predictorImportance

Estimates of predictor importance for regression ensemble

Syntax

imp = predictorImportance(ens)
[imp,ma] = predictorImportance(ens)


Description

imp = predictorImportance(ens) computes estimates of predictor importance for ens by summing these estimates over all weak learners in the ensemble. imp has one element for each input predictor in the data used to train this ensemble. A high value indicates that this predictor is important for ens.

[imp,ma] = predictorImportance(ens) returns a P-by-P matrix with predictive measures of association for P predictors.

Input Arguments

ens	A regression ensemble, created by fitrensemble, or by the compact method.

Output Arguments

imp	A row vector with the same number of elements as the number of predictors (columns) in ens.X. The entries are the estimates of predictor importance, with 0 representing the smallest possible importance.
ma	A P-by-P matrix of predictive measures of association for P predictors. Element ma(I,J) is the predictive measure of association averaged over surrogate splits on predictor J for which predictor I is the optimal split predictor. predictorImportance averages this predictive measure of association over all trees in the ensemble.

Examples

Estimate Predictor Importance

Estimate the predictor importance for all predictor variables in the data.

Load the carsmall data set.

load carsmall

Grow an ensemble of 100 regression trees for MPG using Acceleration, Cylinders, Displacement, Horsepower, Model_Year, and Weight as predictors. Specify tree stumps as the weak learners.

X = [Acceleration Cylinders Displacement Horsepower Model_Year Weight];
t = templateTree('MaxNumSplits',1);
ens = fitrensemble(X,MPG,'Method','LSBoost','Learners',t);

Estimate the predictor importance for all predictor variables.

imp = predictorImportance(ens)

imp = 1×6

    0.0150         0    0.0066    0.1111    0.0437    0.5181

Weight, the last predictor, has the most impact on mileage. The second predictor has importance 0, which means that the number of cylinders has no impact on predictions made with ens.

Predictor Importance and Surrogate Splits

Estimate the predictor importance for all variables in the data and where the regression tree ensemble contains surrogate splits.

Load the carsmall data set.

load carsmall

Grow an ensemble of 100 regression trees for MPG using Acceleration, Cylinders, Displacement, Horsepower, Model_Year, and Weight as predictors. Specify tree stumps as the weak learners, and also identify surrogate splits.

X = [Acceleration Cylinders Displacement Horsepower Model_Year Weight];
t = templateTree('MaxNumSplits',1,'Surrogate','on');
ens = fitrensemble(X,MPG,'Method','LSBoost','Learners',t);

Estimate the predictor importance and predictive measures of association for all predictor variables.

[imp,ma] = predictorImportance(ens)

imp = 1×6

    0.2141    0.3798    0.4369    0.6498    0.3728    0.5700

ma = 6×6

    1.0000    0.0098    0.0102    0.0098    0.0033    0.0067
         0    1.0000         0         0         0         0
    0.0056    0.0084    1.0000    0.0078    0.0022    0.0084
    0.3537    0.4769    0.5834    1.0000    0.1612    0.5827
    0.0061    0.0070    0.0063    0.0064    1.0000    0.0056
    0.0154    0.0296    0.0533    0.0447    0.0070    1.0000

Comparing imp to the results in Estimate Predictor Importance, Horsepower has the greatest impact on mileage, with Weight having the second greatest impact.

More About

Predictor Importance

predictorImportance estimates predictor importance of the predictors for each tree learner in the ensemble ens and returns the weighted average imp computed using ens.TrainedWeight. The output imp has one element for each predictor.

predictorImportance computes importance measures of the predictors in a tree by summing changes in the node risk due to splits on every predictor, and then dividing the sum by the total number of branch nodes. The change in the node risk is the difference between the risk for the parent node and the total risk for the two children. For example, if a tree splits a parent node (for example, node 1) into two child nodes (for example, nodes 2 and 3), then predictorImportance increases the importance of the split predictor by

(R₁ – R₂ – R₃)/N_branch,

where R_i is node risk of node i, and N_branch is the total number of branch nodes. A node risk is defined as a node error weighted by the node probability:

R_i = P_iE_i,

where P_i is the node probability of node i, and E_i is the mean squared error of node i.

The estimates of predictor importance depend on whether you use surrogate splits for training.

• If you use surrogate splits, predictorImportance sums the changes in the node risk over all splits at each branch node, including surrogate splits. If you do not use surrogate splits, then the function takes the sum over the best splits found at each branch node.

• Estimates of predictor importance do not depend on the order of predictors if you use surrogate splits, but do depend on the order if you do not use surrogate splits.

Predictive Measure of Association

The predictive measure of association is a value that indicates the similarity between decision rules that split observations. Among all possible decision splits that are compared to the optimal split (found by growing the tree), the best surrogate decision split yields the maximum predictive measure of association. The second-best surrogate split has the second-largest predictive measure of association.

Suppose x_j and x_k are predictor variables j and k, respectively, and j ≠ k. At node t, the predictive measure of association between the optimal split x_j < u and a surrogate split x_k < v is

$${\lambda }_{jk}=\frac{\text{min}\left(P_L,P_R\right)-\left(1-P_{L_j{L}_k}-P_{R_j{R}_k}\right)}{\text{min}\left(P_L,P_R\right)}.$$

• P_L is the proportion of observations in node t, such that x_j < u. The subscript L stands for the left child of node t.

• P_R is the proportion of observations in node t, such that x_j ≥ u. The subscript R stands for the right child of node t.

• $$P_{L_j{L}_k}$$ is the proportion of observations at node t, such that x_j < u and x_k < v.

• $$P_{R_j{R}_k}$$ is the proportion of observations at node t, such that x_j ≥ u and x_k ≥ v.

• Observations with missing values for x_j or x_k do not contribute to the proportion calculations.

λ_jk is a value in (–∞,1]. If λ_jk > 0, then x_k < v is a worthwhile surrogate split for x_j < u.

Algorithms

Element ma(i,j) is the predictive measure of association averaged over surrogate splits on predictor j for which predictor i is the optimal split predictor. This average is computed by summing positive values of the predictive measure of association over optimal splits on predictor i and surrogate splits on predictor j and dividing by the total number of optimal splits on predictor i, including splits for which the predictive measure of association between predictors i and j is negative.

Extended Capabilities

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).