Create decision tree template

returns
a default decision tree learner template suitable for training ensembles
or error-correcting output code (ECOC) multiclass models. Specify `t`

= templateTree`t`

as
a learner using:

`fitensemble`

for classification or regression ensembles`fitcecoc`

for ECOC model classification

If you specify a default decision tree template, then
the software uses default values for all input arguments during training.
It is good practice to specify the type of decision tree, e.g., for
a classification tree template, specify `'Type','classification'`

.
If you specify the type of decision tree and display `t`

in
the Command Window, then all options except `Type`

appear
empty (`[]`

).

creates
a template with additional options specified by one or more name-value
pair arguments.`t`

= templateTree(`Name,Value`

)

For example, you can specify the algorithm used to find the best split on a categorical predictor, the split criterion, or the number of predictors selected for each split.

If you display `t`

in the Command Window, then
all options appear empty (`[]`

), except those that
you specify using name-value pair arguments. During training, the
software uses default values for empty options.

Create a decision tree template with surrogate splits, and use the template to train an ensemble using sample data.

Load Fisher's iris data set.

```
load fisheriris
```

Create a decision tree template with surrogate splits.

t = templateTree('Surrogate','on')

t = Fit template for Tree. Surrogate: 'on'

All options of the template object are empty except for `Surrogate`

. When you pass `t`

to the training function, the software fills in the empty options with their respective default values.

Specify `t`

as a weak learner for a classification ensemble.

```
Mdl = fitensemble(meas,species,'AdaBoostM2',100,t)
```

Mdl = classreg.learning.classif.ClassificationEnsemble ResponseName: 'Y' CategoricalPredictors: [] ClassNames: {'setosa' 'versicolor' 'virginica'} ScoreTransform: 'none' NumObservations: 150 NumTrained: 100 Method: 'AdaBoostM2' LearnerNames: {'Tree'} ReasonForTermination: 'Terminated normally after completing the reque...' FitInfo: [100x1 double] FitInfoDescription: {2x1 cell}

Display the in-sample (resubstitution) misclassification error.

L = resubLoss(Mdl)

L = 0.0333

Use a trained, boosted regression tree ensemble to predict the fuel economy of a car. Choose the number of cylinders, volume displaced by the cylinders, horsepower, and weight as predictors.

Load the `carsmall`

data set. Set the predictors to `X`

.

load carsmall X = [Cylinders,Displacement,Horsepower,Weight]; xnames = {'Cylinders','Displacement','Horsepower','Weight'};

Specify a regression tree template that uses surrogate splits to impove predictive accuracy in the presence of `NaN`

values.

RegTreeTemp = templateTree('Surrogate','On');

Train the regression tree ensemble using LSBoost and 100 learning cycles.

RegTreeEns = fitensemble(X,MPG,'LSBoost',100,RegTreeTemp,... 'PredictorNames',xnames);

`RegTreeEns`

is a trained `RegressionEnsemble`

regression ensemble.

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.

predMPG = predict(RegTreeEns,[4 200 150 3000])

predMPG = 22.6290

The average fuel economy of a car with these specifications is 21.78 mpg.

You can control the depth of the trees in an ensemble of decision trees. You can also control the tree depth in an ECOC model containing decision tree binary learners using the `MaxNumSplits`

, `MinLeafSize`

, or `MinParentSize`

name-value pair parameters.

When bagging decision trees,

`fitensemble`

grows deep decision trees by default. You can grow shallower trees to reduce model complexity or computation time.When boosting decision trees, fitensemble grows stumps (a tree with one split) by default. You can grow deeper trees for better accuracy.

Load the `carsmall`

data set. Specify the variables `Acceleration`

, `Displacement`

, `Horsepower`

, and `Weight`

as predictors, and `MPG`

as the response.

```
load carsmall
X = [Acceleration Displacement Horsepower Weight];
Y = MPG;
```

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

`1`

for`MaxNumSplits`

. This option grows stumps.`5`

for`MinLeafSize`

`10`

for`MinParentSize`

To search for the optimal number of splits:

Train a set of ensembles. Exponentially increase the maximum number of splits for subsequent ensembles from stump to at most

*n*- 1 splits. Also, decrease the learning rate for each ensemble from 1 to 0.1.Cross validate the ensembles.

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

Compare the cross-validated MSEs. The ensemble with the lowest one performs the best, and indicates the optimal maximum number of splits, number of trees, and learning rate for the data set.

Grow and cross validate a deep classification tree and a stump. Specify to use surrogate splits because the data contain missing values. These serve as benchmarks.

MdlDeep = fitrtree(X,Y,'CrossVal','on','MergeLeaves','off',... 'MinParentSize',1,'Surrogate','on'); MdlStump = fitrtree(X,Y,'MaxNumSplits',1,'CrossVal','on','Surrogate','on');

Train the boosting ensembles using 150 regression trees. Cross validate the ensemble using 5-fold cross validation. Vary the maximum number of splits using the values in the sequence
, where *m* is such that
is no greater than *n* - 1. For each variant, adjust the learning rate to each value in the set {0.1, 0.25, 0.5, 1};

n = size(X,1); m = floor(log2(n - 1)); lr = [0.1 0.25 0.5 1]; maxNumSplits = 2.^(0:m); numTrees = 150; Mdl = cell(numel(maxNumSplits),numel(lr)); rng(1); % For reproducibility for k = 1:numel(lr); for j = 1:numel(maxNumSplits); t = templateTree('MaxNumSplits',maxNumSplits(j),'Surrogate','on'); Mdl{j,k} = fitensemble(X,Y,'LSBoost',numTrees,t,... 'Type','regression','KFold',5,'LearnRate',lr(k)); end; end;

Compute the cross-validated MSE for each ensemble.

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

Plot how the cross-validated classification error behaves as the number of trees in the ensemble increases for a few of the ensembles, the deep tree, and the stump. Plot the curves with respect to learning rate in the same plot, and plot separate plots for varying tree complexities. Choose a subset of tree complexity levels.

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(lr','Learning Rate = %0.2f'));... 'Deep Tree';'Stump';'Min. MSE']); hL.Position(1) = 0.6;

Each curve contains a minimum cross-validated MSE occuring 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) fprintf('\nOptimal Parameter Values:\nNum. Trees = %d',idxNumTrees); fprintf('\nMaxNumSplits = %d\nLearning Rate = %0.2f\n',... maxNumSplits(idxMNS),lr(idxLR))

Min. MSE = 18.46270 Optimal Parameter Values: Num. Trees = 38 MaxNumSplits = 4 Learning Rate = 0.10

Specify optional comma-separated pairs of `Name,Value`

arguments.
`Name`

is the argument
name and `Value`

is the corresponding
value. `Name`

must appear
inside single quotes (`' '`

).
You can specify several name and value pair
arguments in any order as `Name1,Value1,...,NameN,ValueN`

.

`'Surrogate','on','NVarToSample','all'`

specifies
a template with surrogate splits, and uses all available predictors
at each split.`'MaxNumSplits'`

— Maximal number of decision splitspositive integerMaximal number of decision splits (or branch nodes) per tree,
specified as the comma-separated pair consisting of `'MaxNumSplits'`

and
a positive integer. `templateTree`

splits `MaxNumSplits`

or
fewer branch nodes. For more details on splitting behavior, see Algorithms.

For bagged decision trees and decision tree binary learners
in ECOC models, the default is `size(X,1) - 1`

. For
boosted decision trees, the default is `1`

.

**Example: **`'MaxNumSplits',5`

**Data Types: **`single`

| `double`

`'MergeLeaves'`

— Leaf merge flag`'off'`

| `'on'`

Leaf merge flag, specified as the comma-separated pair consisting
of `'MergeLeaves'`

and either `'on'`

or `'off'`

.

When `'on'`

, the decision tree merges leaves
that originate from the same parent node, and that provide a sum of
risk values greater or equal to the risk associated with the parent
node. When `'off'`

, the decision tree does not merge
leaves.

For ensembles models, the default is `'off'`

.
For decision tree binary learners in ECOC models, the default is `'on'`

.

**Example: **`'MergeLeaves','on'`

`'MinLeafSize'`

— Minimum observations per leafpositive integer valueMinimum observations per leaf, specified as the comma-separated
pair consisting of `'MinLeafSize'`

and a positive
integer value. Each leaf has at least `MinLeafSize`

observations
per tree leaf. If you supply both `MinParentSize`

and `MinLeafSize`

,
the decision tree uses the setting that gives larger leaves: ```
MinParentSize
= max(MinParentSize,2*MinLeafSize)
```

.

For boosted and bagged decision trees, the defaults are `1`

for
classification and `5`

for regression. For decision
tree binary learners in ECOC models, the default is `1`

.

**Example: **`'MinLeafSize',2`

`'MinParentSize'`

— Minimum observations per branch nodepositive integer valueMinimum observations per branch node, specified as the comma-separated
pair consisting of `'MinParentSize'`

and a positive
integer value. Each branch node in the tree has at least `MinParentSize`

observations.
If you supply both `MinParentSize`

and `MinLeafSize`

,
the decision tree uses the setting that gives larger leaves: ```
MinParentSize
= max(MinParentSize,2*MinLeafSize)
```

.

For boosted and bagged decision trees, the defaults are `2`

for
classification and `10`

for regression. For decision
tree binary learners in ECOC models, the default is `10`

.

**Example: **`'MinParentSize',4`

`'NumVariablesToSample'`

— Number of predictors to select at random for each splitpositive integer value | `'all'`

Number of predictors to select at random for each split, specified
as the comma-separated pair consisting of `'NumVariablesToSample'`

and
a positive integer value. Alternatively, you can specify `'all'`

to
use all available predictors.

For boosted decision trees and decision tree binary learners
in ECOC models models, the default is `'all'`

. The
default for bagged decision trees is the square root of the number
of predictors for classification, or one third of predictors for regression.

**Example: **`'NumVariablesToSample',3`

`'Prune'`

— Flag to estimate optimal sequence of pruned subtrees`'off'`

(default) | `'on'`

Flag to estimate the optimal sequence of pruned subtrees, specified
as the comma-separated pair consisting of `'Prune'`

and `'on'`

or `'off'`

.

If `Prune`

is `'on'`

, then
the software trains the classification tree learners without pruning
them, but estimates the optimal sequence of pruned subtrees for each
learner in the ensemble or decision tree binary learner in ECOC models.
Otherwise, the software trains the classification tree learners without
estimating the optimal sequence of pruned subtrees.

For ensembles, the default is `'off'`

.

For decision tree binary learners in ECOC models, then the default
is `'on'`

.

**Example: **`'Prune','on'`

`'PruneCriterion'`

— Pruning criterion`'error'`

| `'impurity'`

| `'mse'`

Pruning criterion, specified as the comma-separated pair consisting
of `'PruneCriterion'`

and a pruning criterion string
valid for the tree type.

For classification trees, you can specify

`'error'`

(default) or`'impurity'`

.For regression, you can only specify

`'mse'`

(default).

**Example: **`'PruneCriterion','impurity'`

`'SplitCriterion'`

— Split criterion`'gdi'`

| `'twoing'`

| `'deviance'`

| `'mse'`

Split criterion, specified as the comma-separated pair consisting
of `'SplitCriterion'`

and a split criterion string
valid for the tree type.

For classification trees:

`'gdi'`

for Gini's diversity index (default)`'twoing'`

for the twoing rule`'deviance'`

for maximum deviance reduction (also known as cross entropy)

For regression trees:

`'mse'`

for mean squared error (default)

**Example: **`'SplitCriterion','deviance'`

`'Surrogate'`

— Surrogate decision splits`'off'`

(default) | `'on'`

| `'all'`

| positive integer valueSurrogate decision splits flag, specified as the comma-separated
pair consisting of `'Surrogate'`

and one of `'off'`

, `'on'`

, `'all'`

,
or a positive integer value.

When

`'off'`

, the decision tree does not find surrogate splits at the branch nodes.When

`'on'`

, the decision tree finds at most 10 surrogate splits at each branch node.When set to

`'all'`

, the decision tree finds all surrogate splits at each branch node. The`'all'`

setting can consume considerable time and memory.When set to a positive integer value, the decision tree finds at most the specified number of surrogate splits at each branch node.

Use surrogate splits to improve the accuracy of predictions for data with missing values. This setting also lets you compute measures of predictive association between predictors.

**Example: **`'Surrogate','on'`

`'AlgorithmForCategorical'`

— Algorithm for best categorical predictor split`'Exact'`

| `'PullLeft'`

| `'PCA'`

| `'OVAbyClass'`

Algorithm to find the best split on a categorical predictor
for data with *C* categories for data and *K* ≥
3 classes, specified as the comma-separated pair consisting of `'AlgorithmForCategorical'`

and
one of the following.

`'Exact'` | Consider all 2^{C–1} –
1 combinations. |

`'PullLeft'` | Start with all C categories on the right
branch. Consider moving each category to the left branch as it achieves
the minimum impurity for the K classes among the
remaining categories. From this sequence, choose the split that has
the lowest impurity. |

`'PCA'` | Compute a score for each category using the inner product between
the first principal component of a weighted covariance matrix (of
the centered class probability matrix) and the vector of class probabilities
for that category. Sort the scores in ascending order, and consider
all C — 1 splits. |

`'OVAbyClass'` | Start with all C categories on the right
branch. For each class, order the categories based on their probability
for that class. For the first class, consider moving each category
to the left branch in order, recording the impurity criterion at each
move. Repeat for the remaining classes. From this sequence, choose
the split that has the minimum impurity. |

`ClassificationTree`

selects the optimal subset
of algorithms for each split using the known number of classes and
levels of a categorical predictor. For two classes, `ClassificationTree`

always
performs the exact search. Use the `'AlgorithmForCategorical'`

name-value
pair argument to specify a particular algorithm.

**Example: **`'AlgorithmForCategorical','PCA'`

`'MaxNumCategories'`

— Maximum category levels in split node`10`

(default) | nonnegative scalar valueMaximum category levels in the split node, specified as the
comma-separated pair consisting of `'MaxNumCategories'`

and
a nonnegative scalar value. `ClassificationTree`

splits
a categorical predictor using the exact search algorithm if the predictor
has at most `MaxNumCategories`

levels in the split
node. Otherwise, `ClassificationTree`

finds the best
categorical split using one of the inexact algorithms. Note that passing
a small value can increase computation time and memory overload.

**Example: **`'MaxNumCategories',8`

`'QuadraticErrorTolerance'`

— Quadratic error tolerance`1e-6`

(default) | nonnegative scalar valueQuadratic error tolerance per node, specified as the comma —separated
pair consisting of `'QuadraticErrorTolerance'`

and
a nonnegative scalar value. `RegressionTree`

stops
splitting nodes when the quadratic error per node drops below `QuadraticErrorTolerance*QED`

,
where `QED`

is the quadratic error for the entire
data computed before the decision tree is grown. ```
QED = NORM(Y
- YBAR)
```

, where `YBAR`

is estimated as the
average of the input array `Y`

.

**Example: **`'QuadraticErrorTolerance',1e-4`

`t`

— Decision tree template for classification or regressiontemplate objectDecision tree template for classification or regression suitable
for training ensembles or error-correcting output code (ECOC) multiclass
models, returned as a template object. Pass `t`

to `fitensemble`

or `fitcecoc`

to
specify how to create the decision tree for the ensemble or ECOC model,
respectively.

If you display `t`

in the Command Window, then
all unspecified options appear empty (`[]`

). However,
the software replaces empty options with their corresponding default
values during training.

To accommodate

`MaxNumSplits`

, the software splits all nodes in the current*layer*, and then counts the number of branch nodes. A layer is the set of nodes that are equidistant from the root node. If the number of branch nodes exceeds`MaxNumSplits`

, then the software follows this procedure.Determine how many branch nodes in the current layer need to be unsplit so that there would be at most

`MaxNumSplits`

branch nodes.Sort the branch nodes by their impurity gains.

Unsplit the desired number of least successful branches.

Return the decision tree grown so far.

This procedure aims at producing maximally balanced trees.

The software splits branch nodes layer by layer until at least one of these events occurs.

There are

`MaxNumSplits`

+ 1 branch nodes.A proposed split causes the number of observations in at least one branch node to be fewer than

`MinParentSize`

.A proposed split causes the number of observations in at least one leaf node to be fewer than

`MinLeafSize`

.The algorithm cannot find a good split within a layer (i.e., the pruning criterion (see

`PruneCriterion`

), does not improve for all proposed splits in a layer). A special case of this event is when all nodes are pure (i.e., all observations in the node have the same class).

`MaxNumSplits`

and`MinLeafSize`

do not affect splitting at their default values. Therefore, if you set`'MaxNumSplits'`

, then splitting might stop due to the value of`MinParentSize`

before`MaxNumSplits`

splits occur.

[1] Coppersmith, D., S. J. Hong, and J. R.
M. Hosking. "Partitioning Nominal Attributes in Decision Trees." *Data
Mining and Knowledge Discovery*, Vol. 3, 1999, pp. 197–217.

`ClassificationTree`

| `fitcecoc`

| `fitctree`

| `fitensemble`

| `RegressionTree`

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