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# fitrensemble

Fit ensemble of learners for regression

## Syntax

``Mdl = fitrensemble(Tbl,ResponseVarName)``
``Mdl = fitrensemble(Tbl,formula)``
``Mdl = fitrensemble(Tbl,Y)``
``Mdl = fitrensemble(X,Y)``
``Mdl = fitrensemble(___,Name,Value)``

## 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.

```load carsmall 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 `NaN`s, then the corresponding `Xbinned` values are `NaN`s.

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.

```load carsmall 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 = 24.8521 ```

`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.

Load the data.

`load carsmall`

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 | | runtime | (observed) | (estim.) | | ycles | | | |===================================================================================================================================| | 1 | Best | 2.9743 | 9.9614 | 2.9743 | 2.9743 | Bag | 413 | - | 1 | | 2 | Accept | 6.2619 | 1.5009 | 2.9743 | 3.6147 | LSBoost | 57 | 0.0016067 | 6 | | 3 | Best | 2.9562 | 0.77935 | 2.9562 | 2.9654 | Bag | 32 | - | 2 | | 4 | Accept | 4.1884 | 1.2819 | 2.9562 | 2.9573 | Bag | 55 | - | 40 | | 5 | Accept | 4.1881 | 6.0891 | 2.9562 | 2.957 | LSBoost | 258 | 0.096683 | 50 | | 6 | Accept | 3.1479 | 1.8558 | 2.9562 | 2.9569 | LSBoost | 75 | 0.092525 | 6 | | 7 | Accept | 3.7831 | 0.4946 | 2.9562 | 2.9568 | LSBoost | 17 | 0.08882 | 1 | | 8 | Accept | 4.2759 | 0.38923 | 2.9562 | 2.9564 | LSBoost | 12 | 0.094908 | 1 | | 9 | Accept | 3.124 | 4.3066 | 2.9562 | 2.9564 | LSBoost | 157 | 0.083222 | 8 | | 10 | Accept | 3.6349 | 12.142 | 2.9562 | 2.991 | LSBoost | 494 | 0.070612 | 2 | | 11 | Accept | 3.0526 | 0.26891 | 2.9562 | 2.9615 | Bag | 10 | - | 1 | | 12 | Accept | 3.1469 | 1.7301 | 2.9562 | 2.9627 | LSBoost | 62 | 0.08423 | 4 | | 13 | Accept | 4.7629 | 0.50284 | 2.9562 | 2.9629 | LSBoost | 21 | 0.055294 | 48 | | 14 | Accept | 4.1881 | 4.5671 | 2.9562 | 2.9629 | LSBoost | 213 | 0.99467 | 46 | | 15 | Accept | 3.5337 | 1.597 | 2.9562 | 3.0164 | LSBoost | 63 | 0.73987 | 1 | | 16 | Accept | 3.2481 | 1.3843 | 2.9562 | 2.9689 | LSBoost | 59 | 0.2363 | 6 | | 17 | Accept | 2.9732 | 1.7983 | 2.9562 | 3.0016 | Bag | 83 | - | 1 | | 18 | Best | 2.9361 | 10.498 | 2.9361 | 2.9648 | Bag | 498 | - | 2 | | 19 | Accept | 3.5859 | 2.9294 | 2.9361 | 2.9987 | LSBoost | 123 | 0.1522 | 2 | | 20 | Accept | 2.9406 | 10.689 | 2.9361 | 2.9559 | Bag | 496 | - | 2 | |===================================================================================================================================| | Iter | Eval | Objective | Objective | BestSoFar | BestSoFar | Method | NumLearningC-| LearnRate | MinLeafSize | | | result | | runtime | (observed) | (estim.) | | ycles | | | |===================================================================================================================================| | 21 | Accept | 2.9419 | 10.833 | 2.9361 | 2.9517 | Bag | 491 | - | 2 | | 22 | Accept | 5.7918 | 1.0162 | 2.9361 | 2.9525 | LSBoost | 35 | 0.0094806 | 1 | | 23 | Accept | 5.9584 | 5.718 | 2.9361 | 2.9533 | LSBoost | 272 | 0.0010255 | 50 | | 24 | Accept | 6.4218 | 0.29731 | 2.9361 | 2.9538 | LSBoost | 10 | 0.0010003 | 1 | | 25 | Accept | 4.7662 | 4.7116 | 2.9361 | 2.9531 | LSBoost | 202 | 0.0058776 | 50 | | 26 | Accept | 3.4231 | 4.8838 | 2.9361 | 2.9537 | LSBoost | 208 | 0.99319 | 6 | | 27 | Accept | 5.6829 | 0.34583 | 2.9361 | 2.954 | LSBoost | 13 | 0.02954 | 1 | | 28 | Accept | 4.7428 | 1.5779 | 2.9361 | 2.954 | LSBoost | 72 | 0.01677 | 50 | | 29 | Accept | 5.9554 | 1.8831 | 2.9361 | 2.9544 | LSBoost | 78 | 0.0031809 | 1 | | 30 | Accept | 4.1881 | 1.4461 | 2.9361 | 2.9547 | LSBoost | 62 | 0.35508 | 50 | __________________________________________________________ Optimization completed. MaxObjectiveEvaluations of 30 reached. Total function evaluations: 30 Total elapsed time: 167.549 seconds. Total objective function evaluation time: 107.4784 Best observed feasible point: Method NumLearningCycles LearnRate MinLeafSize ______ _________________ _________ ___________ Bag 498 NaN 2 Observed objective function value = 2.9361 Estimated objective function value = 2.9547 Function evaluation time = 10.4983 Best estimated feasible point (according to models): Method NumLearningCycles LearnRate MinLeafSize ______ _________________ _________ ___________ Bag 491 NaN 2 Estimated objective function value = 2.9547 Estimated function evaluation time = 10.8111 ```
```Mdl = classreg.learning.regr.RegressionBaggedEnsemble ResponseName: 'Y' CategoricalPredictors: [] ResponseTransform: 'none' NumObservations: 94 HyperparameterOptimizationResults: [1×1 BayesianOptimization] NumTrained: 491 Method: 'Bag' LearnerNames: {'Tree'} ReasonForTermination: 'Terminated normally after completing the requested number of training cycles.' FitInfo: [] FitInfoDescription: 'None' Regularization: [] FResample: 1 Replace: 1 UseObsForLearner: [94×491 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.

```load carsmall 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 = 17.01148 ```
`fprintf('\nOptimal Parameter Values:\nNum. Trees = %d',idxNumTrees);`
```Optimal Parameter Values: Num. Trees = 38 ```
```fprintf('\nMaxNumSplits = %d\nLearning Rate = %0.2f\n',... maxNumSplits(idxMNS),learnRate(idxLR))```
```MaxNumSplits = 4 Learning Rate = 0.10 ```

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 = classreg.learning.regr.RegressionEnsemble PredictorNames: {'Cylinders' 'Displacement' 'Horsepower' 'Weight'} ResponseName: 'MPG' CategoricalPredictors: [] ResponseTransform: 'none' NumObservations: 94 NumTrained: 38 Method: 'LSBoost' LearnerNames: {'Tree'} ReasonForTermination: 'Terminated normally after completing the requested number of training cycles.' FitInfo: [38×1 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 | | runtime | (observed) | (estim.) | ycles | | | |====================================================================================================================| | 1 | Best | 3.3989 | 0.702 | 3.3989 | 3.3989 | 26 | 0.072054 | 3 | | 2 | Accept | 6.0978 | 4.5175 | 3.3989 | 3.5582 | 170 | 0.0010295 | 70 | | 3 | Best | 3.2848 | 7.3601 | 3.2848 | 3.285 | 273 | 0.61026 | 6 | | 4 | Accept | 6.1839 | 1.9371 | 3.2848 | 3.2849 | 80 | 0.0016871 | 1 | | 5 | Best | 3.0154 | 0.33314 | 3.0154 | 3.016 | 12 | 0.21457 | 2 | | 6 | Accept | 3.3146 | 0.32962 | 3.0154 | 3.16 | 10 | 0.18164 | 8 | | 7 | Accept | 3.0512 | 0.31776 | 3.0154 | 3.1213 | 10 | 0.26719 | 16 | | 8 | Best | 3.0013 | 0.29751 | 3.0013 | 3.0891 | 10 | 0.27408 | 1 | | 9 | Best | 2.9797 | 0.31657 | 2.9797 | 2.9876 | 10 | 0.28184 | 2 | | 10 | Accept | 3.0646 | 0.59906 | 2.9797 | 3.0285 | 23 | 0.9922 | 1 | | 11 | Accept | 2.9825 | 0.31056 | 2.9797 | 2.978 | 10 | 0.54187 | 1 | | 12 | Best | 2.9526 | 0.31189 | 2.9526 | 2.9509 | 10 | 0.49116 | 1 | | 13 | Best | 2.9281 | 0.98544 | 2.9281 | 2.9539 | 38 | 0.30709 | 1 | | 14 | Accept | 2.944 | 0.60449 | 2.9281 | 2.9305 | 20 | 0.40583 | 1 | | 15 | Best | 2.9128 | 0.50753 | 2.9128 | 2.9237 | 20 | 0.39672 | 1 | | 16 | Best | 2.9077 | 0.58614 | 2.9077 | 2.919 | 21 | 0.38157 | 1 | | 17 | Accept | 3.3932 | 0.36691 | 2.9077 | 2.919 | 10 | 0.97862 | 99 | | 18 | Accept | 6.2938 | 0.33318 | 2.9077 | 2.9204 | 10 | 0.0074886 | 95 | | 19 | Accept | 3.0049 | 0.4842 | 2.9077 | 2.9114 | 15 | 0.45073 | 9 | | 20 | Best | 2.9072 | 2.2091 | 2.9072 | 2.9111 | 87 | 0.152 | 1 | |====================================================================================================================| | Iter | Eval | Objective | Objective | BestSoFar | BestSoFar | NumLearningC-| LearnRate | MaxNumSplits | | | result | | runtime | (observed) | (estim.) | ycles | | | |====================================================================================================================| | 21 | Accept | 2.9217 | 0.53329 | 2.9072 | 2.9154 | 21 | 0.31845 | 1 | | 22 | Best | 2.8994 | 4.8548 | 2.8994 | 2.917 | 189 | 0.098534 | 1 | | 23 | Accept | 2.9055 | 7.6922 | 2.8994 | 2.9166 | 310 | 0.15505 | 1 | | 24 | Accept | 2.9264 | 1.4456 | 2.8994 | 2.9012 | 61 | 0.23387 | 1 | | 25 | Accept | 3.0869 | 0.34623 | 2.8994 | 2.9169 | 10 | 0.48674 | 27 | | 26 | Best | 2.8942 | 7.7285 | 2.8942 | 2.8912 | 319 | 0.11093 | 1 | | 27 | Accept | 3.0175 | 0.97891 | 2.8942 | 2.8889 | 38 | 0.32187 | 4 | | 28 | Accept | 2.9049 | 3.4053 | 2.8942 | 2.8941 | 141 | 0.13325 | 1 | | 29 | Accept | 3.0477 | 0.30373 | 2.8942 | 2.8939 | 10 | 0.28155 | 97 | | 30 | Accept | 3.0563 | 0.3135 | 2.8942 | 2.894 | 10 | 0.53101 | 3 | __________________________________________________________ Optimization completed. MaxObjectiveEvaluations of 30 reached. Total function evaluations: 30 Total elapsed time: 91.847 seconds. Total objective function evaluation time: 51.012 Best observed feasible point: NumLearningCycles LearnRate MaxNumSplits _________________ _________ ____________ 319 0.11093 1 Observed objective function value = 2.8942 Estimated objective function value = 2.894 Function evaluation time = 7.7285 Best estimated feasible point (according to models): NumLearningCycles LearnRate MaxNumSplits _________________ _________ ____________ 319 0.11093 1 Estimated objective function value = 2.894 Estimated function evaluation time = 7.9767 ```
```mdl = classreg.learning.regr.RegressionEnsemble PredictorNames: {'Cylinders' 'Displacement' 'Horsepower' 'Weight'} ResponseName: 'MPG' CategoricalPredictors: [] ResponseTransform: 'none' NumObservations: 94 HyperparameterOptimizationResults: [1×1 BayesianOptimization] NumTrained: 319 Method: 'LSBoost' LearnerNames: {'Tree'} ReasonForTermination: 'Terminated normally after completing the requested number of training cycles.' FitInfo: [319×1 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 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. The variables must be variable names in `Tbl` (`Tbl.Properties.VariableNames`).

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`.

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)`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`.

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 integersAn entry in the vector is the 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 contains logical values, categorical values, a string array, or a cell array of character vectors. If the predictor data is a matrix (`X`), `fitrensemble` assumes all predictors are continuous. To identify any categorical predictors when the data is a matrix, use 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 give the predictor variables in `X` names.

• 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 in 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.

• It is a good practice to specify the predictors for training using either `'PredictorNames'` or `formula` only.

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.

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 `Weights` to sum up to the value of the prior probability in the respective class.

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

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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'`.

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

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

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

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

• `NumVariablesToSample``fitrensemble` 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,

```load carsmall 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'`
`MaxObjectiveEvaluations`Maximum 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`
`NumGridDivisions`For `'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`
`ShowPlots`Logical 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`
`SaveIntermediateResults`Logical 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`
`UseParallel`Logical value indicating whether to run Bayesian optimization in parallel, which requires Parallel Computing Toolbox™. 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.
`CVPartition`A `cvpartition` object, as created by `cvpartition`.`'Kfold',5` if you do not specify any cross-validation field
`Holdout`A scalar in the range `(0,1)` representing the holdout fraction.
`Kfold`An 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` Setting`Resample` Setting
`RegressionBaggedEnsemble`No`'Bag'``'on'`
`RegressionEnsemble`No`'LSBoost'``'off'`
`RegressionPartitionedEnsemble`Yes`'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.

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

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

 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.

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

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

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