Confusion Matrix: Overview and Applications

Table of Contents

• 1. Introduction
• 2. What is a Confusion Matrix?
• 3. Components of the Confusion Matrix
• 4. Evaluation Metrics Derived from the Confusion Matrix
  • 4.1 Accuracy
  • 4.2 Precision
  • 4.3 Recall
  • 4.4 F1-Score
• 5. Confusion Matrix Example in Classification
• 6. References

1. Introduction

The confusion matrix is a tool used to evaluate the performance of classification models. It shows how well the model predicts each class correctly or incorrectly.

2. What is a Confusion Matrix?

A confusion matrix is a table that displays the number of correct and incorrect predictions made by a classification model compared to the actual labels.

3. Components of the Confusion Matrix

In a binary classification problem, the confusion matrix consists of four parts:

• True Positive (TP): Correctly predicted as Positive
• True Negative (TN): Correctly predicted as Negative
• False Positive (FP): Incorrectly predicted as Positive (actually Negative)
• False Negative (FN): Incorrectly predicted as Negative (actually Positive)

4. Evaluation Metrics Derived from the Confusion Matrix

4.1 Accuracy

The ratio of correct predictions over the total number of samples:
\(\text{Accuracy} = \frac{TP + TN}{TP + TN + FP + FN}\)

4.2 Precision

The proportion of correct Positive predictions:
\(\text{Precision} = \frac{TP}{TP + FP}\)

4.3 Recall

The proportion of actual Positives correctly predicted:
\(\text{Recall} = \frac{TP}{TP + FN}\)

4.4 F1-Score

The harmonic mean of Precision and Recall:
\(\text{F1} = 2 \times \frac{\text{Precision} \times \text{Recall}}{\text{Precision} + \text{Recall}}\)

5. Confusion Matrix Example in Classification

Suppose we have a binary classification task with the following results:

Sample Count	Ground Truth	Prediction
1000	Positive	Positive
200	Positive	Negative
1500	Negative	Negative
300	Negative	Positive

We get:

• TP = 1000
• TN = 1500
• FP = 300
• FN = 200

Evaluation metrics:

\[\text{Accuracy} = \frac{1000 + 1500}{1000 + 1500 + 300 + 200} = \frac{2500}{3000} = 0.8333\] \[\text{Precision} = \frac{1000}{1000 + 300} = \frac{1000}{1300} \approx 0.7692\] \[\text{Recall} = \frac{1000}{1000 + 200} = \frac{1000}{1200} \approx 0.8333\] \[\text{F1} = 2 \times \frac{0.7692 \times 0.8333}{0.7692 + 0.8333} \approx 0.8000\]

6. References

[1] Scikit-learn Developers. sklearn.metrics.confusion_matrix — scikit-learn 1.4.2 documentation. 🔗 https://scikit-learn.org/stable/modules/generated/sklearn.metrics.confusion_matrix.html