Multiple Regression
Elias Lobato
31/12/2019
1 Introduction
In this work we have to predict the volume sales in four different product types while assessing the effects service and customer reviews have on sales.
Weāll be using Regression to build machine learning models for this analyses. Once we have determined which algorithm works better on the provided data set, we will predict the sales of four product types from the new products list.
To sum up:
What are we trying to predict?
We need to predict the sales volume for the new products list.
What type of problem is it? Classification or Regression? Binary or Multi-class? Uni-variate or Multi-variate?
It is a multiple regression problem with multiple features.
What type of data do we have?
We have two files, one to build predictive models (existingproducts.csv) and the other one is the data set that will be used to test the model (newproducts).
2 Load libraries
First, we need to activate packages to be used during the project.
library(dplyr)
library(caret)
library(readr)
library(dummies)
library(corrplot)
library(normalr)
library(randomForest) # Random forest in regression problems
library(rknn) # k-NN in regression problems
library(kknn) # k-NN in regression with train() function
library(DMwR) # regr.eval function
library(kernlab) # svm radial kernel
library(gbm)
library(plotly) 3 Import Data
The next step is to upload the data we are going to be working in.
4 Initial exploration of data
With the function glimpse()and summary() we can do the initial exploration of data to know better what type of data we are handling.
## Observations: 80
## Variables: 18
## $ ProductType <chr> "PC", "PC", "PC", "Laptop", "Laptop", "Access...
## $ ProductNum <dbl> 101, 102, 103, 104, 105, 106, 107, 108, 109, ...
## $ Price <dbl> 949.00, 2249.99, 399.00, 409.99, 1079.99, 114...
## $ x5StarReviews <dbl> 3, 2, 3, 49, 58, 83, 11, 33, 16, 10, 21, 75, ...
## $ x4StarReviews <dbl> 3, 1, 0, 19, 31, 30, 3, 19, 9, 1, 2, 25, 8, 6...
## $ x3StarReviews <dbl> 2, 0, 0, 8, 11, 10, 0, 12, 2, 1, 2, 6, 5, 13,...
## $ x2StarReviews <dbl> 0, 0, 0, 3, 7, 9, 0, 5, 0, 0, 4, 3, 0, 8, 7, ...
## $ x1StarReviews <dbl> 0, 0, 0, 9, 36, 40, 1, 9, 2, 0, 15, 3, 1, 16,...
## $ PositiveServiceReview <dbl> 2, 1, 1, 7, 7, 12, 3, 5, 2, 2, 2, 9, 2, 44, 5...
## $ NegativeServiceReview <dbl> 0, 0, 0, 8, 20, 5, 0, 3, 1, 0, 1, 2, 0, 3, 3,...
## $ Recommendproduct <dbl> 0.9, 0.9, 0.9, 0.8, 0.7, 0.3, 0.9, 0.7, 0.8, ...
## $ BestSellersRank <dbl> 1967, 4806, 12076, 109, 268, 64, NA, 2, NA, 1...
## $ ShippingWeight <dbl> 25.80, 50.00, 17.40, 5.70, 7.00, 1.60, 7.30, ...
## $ ProductDepth <dbl> 23.94, 35.00, 10.50, 15.00, 12.90, 5.80, 6.70...
## $ ProductWidth <dbl> 6.62, 31.75, 8.30, 9.90, 0.30, 4.00, 10.30, 6...
## $ ProductHeight <dbl> 16.89, 19.00, 10.20, 1.30, 8.90, 1.00, 11.50,...
## $ ProfitMargin <dbl> 0.15, 0.25, 0.08, 0.08, 0.09, 0.05, 0.05, 0.0...
## $ Volume <dbl> 12, 8, 12, 196, 232, 332, 44, 132, 64, 40, 84...Now we will check for outliers.
To finish with the initial exploration we will check for missing values and in case of finding them we will transform them.
## [1] 15There are missing values, and with the str() function seen above we can see that all missing values correspond to the same attribute. In the next part, Pre-processing we will treat outliers and missing values.
5 Pre-processing
Most data will contain a mixture of numeric and nominal data so we need to understand how to incorporate both when it comes to developing regression models and making predictions.
Categorical variables may be used directly as predictor or predicted variables in a multiple regression model as long as theyāve been converted to binary values. In order to pre-process the sales data as needed we first need to convert all factor or āchrā classes to binary features that contain ā0ā and ā1ā classes. Fortunately, caret has a method for creating these āDummy Variablesā as follows:
# dummify the data
existing_products_2 <- dummyVars(" ~ .", data = existing_products)
existing_products_3 <- data.frame(predict(existing_products_2, newdata = existing_products))Now is time to remove outliers.
# Outliers are an Accesory and a Game console, there arenĀ“t PC, Laptop, Netbook or Smartphone (our objectives). So we can remove them.
existing_products_3 <- existing_products_3[-which(existing_products_3$Volume %in% outliers),]Once data is dummified and we donĀ“t have categorical data we can treat missing values. In this case, all missing values are from the same attribute. So we have decided to delete this attribute.
After dummifying data and omitting outliers and missing values we have to check the correlation among variables and between variables and the dependant variable.
6 Feature Engineering
We will use the cor() function to create a correlation matrix to visualize the correlation between the features.
corrData <- cor(existing_products_3)
corrplot(corrData) # There are so many variables that is difficult to distibguish among them, but it can be check if you call corrData object.
From the corrData we can observe that these values have high correlation with the dependant variable and can lead to overfitting.
x5starReview
Now we have to check if there is high correlation among features, not removing these features can lead to too much noise.
x4starReviem and x3starReview has a correlation of 0.937. In order to decide which is going to be removed we have to check the correlation of each feature with the dependant variable.
x4starReview has greater correlation so x3starReview is going to be removed.
The same happen with x2starReview and x1starReview, where x1starReview is going to be removed.
We have check and removed features that have highest correlation with dependant variables and among them. Now we have to treat these features that have less correlation with dependat variables like, profit margin and physical attributes, that have less than 0.2 correlation.
The variables that have low correlation with dependant variable are:
- Display
- Extended Warranty
- Laptop
- Printer Supplies
- Smartphone
- Software
- Tablet
- Product Depth
- Profit Margin
In the case of dummified data we will not remove ProductTypeXXX because we donĀ“t want to get so biased model.
# Variables with high collinearity with dependant variable will be always deleted.
existing_products_3$x5StarReviews <- NULL
# Multicollinearity among variables will also be applied, reducing noise always is effective with any algorithm.
existing_products_3$x1StarReviews <- NULL
existing_products_3$x3StarReviews <- NULL
existing_products_3$ProductNum <- NULL
# We will create Existing_Products to remove all variables that correlation matrix tell us to delete.
drop_variables <- c("ProductDepth", "ProductHeight", "ProductWidth", "ProfitMargin", "ShippingWeight")
Existing_Products <- existing_products_3[ , !(names(existing_products_3) %in% drop_variables)]7 Training and Testing
Once data is preprocessed, itĀ“s time to create training and testing sets for the predictive model that must be performed.
First, we will split data into two sets, training and testing set with cerateDataPartition() function, that does a stratified random split of the data.
set.seed(123) # seed is a number that you choose for a starting point used to create a sequence of random numbers.
inTrain <- createDataPartition(y = Existing_Products$Volume, p=.75, list = FALSE)
# To partition the data:
training <- Existing_Products[ inTrain,]
testing <- Existing_Products[-inTrain,]8 Modelling
We will run 4 different methods and then we will compare them. The best method is going to be used to predict our new product list Volume.
This algorithms can be compared if all have the same resampling method and the same number of repetitions. To modify resampling method trainControl() function can be used.
8.1 Supported Vector Machine
The first method that is going to be performed is Supported Vector Machine algorithm. To run any algorithm the train() function is used. In this case, the method that will be used is called āsvmRadialā, one among many.
set.seed(123) # this function must every before every fucntion that generates something randomly.
svmGrid <- expand.grid(sigma=0.02541634, C=8)
svmfit <- train(Volume ~ ., data = training, method = "svmRadial", trControl=fitControl, tuneGrid=svmGrid, preProc = c("center", "scale")) #Ƶ This model can be adjusted training first with different tuneLengths and then fixing the best parameters with tuneGrid.Once the training algorithm is performed it has to be tried to predict in testing set.
svm_pred <- predict(svmfit, newdata = testing)
actuals_preds_svm <- data.frame(cbind(actuals=testing$Volume, predicteds=svm_pred)) # make actuals_predicteds dataframe.
actuals_preds_svm## actuals predicteds
## 1 232 356.60341
## 2 132 272.62175
## 3 300 177.75650
## 4 60 126.18329
## 5 1576 1230.77253
## 6 2052 648.13852
## 7 32 -25.62101
## 8 20 119.63613
## 9 1232 1174.19628
## 10 88 239.77103
## 11 1536 540.28954
## 12 836 1596.56955
## 13 904 583.01019
## 14 232 326.43896
## 15 8 -30.56946
## 16 80 124.19303
## 17 0 33.32721
## 18 16 -53.750568.2 Random Forest
Next method is called Random Forest In this case, the method that will be used is called ārangerā.
set.seed(123)
rfGrid <- expand.grid(min.node.size=5, mtry=18, splitrule="variance")
rffit <- train(Volume ~., data = training, method = "ranger", trControl=fitControl, tuneGrid=rfGrid, preProc = c("center", "scale"))Once the training algorithm is performed it has to be tried to predict in testing set.
8.3 k-Nearest Neighbour
Third method is called k- Nearest Neighbour. In this case, the method that will be used is called ākknnā.
set.seed(123)
knnGrid <- expand.grid(distance=2, kernel="optimal", kmax=7)
knnfit <- train(Volume ~ ., data = training,method = "kknn", trControl=fitControl, tuneGrid=knnGrid, preProc = c("center", "scale"))Once the training algorithm is performed it has to be tried to predict in testing set.
8.4 Gradient Boosted Trees
Last method is the Gradient Boosted Tree algorithm. In this case, the method that will be used is called āgbmā.
set.seed(123)
gbtGrid <- expand.grid(shrinkage=0.1, n.trees = 100, n.minobsinnode=10, interaction.depth =4)
gbtfit <- train(Volume ~ ., data = training, method = "gbm", trControl=fitControl, tuneGrid=gbtGrid, preProc = c("center", "scale"))## Iter TrainDeviance ValidDeviance StepSize Improve
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## 100 28424.7429 nan 0.1000 -946.3329Once the training algorithm is performed it has to be tried to predict in testing set.
9 Resamples
After making the predictions using the test set, it is goint to use resamples() function to assess the metrics of the new predictions compared to the Ground Truth.
resamps <- resamples(list(svm = svmfit, rf = rffit, knn = knnfit, gbt = gbtfit))
summary(resamps) # Show the accuracy and kappa of each model. Remember that the number of resamples was fit to 10 in fitcontrol object.##
## Call:
## summary.resamples(object = resamps)
##
## Models: svm, rf, knn, gbt
## Number of resamples: 10
##
## MAE
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## svm 30.37503 219.85782 313.7654 292.61240 407.8387 483.0093 0
## rf 14.99248 34.15706 100.0813 96.95222 151.7929 191.6567 0
## knn 32.26114 188.98975 235.3296 210.16284 248.5188 286.6808 0
## gbt 63.01270 112.17930 154.1390 152.32937 179.9038 263.5565 0
##
## RMSE
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## svm 34.93878 308.25199 564.8230 476.7672 681.9374 714.7242 0
## rf 22.26736 58.75182 183.8599 173.5783 264.5494 361.9389 0
## knn 45.31928 295.99877 340.9257 341.7493 425.5244 564.0861 0
## gbt 91.09679 162.28138 231.7509 226.0522 246.6022 429.1217 0
##
## Rsquared
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## svm 0.3002866 0.4913675 0.7429490 0.7039975 0.9577268 0.9981831 0
## rf 0.7868799 0.9062093 0.9664738 0.9298072 0.9880728 0.9987420 0
## knn 0.5653646 0.6915781 0.7458344 0.7702468 0.8640047 0.9884488 0
## gbt 0.7576783 0.8293145 0.8603423 0.8768935 0.9527890 0.9856353 0diffs <- diff(resamps)
summary(diffs) # Gives the differences of the resamps object. Another way to check which model fits better.##
## Call:
## summary.diff.resamples(object = diffs)
##
## p-value adjustment: bonferroni
## Upper diagonal: estimates of the difference
## Lower diagonal: p-value for H0: difference = 0
##
## MAE
## svm rf knn gbt
## svm 195.66 82.45 140.28
## rf 0.004519 -113.21 -55.38
## knn 0.627520 0.010850 57.83
## gbt 0.150495 0.113840 0.364229
##
## RMSE
## svm rf knn gbt
## svm 303.19 135.02 250.72
## rf 0.002998 -168.17 -52.47
## knn 0.703751 0.019815 115.70
## gbt 0.089718 0.808675 0.149251
##
## Rsquared
## svm rf knn gbt
## svm -0.22581 -0.06625 -0.17290
## rf 0.1204 0.15956 0.05291
## knn 1.0000 0.0673 -0.10665
## gbt 0.5268 0.8341 0.3316Then it will generate a data frame that contains different evaluation parameters of all models.
algorithm_comparison <- data.frame(Model="SVM", MAE=mean(resamps$values$`svm~MAE`), RMSE=mean(resamps$values$`svm~RMSE`), Rsquared=mean(resamps$values$`svm~Rsquared`))
algorithm_comparison <- rbind(algorithm_comparison,data.frame(Model="RF", MAE=mean(resamps$values$`rf~MAE`), RMSE=mean(resamps$values$`rf~RMSE`), Rsquared=mean(resamps$values$`rf~Rsquared`)))
algorithm_comparison <- rbind(algorithm_comparison,data.frame(Model="kNN", MAE=mean(resamps$values$`knn~MAE`), RMSE=mean(resamps$values$`knn~RMSE`), Rsquared=mean(resamps$values$`knn~Rsquared`)))
algorithm_comparison <- rbind(algorithm_comparison,data.frame(Model="GBT", MAE=mean(resamps$values$`gbt~MAE`), RMSE=mean(resamps$values$`gbt~RMSE`), Rsquared=mean(resamps$values$`gbt~Rsquared`)))
algorithm_comparison## Model MAE RMSE Rsquared
## 1 SVM 292.61240 476.7672 0.7039975
## 2 RF 96.95222 173.5783 0.9298072
## 3 kNN 210.16284 341.7493 0.7702468
## 4 GBT 152.32937 226.0522 0.8768935Among used different methods the Random Forest method gives the best results. So the performed predictive model that is going to apply to predict new products sales Volume is going to be Random Forest.
Finally we will train our predictive model in the whole set and not just in our testing set.
rf_prediction_allData <- predict(rffit, newdata = Existing_Products)
actuals_preds_allData_rf <- data.frame(cbind(actuals=Existing_Products$Volume, predicteds=rf_prediction_allData)) # make actuals_predicteds dataframe.
correlation_accuracy_allData_rf <- cor(actuals_preds_allData_rf) # % 96.1 of Accuracy.10 New product
The final step is to apply the performed predictive model to the new products in order to predict Volume of different products.
To apply the predictive model, the data set must suffer the same modifications.
Load data.
Dummify data as before.
# dummify the data
new_products_2 <- dummyVars(" ~ .", data = new_products)
new_products_3 <- data.frame(predict(new_products_2, newdata = new_products))Omit these values that were considered no necessary.
new_products_3$x5StarReviews <- NULL
new_products_3$x1StarReviews <- NULL
new_products_3$x3StarReviews <- NULL
new_products_3$BestSellersRank <- NULL
new_products_3$ProductNum <- NULL
new_prodructs_4 <- new_products_3[ , !(names(new_products_3) %in% drop_variables)]Finally, predictive model will be applied in the modified new products data set. Also, will be generated a csv file containin preditcion rules.
11 New Product analysis
Now itĀ“s time to analyze the four products that we have been asked to asses.
11.0.1 PC
PC_new <- subset(New_Products, ProductType == "PC") # Group PCs of New Products data set.
drop_variables <- cbind(drop_variables, "x5StarReviews", "x3StarReviews","x1StarReviews", "BestSellersRank") # Add to previously generated drop_variables object the attributes that were omitted to generate the predictive model.
PC_new_1 <- PC_new[ , !(names(PC_new) %in% drop_variables)] # Generate a data frame with only the attributes that were used to build the predictive model.
PC_existing <- subset(existing_products, ProductType == "PC") # Group PCs of Existing Products data set befor being dummified.
PC_existing_1 <- PC_existing[ , !(names(PC_existing) %in% drop_variables)] # Generate a data frame with only the attributes that were used to build the predictive model.
PC_analysis <- rbind(PC_new_1,PC_existing_1)
PC_analysis## # A tibble: 6 x 9
## ProductType ProductNum Price x4StarReviews x2StarReviews PositiveService~
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 PC 171 699 26 14 12
## 2 PC 172 860 11 10 7
## 3 PC 101 949 3 0 2
## 4 PC 102 2250. 1 0 1
## 5 PC 103 399 0 0 1
## 6 PC 142 610. 7 0 5
## # ... with 3 more variables: NegativeServiceReview <dbl>,
## # Recommendproduct <dbl>, Volume <dbl>PC_analysis$ProductNum <- as.factor(PC_analysis$ProductNum)
#ggplot(PC_analysis, aes( Volume , color=ProductNum, fill=ProductNum)) + geom_col() + labs(title = "PC") + stat_count()
#analysis <- rbind(PC_new_1, Smartphone_new_1, Netbook_new_1, Laptop_new_1)
#ggplot(PC_analysis, aes( Volume , color=ProductNum, fill=ProductNum)) + geom_col() + labs(title = "PC") + stat_count()11.0.2 Laptop
Laptop_new <- subset(New_Products, ProductType == "Laptop") # Group Laptops of New Products data set.
Laptop_new_1 <- Laptop_new[ , !(names(Laptop_new) %in% drop_variables)] # Generate a data frame with only the attributes that were used to build the predictive model.
Laptop_existing <- subset(existing_products, ProductType == "Laptop") # Group PCs of Existing Products data set befor being dummified.
Laptop_existing_1 <- Laptop_existing[ , !(names(Laptop_existing) %in% drop_variables)] # Generate a data frame with only the attributes that were used to build the predictive model.
Laptop_analysis <- rbind(Laptop_new_1,Laptop_existing_1)
Laptop_analysis## # A tibble: 6 x 9
## ProductType ProductNum Price x4StarReviews x2StarReviews PositiveService~
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 Laptop 173 1199 10 3 11
## 2 Laptop 175 1199 2 1 2
## 3 Laptop 176 1999 1 3 0
## 4 Laptop 104 410. 19 3 7
## 5 Laptop 105 1080. 31 7 7
## 6 Laptop 143 771. 14 5 6
## # ... with 3 more variables: NegativeServiceReview <dbl>,
## # Recommendproduct <dbl>, Volume <dbl>Laptop_analysis$ProductNum <- as.factor(Laptop_analysis$ProductNum)
ggplot(Laptop_analysis, aes(ProductNum , Volume, color=ProductNum, fill=ProductNum)) + geom_point() + labs(title = "Laptop")
11.0.3 Netbook
Netbook_new <- subset(New_Products, ProductType == "Netbook") # Group Laptops of New Products data set.
Netbook_new_1 <- Netbook_new[ , !(names(Netbook_new) %in% drop_variables)] # Generate a data frame with only the attributes that were used to build the predictive model.
Netbook_existing <- subset(existing_products, ProductType == "Netbook") # Group PCs of Existing Products data set befor being dummified.
Netbook_existing_1 <- Netbook_existing[ , !(names(Netbook_existing) %in% drop_variables)] # Generate a data frame with only the attributes that were used to build the predictive model.
Netbook_analysis <- rbind(Netbook_new_1, Netbook_existing_1)
Netbook_analysis## # A tibble: 6 x 9
## ProductType ProductNum Price x4StarReviews x2StarReviews PositiveService~
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 Netbook 178 400. 8 1 2
## 2 Netbook 180 329 112 31 28
## 3 Netbook 181 439 18 22 5
## 4 Netbook 183 330 4 1 1
## 5 Netbook 177 380. 0 1 0
## 6 Netbook 182 350. 10 2 3
## # ... with 3 more variables: NegativeServiceReview <dbl>,
## # Recommendproduct <dbl>, Volume <dbl>Netbook_analysis$ProductNum <- as.factor(Netbook_analysis$ProductNum)
ggplot(Netbook_analysis, aes(ProductNum , Volume, color=ProductNum, fill=ProductNum)) + geom_point() + labs(title = "Netbook")
11.0.4 Smartphone
Smartphone_new <- subset(New_Products, ProductType == "Smartphone") # Group Laptops of New Products data set.
Smartphone_new_1 <- Smartphone_new[ , !(names(Smartphone_new) %in% drop_variables)] # Generate a data frame with only the attributes that were used to build the predictive model.
Smartphone_existing <- subset(existing_products, ProductType == "Smartphone") # Group PCs of Existing Products data set befor being dummified.
Smartphone_existing_1 <- Smartphone_existing[ , !(names(Smartphone_existing) %in% drop_variables)] # Generate a data frame with only the attributes that were used to build the predictive model.
Smartphone_analysis <- rbind(Smartphone_new_1, Smartphone_existing_1)
Smartphone_analysis## # A tibble: 8 x 9
## ProductType ProductNum Price x4StarReviews x2StarReviews PositiveService~
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 Smartphone 193 199 26 16 8
## 2 Smartphone 194 49 26 33 14
## 3 Smartphone 195 149 8 4 4
## 4 Smartphone 196 300 19 20 5
## 5 Smartphone 190 199 1 2 1
## 6 Smartphone 191 200 25 11 9
## 7 Smartphone 192 99 17 2 5
## 8 Smartphone 197 499 28 10 22
## # ... with 3 more variables: NegativeServiceReview <dbl>,
## # Recommendproduct <dbl>, Volume <dbl>Smartphone_analysis$ProductNum <- as.factor(Smartphone_analysis$ProductNum)
ggplot(Smartphone_analysis, aes(ProductNum , Volume, color=ProductNum, fill=ProductNum)) + geom_point() + labs(title = "Smartphone")
12 Assesing of service and customer reviews variables
First we will generate a data set that contains only the variables that we have used to make the predictive model.
drop_variables_asses <- c("Price","ProductDepth", "ProductHeight", "ProductWidth", "ProfitMargin", "ProductNum", "ShippingWeight","Recommendproduct", "BestSellersRank")
existing_products_review <- existing_products[ , !(names(existing_products) %in% drop_variables_asses)]As before, we are going to omit outliers.
12.0.1 Service Reviews
On one hand, we will asses the effect of Service Review variables, PositiveServiceReview and NegativeServiceReview.
PositiveReview <- ggplot(existing_products_review , aes(PositiveServiceReview, Volume)) + geom_point() + geom_smooth(color="red") + labs(title = "Positive Service Review")
NegativeReview <- ggplot(existing_products_review , aes(NegativeServiceReview, Volume)) + geom_point() + geom_smooth(color="blue") + labs(title = "Negative Service Review")
grid.arrange(PositiveReview, NegativeReview, ncol = 2)
It looks that there are some points that donĀ“t follow the almost generated linear model. IN the next lines we can extract them with filter() function.
Positive_outliers <- filter(existing_products_review, existing_products_review$PositiveServiceReview > 200)
Positive_outliers## # A tibble: 9 x 9
## ProductType x5StarReviews x4StarReviews x3StarReviews x2StarReviews
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 Accessories 170 100 23 20
## 2 ExtendedWa~ 308 27 8 3
## 3 ExtendedWa~ 308 27 8 3
## 4 ExtendedWa~ 308 27 8 3
## 5 ExtendedWa~ 308 27 8 3
## 6 ExtendedWa~ 308 27 8 3
## 7 ExtendedWa~ 308 27 8 3
## 8 ExtendedWa~ 308 27 8 3
## 9 ExtendedWa~ 308 27 8 3
## # ... with 4 more variables: x1StarReviews <dbl>, PositiveServiceReview <dbl>,
## # NegativeServiceReview <dbl>, Volume <dbl>12.1 Customer Reviews
On the other hand, we will analyze the asses of Star Reviews vartiables.
x4starReview <- ggplot(existing_products_review , aes(x4StarReviews, Volume)) + geom_point() + geom_smooth() + geom_smooth(color="blue") + labs(title = "4 Star Review")
x2starReview <- ggplot(existing_products_review , aes(x2StarReviews, Volume)) + geom_point() + geom_smooth() + geom_smooth(color="yellow") + labs(title = "2 Star Review")
grid.arrange(x4starReview, x2starReview, ncol = 2)