#III. Tiền xử lý ddữ liệu
water_potability <- read.csv("~/Zalo Received Files/LOP L07_NHÓM 14.csv")
save(water_potability, file="water_potability.rda")
head(water_potability,10)
## ph Hardness Solids Chloramines Sulfate Conductivity Organic_carbon
## 1 NA 204.8905 20791.32 7.300212 368.5164 564.3087 10.379783
## 2 3.716080 129.4229 18630.06 6.635246 NA 592.8854 15.180013
## 3 8.099124 224.2363 19909.54 9.275884 NA 418.6062 16.868637
## 4 8.316766 214.3734 22018.42 8.059332 356.8861 363.2665 18.436524
## 5 9.092223 181.1015 17978.99 6.546600 310.1357 398.4108 11.558279
## 6 5.584087 188.3133 28748.69 7.544869 326.6784 280.4679 8.399735
## 7 10.223862 248.0717 28749.72 7.513408 393.6634 283.6516 13.789695
## 8 8.635849 203.3615 13672.09 4.563009 303.3098 474.6076 12.363817
## 9 NA 118.9886 14285.58 7.804174 268.6469 389.3756 12.706049
## 10 11.180284 227.2315 25484.51 9.077200 404.0416 563.8855 17.927806
## Trihalomethanes Turbidity Potability
## 1 86.99097 2.963135 0
## 2 56.32908 4.500656 0
## 3 66.42009 3.055934 0
## 4 100.34167 4.628771 0
## 5 31.99799 4.075075 0
## 6 54.91786 2.559708 0
## 7 84.60356 2.672989 0
## 8 62.79831 4.401425 0
## 9 53.92885 3.595017 0
## 10 71.97660 4.370562 0
water_potability$X=NULL
water_potability$X.1=NULL
water_potability$X.2=NULL
str(water_potability)
## 'data.frame': 3276 obs. of 10 variables:
## $ ph : num NA 3.72 8.1 8.32 9.09 ...
## $ Hardness : num 205 129 224 214 181 ...
## $ Solids : num 20791 18630 19910 22018 17979 ...
## $ Chloramines : num 7.3 6.64 9.28 8.06 6.55 ...
## $ Sulfate : num 369 NA NA 357 310 ...
## $ Conductivity : num 564 593 419 363 398 ...
## $ Organic_carbon : num 10.4 15.2 16.9 18.4 11.6 ...
## $ Trihalomethanes: num 87 56.3 66.4 100.3 32 ...
## $ Turbidity : num 2.96 4.5 3.06 4.63 4.08 ...
## $ Potability : int 0 0 0 0 0 0 0 0 0 0 ...
colSums(is.na(water_potability))
## ph Hardness Solids Chloramines Sulfate
## 491 0 0 0 781
## Conductivity Organic_carbon Trihalomethanes Turbidity Potability
## 0 0 162 0 0
colMeans(is.na(water_potability))
## ph Hardness Solids Chloramines Sulfate
## 0.14987790 0.00000000 0.00000000 0.00000000 0.23840049
## Conductivity Organic_carbon Trihalomethanes Turbidity Potability
## 0.00000000 0.00000000 0.04945055 0.00000000 0.00000000
ph_1<-mean(water_potability[water_potability$Potability == 0 & water_potability$Hardness <= 150, "ph"],na.rm=T)
ph_1
## [1] 6.721973
ph_2<-mean(water_potability[water_potability$Potability == 0 & water_potability$Hardness > 150, "ph"],na.rm=T)
ph_2
## [1] 7.112512
ph_3<-mean(water_potability[water_potability$Potability == 1 & water_potability$Hardness <= 150, "ph"],na.rm=T)
ph_3
## [1] 7.098154
ph_4<-mean(water_potability[water_potability$Potability == 1 & water_potability$Hardness > 150, "ph"],na.rm=T)
ph_4
## [1] 7.071429
water_potability[ water_potability$Potability == 0 & water_potability$Hardness <= 150 & is.na(water_potability$ph), "ph"] <- ph_1
water_potability[ water_potability$Potability == 0 & water_potability$Hardness > 150 & is.na(water_potability$ph), "ph"] <- ph_2
water_potability[ water_potability$Potability == 1 & water_potability$Hardness <= 150 & is.na(water_potability$ph), "ph"] <- ph_3
water_potability[ water_potability$Potability == 1 & water_potability$Hardness > 150 & is.na(water_potability$ph), "ph"] <- ph_4
Sulfate_1<-mean(water_potability[water_potability$Potability == 0, "Sulfate"],na.rm=T)
Sulfate_1
## [1] 334.5643
Sulfate_2<-mean(water_potability[water_potability$Potability == 1, "Sulfate"],na.rm=T)
Sulfate_2
## [1] 332.567
water_potability[water_potability$Potability == 0 & is.na(water_potability$Sulfate), "Sulfate"] <- Sulfate_1
water_potability[water_potability$Potability == 1 & is.na(water_potability$Sulfate), "Sulfate"] <- Sulfate_2
water_potability[is.na(water_potability$Trihalomethanes),"Trihalomethanes"] <- mean(water_potability$Trihalomethanes,na.rm=T)
colSums(is.na(water_potability))
## ph Hardness Solids Chloramines Sulfate
## 0 0 0 0 0
## Conductivity Organic_carbon Trihalomethanes Turbidity Potability
## 0 0 0 0 0
#IV. Thống kê mô tả
water_potability$Potability<-as.factor(water_potability$Potability)
summary(water_potability)
## ph Hardness Solids Chloramines
## Min. : 0.000 Min. : 47.43 Min. : 320.9 Min. : 0.352
## 1st Qu.: 6.278 1st Qu.:176.85 1st Qu.:15666.7 1st Qu.: 6.127
## Median : 7.104 Median :196.97 Median :20927.8 Median : 7.130
## Mean : 7.081 Mean :196.37 Mean :22014.1 Mean : 7.122
## 3rd Qu.: 7.870 3rd Qu.:216.67 3rd Qu.:27332.8 3rd Qu.: 8.115
## Max. :14.000 Max. :323.12 Max. :61227.2 Max. :13.127
## Sulfate Conductivity Organic_carbon Trihalomethanes
## Min. :129.0 Min. :181.5 Min. : 2.20 Min. : 0.738
## 1st Qu.:317.1 1st Qu.:365.7 1st Qu.:12.07 1st Qu.: 56.648
## Median :334.6 Median :421.9 Median :14.22 Median : 66.396
## Mean :333.8 Mean :426.2 Mean :14.28 Mean : 66.396
## 3rd Qu.:350.4 3rd Qu.:481.8 3rd Qu.:16.56 3rd Qu.: 76.667
## Max. :481.0 Max. :753.3 Max. :28.30 Max. :124.000
## Turbidity Potability
## Min. :1.450 0:1998
## 1st Qu.:3.440 1:1278
## Median :3.955
## Mean :3.967
## 3rd Qu.:4.500
## Max. :6.739
table (water_potability$Potability)
##
## 0 1
## 1998 1278
barplot ( table ( water_potability $ Potability ) , xlab =" Potability ", ylab =" Frequency ", main ="Barplot of Potability ",col =c(" grey "," lightblue "))
par(mfrow=c(1,3))
hist(water_potability$ph,xlab="pH",main="Histogram of pH",col="blue",labels=T,ylim=c(0,1200))
hist(water_potability$Hardness,xlab="Hardness",main="Histogram of Hardness",col="blue",labels=T,ylim=c(0,1000))
hist(water_potability$Solids,xlab="Solids",main="Histogram of Solids",col="blue",labels=T,ylim=c(0,800))
hist(water_potability$Chloramines,xlab="Chloramines",main="Histogram of Chloramines",col="blue",labels=T,ylim=c(0,1000))
hist(water_potability$Sulfate,xlab="Sulfate",main="Histogram of Sulfate",col="blue",labels=T,ylim=c(0,1400))
hist(water_potability$Conductivity,xlab="Conductivity",main="Histogram of Conductivity",col="blue",labels=T,ylim=c(0,800))
hist(water_potability$Organic_carbon,xlab="Organic_carbon",main="Histogram of Organic_carbon",col="blue",labels=T,ylim=c(0,800))
hist(water_potability$Trihalomethanes,xlab="Trihalomethanes",main="Histogram of Trihalomethanes",col="blue",labels=T,ylim=c(0,1000))
hist(water_potability$Turbidity,xlab="Turbidity",main="Histogram of Turbidity",col="blue",labels=T,ylim=c(0,1000))
boxplot ( water_potability $ ph ~ water_potability $ Potability , xlab = " Potability " , ylab = " ph " , main =" pH and Potability " , col = c ( " grey " ," lightblue " ) )
boxplot ( water_potability $ Hardness ~ water_potability $ Potability , xlab = " Potability " , ylab = "Hardness " , main = " Hardness and Potability " , col = c ( " grey " ," lightblue " ) )
boxplot ( water_potability $ Solids ~ water_potability $ Potability , xlab = " Potability " , ylab = "Solids " , main = " Solids and Potability " , col = c ( " grey " ," lightblue " ) )
boxplot ( water_potability $ Chloramines ~ water_potability $ Potability , xlab = " Potability " , ylab =" Chloramines " , main = " Chloramines and Potability " , col = c ( " grey " ," lightblue " ) )
boxplot ( water_potability $ Sulfate ~ water_potability $ Potability , xlab = " Potability " , ylab = "Sulfate " , main = " Sulfate and Potability " , col = c ( " grey " ," lightblue " ) )
boxplot ( water_potability $ Conductivity ~ water_potability $ Potability , xlab = " Potability " , ylab= " Conductivity " , main = " Conductivity and Potability " , col = c ( " grey " ," lightblue " ) )
boxplot ( water_potability $ Organic_carbon ~ water_potability $ Potability , xlab = " Potability " ,ylab = " Organic_carbon " , main = " Organic_carbon and Potability " , col = c ( " grey " ," lightblue " ))
boxplot ( water_potability $ Trihalomethanes ~ water_potability $ Potability , xlab = " Potability " ,ylab = " T ri h al om e th an e s " , main = " Tr i ha lo m et h an es and Potability " , col = c ( " grey " ," lightblue ") )
boxplot ( water_potability $ Turbidity ~ water_potability $ Potability , xlab = " Potability " , ylab = "Turbidity " , main = " Turbidity and Potability " , col = c ( " grey " ," lightblue " ) )
round(cor(water_potability[,1:9]),3)
## ph Hardness Solids Chloramines Sulfate Conductivity
## ph 1.000 0.079 -0.082 -0.032 0.015 0.017
## Hardness 0.079 1.000 -0.047 -0.030 -0.093 -0.024
## Solids -0.082 -0.047 1.000 -0.070 -0.150 0.014
## Chloramines -0.032 -0.030 -0.070 1.000 0.023 -0.020
## Sulfate 0.015 -0.093 -0.150 0.023 1.000 -0.014
## Conductivity 0.017 -0.024 0.014 -0.020 -0.014 1.000
## Organic_carbon 0.040 0.004 0.010 -0.013 0.027 0.021
## Trihalomethanes 0.004 -0.013 -0.009 0.017 -0.026 0.001
## Turbidity -0.036 -0.014 0.020 0.002 -0.010 0.006
## Organic_carbon Trihalomethanes Turbidity
## ph 0.040 0.004 -0.036
## Hardness 0.004 -0.013 -0.014
## Solids 0.010 -0.009 0.020
## Chloramines -0.013 0.017 0.002
## Sulfate 0.027 -0.026 -0.010
## Conductivity 0.021 0.001 0.006
## Organic_carbon 1.000 -0.013 -0.027
## Trihalomethanes -0.013 1.000 -0.022
## Turbidity -0.027 -0.022 1.000
library(corrplot)
## corrplot 0.92 loaded
M=cor(water_potability[,1:9])
corrplot(M, method="circle")
#V. Thống kế suy diễn
library(caTools)
set.seed(8)
split = sample.split(water_potability$Potability, SplitRatio = 0.7)
train_data = subset(water_potability, split == TRUE)
test_data = subset(water_potability, split == FALSE)
dim(train_data)
## [1] 2294 10
dim(test_data)
## [1] 982 10
model = glm(Potability~., family="binomial"(link='logit'), data=train_data)
research = step(model)
## Start: AIC=3070.69
## Potability ~ ph + Hardness + Solids + Chloramines + Sulfate +
## Conductivity + Organic_carbon + Trihalomethanes + Turbidity
##
## Df Deviance AIC
## - ph 1 3050.7 3068.7
## - Trihalomethanes 1 3050.7 3068.7
## - Conductivity 1 3050.8 3068.8
## - Sulfate 1 3050.9 3068.9
## - Turbidity 1 3051.4 3069.4
## - Hardness 1 3052.0 3070.0
## <none> 3050.7 3070.7
## - Chloramines 1 3055.4 3073.4
## - Solids 1 3055.7 3073.7
## - Organic_carbon 1 3056.2 3074.2
##
## Step: AIC=3068.69
## Potability ~ Hardness + Solids + Chloramines + Sulfate + Conductivity +
## Organic_carbon + Trihalomethanes + Turbidity
##
## Df Deviance AIC
## - Trihalomethanes 1 3050.7 3066.7
## - Conductivity 1 3050.8 3066.8
## - Sulfate 1 3050.9 3066.9
## - Turbidity 1 3051.4 3067.4
## - Hardness 1 3052.0 3068.0
## <none> 3050.7 3068.7
## - Chloramines 1 3055.4 3071.4
## - Solids 1 3055.7 3071.7
## - Organic_carbon 1 3056.2 3072.2
##
## Step: AIC=3066.73
## Potability ~ Hardness + Solids + Chloramines + Sulfate + Conductivity +
## Organic_carbon + Turbidity
##
## Df Deviance AIC
## - Conductivity 1 3050.8 3064.8
## - Sulfate 1 3050.9 3064.9
## - Turbidity 1 3051.4 3065.4
## - Hardness 1 3052.0 3066.0
## <none> 3050.7 3066.7
## - Chloramines 1 3055.4 3069.4
## - Solids 1 3055.7 3069.7
## - Organic_carbon 1 3056.2 3070.2
##
## Step: AIC=3064.84
## Potability ~ Hardness + Solids + Chloramines + Sulfate + Organic_carbon +
## Turbidity
##
## Df Deviance AIC
## - Sulfate 1 3051.0 3063.0
## - Turbidity 1 3051.6 3063.6
## - Hardness 1 3052.1 3064.1
## <none> 3050.8 3064.8
## - Chloramines 1 3055.6 3067.6
## - Solids 1 3055.8 3067.8
## - Organic_carbon 1 3056.4 3068.4
##
## Step: AIC=3063.03
## Potability ~ Hardness + Solids + Chloramines + Organic_carbon +
## Turbidity
##
## Df Deviance AIC
## - Turbidity 1 3051.8 3061.8
## - Hardness 1 3052.2 3062.2
## <none> 3051.0 3063.0
## - Chloramines 1 3055.7 3065.7
## - Solids 1 3056.5 3066.5
## - Organic_carbon 1 3056.7 3066.7
##
## Step: AIC=3061.75
## Potability ~ Hardness + Solids + Chloramines + Organic_carbon
##
## Df Deviance AIC
## - Hardness 1 3052.9 3060.9
## <none> 3051.8 3061.8
## - Chloramines 1 3056.4 3064.4
## - Solids 1 3057.2 3065.2
## - Organic_carbon 1 3057.5 3065.5
##
## Step: AIC=3060.93
## Potability ~ Solids + Chloramines + Organic_carbon
##
## Df Deviance AIC
## <none> 3052.9 3060.9
## - Chloramines 1 3057.7 3063.7
## - Solids 1 3058.7 3064.7
## - Organic_carbon 1 3058.9 3064.9
summary(research)
##
## Call:
## glm(formula = Potability ~ Solids + Chloramines + Organic_carbon,
## family = binomial(link = "logit"), data = train_data)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -6.831e-01 2.974e-01 -2.297 0.0216 *
## Solids 1.174e-05 4.907e-06 2.393 0.0167 *
## Chloramines 5.946e-02 2.717e-02 2.189 0.0286 *
## Organic_carbon -3.153e-02 1.297e-02 -2.431 0.0151 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 3068.5 on 2293 degrees of freedom
## Residual deviance: 3052.9 on 2290 degrees of freedom
## AIC: 3060.9
##
## Number of Fisher Scoring iterations: 4
confint(research)
## Waiting for profiling to be done...
## 2.5 % 97.5 %
## (Intercept) -1.267567e+00 -1.012542e-01
## Solids 2.124268e-06 2.137074e-05
## Chloramines 6.308056e-03 1.128470e-01
## Organic_carbon -5.702291e-02 -6.155123e-03
exp(cbind(OR = coef(research), confint(research)))
## Waiting for profiling to be done...
## OR 2.5 % 97.5 %
## (Intercept) 0.5050704 0.2815156 0.9037032
## Solids 1.0000117 1.0000021 1.0000214
## Chloramines 1.0612651 1.0063280 1.1194606
## Organic_carbon 0.9689596 0.9445724 0.9938638
predicted <- predict(research,test_data, type="response")
test_data$predicted<-round(predicted)
head(test_data,10)
## ph Hardness Solids Chloramines Sulfate Conductivity Organic_carbon
## 3 8.099124 224.2363 19909.54 9.275884 334.5643 418.6062 16.868637
## 6 5.584087 188.3133 28748.69 7.544869 326.6784 280.4679 8.399735
## 8 8.635849 203.3615 13672.09 4.563009 303.3098 474.6076 12.363817
## 9 6.721973 118.9886 14285.58 7.804174 268.6469 389.3756 12.706049
## 16 6.347272 186.7329 41065.23 9.629596 364.4877 516.7433 11.539781
## 26 6.514415 198.7674 21218.70 8.670937 323.5963 413.2905 14.900000
## 30 6.721973 148.1531 15193.41 9.046833 307.0118 563.8047 16.568656
## 33 10.433291 117.7912 22326.89 8.161505 307.7075 412.9868 12.890709
## 37 3.641630 183.9087 24752.07 5.538314 286.0596 456.8601 9.034067
## 38 5.618064 304.2359 17281.98 6.101084 334.5643 399.4716 12.265002
## Trihalomethanes Turbidity Potability predicted
## 3 66.42009 3.055934 0 0
## 6 54.91786 2.559708 0 0
## 8 62.79831 4.401425 0 0
## 9 53.92885 3.595017 0 0
## 16 75.07162 4.376348 0 1
## 26 79.84784 5.200885 0 0
## 30 52.67619 6.038185 0 0
## 33 65.73348 5.057311 0 0
## 37 73.59466 3.464353 0 0
## 38 81.58899 2.896547 0 0
Observation<-table(test_data$Potability)
Predicted_value<-table(test_data$predicted)
rbind(Observation,Predicted_value)
## 0 1
## Observation 599 383
## Predicted_value 974 8
table(test_data$Potability, test_data$predicted)
##
## 0 1
## 0 594 5
## 1 380 3
predictions = predict(research, newdata=test_data, type='response')
predicted_classes = ifelse(predicted > 0.5, 1, 0)
accuracy = sum(predicted_classes == test_data$Potability) / nrow(test_data)
print(paste('Accuracy:', accuracy))
## [1] "Accuracy: 0.607942973523422"
library ( ROCR )
ROCRpred=prediction(predictions, test_data$Potability)
ROCRperf=performance( ROCRpred , "tpr", "fpr")
plot(ROCRperf)
as.numeric(performance(ROCRpred, "auc")@y.values)
## [1] 0.4854392
qqnorm(water_potability$ph)
qqline(water_potability$ph)
qqnorm(water_potability$Organic_carbon)
qqline(water_potability$Organic_carbon)
water_potability$ph_category <- cut(water_potability$ph, breaks = c(0, 6.5, 8.5, 14), labels = c("Acidic", "Neutral", "Alkaline"))
water_potability$organic_carbon_category <- cut(water_potability$Organic_carbon, breaks = c(0, 10, 20, 30, Inf), labels = c("Low", "Moderate", "High", "Very High"))
head(water_potability)
## ph Hardness Solids Chloramines Sulfate Conductivity Organic_carbon
## 1 7.112512 204.8905 20791.32 7.300212 368.5164 564.3087 10.379783
## 2 3.716080 129.4229 18630.06 6.635246 334.5643 592.8854 15.180013
## 3 8.099124 224.2363 19909.54 9.275884 334.5643 418.6062 16.868637
## 4 8.316766 214.3734 22018.42 8.059332 356.8861 363.2665 18.436524
## 5 9.092223 181.1015 17978.99 6.546600 310.1357 398.4108 11.558279
## 6 5.584087 188.3133 28748.69 7.544869 326.6784 280.4679 8.399735
## Trihalomethanes Turbidity Potability ph_category organic_carbon_category
## 1 86.99097 2.963135 0 Neutral Moderate
## 2 56.32908 4.500656 0 Acidic Moderate
## 3 66.42009 3.055934 0 Neutral Moderate
## 4 100.34167 4.628771 0 Neutral Moderate
## 5 31.99799 4.075075 0 Alkaline Moderate
## 6 54.91786 2.559708 0 Acidic Low
library(car)
## Loading required package: carData
water_potability$Potability <- as.numeric(water_potability$Potability)
str(water_potability$Potability)
## num [1:3276] 1 1 1 1 1 1 1 1 1 1 ...
leveneTest(Potability ~ ph_category * organic_carbon_category , data = water_potability)
## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 8 2.0831 0.03411 *
## 3266
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova<-aov(Potability~ph_category*organic_carbon_category, data=water_potability)
summary(anova)
## Df Sum Sq Mean Sq F value Pr(>F)
## ph_category 2 3.6 1.7771 7.486 0.000571 ***
## organic_carbon_category 2 0.1 0.0258 0.109 0.897022
## ph_category:organic_carbon_category 4 0.4 0.0876 0.369 0.830876
## Residuals 3266 775.3 0.2374
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 1 observation deleted due to missingness
#VI. Thảo luận và mở rộng
library(rpart)
library(rpart.plot)
fit=rpart(Potability~., data=train_data, method="class")
rpart.plot(fit, extra="auto")
test_data$predicted_2=predict(fit, test_data, type="class")
test_data$predicted_2=as.numeric(as.character(test_data$predicted_2))
head(test_data, 10)
## ph Hardness Solids Chloramines Sulfate Conductivity Organic_carbon
## 3 8.099124 224.2363 19909.54 9.275884 334.5643 418.6062 16.868637
## 6 5.584087 188.3133 28748.69 7.544869 326.6784 280.4679 8.399735
## 8 8.635849 203.3615 13672.09 4.563009 303.3098 474.6076 12.363817
## 9 6.721973 118.9886 14285.58 7.804174 268.6469 389.3756 12.706049
## 16 6.347272 186.7329 41065.23 9.629596 364.4877 516.7433 11.539781
## 26 6.514415 198.7674 21218.70 8.670937 323.5963 413.2905 14.900000
## 30 6.721973 148.1531 15193.41 9.046833 307.0118 563.8047 16.568656
## 33 10.433291 117.7912 22326.89 8.161505 307.7075 412.9868 12.890709
## 37 3.641630 183.9087 24752.07 5.538314 286.0596 456.8601 9.034067
## 38 5.618064 304.2359 17281.98 6.101084 334.5643 399.4716 12.265002
## Trihalomethanes Turbidity Potability predicted predicted_2
## 3 66.42009 3.055934 0 0 0
## 6 54.91786 2.559708 0 0 0
## 8 62.79831 4.401425 0 0 0
## 9 53.92885 3.595017 0 0 0
## 16 75.07162 4.376348 0 1 0
## 26 79.84784 5.200885 0 0 0
## 30 52.67619 6.038185 0 0 0
## 33 65.73348 5.057311 0 0 1
## 37 73.59466 3.464353 0 0 0
## 38 81.58899 2.896547 0 0 0
Predicted_value_2<-table(test_data$predicted_2)
rbind(Observation,Predicted_value_2)
## 0 1
## Observation 599 383
## Predicted_value_2 690 292
table(test_data$Potability, test_data$predicted_2)
##
## 0 1
## 0 537 62
## 1 153 230
predictions_2 = predict(fit, newdata=test_data, type='class')
accuracy = sum(predictions_2 == test_data$Potability) / nrow(test_data)
print(paste('Accuracy:', accuracy))
## [1] "Accuracy: 0.781059063136456"
ROCRpred2 = prediction ( test_data$predicted_2, test_data$Potability )
ROCRperf2 = performance ( ROCRpred2 , "tpr", "fpr")
plot(ROCRperf2)
as.numeric(performance(ROCRpred2,"auc")@y.values )
## [1] 0.7485082