Fit ensemble of learners for regression

returns the trained regression ensemble model object (`Mdl`

= fitrensemble(`Tbl`

,`ResponseVarName`

)`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`

.

applies `Mdl`

= fitrensemble(`Tbl`

,`formula`

)`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`

.

uses additional options specified by one or more `Mdl`

= fitrensemble(___,`Name,Value`

)`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.

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 = 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.

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 | 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.

```
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:

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

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.

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 $$\{{2}^{0},{2}^{1},...,{2}^{m}\}$$.

*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

`Tbl`

— Sample datatable

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`

`ResponseVarName`

— Response variable namename of response variable in

`Tbl`

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`

`formula`

— Explanatory model of response variable and subset of predictor variablescharacter vector | string scalar

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)

`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`

`X`

— Predictor datanumeric matrix

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`

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`

.

`'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.

`'Method'`

— Ensemble aggregation method`'LSBoost'`

(default) | `'Bag'`

Ensemble aggregation method, specified as the comma-separated pair
consisting of `'Method'`

and
`'LSBoost'`

or
`'Bag'`

.

Value | Method | Notes |
---|---|---|

`'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'`

`'NumLearningCycles'`

— Number of ensemble learning cycles`100`

(default) | positive integerNumber 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`

`'Learners'`

— Weak learners to use in ensemble`'tree'`

(default) | tree template object | cell vector of tree template objectsWeak 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)`

`'NPrint'`

— Printout frequency`'off'`

(default) | positive integerPrintout 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`

`'NumBins'`

— Number of bins for numeric predictors`[]`

(empty) (default) | positive integer scalarNumber 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`

`'CategoricalPredictors'`

— Categorical predictors listvector of positive integers | logical vector | character matrix | string array | cell array of character vectors |

`'all'`

Categorical predictors
list, specified as the comma-separated pair consisting of
`'CategoricalPredictors'`

and one of the values in this table.

Value | Description |
---|---|

Vector of positive integers | Each 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 vector | A `true` entry means that the corresponding column of predictor
data (`X` or `Tbl` ) is a categorical
variable. |

Character matrix | Each 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 vectors | Each 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`

`'PredictorNames'`

— Predictor variable namesstring array of unique names | cell array of unique character vectors

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`

`'ResponseName'`

— Response variable name`'Y'`

(default) | character vector | string scalarResponse 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`

`'ResponseTransform'`

— Response transformation`'none'`

(default) | function handleResponse 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`

`'CrossVal'`

— Cross-validation flag`'off'`

(default) | `'on'`

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'`

`'CVPartition'`

— Cross-validation partition`[]`

(default) | `cvpartition`

partition objectCross-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`

.

`'Holdout'`

— Fraction of data for holdout validationscalar value in the range (0,1)

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:

Randomly select and reserve

`p*100`

% of the data as validation data, and train the model using the rest of the data.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`

`'KFold'`

— Number of folds`10`

(default) | positive integer value greater than 1Number 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:

Randomly partition the data into

`k`

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

`k`

– 1 sets.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`

`'Leaveout'`

— Leave-one-out cross-validation flag`'off'`

(default) | `'on'`

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:

Reserve the observation as validation data, and train the model using the other

*n*– 1 observations.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.

`CVPartition`

, `Holdout`

, `KFold`

,
or `Leaveout`

.

**Example: **`'Leaveout','on'`

`'Weights'`

— Observation weightsnumeric vector of positive values | name of variable in

`Tbl`

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(`

, where
* n*,1)

`n`

`X`

or `Tbl`

.**Data Types: **`double`

| `single`

| `char`

| `string`

`'FResample'`

— Fraction of training set to resample`1`

(default) | positive scalar in (0,1]`'Replace'`

— Flag indicating to sample with replacement`'on'`

(default) | `'off'`

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'`

`'Resample'`

— Flag indicating to resample`'off'`

| `'on'`

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`

.

`'LearnRate'`

— Learning rate for shrinkage`1`

(default) | numeric scalar in (0,1]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`

`'OptimizeHyperparameters'`

— Parameters to optimize`'none'`

(default) | `'auto'`

| `'all'`

| string array or cell array of eligible parameter names | vector of `optimizableVariable`

objectsParameters 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'}`

`'HyperparameterOptimizationOptions'`

— Options for optimizationstructure

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 Name | Values | Default |
---|---|---|

`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.
| `'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
| `'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 | `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
| `1` |

`UseParallel` | Logical 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` |

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`

`Mdl`

— Trained regression ensemble model`RegressionBaggedEnsemble`

model object | `RegressionEnsemble`

model object | `RegressionPartitionedEnsemble`

cross-validated model
objectTrained ensemble model, returned as one of the model objects in this table.

Model Object | Specify 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.

`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.

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.

[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.

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

To perform parallel hyperparameter optimization, use the
`'HyperparameterOptimizationOptions', struct('UseParallel',true)`

name-value pair argument in the call to this function.

For more information on parallel hyperparameter optimization, see Parallel Bayesian Optimization.

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

`predict`

| `RegressionBaggedEnsemble`

| `RegressionEnsemble`

| `RegressionPartitionedEnsemble`

| `templateTree`

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