Homework 2

Overview

In this homework assignment, you will work through various classification metrics. You will be asked to create functions in R to carry out the various calculations. You will also investigate some functions in packages that will let you obtain the equivalent results. Finally, you will create graphical output that also can be used to evaluate the output of classification models, such as binary logistic regression.

Download the classification output data set

file <- "https://raw.githubusercontent.com/saayedalam/Data/master/classification-output-data.csv"
data <- read.csv(file)
head(data, 5)

##   pregnant glucose diastolic skinfold insulin  bmi pedigree age class
## 1        7     124        70       33     215 25.5    0.161  37     0
## 2        2     122        76       27     200 35.9    0.483  26     0
## 3        3     107        62       13      48 22.9    0.678  23     1
## 4        1      91        64       24       0 29.2    0.192  21     0
## 5        4      83        86       19       0 29.3    0.317  34     0
##   scored.class scored.probability
## 1            0         0.32845226
## 2            0         0.27319044
## 3            0         0.10966039
## 4            0         0.05599835
## 5            0         0.10049072

The data set has three key columns we will use:

class: the actual class for the observation
scored.class: the predicted class for the observation (based on a threshold of 0.5)
scored.probability: the predicted probability of success for the observation
Use the table() function to get the raw confusion matrix for this scored dataset. Make sure you understand the output. In particular, do the rows represent the actual or predicted class? The columns?

#The rows represent the predicted class and the columns are actual class. 
data %>%
  select(scored.class, class) %>%
  table()

##             class
## scored.class   0   1
##            0 119  30
##            1   5  27

Write a function that takes the data set as a dataframe, with actual and predicted classifications identified, and returns the accuracy of the predictions.
\[Accuracy=\frac{TP+TN}{TP+FP+TN+FN}\]

accurary_predictions <- function(x){
  TP <- sum(x$class == 1 & x$scored.class == 1)
  TN <- sum(x$class == 0 & x$scored.class == 0)
  round((TP + TN)/nrow(x), 4)
}
accurary_predictions(data)

## [1] 0.8066

Write a function that takes the data set as a dataframe, with actual and predicted classifications identified, and returns the classification error rate of the predictions. \[Classification\hspace{.1cm}Error\hspace{.1cm}Rate=\frac{FP+FN}{TP+FP+TN+FN}\]

class_error_rate <- function(x){
  FP <- sum(x$class == 0 & x$scored.class == 1)
  FN <- sum(x$class == 1 & x$scored.class == 0)
  round((FP + FN)/nrow(x), 4)
}
class_error_rate(data)

## [1] 0.1934

#Verify that you get an accuracy and an error rate that sums to one
accurary_predictions(data) + class_error_rate(data)

## [1] 1

Write a function that takes the data set as a dataframe, with actual and predicted classifications identified, and returns the precision of the predictions.
\[Precision=\frac{TP}{TP+FP}\]

precision <- function(x){
  TP <- sum(x$class == 1 & x$scored.class == 1)
  FP <- sum(x$class == 0 & x$scored.class == 1)
  round(TP/(TP + FP), 4)
}
precision(data)

## [1] 0.8438

Write a function that takes the data set as a dataframe, with actual and predicted classifications identified, and returns the sensitivity of the predictions. Sensitivity is also known as recall. \[Sensitivity=\frac{TP}{TP+FN}\]

sensitivity <- function(x){
  TP <- sum(x$class == 1 & x$scored.class == 1)
  FN <- sum(x$class == 1 & x$scored.class == 0)
  round(TP/(TP + FN), 4)
}
sensitivity(data)

## [1] 0.4737

Write a function that takes the data set as a dataframe, with actual and predicted classifications identified, and returns the specificity of the predictions. \[Specificity=\frac{TN}{TN+FP} \]

specificity <- function(x){
  TN <- sum(x$class == 0 & x$scored.class == 0)
  FP <- sum(x$class == 0 & x$scored.class == 1)
  round(TN/(TN + FP), 4)
}
specificity(data)

## [1] 0.9597

Write a function that takes the data set as a dataframe, with actual and predicted classifications identified, and returns the F1 score of the predictions.
\[F1\hspace{.1cm}Score=\frac{2\times Precision\times Sensitivity}{Precision+Sensitivity} \]

f1_score <- function(x){
  (2*precision(x)*sensitivity(x))/(precision(x)+sensitivity(x))
}
f1_score(data)

## [1] 0.6067675

Before we move on, let’s consider a question that was asked: What are the bounds on the F1 score? Show that the F1 score will always be between 0 and 1.
Both Precision and Sensitivity used to calculate F1 score are bounded between 0 and 1. Therefore, F1 score will be between 0 and 1.
Write a function that generates an ROC curve from a data set with a true classification column (class in our example) and a probability column (scored.probability in our example). Your function should return a list that includes the plot of the ROC curve and a vector that contains the calculated area under the curve (AUC). Note that I recommend using a sequence of thresholds ranging from 0 to 1 at 0.01 intervals.

ROC <- function(x, y){
  x <- x[order(y, decreasing = TRUE)]
  TPR <- cumsum(x) / sum(x)
  FPR <- cumsum(!x) / sum(!x)
  xy <- data.frame(TPR, FPR, x)
  
  FPR_df <- c(diff(xy$FPR), 0)
  TPR_df <- c(diff(xy$TPR), 0)
  AUC <- round(sum(xy$TPR * FPR_df) + sum(TPR_df * FPR_df)/2, 4)
  
  plot(xy$FPR, xy$TPR, type = "l",
       main = "ROC Curve",
       xlab = "False Postivie Rate",
       ylab = "True Positive Rate")
  abline(a = 0, b = 1)
  legend(.6, .4, AUC, title = "AUC")
}

ROC(data$class,data$scored.probability)

Use your created R functions and the provided classification output data set to produce all of the classification metrics discussed above.

metrics <- c(accurary_predictions(data), class_error_rate(data), precision(data), sensitivity(data), specificity(data), f1_score(data))
names(metrics) <- c("Accuracy", "Classification Error Rate", "Precision", "Sensitivity", "Specificity", "F1 Score")
kable(metrics, col.names = "Metrics")

	Metrics
Accuracy	0.8066000
Classification Error Rate	0.1934000
Precision	0.8438000
Sensitivity	0.4737000
Specificity	0.9597000
F1 Score	0.6067675

Investigate the caret package. In particular, consider the functions confusionMatrix, sensitivity, and specificity. Apply the functions to the data set. How do the results compare with your own functions?

b <- data %>%
  select(scored.class, class) %>%
  mutate(scored.class = as.factor(scored.class), 
         class = as.factor(class))

c <- confusionMatrix(b$scored.class, b$class, positive = "1")

caret_package <- c(c$overall["Accuracy"], c$byClass["Sensitivity"], c$byClass["Specificity"])
written_function <- c(accurary_predictions(data), sensitivity(data), specificity(data))
d <- cbind(caret_package, written_function)
kable(d)

	caret_package	written_function
Accuracy	0.8066298	0.8066
Sensitivity	0.4736842	0.4737
Specificity	0.9596774	0.9597

Investigate the pROC package. Use it to generate an ROC curve for the data set. How do the results compare with your own functions?

#The results are exactly the same
par(mfrow = c(1, 2))
plot(roc(data$class, data$scored.probability), print.auc = TRUE)
ROC(data$class,data$scored.probability)

Homework 2

Saayed Alam

March 14, 2019

Overview