Weighted Harmonic Mean of Precision and Recall

Easy

In binary classification tasks, evaluating model performance requires balancing two often competing metrics: Precision (how accurate are the positive predictions?) and Recall (how many actual positives were captured?). The F-beta score provides an elegant solution by computing the weighted harmonic mean of these two metrics. Understanding Precision and Recall Before diving into the F-beta score, let's establish the foundational metrics: Precision measures the proportion of positive predictions that are actually correct: $$\text{Precision} = \frac{\text{True Positives}}{\text{True Positives} + \text{False Positives}}$$ Recall (also called Sensitivity or True Positive Rate) measures the proportion of actual positives that were correctly identified: $$\text{Recall} = \frac{\text{True Positives}}{\text{True Positives} + \text{False Negatives}}$$ The Harmonic Mean Approach A simple arithmetic mean of Precision and Recall can be misleading because it doesn't properly penalize extremely low values in either metric. The harmonic mean addresses this by heavily penalizing cases where either Precision or Recall is very low—ensuring that a model cannot achieve a high score by excelling in only one dimension. The F-beta Score Formula The F-beta score extends this concept by introducing a weighting parameter β (beta): $$F_\beta = (1 + \beta^2) \cdot \frac{\text{Precision} \cdot \text{Recall}}{(\beta^2 \cdot \text{Precision}) + \text{Recall}}$$ Interpretation of β: β = 1: Equal weight to Precision and Recall (the classic F1-score) β > 1: More emphasis on Recall (useful when missing positive cases is costly, e.g., disease detection) β < 1: More emphasis on Precision (useful when false positives are expensive, e.g., spam filtering) Common variants include F2 (β=2, recall-focused) and F0.5 (β=0.5, precision-focused). Your Task Implement a function that computes the F-beta score for binary classification. Given arrays of actual labels and predicted labels, along with a beta weighting parameter, calculate and return the F-beta score rounded to three decimal places. Edge Case Handling: If both Precision and Recall are zero (no true positives), the F-beta score should be 0.0 Handle the case where either the denominator in Precision or Recall calculations is zero

Example 1

Input

actual_labels = [1, 0, 1, 1, 0, 1]
predicted_labels = [1, 0, 1, 0, 0, 1]
beta_weight = 1

Output

0.857

Explanation

Let's break down the calculation step by step: **Step 1: Build the Confusion Matrix** • True Positives (TP) = 3 (positions 0, 2, 5 where both actual and predicted are 1) • True Negatives (TN) = 2 (positions 1, 4 where both are 0) • False Positives (FP) = 0 (no cases where actual is 0 but predicted is 1) • False Negatives (FN) = 1 (position 3 where actual is 1 but predicted is 0) **Step 2: Calculate Precision and Recall** • Precision = TP / (TP + FP) = 3 / (3 + 0) = 1.0 • Recall = TP / (TP + FN) = 3 / (3 + 1) = 0.75 **Step 3: Apply F-beta Formula with β = 1** • F1 = (1 + 1²) × (1.0 × 0.75) / ((1² × 1.0) + 0.75) • F1 = 2 × 0.75 / 1.75 = 1.5 / 1.75 ≈ 0.857

Example 2

Input

actual_labels = [1, 1, 1, 0, 0, 0]
predicted_labels = [1, 1, 0, 0, 0, 1]
beta_weight = 1

Output

0.667

Explanation

**Step 1: Build the Confusion Matrix** • True Positives (TP) = 2 (positions 0, 1 where both are 1) • True Negatives (TN) = 2 (positions 3, 4 where both are 0) • False Positives (FP) = 1 (position 5 where actual is 0 but predicted is 1) • False Negatives (FN) = 1 (position 2 where actual is 1 but predicted is 0) **Step 2: Calculate Precision and Recall** • Precision = TP / (TP + FP) = 2 / (2 + 1) = 0.667 • Recall = TP / (TP + FN) = 2 / (2 + 1) = 0.667 **Step 3: Apply F-beta Formula with β = 1** • F1 = 2 × (0.667 × 0.667) / (0.667 + 0.667) • F1 = 2 × 0.444 / 1.334 ≈ 0.667 When Precision equals Recall, the F1-score equals both.

Example 3

Input

actual_labels = [1, 1, 1, 1, 0, 0]
predicted_labels = [1, 0, 1, 1, 0, 1]
beta_weight = 2

Output

0.75

Explanation

**Step 1: Build the Confusion Matrix** • True Positives (TP) = 3 (positions 0, 2, 3 where both are 1) • True Negatives (TN) = 1 (position 4 where both are 0) • False Positives (FP) = 1 (position 5 where actual is 0 but predicted is 1) • False Negatives (FN) = 1 (position 1 where actual is 1 but predicted is 0) **Step 2: Calculate Precision and Recall** • Precision = TP / (TP + FP) = 3 / (3 + 1) = 0.75 • Recall = TP / (TP + FN) = 3 / (3 + 1) = 0.75 **Step 3: Apply F-beta Formula with β = 2** • F2 = (1 + 2²) × (0.75 × 0.75) / ((2² × 0.75) + 0.75) • F2 = 5 × 0.5625 / (3 + 0.75) = 2.8125 / 3.75 = 0.75 With β = 2, Recall is weighted more heavily, but since Precision and Recall are equal here, the result matches both values.

Constraints

1 ≤ length of actual_labels, predicted_labels ≤ 10,000
Both arrays have the same length
All values in actual_labels and predicted_labels are 0 or 1
0.1 ≤ beta_weight ≤ 10
At least one element exists in each array

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Weighted Harmonic Mean of Precision and Recall

Easy

In binary classification tasks, evaluating model performance requires balancing two often competing metrics: Precision (how accurate are the positive predictions?) and Recall (how many actual positives were captured?). The F-beta score provides an elegant solution by computing the weighted harmonic mean of these two metrics. Understanding Precision and Recall Before diving into the F-beta score, let's establish the foundational metrics: Precision measures the proportion of positive predictions that are actually correct: $$\text{Precision} = \frac{\text{True Positives}}{\text{True Positives} + \text{False Positives}}$$ Recall (also called Sensitivity or True Positive Rate) measures the proportion of actual positives that were correctly identified: $$\text{Recall} = \frac{\text{True Positives}}{\text{True Positives} + \text{False Negatives}}$$ The Harmonic Mean Approach A simple arithmetic mean of Precision and Recall can be misleading because it doesn't properly penalize extremely low values in either metric. The harmonic mean addresses this by heavily penalizing cases where either Precision or Recall is very low—ensuring that a model cannot achieve a high score by excelling in only one dimension. The F-beta Score Formula The F-beta score extends this concept by introducing a weighting parameter β (beta): $$F_\beta = (1 + \beta^2) \cdot \frac{\text{Precision} \cdot \text{Recall}}{(\beta^2 \cdot \text{Precision}) + \text{Recall}}$$ Interpretation of β: β = 1: Equal weight to Precision and Recall (the classic F1-score) β > 1: More emphasis on Recall (useful when missing positive cases is costly, e.g., disease detection) β < 1: More emphasis on Precision (useful when false positives are expensive, e.g., spam filtering) Common variants include F2 (β=2, recall-focused) and F0.5 (β=0.5, precision-focused). Your Task Implement a function that computes the F-beta score for binary classification. Given arrays of actual labels and predicted labels, along with a beta weighting parameter, calculate and return the F-beta score rounded to three decimal places. Edge Case Handling: If both Precision and Recall are zero (no true positives), the F-beta score should be 0.0 Handle the case where either the denominator in Precision or Recall calculations is zero

Example 1

Input

actual_labels = [1, 0, 1, 1, 0, 1]
predicted_labels = [1, 0, 1, 0, 0, 1]
beta_weight = 1

Output

0.857

Explanation

Example 2

Input

actual_labels = [1, 1, 1, 0, 0, 0]
predicted_labels = [1, 1, 0, 0, 0, 1]
beta_weight = 1

Output

0.667

Explanation

Example 3

Input

actual_labels = [1, 1, 1, 1, 0, 0]
predicted_labels = [1, 0, 1, 1, 0, 1]
beta_weight = 2

Output

0.75

Explanation

Constraints

1 ≤ length of actual_labels, predicted_labels ≤ 10,000
Both arrays have the same length
All values in actual_labels and predicted_labels are 0 or 1
0.1 ≤ beta_weight ≤ 10
At least one element exists in each array