oobPermutedPredictorImportance

Out-of-bag predictor importance estimates for random forest of regression trees by permutation

Syntax

imp = oobPermutedPredictorImportance(Mdl)

imp = oobPermutedPredictorImportance(Mdl,Name=Value)

Description

imp = oobPermutedPredictorImportance(Mdl) returns out-of-bag, predictor importance estimates by permutation using the random forest of regression trees Mdl. Mdl must be a RegressionBaggedEnsemble model object. imp is a 1-by-p numeric vector, where p is the number of predictor variables in the training data (size(Mdl.X,2)). imp(j) is the predictor importance of the predictor Mdl.PredictorNames(j).

imp = oobPermutedPredictorImportance(Mdl,Name=Value) specifies additional options using one or more name-value arguments. For example, you can specify which trees to use in the predictor importance estimation.

Examples

Estimate Importance of Predictors

Load the carsmall data set. Consider a model that predicts the mean fuel economy of a car given its acceleration, number of cylinders, engine displacement, horsepower, manufacturer, model year, and weight. Consider Cylinders, Mfg, and Model_Year as categorical variables.

load carsmall
Cylinders = categorical(Cylinders);
Mfg = categorical(cellstr(Mfg));
Model_Year = categorical(Model_Year);
X = table(Acceleration,Cylinders,Displacement,Horsepower,Mfg,...
    Model_Year,Weight,MPG);

You can train a random forest of 500 regression trees using the entire data set.

Mdl = fitrensemble(X,'MPG','Method','Bag','NumLearningCycles',500);

fitrensemble uses a default template tree object templateTree() as a weak learner when 'Method' is 'Bag'. In this example, for reproducibility, specify 'Reproducible',true when you create a tree template object, and then use the object as a weak learner.

rng('default') % For reproducibility
t = templateTree('Reproducible',true); % For reproducibiliy of random predictor selections
Mdl = fitrensemble(X,'MPG','Method','Bag','NumLearningCycles',500,'Learners',t);

Mdl is a RegressionBaggedEnsemble model.

Estimate predictor importance measures by permuting out-of-bag observations. Compare the estimates using a bar graph.

imp = oobPermutedPredictorImportance(Mdl);

figure;
bar(imp);
title('Out-of-Bag Permuted Predictor Importance Estimates');
ylabel('Estimates');
xlabel('Predictors');
h = gca;
h.XTickLabel = Mdl.PredictorNames;
h.XTickLabelRotation = 45;
h.TickLabelInterpreter = 'none';

imp is a 1-by-7 vector of predictor importance estimates. Larger values indicate predictors that have a greater influence on predictions. In this case, Weight is the most important predictor, followed by Model_Year.

Unbiased Estimates of Predictor Importance Using Parallel Computing

Load the carsmall data set. Consider a model that predicts the mean fuel economy of a car given its acceleration, number of cylinders, engine displacement, horsepower, manufacturer, model year, and weight. Consider Cylinders, Mfg, and Model_Year as categorical variables.

load carsmall
Cylinders = categorical(Cylinders);
Mfg = categorical(cellstr(Mfg));
Model_Year = categorical(Model_Year);
X = table(Acceleration,Cylinders,Displacement,Horsepower,Mfg,...
    Model_Year,Weight,MPG);

Display the number of categories represented in the categorical variables.

numCylinders = numel(categories(Cylinders))

numCylinders = 3

numMfg = numel(categories(Mfg))

numMfg = 28

numModelYear = numel(categories(Model_Year))

numModelYear = 3

Because there are 3 categories only in Cylinders and Model_Year, the standard CART, predictor-splitting algorithm prefers splitting a continuous predictor over these two variables.

Train a random forest of 500 regression trees using the entire data set. To grow unbiased trees, specify usage of the curvature test for splitting predictors. Because there are missing values in the data, specify usage of surrogate splits. To reproduce random predictor selections, set the seed of the random number generator by using rng and specify 'Reproducible',true.

rng('default'); % For reproducibility
t = templateTree('PredictorSelection','curvature','Surrogate','on', ...
    'Reproducible',true); % For reproducibility of random predictor selections
Mdl = fitrensemble(X,'MPG','Method','bag','NumLearningCycles',500, ...
    'Learners',t);

Estimate predictor importance measures by permuting out-of-bag observations. Perform calculations in parallel.

options = statset('UseParallel',true);
imp = oobPermutedPredictorImportance(Mdl,'Options',options);

Starting parallel pool (parpool) using the 'local' profile ...
Connected to the parallel pool (number of workers: 6).

Compare the estimates using a bar graph.

figure;
bar(imp);
title('Out-of-Bag Permuted Predictor Importance Estimates');
ylabel('Estimates');
xlabel('Predictors');
h = gca;
h.XTickLabel = Mdl.PredictorNames;
h.XTickLabelRotation = 45;
h.TickLabelInterpreter = 'none';

In this case, Model_Year is the most important predictor, followed by Cylinders. Compare these results to the results in Estimate Importance of Predictors.

Input Arguments

Mdl — Random forest of regression trees

RegressionBaggedEnsemble model object

Random forest of regression trees, specified as a RegressionBaggedEnsemble model object created with fitrensemble.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: oobPermutedPredictorImportance(Mdl,Learners=[1 2 3 5]) specifies to use the first, second, third, and fifth learners in Mdl.

Learners — Indices of learners to use in predictor importance estimation

1:Mdl.NumTrained (default) | vector of positive integers

Indices of learners to use in predictor importance estimation, specified as a vector of positive integers. The maximum allowable integer value is Mdl.NumTrained. When oobPermutedPredictorImportance estimates the predictor importance, it includes the learners in Mdl.Trained(learners) only, where learners is the value of Learners.

Example: Learners=[1:2:Mdl.NumTrained]

Data Types: single | double

Options — Parallel computing options

[] (default) | structure array returned by statset

Parallel computing options, specified as a structure array returned by statset. You must have a Parallel Computing Toolbox™ license to specify Options.

oobPermutedPredictorImportance uses only the UseParallel field in the structure array. statset(UseParallel=true) invokes a pool of parallel workers.

Example: Options=statset(UseParallel=true)

Data Types: structure array

More About

Out-of-Bag Predictor Importance Estimates by Permutation

Out-of-bag predictor importance estimates by permutation measure how influential the model's predictor variables are at predicting the response. The influence of a predictor increases with the value of this measure.

If a predictor is influential in prediction, then permuting its values should affect the model error. If a predictor is not influential, then permuting its values should have little to no effect on the model error.

The following process describes the estimation of out-of-bag predictor importance values by permutation. Suppose that R is a random forest of T learners and p is the number of predictors in the training data.

1. For tree t, t = 1,...,T:

   1. Identify the out-of-bag observations and the indices of the predictor variables that were split to grow tree t, s_t ⊆ {1,...,p}.

   2. Estimate the out-of-bag error ε_t.

   3. For each predictor variable x_j, j ∊ s_t:

      1. Randomly permute the observations of x_j.

      2. Estimate the model error, ε_tj, using the out-of-bag observations containing the permuted values of x_j.

      3. Take the difference d_tj = ε_tj – ε_t. Predictor variables not split when growing tree t are attributed a difference of 0.

2. For each predictor variable in the training data, compute the mean, $${\overline{d}}_j$$, and standard deviation, σ_j, of the differences over the learners, j = 1,...,p.

3. The out-of-bag predictor importance by permutation for x_j is $${\overline{d}}_j/{\sigma }_j$$.

Tips

When growing a random forest using fitrensemble:

• Standard CART tends to select split predictors containing many distinct values, e.g., continuous variables, over those containing few distinct values, e.g., categorical variables [3]. If the predictor data set is heterogeneous, or if there are predictors that have relatively fewer distinct values than other variables, then consider specifying the curvature or interaction test.

• Trees grown using standard CART are not sensitive to predictor variable interactions. Also, such trees are less likely to identify important variables in the presence of many irrelevant predictors than the application of the interaction test. Therefore, to account for predictor interactions and identify importance variables in the presence of many irrelevant variables, specify the interaction test [2].

• If the training data includes many predictors and you want to analyze predictor importance, then specify NumVariablesToSample of the templateTree function as "all" for the tree learners of the ensemble. Otherwise, the software might not select some predictors, underestimating their importance.

For more details, see templateTree and Choose Split Predictor Selection Technique.

References

[1] Breiman, L., J. Friedman, R. Olshen, and C. Stone. Classification and Regression Trees. Boca Raton, FL: CRC Press, 1984.

[2] Loh, W.Y. “Regression Trees with Unbiased Variable Selection and Interaction Detection.” Statistica Sinica, Vol. 12, 2002, pp. 361–386.

[3] Loh, W.Y. and Y.S. Shih. “Split Selection Methods for Classification Trees.” Statistica Sinica, Vol. 7, 1997, pp. 815–840.

Extended Capabilities

Automatic Parallel Support

Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

To run in parallel, specify the Options name-value argument in the call to this function and set the UseParallel field of the options structure to true using statset:

Options=statset(UseParallel=true)

For more information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).

Version History

Introduced in R2016b