# Synthesizing Explainability Across Multiple ML Models for Structured Data
## 3 Proposed Method: Weighted Importance Score and Frequency Count (WISFC)
### Aggregating Importance Scores (Weighted Importance Score)  
The first component of _WISFC_ is the Weighted Importance Score (_WIS_). To compute _WIS_, we proceed as follows. Suppose we have \(N\) different trained models or explanation outputs (these could be entirely different model types or the same model type trained with different hyperparameters or on different folds--WISFC works in either case). Each model \(j\) provides an importance value \(I_{j}(f)\) for each feature \(f\). This \(I_{j}(f)\) could be, for example, the mean absolute SHAP value of feature \(f\) for model \(j\), the weight coefficient magnitude in model \(j\), or the feature's permutation importance score--whatever is appropriate for that model. We do not assume all models use the same importance metric since tree-based models and linear models, for instance, have different scales; therefore, as a pre-processing step, we normalize or rank-transform each model's importance outputs into a comparable scale [14]. A simple and robust choice is to convert each model's raw importance scores into ranks (1 = most important feature for that model, 2 = second, etc.) [14]. Alternatively, we can normalize by the sum of importances so that each model contributes a relative importance percentage. Let \(R_{j}(f)\) denote the normalized importance (rank or scaled score) of feature \(f\) according to model \(j\). Next, we assign a weight \(w_{j}\) to each model's importance contribution. In the basic version of WISFC, we can set all weights equal (\(w_{j}=1/N\) for \(j=1,\dots,N\)), which effectively means every model's opinion counts equally--so _WIS_ becomes a simple average importance over models [14]. This unweighted average already embodies an ensemble consensus: features that consistently have high \(I_{j}(f)\) across models will receive a high average, whereas features that are high in one model but low in others become \"diluted.\" However, WISFC also allows for performance-based weighting. If we have reason to trust some models more (for example, if Model A had higher cross-validated accuracy than Model B), we can give Model A a larger weight. A principled choice for weights is to use a SoftMax on performance so that all weights are positive and sum to 1 [1]. For instance, one could set the model weights \(w_{j}\) proportional to their predictive performance using a SoftMax function over model accuracies:  
\[w_{j}\ =\ \frac{\exp(\text{perf}_{j})}{\sum_{k=1}^{N}\exp(\text{perf}_{k})}\,\]  
so that a model with better accuracy (or \(R^{2}\) in regression) attains an exponentially higher weight in the importance aggregation [1]. This mirrors an existing approach [1], which demonstrates that incorporating model accuracy in the importance ranking improved the stability of the results. One must be careful here to use a performance metric evaluated on a validation set or via cross-validation (not the training set) to avoid overweight an overfit model. Finally, the Weighted Importance Score for feature \(f\) is computed as:  
\[WIS(f)\ =\ \sum_{j=1}^{N}w_{j}\,R_{j}(f)\,,\]  
i.e., a weighted sum (or average) of the feature's importance across all models. If we used rank-based \(R_{j}(f)\), then \(WIS(f)\) can be interpreted as (inversely) the average rank of feature \(f\). A lower \(WIS\) means the feature tended to rank high (1 st, 2 nd) across models, while a higher \(WIS\) means the feature often ranked lower (20 th, 30 th, etc.). We may invert it if we want a \"higher is better\" convention. If we used normalized scores, \(WIS(f)\) might range between 0 and 1 (if using importance percentages) or some relative scale. The key is that \(WIS(f)\) aggregates the strength of feature \(f\)'s importance as seen by all models collectively. This fulfills part of RQ4 by providing a quantitative, reproducible way to combine different models' feature rankings, as opposed to eyeballing or narratively comparing them.