Binary Overlap Coefficient Calculation

The Binary Overlap Coefficient, also known as the Sørensen-Dice similarity index, is a fundamental metric used to quantify the degree of agreement between two binary classification results. This coefficient is particularly prevalent in computer vision, medical imaging, and any domain where measuring the spatial or categorical overlap between predictions and ground truth is essential.

Mathematical Foundation

Given two binary arrays—one representing the actual (ground truth) labels and another representing the predicted labels—the Binary Overlap Coefficient measures the harmonic mean of their precision and recall characteristics. The formula for computing this coefficient is:

$$\text{Overlap Coefficient} = \frac{2 \times |A \cap P|}{|A| + |P|}$$

Where:

• |A ∩ P| represents the count of true positive instances where both actual and predicted values are 1 (the intersection)
• |A| represents the total count of positive instances in the actual labels
• |P| represents the total count of positive instances in the predicted labels

The coefficient produces a value in the range [0, 1]:

• A value of 0 indicates no overlap whatsoever between predictions and actual labels
• A value of 1 indicates perfect agreement, where all positive predictions match all actual positive instances

Key Characteristics

This metric is particularly attractive because:

1. It equally weights precision and recall contributions
2. It is symmetric—swapping actual and predicted arrays yields the same result
3. It handles class imbalance more gracefully than simple accuracy metrics
4. It focuses exclusively on the positive class, making it ideal for tasks where the positive class is of primary interest

Edge Case Considerations

When both arrays contain no positive instances (all zeros), the coefficient is typically defined as 0.0 to handle the mathematical edge case of division by zero in a semantically meaningful way.

Your Task

Implement a function that computes the Binary Overlap Coefficient between two binary label arrays. The function should:

• Accept two binary arrays of equal length representing actual and predicted labels
• Calculate and return the overlap coefficient as a float rounded to 3 decimal places
• Handle edge cases appropriately, returning 0.0 when there are no positive labels in either or both arrays