predictorImportance

Estimates of predictor importance for classification ensemble of decision trees

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, when the learners in ens contain surrogate splits. See More About.

Input Arguments

ens	A classification ensemble of decision trees, created by fitcensemble, 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 variables in the Fisher iris data.

Load Fisher's iris data set.

load fisheriris

Train a classification ensemble using AdaBoostM2. Specify tree stumps as the weak learners.

t = templateTree('MaxNumSplits',1);
ens = fitcensemble(meas,species,'Method','AdaBoostM2','Learners',t);

Estimate the predictor importance for all predictor variables.

imp = predictorImportance(ens)

imp = 1×4

    0.0004    0.0016    0.1266    0.0324

The first two predictors are not very important in the ensemble.

Predictor Importance and Surrogate Splits

Estimate the predictor importance for all variables in the Fisher iris data for an ensemble where the trees contain surrogate splits.

Load Fisher's iris data set.

load fisheriris

Grow an ensemble of 100 classification trees using AdaBoostM2. Specify tree stumps as the weak learners, and also identify surrogate splits.

t = templateTree('MaxNumSplits',1,'Surrogate','on');
ens = fitcensemble(meas,species,'Method','AdaBoostM2','Learners',t);

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

[imp,ma] = predictorImportance(ens)

imp = 1×4

    0.0674    0.0417    0.1582    0.1537

ma = 4×4

    1.0000         0         0         0
    0.0115    1.0000    0.0022    0.0054
    0.3186    0.2137    1.0000    0.6391
    0.0392    0.0073    0.1137    1.0000

The first two predictors show much more importance than the analysis in Estimate Predictor Importance.

More About

Predictor Importance

predictorImportance estimates predictor importance 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 the node risk of node i, and N_branch is the total number of branch nodes. A node risk is defined as a node error or node impurity weighted by the node probability:

R_i = P_iE_i,

where P_i is the node probability of node i, and E_i is either the node error (for a tree grown by minimizing the twoing criterion) or node impurity (for a tree grown by minimizing an impurity criterion, such as the Gini index or deviance) 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.

Impurity and Node Error

A decision tree splits nodes based on either impurity or node error.

Impurity means one of several things, depending on your choice of the SplitCriterion name-value pair argument:

• Gini's Diversity Index (gdi) — The Gini index of a node is

  $$1-{\displaystyle \sum _i{p}^2(i)},$$

  where the sum is over the classes i at the node, and p(i) is the observed fraction of classes with class i that reach the node. A node with just one class (a pure node) has Gini index 0; otherwise the Gini index is positive. So the Gini index is a measure of node impurity.

• Deviance ('deviance') — With p(i) defined the same as for the Gini index, the deviance of a node is

  $$-{\displaystyle \sum _{i}p(i){\log }_{2}p(i)}.$$

  A pure node has deviance 0; otherwise, the deviance is positive.

• Twoing rule ('twoing') — Twoing is not a purity measure of a node, but is a different measure for deciding how to split a node. Let L(i) denote the fraction of members of class i in the left child node after a split, and R(i) denote the fraction of members of class i in the right child node after a split. Choose the split criterion to maximize

  $$P(L)P(R){\left({\displaystyle \sum _i\left|L(i)-R(i)\right|}\right)}^2,$$

  where P(L) and P(R) are the fractions of observations that split to the left and right respectively. If the expression is large, the split made each child node purer. Similarly, if the expression is small, the split made each child node similar to each other, and therefore similar to the parent node. The split did not increase node purity.

• Node error — The node error is the fraction of misclassified classes at a node. If j is the class with the largest number of training samples at a node, the node error is

  1 – p(j).

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).