Feature Permutation-Based Metric#

Explainability Ease Score Metric#

The Explainability Ease Score (E) metric evaluates the simplicity of interpreting the partial dependence curve of a feature. This metric quantifies how easily these curves can be interpreted, facilitating better model explainability and transparency.

Methodology#

To compute the Explainability Ease Score, follow these steps:

  1. Compute the Partial Dependence Curve: - Calculate the partial dependence of the feature of interest. This curve shows the average effect of the feature on the predicted outcome.

  2. Compute the Second Derivative: - Compute the numerical second derivative of the partial dependence curve. The second derivative represents the rate of change of the slope of the curve. - If the curve is linear, the second derivative will be zero. Non-linear curves will have a non-zero second derivative.

  3. Normalize the Second Derivative: - Normalize the second derivative vector by computing the norm of the vector containing the absolute values of the second derivative at multiple points along the feature’s domain.

  4. Compare Tangents: - Divide the partial dependence curve into three sections and compute the average slope for each section. - Calculate the cosine similarity between the slopes of consecutive sections.

  5. Compute Feature Scores: - Based on the cosine similarity of the slopes, assign scores to each feature. A feature with more consistent slopes (higher cosine similarity) indicates easier interpretability.

  6. Calculate the Explainability Ease Score: - The Explainability Ease Score is computed as the weighted average of the feature scores, normalized to a range from 0 to 1. - A higher score (close to 1) indicates easier interpretability, while a lower score (close to 0) indicates more complex interpretability.

Mathematical Representation#

Let \(f_{x''}\) denote the second derivative of the partial dependence curve for feature \(x\), computed over \(q\) points in the domain of \(x\). The Explainability Ease Score is given by:

\[E_{x} := \| \vec{v_{x}} \|, \quad \text{where} \quad \vec{v_{x}} := \left\{ |f_{x''}(x_{1})|, |f_{x''}(x_{2})|, \ldots, |f_{x''}(x_{q})| \right\}\]

The score for each feature is then computed based on the similarity of the slopes in different sections of the partial dependence curve. Let \(S_i\) be the slope of section \(i\), and \(\cos(S_i, S_{i+1})\) be the cosine similarity between consecutive sections. The feature score is computed as:

\[\text{Feature Score} = \sum_{i=1}^{n-1} \cos(S_i, S_{i+1})\]

The final Explainability Ease Score ( E ) is the normalized weighted average of the feature scores:

\[E = \frac{1}{n} \sum_{j=1}^{n} \text{Feature Score}_j\]

where ( n ) is the number of features.

Interpretation#

The score ( E ) ranges from 0 to 1, where:

  • High Score : Indicates a simple, linear relationship between the feature and the predicted outcome. These curves are easy to interpret.

  • Low EScore : Indicates a complex, non-linear relationship. These curves are harder to interpret and may require more sophisticated analysis to understand.sis to understand.