# fitrensemble

Fit ensemble of learners for regression

## Description

example

Mdl = fitrensemble(Tbl,ResponseVarName) returns the trained regression ensemble model object (Mdl) that contains the results of boosting 100 regression trees using LSBoost and the predictor and response data in the table Tbl. ResponseVarName is the name of the response variable in Tbl.

example

Mdl = fitrensemble(Tbl,formula) applies formula to fit the model to the predictor and response data in the table Tbl. formula is an explanatory model of the response and a subset of predictor variables in Tbl used to fit Mdl. For example, 'Y~X1+X2+X3' fits the response variable Tbl.Y as a function of the predictor variables Tbl.X1, Tbl.X2, and Tbl.X3.

example

Mdl = fitrensemble(Tbl,Y) treats all variables in the table Tbl as predictor variables. Y is the vector of responses that is not in Tbl.

example

Mdl = fitrensemble(X,Y) uses the predictor data in the matrix X and response data in the vector Y.

example

Mdl = fitrensemble(___,Name,Value) uses additional options specified by one or more Name,Value pair arguments and any of the input arguments in the previous syntaxes. For example, you can specify the number of learning cycles, the ensemble aggregation method, or to implement 10-fold cross-validation.

## Examples

collapse all

Create a regression ensemble that predicts the fuel economy of a car given the number of cylinders, volume displaced by the cylinders, horsepower, and weight. Then, train another ensemble using fewer predictors. Compare the in-sample predictive accuracies of the ensembles.

Load the carsmall data set. Store the variables to be used in training in a table.

Tbl = table(Cylinders,Displacement,Horsepower,Weight,MPG);

Train a regression ensemble.

Mdl1 = fitrensemble(Tbl,'MPG');

Mdl1 is a RegressionEnsemble model. Some notable characteristics of Mdl1 are:

• The ensemble aggregation algorithm is 'LSBoost'.

• Because the ensemble aggregation method is a boosting algorithm, regression trees that allow a maximum of 10 splits compose the ensemble.

• One hundred trees compose the ensemble.

Because MPG is a variable in the MATLAB® Workspace, you can obtain the same result by entering

Mdl1 = fitrensemble(Tbl,MPG);

Use the trained regression ensemble to predict the fuel economy for a four-cylinder car with a 200-cubic inch displacement, 150 horsepower, and weighing 3000 lbs.

pMPG = predict(Mdl1,[4 200 150 3000])
pMPG = 25.6467

Train a new ensemble using all predictors in Tbl except Displacement.

formula = 'MPG ~ Cylinders + Horsepower + Weight';
Mdl2 = fitrensemble(Tbl,formula);

Compare the resubstitution MSEs between Mdl1 and Mdl2.

mse1 = resubLoss(Mdl1)
mse1 = 0.3096
mse2 = resubLoss(Mdl2)
mse2 = 0.5861

The in-sample MSE for the ensemble that trains on all predictors is lower.

Train an ensemble of boosted regression trees by using fitrensemble. Reduce training time by specifying the 'NumBins' name-value pair argument to bin numeric predictors. After training, you can reproduce binned predictor data by using the BinEdges property of the trained model and the discretize function.

Generate a sample data set.

rng('default') % For reproducibility
N = 1e6;
X1 = randi([-1,5],[N,1]);
X2 = randi([5,10],[N,1]);
X3 = randi([0,5],[N,1]);
X4 = randi([1,10],[N,1]);
X = [X1 X2 X3 X4];
y = X1 + X2 + X3 + X4 + normrnd(0,1,[N,1]);

Train an ensemble of boosted regression trees using least-squares boosting (LSBoost, the default value). Time the function for comparison purposes.

tic
Mdl1 = fitrensemble(X,y);
toc
Elapsed time is 78.662954 seconds.

Speed up training by using the 'NumBins' name-value pair argument. If you specify the 'NumBins' value as a positive integer scalar, then the software bins every numeric predictor into a specified number of equiprobable bins, and then grows trees on the bin indices instead of the original data. The software does not bin categorical predictors.

tic
Mdl2 = fitrensemble(X,y,'NumBins',50);
toc
Elapsed time is 43.353208 seconds.

The process is about two times faster when you use binned data instead of the original data. Note that the elapsed time can vary depending on your operating system.

Compare the regression errors by resubstitution.

rsLoss = resubLoss(Mdl1)
rsLoss = 1.0134
rsLoss2 = resubLoss(Mdl2)
rsLoss2 = 1.0133

In this example, binning predictor values reduces training time without a significant loss of accuracy. In general, when you have a large data set like the one in this example, using the binning option speeds up training but causes a potential decrease in accuracy. If you want to reduce training time further, specify a smaller number of bins.

Reproduce binned predictor data by using the BinEdges property of the trained model and the discretize function.

X = Mdl2.X; % Predictor data
Xbinned = zeros(size(X));
edges = Mdl2.BinEdges;
% Find indices of binned predictors.
idxNumeric = find(~cellfun(@isempty,edges));
if iscolumn(idxNumeric)
idxNumeric = idxNumeric';
end
for j = idxNumeric
x = X(:,j);
% Convert x to array if x is a table.
if istable(x)
x = table2array(x);
end
% Group x into bins by using the discretize function.
xbinned = discretize(x,[-inf; edges{j}; inf]);
Xbinned(:,j) = xbinned;
end

Xbinned contains the bin indices, ranging from 1 to the number of bins, for numeric predictors. Xbinned values are 0 for categorical predictors. If X contains NaNs, then the corresponding Xbinned values are NaNs.

Estimate the generalization error of an ensemble of boosted regression trees.

Load the carsmall data set. Choose the number of cylinders, volume displaced by the cylinders, horsepower, and weight as predictors of fuel economy.

X = [Cylinders Displacement Horsepower Weight];

Cross-validate an ensemble of regression trees using 10-fold cross-validation. Using a decision tree template, specify that each tree should be a split once only.

rng(1); % For reproducibility
t = templateTree('MaxNumSplits',1);
Mdl = fitrensemble(X,MPG,'Learners',t,'CrossVal','on');

Mdl is a RegressionPartitionedEnsemble model.

Plot the cumulative, 10-fold cross-validated, mean-squared error (MSE). Display the estimated generalization error of the ensemble.

kflc = kfoldLoss(Mdl,'Mode','cumulative');
figure;
plot(kflc);
ylabel('10-fold cross-validated MSE');
xlabel('Learning cycle');

estGenError = kflc(end)
estGenError = 26.2356

kfoldLoss returns the generalization error by default. However, plotting the cumulative loss allows you to monitor how the loss changes as weak learners accumulate in the ensemble.

The ensemble achieves an MSE of around 23.5 after accumulating about 30 weak learners.

If you are satisfied with the generalization error of the ensemble, then, to create a predictive model, train the ensemble again using all of the settings except cross-validation. However, it is good practice to tune hyperparameters such as the maximum number of decision splits per tree and the number of learning cycles..

This example shows how to optimize hyperparameters automatically using fitrensemble. The example uses the carsmall data.

You can find hyperparameters that minimize five-fold cross-validation loss by using automatic hyperparameter optimization.

Mdl = fitrensemble([Horsepower,Weight],MPG,'OptimizeHyperparameters','auto')

In this example, for reproducibility, set the random seed and use the 'expected-improvement-plus' acquisition function. Also, for reproducibility of random forest algorithm, specify the 'Reproducible' name-value pair argument as true for tree learners.

rng('default')
t = templateTree('Reproducible',true);
Mdl = fitrensemble([Horsepower,Weight],MPG,'OptimizeHyperparameters','auto','Learners',t, ...
'HyperparameterOptimizationOptions',struct('AcquisitionFunctionName','expected-improvement-plus'))
|===================================================================================================================================|
| Iter | Eval   | Objective:  | Objective   | BestSoFar   | BestSoFar   |       Method | NumLearningC-|    LearnRate |  MinLeafSize |
|      | result | log(1+loss) | runtime     | (observed)  | (estim.)    |              | ycles        |              |              |
|===================================================================================================================================|
|    1 | Best   |      2.9718 |      9.4175 |      2.9718 |      2.9718 |          Bag |          413 |            - |            1 |
|    2 | Accept |      6.2619 |      1.5813 |      2.9718 |      3.6127 |      LSBoost |           57 |    0.0016067 |            6 |
|    3 | Accept |      2.9975 |     0.77235 |      2.9718 |      2.9847 |          Bag |           32 |            - |            2 |
|    4 | Accept |      4.1897 |      1.2345 |      2.9718 |      2.9712 |          Bag |           55 |            - |           40 |
|    5 | Accept |      6.3321 |      1.3497 |      2.9718 |      2.9707 |      LSBoost |           55 |     0.001005 |            2 |
|    6 | Best   |      2.9689 |     0.88428 |      2.9689 |      2.9698 |          Bag |           39 |            - |            1 |
|    7 | Accept |      3.0113 |     0.53827 |      2.9689 |      2.9833 |          Bag |           23 |            - |            1 |
|    8 | Accept |      2.9823 |      10.275 |      2.9689 |      2.9832 |          Bag |          496 |            - |            1 |
|    9 | Accept |      4.1881 |     0.29182 |      2.9689 |      2.9833 |      LSBoost |           12 |        0.883 |           50 |
|   10 | Accept |      3.6685 |       8.304 |      2.9689 |      2.9832 |      LSBoost |          398 |      0.97772 |            1 |
|   11 | Accept |      3.4414 |     0.36526 |      2.9689 |      2.9833 |      LSBoost |           14 |      0.13404 |            1 |
|   12 | Accept |      4.1881 |      1.7024 |      2.9689 |      2.9832 |      LSBoost |           84 |     0.079388 |           49 |
|   13 | Accept |      5.6912 |     0.63921 |      2.9689 |      2.9832 |      LSBoost |           27 |     0.014186 |            1 |
|   14 | Accept |      3.5833 |      4.2041 |      2.9689 |      2.9831 |      LSBoost |          198 |      0.29995 |            3 |
|   15 | Accept |      5.7781 |     0.85679 |      2.9689 |      2.9829 |      LSBoost |           37 |     0.010476 |           50 |
|   16 | Accept |      6.4093 |     0.38185 |      2.9689 |      2.9828 |      LSBoost |           18 |    0.0010034 |           50 |
|   17 | Accept |      4.1881 |      2.0801 |      2.9689 |      2.9827 |      LSBoost |          100 |      0.26658 |           50 |
|   18 | Accept |      5.2307 |     0.30503 |      2.9689 |       2.983 |      LSBoost |           12 |     0.051319 |            4 |
|   19 | Accept |      5.9165 |      1.3811 |      2.9689 |      2.9835 |      LSBoost |           59 |    0.0045433 |            1 |
|   20 | Accept |       3.533 |      1.3538 |      2.9689 |      2.9829 |      LSBoost |           60 |       0.3064 |            1 |
|===================================================================================================================================|
| Iter | Eval   | Objective:  | Objective   | BestSoFar   | BestSoFar   |       Method | NumLearningC-|    LearnRate |  MinLeafSize |
|      | result | log(1+loss) | runtime     | (observed)  | (estim.)    |              | ycles        |              |              |
|===================================================================================================================================|
|   21 | Accept |      6.3754 |     0.27538 |      2.9689 |      2.9829 |      LSBoost |           10 |    0.0037054 |           50 |
|   22 | Accept |      5.7958 |     0.31299 |      2.9689 |      2.9827 |      LSBoost |           13 |     0.028703 |           50 |
|   23 | Accept |      3.2511 |      0.5558 |      2.9689 |      2.9828 |      LSBoost |           23 |      0.57224 |            5 |
|   24 | Accept |      3.5298 |       7.504 |      2.9689 |      2.9726 |      LSBoost |          367 |      0.56349 |            1 |
|   25 | Best   |      2.9188 |      4.3433 |      2.9188 |      2.9219 |          Bag |          216 |            - |            2 |
|   26 | Accept |      3.5858 |     0.25552 |      2.9188 |      2.9586 |      LSBoost |           10 |      0.67582 |            1 |
|   27 | Accept |      2.9384 |      10.206 |      2.9188 |      2.9444 |          Bag |          500 |            - |            2 |
|   28 | Accept |      2.9322 |       11.78 |      2.9188 |       2.939 |          Bag |          499 |            - |            2 |
|   29 | Accept |       2.937 |      11.726 |      2.9188 |      2.9378 |          Bag |          498 |            - |            2 |
|   30 | Accept |      3.6531 |     0.31678 |      2.9188 |      2.9379 |      LSBoost |           11 |      0.13843 |           11 |

__________________________________________________________
Optimization completed.
MaxObjectiveEvaluations of 30 reached.
Total function evaluations: 30
Total elapsed time: 141.9069 seconds.
Total objective function evaluation time: 95.1937

Best observed feasible point:
Method    NumLearningCycles    LearnRate    MinLeafSize
______    _________________    _________    ___________

Bag             216              NaN            2

Observed objective function value = 2.9188
Estimated objective function value = 2.9466
Function evaluation time = 4.3433

Best estimated feasible point (according to models):
Method    NumLearningCycles    LearnRate    MinLeafSize
______    _________________    _________    ___________

Bag             500              NaN            2

Estimated objective function value = 2.9379
Estimated function evaluation time = 11.0122
Mdl =
RegressionBaggedEnsemble
ResponseName: 'Y'
CategoricalPredictors: []
ResponseTransform: 'none'
NumObservations: 94
HyperparameterOptimizationResults: [1×1 BayesianOptimization]
NumTrained: 500
Method: 'Bag'
LearnerNames: {'Tree'}
ReasonForTermination: 'Terminated normally after completing the requested number of training cycles.'
FitInfo: []
FitInfoDescription: 'None'
Regularization: []
FResample: 1
Replace: 1
UseObsForLearner: [94×500 logical]

Properties, Methods

The optimization searched over the methods for regression (Bag and LSBoost), over NumLearningCycles, over the LearnRate for LSBoost, and over the tree learner MinLeafSize. The output is the ensemble regression with the minimum estimated cross-validation loss.

One way to create an ensemble of boosted regression trees that has satisfactory predictive performance is to tune the decision tree complexity level using cross-validation. While searching for an optimal complexity level, tune the learning rate to minimize the number of learning cycles as well.

This example manually finds optimal parameters by using the cross-validation option (the 'KFold' name-value pair argument) and the kfoldLoss function. Alternatively, you can use the 'OptimizeHyperparameters' name-value pair argument to optimize hyperparameters automatically. See Optimize Regression Ensemble.

Load the carsmall data set. Choose the number of cylinders, volume displaced by the cylinders, horsepower, and weight as predictors of fuel economy.

Tbl = table(Cylinders,Displacement,Horsepower,Weight,MPG);

The default values of the tree depth controllers for boosting regression trees are:

• 10 for MaxNumSplits.

• 5 for MinLeafSize

• 10 for MinParentSize

To search for the optimal tree-complexity level:

1. Cross-validate a set of ensembles. Exponentially increase the tree-complexity level for subsequent ensembles from decision stump (one split) to at most n - 1 splits. n is the sample size. Also, vary the learning rate for each ensemble between 0.1 to 1.

2. Estimate the cross-validated mean-squared error (MSE) for each ensemble.

3. For tree-complexity level $j$, $j=1...J$, compare the cumulative, cross-validated MSE of the ensembles by plotting them against number of learning cycles. Plot separate curves for each learning rate on the same figure.

4. Choose the curve that achieves the minimal MSE, and note the corresponding learning cycle and learning rate.

Cross-validate a deep regression tree and a stump. Because the data contain missing values, use surrogate splits. These regression trees serve as benchmarks.

rng(1) % For reproducibility
MdlDeep = fitrtree(Tbl,'MPG','CrossVal','on','MergeLeaves','off', ...
'MinParentSize',1,'Surrogate','on');
MdlStump = fitrtree(Tbl,'MPG','MaxNumSplits',1,'CrossVal','on', ...
'Surrogate','on');

Cross-validate an ensemble of 150 boosted regression trees using 5-fold cross-validation. Using a tree template:

• Vary the maximum number of splits using the values in the sequence $\left\{{2}^{0},{2}^{1},...,{2}^{m}\right\}$. m is such that ${2}^{m}$ is no greater than n - 1.

• Turn on surrogate splits.

For each variant, adjust the learning rate using each value in the set {0.1, 0.25, 0.5, 1}.

n = size(Tbl,1);
m = floor(log2(n - 1));
learnRate = [0.1 0.25 0.5 1];
numLR = numel(learnRate);
maxNumSplits = 2.^(0:m);
numMNS = numel(maxNumSplits);
numTrees = 150;
Mdl = cell(numMNS,numLR);

for k = 1:numLR
for j = 1:numMNS
t = templateTree('MaxNumSplits',maxNumSplits(j),'Surrogate','on');
Mdl{j,k} = fitrensemble(Tbl,'MPG','NumLearningCycles',numTrees, ...
'Learners',t,'KFold',5,'LearnRate',learnRate(k));
end
end

Estimate the cumulative, cross-validated MSE of each ensemble.

kflAll = @(x)kfoldLoss(x,'Mode','cumulative');
errorCell = cellfun(kflAll,Mdl,'Uniform',false);
error = reshape(cell2mat(errorCell),[numTrees numel(maxNumSplits) numel(learnRate)]);
errorDeep = kfoldLoss(MdlDeep);
errorStump = kfoldLoss(MdlStump);

Plot how the cross-validated MSE behaves as the number of trees in the ensemble increases. Plot the curves with respect to learning rate on the same plot, and plot separate plots for varying tree-complexity levels. Choose a subset of tree complexity levels to plot.

mnsPlot = [1 round(numel(maxNumSplits)/2) numel(maxNumSplits)];
figure;
for k = 1:3
subplot(2,2,k)
plot(squeeze(error(:,mnsPlot(k),:)),'LineWidth',2)
axis tight
hold on
h = gca;
plot(h.XLim,[errorDeep errorDeep],'-.b','LineWidth',2)
plot(h.XLim,[errorStump errorStump],'-.r','LineWidth',2)
plot(h.XLim,min(min(error(:,mnsPlot(k),:))).*[1 1],'--k')
h.YLim = [10 50];
xlabel('Number of trees')
ylabel('Cross-validated MSE')
title(sprintf('MaxNumSplits = %0.3g', maxNumSplits(mnsPlot(k))))
hold off
end
hL = legend([cellstr(num2str(learnRate','Learning Rate = %0.2f')); ...
'Deep Tree';'Stump';'Min. MSE']);
hL.Position(1) = 0.6;

Each curve contains a minimum cross-validated MSE occurring at the optimal number of trees in the ensemble.

Identify the maximum number of splits, number of trees, and learning rate that yields the lowest MSE overall.

[minErr,minErrIdxLin] = min(error(:));
[idxNumTrees,idxMNS,idxLR] = ind2sub(size(error),minErrIdxLin);
fprintf('\nMin. MSE = %0.5f',minErr)
Min. MSE = 16.77593
fprintf('\nOptimal Parameter Values:\nNum. Trees = %d',idxNumTrees);
Optimal Parameter Values:
Num. Trees = 78
fprintf('\nMaxNumSplits = %d\nLearning Rate = %0.2f\n',...
maxNumSplits(idxMNS),learnRate(idxLR))
MaxNumSplits = 1
Learning Rate = 0.25

Create a predictive ensemble based on the optimal hyperparameters and the entire training set.

tFinal = templateTree('MaxNumSplits',maxNumSplits(idxMNS),'Surrogate','on');
MdlFinal = fitrensemble(Tbl,'MPG','NumLearningCycles',idxNumTrees, ...
'Learners',tFinal,'LearnRate',learnRate(idxLR))
MdlFinal =
RegressionEnsemble
PredictorNames: {'Cylinders'  'Displacement'  'Horsepower'  'Weight'}
ResponseName: 'MPG'
CategoricalPredictors: []
ResponseTransform: 'none'
NumObservations: 94
NumTrained: 78
Method: 'LSBoost'
LearnerNames: {'Tree'}
ReasonForTermination: 'Terminated normally after completing the requested number of training cycles.'
FitInfo: [78×1 double]
FitInfoDescription: {2×1 cell}
Regularization: []

Properties, Methods

MdlFinal is a RegressionEnsemble. To predict the fuel economy of a car given its number of cylinders, volume displaced by the cylinders, horsepower, and weight, you can pass the predictor data and MdlFinal to predict.

Instead of searching optimal values manually by using the cross-validation option ('KFold') and the kfoldLoss function, you can use the 'OptimizeHyperparameters' name-value pair argument. When you specify 'OptimizeHyperparameters', the software finds optimal parameters automatically using Bayesian optimization. The optimal values obtained by using 'OptimizeHyperparameters' can be different from those obtained using manual search.

t = templateTree('Surrogate','on');
mdl = fitrensemble(Tbl,'MPG','Learners',t, ...
'OptimizeHyperparameters',{'NumLearningCycles','LearnRate','MaxNumSplits'})
|====================================================================================================================|
| Iter | Eval   | Objective:  | Objective   | BestSoFar   | BestSoFar   | NumLearningC-|    LearnRate | MaxNumSplits |
|      | result | log(1+loss) | runtime     | (observed)  | (estim.)    | ycles        |              |              |
|====================================================================================================================|
|    1 | Best   |      3.3974 |     0.67187 |      3.3974 |      3.3974 |           26 |     0.072054 |            3 |
|    2 | Accept |      6.0976 |      4.2306 |      3.3974 |      3.5568 |          170 |    0.0010295 |           70 |
|    3 | Best   |      3.2885 |      6.5786 |      3.2885 |      3.2887 |          273 |      0.61026 |            6 |
|    4 | Accept |      6.1839 |      1.8593 |      3.2885 |      3.2885 |           80 |    0.0016871 |            1 |
|    5 | Best   |      3.1395 |     0.26569 |      3.1395 |      3.1394 |           10 |      0.21358 |            2 |
|    6 | Accept |      3.5817 |     0.28615 |      3.1395 |       3.303 |           10 |       0.1666 |            1 |
|    7 | Best   |      3.1268 |     0.31144 |      3.1268 |      3.2143 |           10 |      0.99816 |            3 |
|    8 | Best   |      3.0582 |     0.31361 |      3.0582 |      3.0927 |           10 |      0.97817 |            3 |
|    9 | Best   |      3.0005 |      0.3115 |      3.0005 |      3.0084 |           10 |      0.38895 |            3 |
|   10 | Best   |      2.9744 |     0.29252 |      2.9744 |      2.9894 |           10 |      0.39702 |            3 |
|   11 | Best   |      2.9704 |     0.26561 |      2.9704 |      2.9873 |           10 |      0.34289 |            5 |
|   12 | Accept |      3.2964 |     0.27885 |      2.9704 |      2.9628 |           10 |      0.91391 |           98 |
|   13 | Accept |      3.2164 |     0.32053 |      2.9704 |      2.9629 |           10 |      0.20551 |           20 |
|   14 | Accept |      3.2572 |     0.30051 |      2.9704 |      2.9883 |           10 |      0.95514 |           13 |
|   15 | Best   |      2.9374 |     0.28596 |      2.9374 |      2.9544 |           10 |      0.28703 |            5 |
|   16 | Accept |      2.9642 |     0.26404 |      2.9374 |      2.9571 |           10 |      0.26332 |            5 |
|   17 | Accept |      2.9396 |     0.34495 |      2.9374 |      2.9528 |           10 |      0.28684 |            5 |
|   18 | Accept |      2.9659 |     0.29281 |      2.9374 |      2.9549 |           10 |      0.25925 |            5 |
|   19 | Accept |      2.9378 |     0.29386 |      2.9374 |      2.9532 |           10 |      0.34163 |            4 |
|   20 | Accept |      5.8728 |     0.33079 |      2.9374 |      2.9514 |           10 |     0.028687 |           98 |
|====================================================================================================================|
| Iter | Eval   | Objective:  | Objective   | BestSoFar   | BestSoFar   | NumLearningC-|    LearnRate | MaxNumSplits |
|      | result | log(1+loss) | runtime     | (observed)  | (estim.)    | ycles        |              |              |
|====================================================================================================================|
|   21 | Accept |      3.1123 |     0.30967 |      2.9374 |       2.951 |           10 |      0.95322 |            1 |
|   22 | Accept |      3.1405 |     0.27687 |      2.9374 |      2.9423 |           10 |      0.20618 |            6 |
|   23 | Accept |      2.9494 |     0.59109 |      2.9374 |      2.9415 |           24 |      0.30598 |            5 |
|   24 | Best   |       2.906 |     0.28183 |       2.906 |      2.9064 |           10 |      0.32881 |            1 |
|   25 | Accept |      2.9436 |     0.30565 |       2.906 |      2.9222 |           10 |      0.31006 |            2 |
|   26 | Best   |      2.8822 |     0.25146 |      2.8822 |      2.8912 |           10 |      0.36677 |            1 |
|   27 | Accept |      2.9123 |     0.29838 |      2.8822 |      2.8952 |           10 |      0.39598 |            1 |
|   28 | Accept |      4.6089 |     0.28759 |      2.8822 |       2.901 |           10 |     0.093135 |           98 |
|   29 | Accept |      2.8917 |     0.37785 |      2.8822 |      2.9009 |           15 |      0.33056 |            1 |
|   30 | Accept |      2.9289 |      1.1702 |      2.8822 |      2.9008 |           50 |      0.14154 |            1 |

__________________________________________________________
Optimization completed.
MaxObjectiveEvaluations of 30 reached.
Total function evaluations: 30
Total elapsed time: 60.5324 seconds.
Total objective function evaluation time: 22.2498

Best observed feasible point:
NumLearningCycles    LearnRate    MaxNumSplits
_________________    _________    ____________

10             0.36677          1

Observed objective function value = 2.8822
Estimated objective function value = 2.9008
Function evaluation time = 0.25146

Best estimated feasible point (according to models):
NumLearningCycles    LearnRate    MaxNumSplits
_________________    _________    ____________

10             0.36677          1

Estimated objective function value = 2.9008
Estimated function evaluation time = 0.291
mdl =
RegressionEnsemble
PredictorNames: {'Cylinders'  'Displacement'  'Horsepower'  'Weight'}
ResponseName: 'MPG'
CategoricalPredictors: []
ResponseTransform: 'none'
NumObservations: 94
HyperparameterOptimizationResults: [1×1 BayesianOptimization]
NumTrained: 10
Method: 'LSBoost'
LearnerNames: {'Tree'}
ReasonForTermination: 'Terminated normally after completing the requested number of training cycles.'
FitInfo: [10×1 double]
FitInfoDescription: {2×1 cell}
Regularization: []

Properties, Methods

## Input Arguments

collapse all

Sample data used to train the model, specified as a table. Each row of Tbl corresponds to one observation, and each column corresponds to one predictor variable. Tbl can contain one additional column for the response variable. Multi-column variables and cell arrays other than cell arrays of character vectors are not allowed.

• If Tbl contains the response variable and you want to use all remaining variables as predictors, then specify the response variable using ResponseVarName.

• If Tbl contains the response variable, and you want to use a subset of the remaining variables only as predictors, then specify a formula using formula.

• If Tbl does not contain the response variable, then specify the response data using Y. The length of response variable and the number of rows of Tbl must be equal.

Note

To save memory and execution time, supply X and Y instead of Tbl.

Data Types: table

Response variable name, specified as the name of the response variable in Tbl.

You must specify ResponseVarName as a character vector or string scalar. For example, if Tbl.Y is the response variable, then specify ResponseVarName as 'Y'. Otherwise, fitrensemble treats all columns of Tbl as predictor variables.

Data Types: char | string

Explanatory model of the response variable and a subset of the predictor variables, specified as a character vector or string scalar in the form 'Y~X1+X2+X3'. In this form, Y represents the response variable, and X1, X2, and X3 represent the predictor variables.

To specify a subset of variables in Tbl as predictors for training the model, use a formula. If you specify a formula, then the software does not use any variables in Tbl that do not appear in formula.

The variable names in the formula must be both variable names in Tbl (Tbl.Properties.VariableNames) and valid MATLAB® identifiers.

You can verify the variable names in Tbl by using the isvarname function. The following code returns logical 1 (true) for each variable that has a valid variable name.

cellfun(@isvarname,Tbl.Properties.VariableNames)
If the variable names in Tbl are not valid, then convert them by using the matlab.lang.makeValidName function.
Tbl.Properties.VariableNames = matlab.lang.makeValidName(Tbl.Properties.VariableNames);

Data Types: char | string

Predictor data, specified as numeric matrix.

Each row corresponds to one observation, and each column corresponds to one predictor variable.

The length of Y and the number of rows of X must be equal.

To specify the names of the predictors in the order of their appearance in X, use the PredictorNames name-value pair argument.

Data Types: single | double

Response, specified as a numeric vector. Each element in Y is the response to the observation in the corresponding row of X or Tbl. The length of Y and the number of rows of X or Tbl must be equal.

Data Types: single | double

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'NumLearningCycles',500,'Method','Bag','Learners',templateTree(),'CrossVal','on' cross-validates an ensemble of 500 bagged regression trees using 10-fold cross-validation.

Note

You cannot use any cross-validation name-value pair argument along with the 'OptimizeHyperparameters' name-value pair argument. You can modify the cross-validation for 'OptimizeHyperparameters' only by using the 'HyperparameterOptimizationOptions' name-value pair argument.

General Ensemble Options

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Ensemble aggregation method, specified as the comma-separated pair consisting of 'Method' and 'LSBoost' or 'Bag'.

ValueMethodNotes
'LSBoost'Least-squares boosting (LSBoost)You can specify the learning rate for shrinkage by using the 'LearnRate' name-value pair argument.
'Bag'Bootstrap aggregation (bagging, for example, random forest[2])fitrensemble uses bagging with random predictor selections at each split (random forest) by default. To use bagging without the random selections, use tree learners whose 'NumVariablesToSample' value is 'all'.

For details about ensemble aggregation algorithms and examples, see Algorithms, Ensemble Algorithms, and Choose an Applicable Ensemble Aggregation Method.

Example: 'Method','Bag'

Number of ensemble learning cycles, specified as the comma-separated pair consisting of 'NumLearningCycles' and a positive integer. At every learning cycle, the software trains one weak learner for every template object in Learners. Consequently, the software trains NumLearningCycles*numel(Learners) learners.

The software composes the ensemble using all trained learners and stores them in Mdl.Trained.

For more details, see Tips.

Example: 'NumLearningCycles',500

Data Types: single | double

Weak learners to use in the ensemble, specified as the comma-separated pair consisting of 'Learners' and 'tree', a tree template object, or a cell vector of tree template objects.

• 'tree' (default) — fitrensemble uses default regression tree learners, which is the same as using templateTree(). The default values of templateTree() depend on the value of 'Method'.

• For bagged decision trees, the maximum number of decision splits ('MaxNumSplits') is n–1, where n is the number of observations. The number of predictors to select at random for each split ('NumVariablesToSample') is one third of the number of predictors. Therefore, fitrensemble grows deep decision trees. You can grow shallower trees to reduce model complexity or computation time.

• For boosted decision trees, 'MaxNumSplits' is 10 and 'NumVariablesToSample' is 'all'. Therefore, fitrensemble grows shallow decision trees. You can grow deeper trees for better accuracy.

See templateTree for the default settings of a weak learner.

• Tree template object — fitrensemble uses the tree template object created by templateTree. Use the name-value pair arguments of templateTree to specify settings of the tree learners.

• Cell vector of m tree template objects — fitrensemble grows m regression trees per learning cycle (see NumLearningCycles). For example, for an ensemble composed of two types of regression trees, supply {t1 t2}, where t1 and t2 are regression tree template objects returned by templateTree.

To obtain reproducible results, you must specify the 'Reproducible' name-value pair argument of templateTree as true if 'NumVariablesToSample' is not 'all'.

For details on the number of learners to train, see NumLearningCycles and Tips.

Example: 'Learners',templateTree('MaxNumSplits',5)

Printout frequency, specified as the comma-separated pair consisting of 'NPrint' and a positive integer or 'off'.

To track the number of weak learners or folds that fitrensemble trained so far, specify a positive integer. That is, if you specify the positive integer m:

• Without also specifying any cross-validation option (for example, CrossVal), then fitrensemble displays a message to the command line every time it completes training m weak learners.

• And a cross-validation option, then fitrensemble displays a message to the command line every time it finishes training m folds.

If you specify 'off', then fitrensemble does not display a message when it completes training weak learners.

Tip

When training an ensemble of many weak learners on a large data set, specify a positive integer for NPrint.

Example: 'NPrint',5

Data Types: single | double | char | string

Number of bins for numeric predictors, specified as the comma-separated pair consisting of 'NumBins' and a positive integer scalar.

• If the 'NumBins' value is empty (default), then the software does not bin any predictors.

• If you specify the 'NumBins' value as a positive integer scalar, then the software bins every numeric predictor into a specified number of equiprobable bins, and then grows trees on the bin indices instead of the original data.

• If the 'NumBins' value exceeds the number (u) of unique values for a predictor, then fitrensemble bins the predictor into u bins.

• fitrensemble does not bin categorical predictors.

When you use a large training data set, this binning option speeds up training but causes a potential decrease in accuracy. You can try 'NumBins',50 first, and then change the 'NumBins' value depending on the accuracy and training speed.

A trained model stores the bin edges in the BinEdges property.

Example: 'NumBins',50

Data Types: single | double

Categorical predictors list, specified as the comma-separated pair consisting of 'CategoricalPredictors' and one of the values in this table.

ValueDescription
Vector of positive integersEach entry in the vector is an index value corresponding to the column of the predictor data (X or Tbl) that contains a categorical variable.
Logical vectorA true entry means that the corresponding column of predictor data (X or Tbl) is a categorical variable.
Character matrixEach row of the matrix is the name of a predictor variable. The names must match the entries in PredictorNames. Pad the names with extra blanks so each row of the character matrix has the same length.
String array or cell array of character vectorsEach element in the array is the name of a predictor variable. The names must match the entries in PredictorNames.
'all'All predictors are categorical.

By default, if the predictor data is in a table (Tbl), fitrensemble assumes that a variable is categorical if it is a logical vector, unordered categorical vector, character array, string array, or cell array of character vectors. If the predictor data is a matrix (X), fitrensemble assumes that all predictors are continuous. To identify any other predictors as categorical predictors, specify them by using the 'CategoricalPredictors' name-value pair argument.

Example: 'CategoricalPredictors','all'

Data Types: single | double | logical | char | string | cell

Predictor variable names, specified as the comma-separated pair consisting of 'PredictorNames' and a string array of unique names or cell array of unique character vectors. The functionality of 'PredictorNames' depends on the way you supply the training data.

• If you supply X and Y, then you can use 'PredictorNames' to assign names to the predictor variables in X.

• The order of the names in PredictorNames must correspond to the column order of X. That is, PredictorNames{1} is the name of X(:,1), PredictorNames{2} is the name of X(:,2), and so on. Also, size(X,2) and numel(PredictorNames) must be equal.

• By default, PredictorNames is {'x1','x2',...}.

• If you supply Tbl, then you can use 'PredictorNames' to choose which predictor variables to use in training. That is, fitrensemble uses only the predictor variables in PredictorNames and the response variable during training.

• PredictorNames must be a subset of Tbl.Properties.VariableNames and cannot include the name of the response variable.

• By default, PredictorNames contains the names of all predictor variables.

• A good practice is to specify the predictors for training using either 'PredictorNames' or formula, but not both.

Example: 'PredictorNames',{'SepalLength','SepalWidth','PetalLength','PetalWidth'}

Data Types: string | cell

Response variable name, specified as the comma-separated pair consisting of 'ResponseName' and a character vector or string scalar.

• If you supply Y, then you can use 'ResponseName' to specify a name for the response variable.

• If you supply ResponseVarName or formula, then you cannot use 'ResponseName'.

Example: 'ResponseName','response'

Data Types: char | string

Response transformation, specified as the comma-separated pair consisting of 'ResponseTransform' and either 'none' or a function handle. The default is 'none', which means @(y)y, or no transformation. For a MATLAB function or a function you define, use its function handle. The function handle must accept a vector (the original response values) and return a vector of the same size (the transformed response values).

Example: Suppose you create a function handle that applies an exponential transformation to an input vector by using myfunction = @(y)exp(y). Then, you can specify the response transformation as 'ResponseTransform',myfunction.

Data Types: char | string | function_handle

Cross-Validation Options

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Cross-validation flag, specified as the comma-separated pair consisting of 'Crossval' and 'on' or 'off'.

If you specify 'on', then the software implements 10-fold cross-validation.

To override this cross-validation setting, use one of these name-value pair arguments: CVPartition, Holdout, KFold, or Leaveout. To create a cross-validated model, you can use one cross-validation name-value pair argument at a time only.

Alternatively, cross-validate later by passing Mdl to crossval or crossval.

Example: 'Crossval','on'

Cross-validation partition, specified as the comma-separated pair consisting of 'CVPartition' and a cvpartition partition object created by cvpartition. The partition object specifies the type of cross-validation and the indexing for the training and validation sets.

To create a cross-validated model, you can use one of these four name-value pair arguments only: CVPartition, Holdout, KFold, or Leaveout.

Example: Suppose you create a random partition for 5-fold cross-validation on 500 observations by using cvp = cvpartition(500,'KFold',5). Then, you can specify the cross-validated model by using 'CVPartition',cvp.

Fraction of the data used for holdout validation, specified as the comma-separated pair consisting of 'Holdout' and a scalar value in the range (0,1). If you specify 'Holdout',p, then the software completes these steps:

1. Randomly select and reserve p*100% of the data as validation data, and train the model using the rest of the data.

2. Store the compact, trained model in the Trained property of the cross-validated model.

To create a cross-validated model, you can use one of these four name-value pair arguments only: CVPartition, Holdout, KFold, or Leaveout.

Example: 'Holdout',0.1

Data Types: double | single

Number of folds to use in a cross-validated model, specified as the comma-separated pair consisting of 'KFold' and a positive integer value greater than 1. If you specify 'KFold',k, then the software completes these steps:

1. Randomly partition the data into k sets.

2. For each set, reserve the set as validation data, and train the model using the other k – 1 sets.

3. Store the k compact, trained models in the cells of a k-by-1 cell vector in the Trained property of the cross-validated model.

To create a cross-validated model, you can use one of these four name-value pair arguments only: CVPartition, Holdout, KFold, or Leaveout.

Example: 'KFold',5

Data Types: single | double

Leave-one-out cross-validation flag, specified as the comma-separated pair consisting of 'Leaveout' and 'on' or 'off'. If you specify 'Leaveout','on', then, for each of the n observations (where n is the number of observations excluding missing observations, specified in the NumObservations property of the model), the software completes these steps:

1. Reserve the observation as validation data, and train the model using the other n – 1 observations.

2. Store the n compact, trained models in the cells of an n-by-1 cell vector in the Trained property of the cross-validated model.

To create a cross-validated model, you can use one of these four name-value pair arguments only: CVPartition, Holdout, KFold, or Leaveout.

Example: 'Leaveout','on'

Other Regression Options

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Observation weights, specified as the comma-separated pair consisting of 'Weights' and a numeric vector of positive values or name of a variable in Tbl. The software weighs the observations in each row of X or Tbl with the corresponding value in Weights. The size of Weights must equal the number of rows of X or Tbl.

If you specify the input data as a table Tbl, then Weights can be the name of a variable in Tbl that contains a numeric vector. In this case, you must specify Weights as a character vector or string scalar. For example, if the weights vector W is stored as Tbl.W, then specify it as 'W'. Otherwise, the software treats all columns of Tbl, including W, as predictors or the response when training the model.

The software normalizes the values of Weights to sum to 1.

By default, Weights is ones(n,1), where n is the number of observations in X or Tbl.

Data Types: double | single | char | string

Sampling Options

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Fraction of the training set to resample for every weak learner, specified as the comma-separated pair consisting of 'FResample' and a positive scalar in (0,1].

To use 'FResample', specify 'bag' for Method or set Resample to 'on'.

Example: 'FResample',0.75

Data Types: single | double

Flag indicating sampling with replacement, specified as the comma-separated pair consisting of 'Replace' and 'off' or 'on'.

• For 'on', the software samples the training observations with replacement.

• For 'off', the software samples the training observations without replacement. If you set Resample to 'on', then the software samples training observations assuming uniform weights. If you also specify a boosting method, then the software boosts by reweighting observations.

Unless you set Method to 'bag' or set Resample to 'on', Replace has no effect.

Example: 'Replace','off'

Flag indicating to resample, specified as the comma-separated pair consisting of 'Resample' and 'off' or 'on'.

• If Method is a boosting method, then:

• 'Resample','on' specifies to sample training observations using updated weights as the multinomial sampling probabilities.

• 'Resample','off'(default) specifies to reweight observations at every learning iteration.

• If Method is 'bag', then 'Resample' must be 'on'. The software resamples a fraction of the training observations (see FResample) with or without replacement (see Replace).

If you specify to resample using Resample, then it is good practice to resample to entire data set. That is, use the default setting of 1 for FResample.

LSBoost Method Options

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Learning rate for shrinkage, specified as the comma-separated pair consisting of a numeric scalar in the interval (0,1].

To train an ensemble using shrinkage, set LearnRate to a value less than 1, for example, 0.1 is a popular choice. Training an ensemble using shrinkage requires more learning iterations, but often achieves better accuracy.

Example: 'LearnRate',0.1

Data Types: single | double

Hyperparameter Optimization Options

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Parameters to optimize, specified as the comma-separated pair consisting of 'OptimizeHyperparameters' and one of the following:

• 'none' — Do not optimize.

• 'auto' — Use {'Method','NumLearningCycles','LearnRate'} along with the default parameters for the specified Learners:

• Learners = 'tree' (default) — {'MinLeafSize'}

Note

For hyperparameter optimization, Learners must be a single argument, not a string array or cell array.

• 'all' — Optimize all eligible parameters.

• String array or cell array of eligible parameter names

• Vector of optimizableVariable objects, typically the output of hyperparameters

The optimization attempts to minimize the cross-validation loss (error) for fitrensemble by varying the parameters. To control the cross-validation type and other aspects of the optimization, use the HyperparameterOptimizationOptions name-value pair.

Note

'OptimizeHyperparameters' values override any values you set using other name-value pair arguments. For example, setting 'OptimizeHyperparameters' to 'auto' causes the 'auto' values to apply.

The eligible parameters for fitrensemble are:

• Method — Eligible methods are 'Bag' or 'LSBoost'.

• NumLearningCyclesfitrensemble searches among positive integers, by default log-scaled with range [10,500].

• LearnRatefitrensemble searches among positive reals, by default log-scaled with range [1e-3,1].

• MinLeafSizefitrensemble searches among integers log-scaled in the range [1,max(2,floor(NumObservations/2))].

• MaxNumSplitsfitrensemble searches among integers log-scaled in the range [1,max(2,NumObservations-1)].

• NumVariablesToSamplefitrensemble searches among integers in the range [1,max(2,NumPredictors)].

Set nondefault parameters by passing a vector of optimizableVariable objects that have nondefault values. For example,

params = hyperparameters('fitrensemble',[Horsepower,Weight],MPG,'Tree');
params(4).Range = [1,20];

Pass params as the value of OptimizeHyperparameters.

By default, iterative display appears at the command line, and plots appear according to the number of hyperparameters in the optimization. For the optimization and plots, the objective function is log(1 + cross-validation loss) for regression and the misclassification rate for classification. To control the iterative display, set the Verbose field of the 'HyperparameterOptimizationOptions' name-value pair argument. To control the plots, set the ShowPlots field of the 'HyperparameterOptimizationOptions' name-value pair argument.

For an example, see Optimize Regression Ensemble.

Example: 'OptimizeHyperparameters',{'Method','NumLearningCycles','LearnRate','MinLeafSize','MaxNumSplits'}

Options for optimization, specified as the comma-separated pair consisting of 'HyperparameterOptimizationOptions' and a structure. This argument modifies the effect of the OptimizeHyperparameters name-value pair argument. All fields in the structure are optional.

Field NameValuesDefault
Optimizer
• 'bayesopt' — Use Bayesian optimization. Internally, this setting calls bayesopt.

• 'gridsearch' — Use grid search with NumGridDivisions values per dimension.

• 'randomsearch' — Search at random among MaxObjectiveEvaluations points.

'gridsearch' searches in a random order, using uniform sampling without replacement from the grid. After optimization, you can get a table in grid order by using the command sortrows(Mdl.HyperparameterOptimizationResults).

'bayesopt'
AcquisitionFunctionName

• 'expected-improvement-per-second-plus'

• 'expected-improvement'

• 'expected-improvement-plus'

• 'expected-improvement-per-second'

• 'lower-confidence-bound'

• 'probability-of-improvement'

Acquisition functions whose names include per-second do not yield reproducible results because the optimization depends on the runtime of the objective function. Acquisition functions whose names include plus modify their behavior when they are overexploiting an area. For more details, see Acquisition Function Types.

'expected-improvement-per-second-plus'
MaxObjectiveEvaluationsMaximum number of objective function evaluations.30 for 'bayesopt' or 'randomsearch', and the entire grid for 'gridsearch'
MaxTime

Time limit, specified as a positive real. The time limit is in seconds, as measured by tic and toc. Run time can exceed MaxTime because MaxTime does not interrupt function evaluations.

Inf
NumGridDivisionsFor 'gridsearch', the number of values in each dimension. The value can be a vector of positive integers giving the number of values for each dimension, or a scalar that applies to all dimensions. This field is ignored for categorical variables.10
ShowPlotsLogical value indicating whether to show plots. If true, this field plots the best objective function value against the iteration number. If there are one or two optimization parameters, and if Optimizer is 'bayesopt', then ShowPlots also plots a model of the objective function against the parameters.true
SaveIntermediateResultsLogical value indicating whether to save results when Optimizer is 'bayesopt'. If true, this field overwrites a workspace variable named 'BayesoptResults' at each iteration. The variable is a BayesianOptimization object.false
Verbose

Display to the command line.

• 0 — No iterative display

• 1 — Iterative display

• 2 — Iterative display with extra information

For details, see the bayesopt Verbose name-value pair argument.

1
UseParallelLogical value indicating whether to run Bayesian optimization in parallel, which requires Parallel Computing Toolbox™. Due to the nonreproducibility of parallel timing, parallel Bayesian optimization does not necessarily yield reproducible results. For details, see Parallel Bayesian Optimization.false
Repartition

Logical value indicating whether to repartition the cross-validation at every iteration. If false, the optimizer uses a single partition for the optimization.

true usually gives the most robust results because this setting takes partitioning noise into account. However, for good results, true requires at least twice as many function evaluations.

false
Use no more than one of the following three field names.
CVPartitionA cvpartition object, as created by cvpartition.'Kfold',5 if you do not specify any cross-validation field
HoldoutA scalar in the range (0,1) representing the holdout fraction.
KfoldAn integer greater than 1.

Example: 'HyperparameterOptimizationOptions',struct('MaxObjectiveEvaluations',60)

Data Types: struct

## Output Arguments

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Trained ensemble model, returned as one of the model objects in this table.

Model ObjectSpecify Any Cross-Validation Options?Method SettingResample Setting
RegressionBaggedEnsembleNo'Bag''on'
RegressionEnsembleNo'LSBoost''off'
RegressionPartitionedEnsembleYes'LSBoost' or 'Bag''off' or 'on'

The name-value pair arguments that control cross-validation are CrossVal, Holdout, KFold, Leaveout, and CVPartition.

To reference properties of Mdl, use dot notation. For example, to access or display the cell vector of weak learner model objects for an ensemble that has not been cross-validated, enter Mdl.Trained at the command line.

## Tips

• NumLearningCycles can vary from a few dozen to a few thousand. Usually, an ensemble with good predictive power requires from a few hundred to a few thousand weak learners. However, you do not have to train an ensemble for that many cycles at once. You can start by growing a few dozen learners, inspect the ensemble performance and then, if necessary, train more weak learners using resume.

• Ensemble performance depends on the ensemble setting and the setting of the weak learners. That is, if you specify weak learners with default parameters, then the ensemble can perform poorly. Therefore, like ensemble settings, it is good practice to adjust the parameters of the weak learners using templates, and to choose values that minimize generalization error.

• If you specify to resample using Resample, then it is good practice to resample to entire data set. That is, use the default setting of 1 for FResample.

• After training a model, you can generate C/C++ code that predicts responses for new data. Generating C/C++ code requires MATLAB Coder™. For details, see Introduction to Code Generation.

## Algorithms

• For details of ensemble aggregation algorithms, see Ensemble Algorithms.

• If you specify 'Method','LSBoost', then the software grows shallow decision trees by default. You can adjust tree depth by specifying the MaxNumSplits, MinLeafSize, and MinParentSize name-value pair arguments using templateTree.

• For dual-core systems and above, fitrensemble parallelizes training using Intel® Threading Building Blocks (TBB). For details on Intel TBB, see https://software.intel.com/en-us/intel-tbb.

## References

[1] Breiman, L. “Bagging Predictors.” Machine Learning. Vol. 26, pp. 123–140, 1996.

[2] Breiman, L. “Random Forests.” Machine Learning. Vol. 45, pp. 5–32, 2001.

[3] Freund, Y. and R. E. Schapire. “A Decision-Theoretic Generalization of On-Line Learning and an Application to Boosting.” J. of Computer and System Sciences, Vol. 55, pp. 119–139, 1997.

[4] Friedman, J. “Greedy function approximation: A gradient boosting machine.” Annals of Statistics, Vol. 29, No. 5, pp. 1189–1232, 2001.

[5] Hastie, T., R. Tibshirani, and J. Friedman. The Elements of Statistical Learning section edition, Springer, New York, 2008.