Stability Metrics#

Feature Stability metrics are designed to evaluate the consistency of feature importance across different instances and features in a dataset. These metrics help quantify the robustness and reliability of feature importance, facilitating better model explainability and transparency.

Feature Stability#

Methodology#

The Feature Stability metric measures the stability of individual feature importances across different instances. It focuses on the consistency of the importance of a specific feature throughout the dataset.

Mathematical Representation#

Let \(\mathbf{I} = \{I_1, I_2, \ldots, I_n\}\) be the set of local feature importances for \(n\) instances, where each \(I_i\) is a vector of feature importances.

  1. Normalization of Data: Each vector \(I_i\) is normalized so that the sum of its elements equals 1:

    \[I_{i,j} \leftarrow \frac{I_{i,j}}{\sum_{k=1}^{m} I_{i,k}} \quad \text{for } i = 1, 2, \ldots, n \text{ and } j = 1, 2, \ldots, m\]

    where \(m\) is the number of features.

  2. Computation of Importance Distributions: The importance distribution \(\mathbf{D}\) of features is computed by finding the density distribution of feature importance vectors. This is done by evaluating the proximity of these vectors to a set of synthetic samples generated from a Dirichlet distribution:

    \[\mathbf{D} = \left( d_1, d_2, \ldots, d_{m} \right)\]

    where \(d_j\) represents the density estimate for feature \(j\).

  3. Calculation of Feature Stability: Feature Stability is computed using one of the following strategies:

    • Variance Strategy: The stability is determined by the ratio of the standard deviation to the maximum density:

      \[\textrm{FS} = 1 - \frac{\sigma_D}{\max(D)}\]

      where \(\sigma_D\) represents the standard deviation of the density distribution \(\mathbf{D}\).

    • Entropy Strategy: Alternatively, the stability can be computed based on the Jensen-Shannon divergence between the distribution \(\mathbf{D}\) and a uniform distribution:

      \[\textrm{FS} = 1 - \text{JSD}\left(\mathbf{D} \| \mathbf{U} \right)\]

      where \(\mathbf{U}\) is the uniform distribution, and \(\text{JSD}\) denotes the Jensen-Shannon divergence.

Interpretation#

  • High value: Indicates that the importance of a specific feature is consistent across instances. This suggests that the feature is robust and its relationship with the model’s target is reliable.

  • Low value: Indicates that the importance of a feature varies significantly between instances. This may suggest that the feature is less reliable and its relationship with the model’s target may be weak.

The Feature Stability metric provides a quantitative measure of the consistency of feature importances across different instances by evaluating the dispersion of these importances using either variance-based or entropy-based methods.