Computing the F1 Score

Solution & Explanation

What is the F1 Score?

The F1 score is a widely-used metric in machine learning and data science for evaluating the performance of classification models, particularly in binary classification problems. It serves as a balanced measure that considers both precision and recall, offering insights into the model's accuracy in classifying positive cases. The F1 score ranges from 0 to 1, with a value closer to 1 indicating better model performance.

Key Metrics:

• Precision: Measures the accuracy of positive predictions. It is the ratio of true positive predictions to the total number of positive predictions made by the model.

  $\text{Precision} = \frac{\text{True Positives (TP)}}{\text{True Positives (TP)} + \text{False Positives (FP)}}$

• Recall: Also known as the true positive rate or sensitivity, it measures the ability of the model to identify all relevant instances. It is the ratio of true positive predictions to the actual number of positive instances.

  $\text{Recall} = \frac{\text{True Positives (TP)}}{\text{True Positives (TP)} + \text{False Negatives (FN)}}$

Calculating the F1 Score:

The F1 score is calculated as the harmonic mean of precision and recall. The harmonic mean is particularly useful in scenarios where you want to find a balance between two rates, such as precision and recall.

• Formula:

  $F1 = 2 \times \frac{\text{Precision} \times \text{Recall}}{\text{Precision} + \text{Recall}}$

• Simplified Version:

  $F1 = \frac{2 \times TP}{2 \times TP + FP + FN}$

Why Use the F1 Score?

• Imbalanced Datasets: In cases where the dataset is imbalanced (e.g., fraud detection, spam filtering), accuracy alone can be misleading. The F1 score provides a more balanced view by considering both false positives and false negatives.

• Trade-off Between Precision and Recall: Different applications may prioritize precision over recall or vice versa, depending on the cost of false positives and false negatives. The F1 score helps in finding a middle ground.

Example Calculation:

Suppose a model predicts 100 instances as positive, out of which 80 are true positives, and there are 20 false positives. The actual number of positive instances is 90.

• Precision: $\text{Precision} = \frac{80}{80 + 20} = 0.8$

• Recall: $\text{Recall} = \frac{80}{80 + 10} = 0.89$

• F1 Score: $F1 = 2 \times \frac{0.8 \times 0.89}{0.8 + 0.89} \approx 0.84$

In this example, the F1 score of 0.84 indicates a well-performing model with a good balance between precision and recall, suitable for the given classification task.