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GO Classes DPP | ML |Classification

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A bank wants to predict whether a customer will default on a loan using a classification model. The bank collects data from past customers, which includes the following features:

  • $x_1$: Credit score (scaled between $0$ and $1$)
     
  • $x_2$: Monthly income (in $\$ 1000 \mathrm{~s}$)
     
  • $x_3$: Number of late payments in the past year

The target variable $y$ represents whether the customer defaulted ( 1 for default, 0 for no default). The data is provided in the table below:

\begin{array}{|c|c|} \hline \mathbf{x} & \mathbf{y} \\ \hline [0.9,5,1] & 0 \\ \hline [0.4,3,5] & 1 \\ \hline [0.8,4,2] & 0 \\ \hline [0.2,2,7] & 1 \\ \hline [0.7,6,0] & 0 \\ \hline \end{array}

The bank uses the following linear combination to compute $z$:

$$
z=0.5 x_1+0.3 x_2-0.2 x_3
$$


The step function $u(z)$ is used to classify the outcome:

$$
u(z)= \begin{cases}1 & \text { if } z \geq 1.5 \\ 0 & \text { otherwise }\end{cases}
$$


If the threshold in the step function is changed to $z \geq 1.0$, what will happen to the misclassification rate?

  1. Increase
     
  2. Decrease
     
  3. Remain the same.
     
  4. Cannot be determined

2 Answers

1 1 vote

To determine what will happen to the misclassification rate, we need to calculate it for both the original and the new threshold.

The formula to calculate the score $z$ is: $z=0.5 x_1+0.3 x_2-0.2 x_3$
First, let's calculate the $z$ score for each customer in the dataset.

  • Customer 1: $[0.9,5,1]$

$$
z=0.5(0.9)+0.3(5)-0.2(1)=0.45+1.5-0.2=1.75
$$

  • Customer 2: $[0.4,3,5]$

$$
z=0.5(0.4)+0.3(3)-0.2(5)=0.2+0.9-1.0=0.1
$$

  • Customer 3: $[0.8,4,2]$

$$
z=0.5(0.8)+0.3(4)-0.2(2)=0.4+1.2-0.4=1.2
$$

  • Customer 4: $[0.2,2,7]$

$$
z=0.5(0.2)+0.3(2)-0.2(7)=0.1+0.6-1.4=-0.7
$$

  • Customer 5: $[0.7,6,0]$

$$
z=0.5(0.7)+0.3(6)-0.2(0)=0.35+1.8-0=2.15
$$


Step 1: Calculate the Original Misclassification Rate (Threshold $\geq 1.5$ )
Let's find the model's prediction for each customer using the original rule (Predict 1 if $z \geq 1.5$, otherwise 0 ) and compare it to the actual outcome $y$.

\[
\begin{array}{|c|c|c|c|c|}
\hline \textbf{Customer} & \textbf{z-score} & \textbf{Actual (y)} & \textbf{Prediction } (z \geq 1.5) & \textbf{Correct?} \\
\hline 1 & 1.75 & 0 & 1 & No \\
\hline 2 & 0.1 & 1 & 0 & No \\
\hline 3 & 1.2 & 0 & 0 & Yes \\
\hline 4 & -0.7 & 1 & 0 & No \\
\hline 5 & 2.15 & 0 & 1 & No \\
\hline
\end{array}
\]

With the original threshold, there are $\mathbf{4}$ misclassifications out of $\mathbf{5}$ data points.
The misclassification rate is $4 / 5 = 80\%$.
 

Step 2: Calculate the New Misclassification Rate (Threshold $\geq 1.0$ )
Now, let's use the new rule (Predict 1 if $z \geq 1.0$, otherwise 0 ) and find the new number of misclassifications.

\[
\begin{array}{|c|c|c|c|c|}
\hline \textbf{Customer} & \textbf{z-score} & \textbf{Actual (y)} & \textbf{Prediction } (z \geq 1.0) & \textbf{Correct?} \\
\hline 1 & 1.75 & 0 & 1 & No \\
\hline 2 & 0.1 & 1 & 0 & No \\
\hline 3 & 1.2 & 0 & 1 & No \\
\hline 4 & -0.7 & 1 & 0 & No \\
\hline 5 & 2.15 & 0 & 1 & No \\
\hline
\end{array}
\]

With the new threshold, there are $\mathbf{5}$ misclassifications out of 5 data points. The new misclassification rate is $5 / 5=100\%$.

The original misclassification rate was $80 \%$, and the new rate is $100\%$. Therefore, the misclassification rate will increase.

The correct option is A. Increase.

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