## 0.8039996258 -0.0140308456 -0.0273338465 0.0558433723 0.7050304989
## 601 602 603 604 605
## 0.6470821659 1.2027324934 0.8861110795 0.1087285308 0.9057355494
## 606 607 608 609 610
## 1.0484188296 0.1442512826 0.0565111483 0.8572880243 0.2914198999
## 611 612 613 614 615
## 0.9330884005 0.9791520856 0.1106257150 -0.0131179752 0.0204894681
## 616 617 618 619 620
## 1.0898087996 1.1606041672 0.0140374076 1.0754953922 0.0448437200
## 621 622 623 624 625
## 0.1394510047 0.0212298127 0.9813413431 0.3978035661 0.1230291921
## 626 627 628 629 630
## 0.5207382047 0.1087156160 1.0475289757 1.1001824882 -0.0029897952
## 631 632 633 634 635
## 0.0864500231 -0.0275551677 0.0541029394 1.0499357828 0.9976890691
## 636 637 638 639 640
## 0.9869308671 1.1036948360 0.1045083679 0.6295731297 0.1149770909
## 641 642 643 644 645
## 0.9407271435 0.1448074419 -0.0147364253 0.3227251809 0.0434337767
## 646 647 648 649 650
## 0.9842145287 1.1933991883 0.8969816089 0.1971089003 -0.0028795868
## 651 652 653 654 655
## 0.2200826324 -0.0205724820 -0.0295585198 0.0865368045 0.0071488145
## 656 657 658 659 660
## 1.1859451394 0.1396102895 0.1599304074 0.0944639923 0.0036668240
## 661 662 663 664 665
## 0.8227052126 0.4215744349 1.2136902530 0.9140031170 0.0089020649
## 666 667 668 669 670
## 0.7118340776 -0.0116813462 -0.0473284574 0.3770626184 0.7241405881
## 671 672 673 674 675
## 0.0750885369 0.6647951677 0.0705314119 0.9742047637 0.0433371880
## 676 677 678 679 680
## -0.0095508436 0.8754167634 0.8076951745 0.3526023254 1.0369623453
## 681 682 683 684 685
## 0.0869394001 0.9321230389 1.0870932064 0.1292345309 1.1025036316
## 686 687 688 689 690
## -0.0025921121 -0.0257890274 0.0552665974 0.0143900894 -0.0077805132
## 691 692 693 694 695
## 0.0640531055 -0.0211939920 0.8376807252 0.8499741644 0.1994146636
## 696 697 698 699 700
## -0.0039900749 1.0428734391 0.0244208027 0.1370770269 -0.0092706431
## 701 702 703 704 705
## 0.3984741566 0.8367630783 0.0534510431 0.0496473870 0.0747171469
## 706 707 708 709 710
## 0.0247892778 0.5055413020 0.0170249485 0.7477016207 0.7385127678
## 711 712 713 714 715
## 1.1731927882 0.0983072415 0.3012826681 1.1850025852 0.1245830630
## 716 717 718 719 720
## 0.9827642121 1.0357695643 0.0861476392 0.4328907319 0.8751941486
## 721 722 723 724 725
## -0.0470285193 0.9685942913 0.9104308316 0.4523087681 0.0109177349
## 726 727 728 729 730
## -0.0230944155 0.0885392680 -0.0477406481 0.2273191204 -0.0179458478
## 731 732 733 734 735
## 0.3105934941 0.0887099669 0.1409280294 1.1946803743 0.7617051719
## 736 737 738 739 740
## -0.0194907558 -0.0375000293 0.9508011022 0.5123628554 0.0261486539
## 741 742 743 744 745
## 0.9848014105 0.1915959224 0.1725794494 0.9634969799 0.7764986789
## 746 747 748 749 750
## 0.9618466916 0.0235263590 1.0659094289 0.9628821087 0.6384750557
## 751 752 753 754 755
## 0.8781945051 -0.0526036251 0.1970365078 0.1638281297 0.0659212084
## 756 757 758 759 760
## 0.1133810518 0.9538079021 0.9230093725 0.7905888754 0.4389330614
## 761 762 763 764 765
## 0.0389820674 0.1782139961 1.1193311022 0.8668052845 1.1993900151
## 766 767 768 769 770
## 1.0140207093 0.6336924157 1.1349334170 0.7969702756 0.2195017171
## 771 772 773 774 775
## 0.0400333515 0.3672939189 0.0320143746 0.5625663276 1.0072049491
## 776 777 778 779 780
## 1.1457076552 0.5992759593 0.0276950668 0.9180181829 0.0017969427
## 781 782 783 784 785
## 0.2438018160 0.8042920965 -0.0399758629 0.2396784018 1.0603463489
## 786 787 788 789 790
## 1.1353896968 0.3598991090 0.3073423716 0.0786546307 1.0513238724
## 791 792 793 794 795
## 0.9713639111 1.1396477832 0.3032206807 0.9422517105 1.1999338616
## 796 797 798 799 800
## 0.0794056513 0.0863420168 -0.0339589809 0.3403469753 0.0391831739
## 801 802 803 804 805
## 0.8367538984 1.0575330786 1.1889804784 1.2343454817 1.1545275661
## 806 807 808 809 810
## 0.1503018248 0.8843890824 1.1635752379 1.0781744947 1.0012052425
## 811 812 813 814 815
## 0.9369661326 -0.0391831251 0.2113073917 0.0930517067 -0.0308780828
## 816 817 818 819 820
## 0.0067103797 1.0755871265 0.7706673194 0.0152419163 0.1522920192
## 821 822 823 824 825
## 1.0505215596 -0.0518739256 -0.0172671055 0.0380800356 0.0632493632
## 826 827 828 829 830
## 0.1106986861 -0.0329246397 1.1905929553 0.8862853586 1.0408045339
## 831 832 833 834 835
## 0.9199147761 1.0199277226 1.1407161845 1.0311207218 0.0556387721
## 836 837 838 839 840
## 0.0606048407 1.1668968861 1.0596314455 1.2058822505 1.1016349582
## 841 842 843 844 845
## 0.7811688304 1.1030225651 0.1176482501 0.0258489054 0.0215228747
## 846 847 848 849 850
## 1.0241756534 1.0987469582 0.2312932678 0.0122165774 1.0750850125
## 851 852 853 854 855
## -0.0502260809 1.0332998620 1.1171670888 0.0432738801 0.0740078612
## 856 857 858 859 860
## 0.4679588977 -0.0323230817 0.0675234635 1.1106023077 -0.0037448784
## 861 862 863 864 865
## 0.2499006909 0.0893727303 0.2843146154 -0.0150840873 0.1242929912
## 866 867 868 869 870
## 1.0879957730 0.0729026193 0.0709995522 0.0727587968 -0.0111810814
## 871 872 873 874 875
## 0.6514501222 -0.0308412178 0.1579767668 -0.0362301258 0.2847263043
## 876 877 878 879 880
## 0.9226743052 0.9487991567 0.2607934788 -0.0016034943 0.1361152375
## 881 882 883 884 885
## 1.0259617575 -0.0476945728 0.0810199713 0.8874544702 0.0121155410
## 886 887 888 889 890
## 1.2361359660 1.1954377799 1.0605867544 0.0407574310 1.0153153877
## 891 892 893 894 895
## -0.0396888431 0.5486825759 0.7592953645 0.0427825768 0.0939242238
## 896 897 898 899 900
## 0.1808249616 -0.0501276590 0.7595111795 1.0896016449 1.1586368815
## 901 902 903 904 905
## 1.1533603800 0.8953725923 1.1282477240 0.1635492675 0.0404563855
## 906 907 908 909 910
## -0.0091313085 0.8311230671 0.0800277472 1.1113463711 0.0328491716
## 911 912 913 914 915
## 1.0874046759 0.9628444503 1.1515209715 0.0190213515 1.1051316503
## 916 917 918 919 920
## 0.9950264165 1.0042691456 0.0502638000 0.0196598658 0.2966383495
## 921 922 923 924 925
## -0.0624007248 0.9660668471 1.1444822753 1.0983128375 0.7437997133
## 926 927 928 929 930
## 1.1776988782 0.1351803332 0.2703698152 0.1285084829 0.7183450762
## 931 932 933 934 935
## -0.0553228249 1.0000596426 0.8264720894 1.0526590016 -0.0603267311
## 936 937 938 939 940
## 0.0987226084 1.0819556291 0.9579860233 0.9695679351 0.4605785465
## 941 942 943 944 945
## 1.2344721750 0.4805360809 0.7553108033 1.0134206114 0.8644202527
## 946 947 948 949 950
## 0.0731385908 0.1916913609 0.9435439601 0.4037493327 0.2870776789
## 951 952 953 954 955
## 0.6852359443 0.9103381717 0.4804487411 0.8807465982 0.2311376513
## 956 957 958 959 960
## 1.1626417580 0.8870185038 0.0005105901 -0.0430815797 0.2905373274
## 961 962 963 964 965
## 0.9192475132 0.1020508901 0.1370215778 0.1633168614 0.0910256676
## 966 967 968 969 970
## 0.9783088179 1.0321595658 0.2360529960 1.1818904032 0.4630651697
## 971 972 973 974 975
## 0.9273214585 1.2407846885 1.1162470605 0.0499153412 1.1815762054
## 976 977 978 979 980
## 1.1866398516 1.0690888392 0.9374717662 0.3304009239 -0.0186151321
## 981 982 983 984 985
## 0.9024876509 0.1105022062 0.7543495764 -0.0485895204 0.0143476999
## 986 987 988 989 990
## 0.7055279684 -0.0266310544 1.0523621113 0.1859560200 -0.0253665479
## 991 992 993 994 995
## 1.1510376126 1.1715876447 0.7112068269 0.1085834501 0.6860781611
## 996 997 998 999 1000
## 0.0923456762 0.9173425996 1.1978601695 0.5058612621 0.8283149091error <- pred2 - advert$Clicked.on.Ad
error## 1 2 3 4 5
## -3.039961e-03 1.050918e-01 2.516126e-02 -2.626870e-02 9.093394e-02
## 6 7 8 9 10
## 1.706122e-01 2.549947e-02 3.747042e-02 1.980607e-02 2.693166e-01
## 11 12 13 14 15
## 2.021440e-01 -3.602545e-02 -7.655232e-02 4.612909e-02 9.444986e-02
## 16 17 18 19 20
## -3.725058e-01 8.999803e-02 1.862989e-01 1.037990e-01 -1.590771e-01
## 21 22 23 24 25
## -4.900385e-03 -3.532248e-02 1.619363e-01 1.694811e-02 -1.727043e-01
## 26 27 28 29 30
## 6.119477e-02 -1.582788e-02 5.808250e-02 -1.261230e-01 1.302828e-01
## 31 32 33 34 35
## 1.997156e-02 6.788293e-02 3.444813e-03 -5.165076e-01 8.490826e-02
## 36 37 38 39 40
## 4.560030e-02 -3.499303e-02 1.630126e-01 8.100725e-02 1.234925e-01
## 41 42 43 44 45
## 4.618448e-02 4.424672e-02 7.355084e-02 1.434151e-02 5.680735e-02
## 46 47 48 49 50
## 3.911650e-02 2.041424e-01 1.298032e-01 8.342233e-02 1.280129e-01
## 51 52 53 54 55
## 1.499917e-01 -2.337470e-02 -7.662391e-02 1.284778e-02 -9.400772e-02
## 56 57 58 59 60
## 1.918821e-01 -4.724124e-01 1.562358e-01 -4.351180e-02 1.512418e-01
## 61 62 63 64 65
## 1.828989e-01 8.841739e-02 3.018107e-01 1.603295e-01 -4.119285e-02
## 66 67 68 69 70
## 2.554239e-01 -3.262503e-02 -2.430234e-01 4.078827e-02 -2.973132e-01
## 71 72 73 74 75
## -1.207230e-01 1.934922e-01 -3.349871e-01 1.127418e-01 5.222554e-02
## 76 77 78 79 80
## 1.618816e-01 -1.161339e-02 -6.155304e-02 1.192336e-01 6.344522e-02
## 81 82 83 84 85
## 4.612658e-02 2.866672e-02 -7.838826e-02 -2.041231e-02 2.388611e-01
## 86 87 88 89 90
## 7.623165e-02 1.172610e-01 -5.923818e-02 2.506349e-02 9.436136e-02
## 91 92 93 94 95
## -3.808326e-01 -3.589256e-01 3.671282e-02 -4.455480e-02 -2.004438e-02
## 96 97 98 99 100
## 1.302182e-02 8.346203e-02 -7.171320e-02 9.901072e-02 3.818220e-02
## 101 102 103 104 105
## 1.522493e-01 1.282534e-01 1.149975e-01 1.636178e-01 2.081156e-01
## 106 107 108 109 110
## 4.181740e-03 3.866869e-02 -7.587449e-02 3.913804e-02 8.117250e-02
## 111 112 113 114 115
## -3.214203e-01 -6.817330e-01 1.224676e-01 2.338813e-01 8.634962e-02
## 116 117 118 119 120
## -4.813240e-02 -4.085927e-01 -8.068339e-02 -6.092827e-01 -5.440356e-01
## 121 122 123 124 125
## 8.572679e-03 3.212707e-01 3.345106e-02 -2.708242e-01 -2.089492e-01
## 126 127 128 129 130
## -3.802811e-02 -3.523231e-01 1.144261e-01 5.110521e-02 8.459143e-02
## 131 132 133 134 135
## 8.611224e-02 5.498641e-02 -4.392316e-01 3.547530e-02 1.773520e-01
## 136 137 138 139 140
## 5.502910e-02 1.329894e-01 1.575462e-01 2.919600e-01 4.625411e-01
## 141 142 143 144 145
## 1.040320e-01 -9.611364e-02 -8.224778e-02 1.573987e-02 1.929882e-01
## 146 147 148 149 150
## 1.813717e-01 -6.174287e-02 -1.205903e-01 -1.725590e-01 -2.417733e-01
## 151 152 153 154 155
## 4.202382e-01 2.518392e-01 1.924002e-01 4.048914e-02 1.579902e-01
## 156 157 158 159 160
## -7.278921e-03 3.419118e-02 -4.304524e-01 3.328105e-02 -6.375595e-01
## 161 162 163 164 165
## 1.475679e-01 -2.549082e-02 -2.611442e-02 3.276542e-01 1.624632e-01
## 166 167 168 169 170
## -1.859110e-01 1.408670e-02 -1.560221e-02 8.978089e-02 6.278486e-02
## 171 172 173 174 175
## 1.703668e-01 2.779196e-01 9.425452e-02 3.212357e-01 -2.392516e-01
## 176 177 178 179 180
## 4.656833e-02 9.271856e-02 1.486651e-01 6.508320e-02 1.694194e-02
## 181 182 183 184 185
## 1.641295e-01 -6.808733e-01 -2.382250e-02 5.125288e-02 2.217075e-02
## 186 187 188 189 190
## 1.932857e-01 1.311662e-01 -4.002413e-02 -5.737610e-01 1.388350e-01
## 191 192 193 194 195
## 1.443438e-02 9.677563e-03 1.123283e-01 1.716818e-01 -3.832088e-02
## 196 197 198 199 200
## -2.679896e-01 1.285346e-02 -4.184391e-02 1.241003e-01 1.346520e-01
## 201 202 203 204 205
## 1.973208e-01 7.112659e-03 9.575137e-02 2.482316e-01 7.614585e-02
## 206 207 208 209 210
## -5.604668e-02 3.518503e-02 1.184753e-01 7.283243e-02 1.730926e-01
## 211 212 213 214 215
## 1.477696e-02 -1.831426e-01 -2.318143e-02 -4.694395e-01 -4.701890e-03
## 216 217 218 219 220
## -3.534200e-01 1.039675e-01 5.246375e-02 -2.151148e-01 5.160782e-02
## 221 222 223 224 225
## -5.085466e-02 9.485112e-02 -1.368925e-01 -4.062566e-01 2.630972e-01
## 226 227 228 229 230
## -4.764971e-01 -6.410921e-02 6.662578e-02 1.618875e-01 1.082768e-01
## 231 232 233 234 235
## 8.894099e-02 7.378335e-02 -3.788792e-01 -1.020930e+00 -1.332042e-01
## 236 237 238 239 240
## 2.290657e-01 -3.727721e-01 4.471550e-01 -4.160721e-01 -5.157660e-03
## 241 242 243 244 245
## -2.746522e-01 7.768959e-02 8.591890e-02 2.794905e-02 2.655321e-01
## 246 247 248 249 250
## 3.467473e-01 5.055321e-02 -7.991070e-01 1.010250e-01 -2.816081e-01
## 251 252 253 254 255
## 2.830670e-02 -1.038853e-01 3.593834e-02 -3.045878e-01 -3.405805e-02
## 256 257 258 259 260
## 5.799223e-02 -5.102824e-02 4.291335e-02 -1.005109e-02 -2.669276e-01
## 261 262 263 264 265
## 5.481884e-02 -7.726309e-02 -3.618222e-01 1.708993e-01 -3.894674e-02
## 266 267 268 269 270
## -2.497299e-01 -4.446720e-01 2.659472e-02 -1.809054e-01 6.168454e-02
## 271 272 273 274 275
## -7.196913e-02 7.067380e-02 -4.885738e-02 3.190517e-01 -1.695539e-02
## 276 277 278 279 280
## -4.614776e-02 -8.502194e-03 1.947671e-01 3.371650e-01 7.792567e-02
## 281 282 283 284 285
## 2.068810e-01 -1.718028e-01 8.564984e-02 1.381550e-02 1.277967e-01
## 286 287 288 289 290
## 8.930910e-02 -1.439645e-01 2.529075e-02 -5.779750e-02 -5.571287e-01
## 291 292 293 294 295
## -4.229000e-01 1.230483e-02 -3.727426e-01 -4.343755e-02 3.732463e-01
## 296 297 298 299 300
## 3.651010e-05 -1.069153e-02 2.888940e-02 5.824113e-03 8.284081e-02
## 301 302 303 304 305
## -5.129871e-03 -3.140227e-01 -4.443750e-02 -4.342037e-01 1.195859e-01
## 306 307 308 309 310
## -7.984007e-01 -3.381794e-03 -4.356662e-02 1.211530e-01 8.154197e-02
## 311 312 313 314 315
## 2.176239e-02 1.512934e-01 -6.237781e-01 -2.865207e-02 -1.289134e-02
## 316 317 318 319 320
## -1.722927e-01 2.295134e-01 9.433237e-03 1.368164e-01 7.200446e-02
## 321 322 323 324 325
## -1.845747e-01 1.767429e-01 2.205078e-01 5.096430e-02 -2.292116e-02
## 326 327 328 329 330
## -6.743242e-02 -8.053107e-02 -2.641198e-02 9.442341e-02 -4.918280e-01
## 331 332 333 334 335
## 7.944063e-02 1.086439e-02 -1.124937e-01 -6.216710e-02 1.497031e-01
## 336 337 338 339 340
## 5.822852e-02 1.361189e-01 -1.739881e-02 8.774117e-02 1.623606e-01
## 341 342 343 344 345
## 4.930742e-03 -1.290125e-01 6.106203e-03 3.914823e-02 -2.875552e-01
## 346 347 348 349 350
## 2.303546e-01 1.814579e-01 -6.368574e-02 2.503454e-01 -6.168580e-01
## 351 352 353 354 355
## -2.036374e-02 9.620206e-02 -2.596915e-02 9.231839e-02 -1.321456e-01
## 356 357 358 359 360
## -2.006662e-02 1.607297e-01 1.379134e-01 7.203595e-02 2.814239e-02
## 361 362 363 364 365
## -4.890703e-02 -8.127421e-02 9.929436e-02 -4.537981e-01 1.152268e-01
## 366 367 368 369 370
## -2.539341e-01 1.538945e-01 1.908943e-02 6.557921e-02 1.002815e-01
## 371 372 373 374 375
## 1.167556e-01 1.840935e-01 4.825474e-02 4.609740e-02 1.138961e-01
## 376 377 378 379 380
## 1.202052e-01 -1.968200e-03 -2.426183e-02 -3.351660e-01 -4.566673e-02
## 381 382 383 384 385
## -2.540225e-02 1.374412e-01 2.633809e-01 1.025379e-02 -8.052639e-02
## 386 387 388 389 390
## -1.789354e-03 1.090201e-01 2.168712e-01 1.218668e-01 -2.413590e-02
## 391 392 393 394 395
## -3.283674e-02 9.596810e-02 6.747763e-02 2.307568e-01 3.798179e-02
## 396 397 398 399 400
## 1.843983e-01 1.208963e-01 -6.705949e-01 1.694122e-01 8.052467e-02
## 401 402 403 404 405
## 3.935017e-02 -4.664814e-02 2.008352e-01 6.127706e-02 1.262548e-01
## 406 407 408 409 410
## -2.891966e-02 -1.318424e-01 -4.516021e-02 -6.035471e-01 1.536412e-01
## 411 412 413 414 415
## -9.378515e-02 4.515495e-02 2.728025e-01 1.238172e-01 4.076840e-02
## 416 417 418 419 420
## 2.154918e-01 -3.185320e-01 5.108415e-02 1.785974e-01 4.012706e-01
## 421 422 423 424 425
## 4.931865e-02 6.997620e-02 -8.053253e-02 -2.479819e-02 7.245330e-04
## 426 427 428 429 430
## -1.381792e-01 -3.475426e-01 -5.095096e-02 -2.178300e-01 -1.520098e-02
## 431 432 433 434 435
## 6.197359e-02 9.769820e-03 -7.249916e-01 1.133467e-02 6.494216e-03
## 436 437 438 439 440
## -2.346669e-01 2.689999e-01 3.849970e-02 7.147196e-02 2.939554e-01
## 441 442 443 444 445
## -1.899605e-01 -4.802878e-02 7.784306e-02 8.006086e-02 7.462078e-02
## 446 447 448 449 450
## -3.273339e-02 -3.913409e-01 1.382422e-01 1.312540e-01 3.032123e-01
## 451 452 453 454 455
## -2.208668e-01 -1.784374e-01 -5.728231e-02 1.796222e-01 -5.232661e-01
## 456 457 458 459 460
## 5.785377e-02 8.995686e-02 5.392375e-02 -2.721852e-01 1.399431e-02
## 461 462 463 464 465
## 1.732491e-01 3.784717e-02 2.427713e-01 -5.926240e-04 1.815818e-01
## 466 467 468 469 470
## -2.051165e-01 -4.416914e-01 -2.028296e-01 6.351559e-02 1.006810e-01
## 471 472 473 474 475
## -2.601397e-01 1.496350e-01 7.730104e-02 -2.420138e-02 4.034899e-02
## 476 477 478 479 480
## 3.458900e-02 5.828608e-02 1.446996e-01 3.767768e-02 -2.179886e-01
## 481 482 483 484 485
## 3.020746e-01 1.683178e-01 1.017050e-01 2.220277e-02 9.158631e-02
## 486 487 488 489 490
## -2.617954e-01 3.383238e-01 3.999633e-02 1.591021e-01 -4.813383e-02
## 491 492 493 494 495
## 1.852264e-01 5.704277e-03 3.124953e-01 -1.533857e-01 -1.677523e-01
## 496 497 498 499 500
## -1.509441e-02 9.543489e-04 1.785629e-01 2.568768e-01 -2.503524e-01
## 501 502 503 504 505
## 2.406178e-03 -4.644322e-02 8.497440e-02 3.202582e-02 5.062899e-02
## 506 507 508 509 510
## 7.553971e-02 -4.178460e-02 9.827022e-03 -4.256430e-01 -4.411724e-02
## 511 512 513 514 515
## -1.706123e-01 2.778206e-02 -4.422612e-03 7.085875e-02 -2.513840e-02
## 516 517 518 519 520
## 4.148135e-02 -1.036045e-03 -4.489094e-01 1.256655e-01 1.254562e-01
## 521 522 523 524 525
## -4.732156e-02 -2.255908e-01 7.267017e-02 -2.839140e-01 -3.891984e-02
## 526 527 528 529 530
## 4.657110e-01 -3.720656e-02 3.012657e-01 1.713040e-01 3.415090e-01
## 531 532 533 534 535
## -8.506887e-02 1.549566e-01 1.622024e-01 8.179753e-02 7.086358e-02
## 536 537 538 539 540
## 4.797994e-02 2.213671e-01 1.727521e-01 -5.365694e-02 2.608776e-02
## 541 542 543 544 545
## 1.886947e-02 -2.267977e-02 1.242623e-01 2.639944e-02 4.483322e-02
## 546 547 548 549 550
## 1.785452e-02 3.187959e-02 -3.409719e-02 -5.150897e-02 1.823483e-02
## 551 552 553 554 555
## 2.527041e-02 2.717564e-02 8.436403e-02 2.402848e-02 4.134259e-02
## 556 557 558 559 560
## 6.719811e-02 8.199108e-02 -2.815487e-02 3.083346e-01 -4.329571e-02
## 561 562 563 564 565
## 2.009497e-02 1.951483e-01 1.897388e-01 4.748986e-01 -2.173698e-01
## 566 567 568 569 570
## 1.485124e-01 1.577322e-01 -5.737397e-02 3.527097e-01 9.524977e-04
## 571 572 573 574 575
## 1.766745e-01 -1.645419e-02 -2.816640e-02 -5.336371e-01 1.775305e-01
## 576 577 578 579 580
## 1.194285e-01 1.119464e-01 -3.835094e-02 1.514160e-01 9.163219e-02
## 581 582 583 584 585
## 6.729198e-03 -2.847393e-02 -1.232602e-01 -4.190180e-01 -4.739897e-01
## 586 587 588 589 590
## 1.048294e-01 -4.924728e-02 2.391455e-01 -4.767872e-02 -1.291096e-01
## 591 592 593 594 595
## -4.405027e-01 2.382332e-03 7.323014e-02 1.291513e-01 -1.314843e-01
## 596 597 598 599 600
## -1.960004e-01 -1.403085e-02 -2.733385e-02 5.584337e-02 -2.949695e-01
## 601 602 603 604 605
## -3.529178e-01 2.027325e-01 -1.138889e-01 1.087285e-01 -9.426445e-02
## 606 607 608 609 610
## 4.841883e-02 1.442513e-01 5.651115e-02 -1.427120e-01 -7.085801e-01
## 611 612 613 614 615
## -6.691160e-02 -2.084791e-02 1.106257e-01 -1.311798e-02 2.048947e-02
## 616 617 618 619 620
## 8.980880e-02 1.606042e-01 1.403741e-02 7.549539e-02 4.484372e-02
## 621 622 623 624 625
## 1.394510e-01 2.122981e-02 -1.865866e-02 3.978036e-01 1.230292e-01
## 626 627 628 629 630
## -4.792618e-01 1.087156e-01 4.752898e-02 1.001825e-01 -2.989795e-03
## 631 632 633 634 635
## 8.645002e-02 -2.755517e-02 5.410294e-02 4.993578e-02 -2.310931e-03
## 636 637 638 639 640
## -1.306913e-02 1.036948e-01 1.045084e-01 -3.704269e-01 1.149771e-01
## 641 642 643 644 645
## -5.927286e-02 1.448074e-01 -1.473643e-02 3.227252e-01 4.343378e-02
## 646 647 648 649 650
## -1.578547e-02 1.933992e-01 -1.030184e-01 1.971089e-01 -2.879587e-03
## 651 652 653 654 655
## 2.200826e-01 -2.057248e-02 -2.955852e-02 8.653680e-02 7.148814e-03
## 656 657 658 659 660
## 1.859451e-01 1.396103e-01 1.599304e-01 9.446399e-02 3.666824e-03
## 661 662 663 664 665
## -1.772948e-01 -5.784256e-01 2.136903e-01 -8.599688e-02 8.902065e-03
## 666 667 668 669 670
## -2.881659e-01 -1.168135e-02 -4.732846e-02 -6.229374e-01 -2.758594e-01
## 671 672 673 674 675
## 7.508854e-02 -3.352048e-01 7.053141e-02 -2.579524e-02 4.333719e-02
## 676 677 678 679 680
## -9.550844e-03 -1.245832e-01 -1.923048e-01 3.526023e-01 3.696235e-02
## 681 682 683 684 685
## 8.693940e-02 -6.787696e-02 8.709321e-02 1.292345e-01 1.025036e-01
## 686 687 688 689 690
## -2.592112e-03 -2.578903e-02 5.526660e-02 1.439009e-02 -7.780513e-03
## 691 692 693 694 695
## 6.405311e-02 -2.119399e-02 -1.623193e-01 -1.500258e-01 1.994147e-01
## 696 697 698 699 700
## -3.990075e-03 4.287344e-02 2.442080e-02 1.370770e-01 -9.270643e-03
## 701 702 703 704 705
## 3.984742e-01 -1.632369e-01 -9.465490e-01 4.964739e-02 7.471715e-02
## 706 707 708 709 710
## 2.478928e-02 -4.944587e-01 1.702495e-02 -2.522984e-01 -2.614872e-01
## 711 712 713 714 715
## 1.731928e-01 9.830724e-02 3.012827e-01 1.850026e-01 1.245831e-01
## 716 717 718 719 720
## -1.723579e-02 3.576956e-02 8.614764e-02 4.328907e-01 -1.248059e-01
## 721 722 723 724 725
## -4.702852e-02 -3.140571e-02 -8.956917e-02 4.523088e-01 1.091773e-02
## 726 727 728 729 730
## -2.309442e-02 8.853927e-02 -4.774065e-02 2.273191e-01 -1.794585e-02
## 731 732 733 734 735
## 3.105935e-01 8.870997e-02 1.409280e-01 1.946804e-01 -2.382948e-01
## 736 737 738 739 740
## -1.949076e-02 -3.750003e-02 -4.919890e-02 -4.876371e-01 2.614865e-02
## 741 742 743 744 745
## -1.519859e-02 1.915959e-01 1.725794e-01 -3.650302e-02 -2.235013e-01
## 746 747 748 749 750
## -3.815331e-02 -9.764736e-01 6.590943e-02 -3.711789e-02 6.384751e-01
## 751 752 753 754 755
## -1.218055e-01 -5.260363e-02 1.970365e-01 1.638281e-01 6.592121e-02
## 756 757 758 759 760
## 1.133811e-01 -4.619210e-02 -7.699063e-02 -2.094111e-01 -5.610669e-01
## 761 762 763 764 765
## 3.898207e-02 1.782140e-01 1.193311e-01 -1.331947e-01 1.993900e-01
## 766 767 768 769 770
## 1.402071e-02 -3.663076e-01 1.349334e-01 -2.030297e-01 2.195017e-01
## 771 772 773 774 775
## 4.003335e-02 3.672939e-01 3.201437e-02 -4.374337e-01 7.204949e-03
## 776 777 778 779 780
## 1.457077e-01 -4.007240e-01 2.769507e-02 -8.198182e-02 1.796943e-03
## 781 782 783 784 785
## -7.561982e-01 -1.957079e-01 -3.997586e-02 2.396784e-01 6.034635e-02
## 786 787 788 789 790
## 1.353897e-01 3.598991e-01 -6.926576e-01 7.865463e-02 5.132387e-02
## 791 792 793 794 795
## -2.863609e-02 1.396478e-01 3.032207e-01 -5.774829e-02 1.999339e-01
## 796 797 798 799 800
## 7.940565e-02 8.634202e-02 -3.395898e-02 3.403470e-01 3.918317e-02
## 801 802 803 804 805
## -1.632461e-01 5.753308e-02 1.889805e-01 2.343455e-01 1.545276e-01
## 806 807 808 809 810
## 1.503018e-01 -1.156109e-01 1.635752e-01 7.817449e-02 1.205243e-03
## 811 812 813 814 815
## -6.303387e-02 -3.918313e-02 2.113074e-01 9.305171e-02 -3.087808e-02
## 816 817 818 819 820
## 6.710380e-03 7.558713e-02 -2.293327e-01 1.524192e-02 1.522920e-01
## 821 822 823 824 825
## 5.052156e-02 -5.187393e-02 -1.017267e+00 3.808004e-02 6.324936e-02
## 826 827 828 829 830
## 1.106987e-01 -3.292464e-02 1.905930e-01 -1.137146e-01 4.080453e-02
## 831 832 833 834 835
## -8.008522e-02 1.992772e-02 1.407162e-01 3.112072e-02 5.563877e-02
## 836 837 838 839 840
## 6.060484e-02 1.668969e-01 5.963145e-02 2.058823e-01 1.016350e-01
## 841 842 843 844 845
## -2.188312e-01 1.030226e-01 1.176483e-01 2.584891e-02 2.152287e-02
## 846 847 848 849 850
## 2.417565e-02 9.874696e-02 2.312933e-01 1.221658e-02 7.508501e-02
## 851 852 853 854 855
## -5.022608e-02 3.329986e-02 1.171671e-01 4.327388e-02 -9.259921e-01
## 856 857 858 859 860
## -5.320411e-01 -3.232308e-02 6.752346e-02 1.106023e-01 -3.744878e-03
## 861 862 863 864 865
## -7.500993e-01 8.937273e-02 2.843146e-01 -1.508409e-02 1.242930e-01
## 866 867 868 869 870
## 8.799577e-02 7.290262e-02 7.099955e-02 7.275880e-02 -1.118108e-02
## 871 872 873 874 875
## -3.485499e-01 -3.084122e-02 1.579768e-01 -3.623013e-02 2.847263e-01
## 876 877 878 879 880
## -7.732569e-02 -5.120084e-02 2.607935e-01 -1.603494e-03 1.361152e-01
## 881 882 883 884 885
## 2.596176e-02 -4.769457e-02 8.101997e-02 -1.125455e-01 1.211554e-02
## 886 887 888 889 890
## 2.361360e-01 1.954378e-01 6.058675e-02 4.075743e-02 1.531539e-02
## 891 892 893 894 895
## -3.968884e-02 -4.513174e-01 -2.407046e-01 4.278258e-02 9.392422e-02
## 896 897 898 899 900
## 1.808250e-01 -5.012766e-02 -2.404888e-01 8.960164e-02 1.586369e-01
## 901 902 903 904 905
## 1.533604e-01 -1.046274e-01 1.282477e-01 1.635493e-01 4.045639e-02
## 906 907 908 909 910
## -9.131309e-03 -1.688769e-01 8.002775e-02 1.113464e-01 3.284917e-02
## 911 912 913 914 915
## 8.740468e-02 -3.715555e-02 1.515210e-01 1.902135e-02 1.051317e-01
## 916 917 918 919 920
## -4.973584e-03 4.269146e-03 5.026380e-02 1.965987e-02 2.966383e-01
## 921 922 923 924 925
## -6.240072e-02 -3.393315e-02 1.444823e-01 9.831284e-02 -2.562003e-01
## 926 927 928 929 930
## 1.776989e-01 1.351803e-01 2.703698e-01 1.285085e-01 -2.816549e-01
## 931 932 933 934 935
## -5.532282e-02 5.964256e-05 -1.735279e-01 5.265900e-02 -6.032673e-02
## 936 937 938 939 940
## 9.872261e-02 8.195563e-02 -4.201398e-02 -3.043206e-02 4.605785e-01
## 941 942 943 944 945
## 2.344722e-01 -5.194639e-01 -2.446892e-01 1.342061e-02 -1.355797e-01
## 946 947 948 949 950
## 7.313859e-02 1.916914e-01 -5.645604e-02 -5.962507e-01 -7.129223e-01
## 951 952 953 954 955
## -3.147641e-01 -8.966183e-02 -5.195513e-01 -1.192534e-01 2.311377e-01
## 956 957 958 959 960
## 1.626418e-01 -1.129815e-01 5.105901e-04 -4.308158e-02 2.905373e-01
## 961 962 963 964 965
## -8.075249e-02 1.020509e-01 1.370216e-01 1.633169e-01 9.102567e-02
## 966 967 968 969 970
## -2.169118e-02 3.215957e-02 2.360530e-01 1.818904e-01 -5.369348e-01
## 971 972 973 974 975
## -7.267854e-02 2.407847e-01 1.162471e-01 4.991534e-02 1.815762e-01
## 976 977 978 979 980
## 1.866399e-01 6.908884e-02 -6.252823e-02 3.304009e-01 -1.861513e-02
## 981 982 983 984 985
## -9.751235e-02 1.105022e-01 -2.456504e-01 -4.858952e-02 1.434770e-02
## 986 987 988 989 990
## -2.944720e-01 -2.663105e-02 5.236211e-02 1.859560e-01 -2.536655e-02
## 991 992 993 994 995
## 1.510376e-01 1.715876e-01 -2.887932e-01 1.085835e-01 -3.139218e-01
## 996 997 998 999 1000
## -9.076543e-01 -8.265740e-02 1.978602e-01 5.058613e-01 -1.716851e-01rmse_xval <- sqrt(mean(error^2))
rmse_xval## [1] 0.2102259K-Nearest Neighbours
Features in our dataser have different ranges when compared to other features. If the distance formula was applied to unmodified features, there is a potential for the features with larger ranges to dominate or mask the features with smaller ranges. Because of this, it is important to prepare the data with feature scaling.
# Randomizing our data for better results
random <- runif(1000, 1:4)## Warning in runif(1000, 1:4): NAs producedadvert_random <- advert[order(random),]Let’s select the first 6 rows from ad_random
head(advert_random)## Daily.Time.Spent.on.Site Age Area.Income Daily.Internet.Usage
## 1 68.95 35 61833.90 256.09
## 5 68.37 35 73889.99 225.58
## 9 74.53 30 68862.00 221.51
## 13 69.57 48 51636.92 113.12
## 17 55.39 37 23936.86 129.41
## 21 77.22 30 64802.33 224.44
## Ad.Topic.Line City Male
## 1 Cloned 5thgeneration orchestration Wrightburgh 0
## 5 Robust logistical utilization South Manuel 0
## 9 Configurable coherent function West Colin 1
## 13 Centralized content-based focus group West Katiefurt 1
## 17 Customizable multi-tasking website West Dylanberg 0
## 21 Object-based reciprocal knowledgebase Port Jacqueline 1
## Country Timestamp Clicked.on.Ad
## 1 Tunisia 2016-03-27 00:53:11 0
## 5 Iceland 2016-06-03 03:36:18 0
## 9 Grenada 2016-04-18 09:33:42 0
## 13 Egypt 2016-06-03 01:14:41 1
## 17 Palestinian Territory 2016-01-30 19:20:41 1
## 21 Cameroon 2016-01-05 07:52:48 0Using the Max-Min Normalization, we’ll be able to normalize our dataset.
normal <- function(x) (
return(((x-min(x)) / (max(x) - min(x))))
)
normal(1:4)## [1] 0.0000000 0.3333333 0.6666667 1.0000000advert_new <- as.data.frame(lapply(advert_random[1:4], normal))
summary(advert_new)## Daily.Time.Spent.on.Site Age Area.Income
## Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:0.3189 1st Qu.:0.2381 1st Qu.:0.5044
## Median :0.6054 Median :0.3810 Median :0.6568
## Mean :0.5507 Mean :0.4050 Mean :0.6261
## 3rd Qu.:0.7810 3rd Qu.:0.5476 3rd Qu.:0.7860
## Max. :1.0000 Max. :1.0000 Max. :1.0000
## Daily.Internet.Usage
## Min. :0.0000
## 1st Qu.:0.2061
## Median :0.4743
## Mean :0.4554
## 3rd Qu.:0.6902
## Max. :1.0000Creating test and train datasets
train <- advert_new[1:800,]
test <- advert_new[801:1000,]
train_sp <- advert_random[1:800,10]
test_sp <- advert_random[801:1000,10]Calling the class pacckage that consists of the KNN algorithm. Our confusion matrix is the table (test_sp,model)
library(class)
require(class)
model <- knn(train = train, test = test, cl = train_sp, k = 10)
table(factor(model))##
## 0 1
## 101 99table(test_sp,model)## model
## test_sp 0 1
## 0 92 1
## 1 9 98Out of 200 observations, the confusion matrix predicted 190 observations giving us an accuracy of 95%
Decision Tress
Complex decisions are often made simpler by using decision trees which breaks the decision down into smallerand much simpler ones using the divide and conquer strategy. Decision Trees basically identify a set of if-else conditions that split data according to the value of the features. We will be implementing the decesion ree model into our dataset.
Calling the necessary libraries:
library(rpart.plot)## Loading required package: rpartlibrary(mlbench)
library(rpart)Data partition To predict the class using rpart () function for the class method. rpart () uses the Gini index measure to split the nodes.
dt <- rpart(Clicked.on.Ad ~ Daily.Time.Spent.on.Site + Age + Area.Income + Daily.Internet.Usage, data = input, method ="class")
rpart.plot(dt) 
Searching for feature importance
data.frame(dt$variable.importance)## dt.variable.importance
## Daily.Internet.Usage 339.7809
## Daily.Time.Spent.on.Site 279.0247
## Age 126.2649
## Area.Income 119.2524pr <- predict(dt, input, type = "class")
table(pr, advert$Clicked.on.Ad)##
## pr 0 1
## 0 485 28
## 1 15 472Out of a total of 1000, the decision tree algorithm predicts 957 correct observationsa. The model will therefore achieve an accuracy of 95.7%
To train our model;
library(caret)
set.seed(15)
model <- train(Clicked.on.Ad ~ Daily.Time.Spent.on.Site + Age + Area.Income + Daily.Internet.Usage ,data = input,method = "ranger") ## Warning in train.default(x, y, weights = w, ...): You are trying to do
## regression and your outcome only has two possible values Are you trying to do
## classification? If so, use a 2 level factor as your outcome column.model## Random Forest
##
## 1000 samples
## 4 predictor
##
## No pre-processing
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 1000, 1000, 1000, 1000, 1000, 1000, ...
## Resampling results across tuning parameters:
##
## mtry splitrule RMSE Rsquared MAE
## 2 variance 0.1773138 0.8741314 0.06594256
## 2 extratrees 0.1728051 0.8812118 0.07273495
## 3 variance 0.1831404 0.8657094 0.06243689
## 3 extratrees 0.1737184 0.8794404 0.06842673
## 4 variance 0.1930556 0.8510197 0.06265898
## 4 extratrees 0.1755956 0.8766744 0.06692780
##
## Tuning parameter 'min.node.size' was held constant at a value of 5
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were mtry = 2, splitrule = extratrees
## and min.node.size = 5.plot(model)
Support Vector Machines
library(caret)
intrain <- createDataPartition(y = advert$Clicked.on.Ad, p= 0.7, list = FALSE)
training <- advert[intrain,]
testing <- advert[-intrain,]Checking the dimension of the training and testing dataframe.
dim(training);## [1] 700 10dim(testing);## [1] 300 10Let’s factorize our target variable for accurate results.
training[["Clicked.on.Ad"]] = factor(training[["Clicked.on.Ad"]])Using the traincontrol() method to control the computational overheads
trctrl <- trainControl(method = "repeatedcv", number = 10, repeats = 5)svm_Linear <- train(Clicked.on.Ad ~ Daily.Time.Spent.on.Site + Age + Area.Income +Daily.Internet.Usage , data = training, method = "svmLinear",
trControl=trctrl,
preProcess = c("center", "scale"),
tuneLength = 10)Results of the training model
svm_Linear## Support Vector Machines with Linear Kernel
##
## 700 samples
## 4 predictor
## 2 classes: '0', '1'
##
## Pre-processing: centered (4), scaled (4)
## Resampling: Cross-Validated (10 fold, repeated 5 times)
## Summary of sample sizes: 630, 630, 630, 630, 630, 630, ...
## Resampling results:
##
## Accuracy Kappa
## 0.9697143 0.9394286
##
## Tuning parameter 'C' was held constant at a value of 1We’ll use the predict() method to predict results
test_pred <- predict(svm_Linear, newdata = testing)
test_pred## [1] 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1
## [38] 1 1 0 1 1 0 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 0 1 0 0 1 0 1
## [75] 0 1 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0
## [112] 1 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 0 1 0 0 1 0 0
## [149] 1 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 1
## [186] 0 1 1 0 1 1 1 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 0 1 0 1 0 1 0 1 1 0 1
## [223] 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 1 0 1 0 1 0 0 0
## [260] 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1
## [297] 1 1 1 1
## Levels: 0 1Checking the accuracy of our model using a confusion matrix.
confusionMatrix(table(test_pred, testing$Clicked.on.Ad))## Confusion Matrix and Statistics
##
##
## test_pred 0 1
## 0 147 9
## 1 3 141
##
## Accuracy : 0.96
## 95% CI : (0.9312, 0.9792)
## No Information Rate : 0.5
## P-Value [Acc > NIR] : <2e-16
##
## Kappa : 0.92
##
## Mcnemar's Test P-Value : 0.1489
##
## Sensitivity : 0.9800
## Specificity : 0.9400
## Pos Pred Value : 0.9423
## Neg Pred Value : 0.9792
## Prevalence : 0.5000
## Detection Rate : 0.4900
## Detection Prevalence : 0.5200
## Balanced Accuracy : 0.9600
##
## 'Positive' Class : 0
## The SVM model achieves an accuracy level of 96%.
Conclusion and Recommendations
- Our data shows rhat most of the respodents fall between the 25-41 age bracket, the course should be tailor made in order to attract the age more people within the age bracket. The olderst respondent is noted to be of 19 years while the youngest is 61 years.
- Our client should consider targetting people with an income of betwen 50,000 to 70,000 since they seem to be more interested and also in a position to afford the course
- Time spent by most people on the site is 70 - 85 minutes. So, to the course offered should span at around the sme time of even shoeter, in order to keep people more interested.
- Since SVM has given the highest accuracy level, the client should implement it.
- To figure out which clients will clieck on her adverts, the client should prioritize the following features: Daily Internet Udage - Daily Time Spent on SIte - Age - Income distribution of the area