Source-linked AI summary
A Simple and Effective Model-Based Variable Importance Measure
Brandon M. Greenwell, Bradley C. Boehmke, Andrew J. McCarthy
TL;DR
Variable-importance scores are difficult to define for fitted models and can depend on the fitting algorithm. The paper presents a partial-dependence-based approach intended for broad algorithmic use, with illustrations spanning multiple model types and simulated or real examples.
Problem
Variable-importance scores are difficult to define for fitted models, and their interpretation often depends on the model fitting algorithm used.
Method
The paper proposes a variable-importance approach using scores based on partial dependence functions.
Results
The approach clearly labels all five predictors in the true model as the most important features in Friedman’s regression problem.
Takeaways & Limitations
The proposed variable-importance measure is described as suitable for use with any algorithm.
Takeaways & Limitations
Outliers in the feature space can cause abnormalities, and more research is needed to determine the approach’s utility.
Abstract
from arXiv · showhide
In the era of "big data", it is becoming more of a challenge to not only build state-of-the-art predictive models, but also gain an understanding of what's really going on in the data. For example, it is often of interest to know which, if any, of the predictors in a fitted model are relatively influential on the predicted outcome. Some modern algorithms---like random forests and gradient boosted decision trees---have a natural way of quantifying the importance or relative influence of each feature. Other algorithms---like naive Bayes classifiers and support vector machines---are not capable of doing so and model-free approaches are generally used to measure each predictor's importance. In this paper, we propose a standardized, model-based approach to measuring predictor importance across the growing spectrum of supervised learning algorithms. Our proposed method is illustrated through both simulated and real data examples. The R code to reproduce all of the figures in this paper is available in the supplementary materials.
1 Introduction
Variable importance is difficult to define consistently for complex supervised models, and its interpretation can depend on the fitting algorithm. The paper proposes a standardized, model-based measure using partial dependence plots (PDPs) that applies across supervised learning algorithms.
- Variable importance scores are difficult to define, and their interpretation often depends on the model fitting algorithm used.
- The proposed method computes model-based variable importance scores using partial dependence plots.
- The procedure standardizes variable-importance quantification across the growing spectrum of supervised learning algorithms.
- The method applies to any trained supervised learning algorithm when predictions on new data can be obtained.
- The approach remains applicable to stacked ensembles because predictions can be generated from the super learner.
- The method can be modified to quantify potential interaction effects and requires analysts to inspect estimated feature–target relationships alongside importance.
2 Background
The background surveys why variable importance and functional interpretation are difficult for complex predictive models, then reviews model-based, filter-based, and partial-dependence approaches. It motivates a model-based measure that can support interpretation across supervised learning algorithms.
- Model-based approaches: Existing model-based importance measures differ across algorithms, including tree improvement scores, random-forest permutation scores, regression statistics, MARS GCV scores, and neural-network connection-weight methods.Tree ensembles average improvement or permutation-based quantities, while neural-network methods use weighted connections.
- Filter-based approaches: Filter-based approaches measure predictor relevance separately from the fitted model and may not account for the other predictors.Examples include AUC for classification and pseudo-R2 from a flexible univariate regression for numeric predictors.
- Motivation: Modern learning algorithms can achieve strong predictive performance, but many provide limited interpretability and no natural variable-importance score.This creates a need to extract both influential predictors and their relationships with the outcome.
- Partial dependence plots: Partial dependence plots summarize the relationship between one or a small subset of predictors and the response while accounting for the average effect of other predictors.They can also rank and score predictors by relative influence and are used to interpret black-box models such as random forests and support vector machines.
- Proposed approach and illustration: The paper defines a partial-dependence-based variable-importance measure and illustrates it on Ames housing data, where Gr Liv Area is highly associated with log sales price and several predictors show strong nonlinear relationships.The Ames data contain 79 predictors, and the cited results identify Gr Liv Area, Overall Qual, and Neighborhood as important nonlinear predictors.
3 A partial dependence-based variable importance mea-
The paper measures variable importance from the flatness of a predictor’s partial dependence function, using variability measures tailored to continuous or categorical predictors. The same partial-dependence framework also quantifies potential interactions and matches the intended importance patterns in linear and Ames-model examples.
- Definition: Variable importance is defined from the variability, or “flatness,” of a predictor’s partial dependence function.The framework allows any flatness measure, with sample standard deviation for continuous predictors and a range-based statistic for categorical predictors.
- Definition: For categorical predictors, the range statistic is divided by four to estimate standard deviation for small to moderate samples.
- Scope: The fitted-model importance measure requires properly tuning and training the model because it depends on the fitted model.
- Ames example: In the Ames GBM example, the partial-dependence algorithm closely matches GBM results, with Overall Qual, Neighborhood, and Gr Liv Area remaining the most important predictors.Neighborhood and Gr Liv Area swap positions in the ranking.
- Linear models: In a linear-model simulation, the metric estimates the known influence ratio closely: X2 is roughly 1.67 times more influential than X1, versus an estimated ratio of approximately 1.65.The estimated importance scores are i(X1) = 1.4828203 and i(X2) = 0.8961719.
- Interaction effects: The interaction statistic varies importance scores across fixed values of the other predictor, with large values indicating potential interaction effects.For a pair of predictors, the procedure computes conditional importance scores in both directions and averages their standard deviations.
4 Friedman’s regression problem
The Friedman regression experiment tests the proposed variable-importance and interaction measures on neural-network fits with known active predictors and interactions. The proposed method identifies the true predictors and the x1–x2 interaction, while the compared H-statistic misses that interaction in the reported GBM example.
- Setup: The simulated Friedman regression uses ten independent predictors, but only five enter the true response model.The response includes an x1–x2 interaction, nonlinear x3, and linear x4 and x5 effects.
- Setup: The neural network has one hidden layer with eight units and weight decay 0.01, fitted to 500 simulated observations with parameters selected by 5-fold cross-validation.
- Variable importance: Garson’s and Olden’s importance methods incorrectly rank inactive predictors above active predictors in the Friedman data.Garson ranks x8 above active x5, while Olden ranks x10 above active x2.
- Variable importance: The proposed method clearly labels all five predictors in the true model as the most important features.
- Interaction effects: The proposed interaction statistic identifies x1 and x2 as interacting, the only interaction present in the true model.The top interaction statistics are displayed in Figure 5.
- Interaction effects: In the GBM comparison, Friedman’s H-statistic fails to detect the true x1–x2 interaction and instead ranks (x8, x9) and (x7, x10) strongest.
5 Application to model stacking
The paper applies partial-dependence-based importance to a stacked ensemble, where direct base-learner importance scores cannot be combined into a super-learner score. The resulting super-learner importance agrees with the individual models on the three leading Ames predictors.
- Results: The GBM achieves a cross-validated R2 of 91.54%, while ensembles can often attain better predictive performance.
- Stacked ensemble: The stacked ensemble generates predictions by feeding RF and GBM learner outputs into the GLM metalearner.
- Variable importance: Because the base learners’ importance scores cannot be constructed for the super learner, the paper computes partial dependence functions and applies Equation (4).
- Results: All models agree on the top three predictors of Log Sale Price: Overall Qual, Neighborhood, and Gr Liv Area.
6 Application to automatic machine learning
The paper extends its importance method to an automated stacked ensemble, addressing AutoML’s lack of automatic interpretation. Partial dependence and importance results identify the strongest predictors and their relationships with scaled sound pressure level.
- Motivation: AutoML can automate model selection and tuning but does not automatically provide useful interpretations of predictor importance or functional relationships.
- Data and task: The airfoil self-noise task predicts scaled sound pressure level (dB) from five aerodynamic and acoustic predictors.
- Modeling: H2O AutoML trains RF, extremely-randomized forest, GBM, DNN, GLM, and stacked-ensemble models using 10-fold cross-validation and RMSE.
- Results: The final stacked ensemble achieves a 10-fold cross-validated RMSE and R-squared of 1.43.
- Results: Partial dependence plots suggest frequency and suction side displacement thickness have strong monotonically decreasing relationships with scaled sound pressure level.
- Results: These two predictors appear more influential than the other three, and importance scores for all five predictors are computed using Equation (4).
7 Discussion
The discussion presents the measure as model-based, consistent across supervised learning algorithms, and applicable when new predictions are available. It also identifies training, outlier sensitivity, computational cost, and remaining validation needs as boundaries.
- Properties: The proposed measure is intended for any supervised learning algorithm that can produce new predictions and accounts for all model features.
- Properties: It is designed to retain the same interpretation across learning algorithms and may help identify possible interaction effects.
- Assumptions: Because the method is model-based, the fitted model must be properly trained and tuned to achieve optimum performance.
- Limitations: The authors state that more research is needed to determine the approach’s utility.
- Limitations: Outliers in feature space can cause abnormally large fluctuations in partial dependence values, motivating robust alternatives such as median absolute deviation.
- Limitations: Algorithm 1 imposes computational burden on large data sets, although methods discussed by Greenwell can mitigate it.