ClassificationTree

Binary decision tree for multiclass classification

Description

A ClassificationTree object represents a decision tree with binary splits for classification. An object of this class can predict responses for new data using predict. The object contains the data used for training, so it can also compute resubstitution predictions using resubPredict.

Creation

Create a ClassificationTree object by using fitctree.

Properties

BinEdges — Bin edges for numeric predictors

cell array of p numeric vectors

This property is read-only.

Bin edges for numeric predictors, specified as a cell array of p numeric vectors, where p is the number of predictors. Each vector includes the bin edges for a numeric predictor. The element in the cell array for a categorical predictor is empty because the software does not bin categorical predictors.

The software bins numeric predictors only if you specify the 'NumBins' name-value argument as a positive integer scalar when training a model with tree learners. The BinEdges property is empty if the 'NumBins' value is empty (default).

You can reproduce the binned predictor data Xbinned by using the BinEdges property of the trained model mdl.

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

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

CategoricalPredictors — Indices of categorical predictors

vector of positive integers | []

This property is read-only.

Categorical predictor indices, specified as a vector of positive integers. CategoricalPredictors contains index values indicating that the corresponding predictors are categorical. The index values are between 1 and p, where p is the number of predictors used to train the model. If none of the predictors are categorical, then this property is empty ([]).

Data Types: single | double

CategoricalSplit — Categorical splits

n-by-2 cell array

This property is read-only.

Categorical splits, returned as an n-by-2 cell array, where n is the number of categorical splits in tree. Each row in CategoricalSplit gives left and right values for a categorical split. For each branch node with categorical split j based on a categorical predictor variable z, the left child is chosen if z is in CategoricalSplit(j,1) and the right child is chosen if z is in CategoricalSplit(j,2). The splits are in the same order as nodes of the tree. Nodes for these splits can be found by running cuttype and selecting 'categorical' cuts from top to bottom.

Data Types: cell

Children — Numbers of the child nodes for each node

n-by-2 array

This property is read-only.

Numbers of the child nodes for each node in the tree, returned as an n-by-2 array, where n is the number of nodes. Leaf nodes have child node 0.

Data Types: double

ClassCount — Class counts

n-by-k array

This property is read-only.

Class counts for the nodes in tree, returned as an n-by-k array, where n is the number of nodes and k is the number of classes. For any node number i, the class counts ClassCount(i,:) are counts of observations (from the data used in fitting the tree) from each class satisfying the conditions for node i.

Data Types: double

ClassNames — List of elements in Y with duplicates removed

categorical array | cell array of character vectors | character array | logical vector | numeric vector

This property is read-only.

List of the elements in Y with duplicates removed, returned as a categorical array, cell array of character vectors, character array, logical vector, or a numeric vector. ClassNames has the same data type as the data in the argument Y. (The software treats string arrays as cell arrays of character vectors.)

Data Types: double | logical | char | cell | categorical

ClassProbability — Class probabilities

n-by-k array

This property is read-only.

Class probabilities for the nodes in tree, returned as an n-by-k array, where n is the number of nodes and k is the number of classes. For any node number i, the class probabilities ClassProbability(i,:) are the estimated probabilities for each class for a point satisfying the conditions for node i.

Data Types: double

Cost — Cost of classifying a point into class j when its true class is i

square matrix

Cost of classifying a point into class j when its true class is i, returned as a square matrix. The rows of Cost correspond to the true class and the columns correspond to the predicted class. The order of the rows and columns of Cost corresponds to the order of the classes in ClassNames. The number of rows and columns in Cost is the number of unique classes in the response.

Data Types: double

CutCategories — Categories used at branches

n-by-2 cell array

This property is read-only.

Categories used at branches in tree, returned as an n-by-2 cell array, where n is the number of nodes. For each branch node i based on a categorical predictor variable X, the left child is chosen if X is among the categories listed in CutCategories{i,1}, and the right child is chosen if X is among those listed in CutCategories{i,2}. Both columns of CutCategories are empty for branch nodes based on continuous predictors and for leaf nodes.

CutPoint contains the cut points for 'continuous' cuts, and CutCategories contains the set of categories.

Data Types: cell

CutPoint — Values used as cut points

n-element vector

This property is read-only.

Values used as cut points in tree, returned as an n-element vector, where n is the number of nodes. For each branch node i based on a continuous predictor variable X, the left child is chosen if X<CutPoint(i) and the right child is chosen if X>=CutPoint(i). CutPoint is NaN for branch nodes based on categorical predictors and for leaf nodes.

CutPoint contains the cut points for 'continuous' cuts, and CutCategories contains the set of categories.

Data Types: double

CutPredictor — Names of the variables used for branching in each node

cell array

This property is read-only.

Names of the variables used for branching in each node in tree, returned as an n-element cell array, where n is the number of nodes. These variables are sometimes known as cut variables. For leaf nodes, CutPredictor contains an empty character vector.

CutPoint contains the cut points for 'continuous' cuts, and CutCategories contains the set of categories.

Data Types: cell

CutPredictorIndex — Indices of variables used for branching in each node

n-element array

This property is read-only.

Indices of variables used for branching in each node in tree, returned as an n-element array, where n is the number of nodes. For more information, see CutPredictor.

Data Types: double

CutType — Type of cut at each node

n-element cell array

This property is read-only.

Type of cut at each node in tree, returned as an n-element cell array, where n is the number of nodes. For each node i, CutType{i} is:

• 'continuous' — If the cut is defined in the form X < v for a variable X and cut point v.

• 'categorical' — If the cut is defined by whether a variable X takes a value in a set of categories.

• '' — If i is a leaf node.

CutPoint contains the cut points for 'continuous' cuts, and CutCategories contains the set of categories.

Data Types: cell

ExpandedPredictorNames — Expanded predictor names

cell array of character vectors

This property is read-only.

Expanded predictor names, returned as a cell array of character vectors.

If the model uses encoding for categorical variables, then ExpandedPredictorNames includes the names that describe the expanded variables. Otherwise, ExpandedPredictorNames is the same as PredictorNames.

Data Types: cell

HyperparameterOptimizationResults — Description of cross-validation optimization of hyperparameters

BayesianOptimization object | table of hyperparameters and associated values

This property is read-only.

Description of the cross-validation optimization of hyperparameters, returned as a BayesianOptimization object or a table of hyperparameters and associated values. Nonempty when the OptimizeHyperparameters name-value pair is nonempty at creation. Value depends on the setting of the HyperparameterOptimizationOptions name-value pair at creation:

• 'bayesopt' (default) — Object of class BayesianOptimization

• 'gridsearch' or 'randomsearch' — Table of hyperparameters used, observed objective function values (cross-validation loss), and rank of observations from lowest (best) to highest (worst)

IsBranchNode — Indicator of branch nodes

logical vector

This property is read-only.

Indicator of branch nodes, returned as an n-element logical vector that is true for each branch node and false for each leaf node of tree.

Data Types: logical

ModelParameters — Parameters used in training tree

TreeParams object

This property is read-only.

Parameters used in training tree, returned as a TreeParams object. To display all parameter values, enter tree.ModelParameters. To access a particular parameter, use dot notation.

NodeClass — Name of most probably class in each node

cell array

This property is read-only.

Name of most probably class in each node of tree, returned as a cell array with n elements, where n is the number of nodes in the tree. Each element of this array is a character vector equal to one of the class names in ClassNames.

Data Types: cell

NodeError — Misclassification probability for each node

n-element vector

This property is read-only.

Misclassification probability for each node in tree, returned as an n-element vector, where n is the number of nodes in the tree.

Data Types: double

NodeProbability — Proportion of observations in original data that satisfy the conditions for the node

n-element vector

This property is read-only.

Proportion of observations in original data that satisfy the conditions for each node in tree, returned as an n-element vector, where n is the number of nodes in the tree. The NodeProbability values are adjusted for any prior probabilities assigned to each class.

Data Types: double

NodeRisk — Impurity of nodes

n-element vector

This property is read-only.

Impurity of each node in tree, weighted by the node probability, returned as an n-element vector, where n is the number of nodes in the tree. The measure of impurity is the Gini index or deviance for the node, weighted by the node probability. If the tree is grown by twoing, the risk for each node is zero.

Data Types: double

NodeSize — Size of nodes

n-element vector

This property is read-only.

Size of the nodes in tree, returned as an n-element vector, where n is the number of nodes in the tree. The size of a node is the number of observations from the data used to create the tree that satisfy the conditions for the node.

Data Types: double

NumNodes — Number of nodes

positive integer

This property is read-only.

The number of nodes in tree, returned as a positive integer.

Data Types: double

NumObservations — Number of observations in the training data

positive integer

This property is read-only.

Number of observations in the training data, returned as a positive integer. NumObservations can be less than the number of rows of input data when there are missing values in the input data or response data.

Data Types: double

Parent — Number of parents of nodes

n-element vector

This property is read-only.

Number of parents of each node in tree, returned as an n-element integer vector, where n is the number of nodes in the tree. The parent of the root node is 0.

Data Types: double

PredictorNames — Predictor names

cell array of character vectors

This property is read-only.

Predictor names, specified as a cell array of character vectors. The order of the entries in PredictorNames is the same as in the training data.

Data Types: cell

Prior — Prior probabilities for each class

m-element vector

Prior probabilities for each class, returned as an m-element vector, where m is the number of unique classes in the response. The order of the elements of Prior corresponds to the order of the classes in ClassNames.

Data Types: double

PruneAlpha — Alpha values for pruning the tree

real vector

Alpha values for pruning the tree, returned as a real vector with one element per pruning level. If the pruning level ranges from 0 to M, then PruneAlpha has M + 1 elements sorted in ascending order. PruneAlpha(1) is for pruning level 0 (no pruning), PruneAlpha(2) is for pruning level 1, and so on.

For the meaning of the ɑ values, see How Decision Trees Create a Pruning Sequence.

Data Types: double

PruneList — Pruning levels of each node in tree

integer vector

Pruning levels of each node in the tree, returned as an integer vector with NumNodes elements. The pruning levels range from 0 (no pruning) to M, where M is the distance between the deepest leaf and the root node.

For details, see Pruning.

Data Types: double

ResponseName — Name of the response variable

character vector

This property is read-only.

Name of the response variable, returned as a character vector.

Data Types: char

RowsUsed — Rows of the original predictor data X used for fitting

logical vector

This property is read-only.

Rows of the original predictor data X used for fitting, returned as an n-element logical vector, where n is the number of rows of X. If the software uses all rows of X for constructing the object, then RowsUsed is an empty array ([]).

Data Types: logical

ScoreTransform — Function for transforming scores

function handle | name of a built-in transformation function | 'none'

Function for transforming scores, specified as a function handle or the name of a built-in transformation function. 'none' means no transformation; equivalently, 'none' means @(x)x. For a list of built-in transformation functions and the syntax of custom transformation functions, see fitctree.

Add or change a ScoreTransform function using dot notation:

ctree.ScoreTransform = 'function'
% or
ctree.ScoreTransform = @function

Data Types: char | string | function_handle

SurrogateCutCategories — Categories used for surrogate splits

n-element cell array

This property is read-only.

Categories used for surrogate splits, returned as an n-element cell array, where n is the number of nodes in tree. For each node k, SurrogateCutCategories{k} is a cell array. The length of SurrogateCutCategories{k} is equal to the number of surrogate predictors found at this node. Every element of SurrogateCutCategories{k} is either an empty character vector for a continuous surrogate predictor, or is a two-element cell array with categories for a categorical surrogate predictor. The first element of this two-element cell array lists categories assigned to the left child by this surrogate split and the second element of this two-element cell array lists categories assigned to the right child by this surrogate split. The order of the surrogate split variables at each node is matched to the order of variables in SurrogateCutVar. The optimal-split variable at this node does not appear. For nonbranch (leaf) nodes, SurrogateCutCategories contains an empty cell.

Data Types: cell

SurrogateCutFlip — Numeric cut assignments used for surrogate splits

n-element cell array

This property is read-only.

Numeric cut assignments used for surrogate splits in tree, returned as an n-element cell array, where n is the number of nodes in tree. For each node k, SurrogateCutFlip{k} is a numeric vector. The length of SurrogateCutFlip{k} is equal to the number of surrogate predictors found at this node. Every element of SurrogateCutFlip{k} is either zero for a categorical surrogate predictor, or a numeric cut assignment for a continuous surrogate predictor. The numeric cut assignment can be either –1 or +1. For every surrogate split with a numeric cut C based on a continuous predictor variable Z, the left child is chosen if Z<C and the cut assignment for this surrogate split is +1, or if Z≥C and the cut assignment for this surrogate split is –1. Similarly, the right child is chosen if Z≥C and the cut assignment for this surrogate split is +1, or if Z<C and the cut assignment for this surrogate split is –1. The order of the surrogate split variables at each node is matched to the order of variables in SurrogateCutPredictor. The optimal-split variable at this node does not appear. For nonbranch (leaf) nodes, SurrogateCutFlip contains an empty array.

Data Types: cell

SurrogateCutPoint — Numeric values used for surrogate splits

n-element cell array

This property is read-only.

Numeric values used for surrogate splits in tree, returned as an n-element cell array, where n is the number of nodes in tree. For each node k, SurrogateCutPoint{k} is a numeric vector. The length of SurrogateCutPoint{k} is equal to the number of surrogate predictors found at this node. Every element of SurrogateCutPoint{k} is either NaN for a categorical surrogate predictor, or a numeric cut for a continuous surrogate predictor. For every surrogate split with a numeric cut C based on a continuous predictor variable Z, the left child is chosen if Z<C and SurrogateCutFlip for this surrogate split is +1, or if Z≥C and SurrogateCutFlip for this surrogate split is –1. Similarly, the right child is chosen if Z≥C and SurrogateCutFlip for this surrogate split is +1, or if Z<C and SurrogateCutFlip for this surrogate split is –1. The order of the surrogate split variables at each node is matched to the order of variables returned by SurrogateCutPredictor. The optimal-split variable at this node does not appear. For nonbranch (leaf) nodes, SurrogateCutPoint contains an empty cell.

Data Types: cell

SurrogateCutPredictor — Names of variables used for surrogate splits in each node

n-element cell array

This property is read-only.

Names of the variables used for surrogate splits in each node in tree, returned as an n-element cell array, where n is the number of nodes in tree. Every element of SurrogateCutPredictor is a cell array with the names of the surrogate split variables at this node. The variables are sorted by the predictive measure of association with the optimal predictor in the descending order, and only variables with the positive predictive measure are included. The optimal-split variable at this node does not appear. For nonbranch (leaf) nodes, SurrogateCutPredictor contains an empty cell.

Data Types: cell

SurrogateCutType — Types of surrogate splits at each node

n-element cell array

This property is read-only.

Types of surrogate splits at each node in tree, returned as an n-element cell array, where n is the number of nodes in tree. For each node k, SurrogateCutType{k} is a cell array with the types of the surrogate split variables at this node. The variables are sorted by the predictive measure of association with the optimal predictor in the descending order, and only variables with the positive predictive measure are included. The order of the surrogate split variables at each node is matched to the order of variables in SurrogateCutPredictor. The optimal-split variable at this node does not appear. For nonbranch (leaf) nodes, SurrogateCutType contains an empty cell. A surrogate split type can be either 'continuous' if the cut is defined in the form Z<V for a variable Z and cut point V or 'categorical' if the cut is defined by whether Z takes a value in a set of categories.

Data Types: cell

SurrogatePredictorAssociation — Predictive measures of association for surrogate splits

n-element cell array

This property is read-only.

Predictive measures of association for surrogate splits in tree, returned as an n-element cell array, where n is the number of nodes in tree. For each node k, SurrogatePredictorAssociation{k} is a numeric vector. The length of SurrogatePredictorAssociation{k} is equal to the number of surrogate predictors found at this node. Every element of SurrogatePredictorAssociation{k} gives the predictive measure of association between the optimal split and this surrogate split. The order of the surrogate split variables at each node is the order of variables in SurrogateCutPredictor. The optimal-split variable at this node does not appear. For nonbranch (leaf) nodes, SurrogatePredictorAssociation contains an empty cell.

Data Types: cell

W — Scaled weights in tree

numeric vector

This property is read-only.

Scaled weights in tree, returned as a numeric vector. W has length n, the number of rows in the training data.

Data Types: double

X — Predictor values

real matrix | table

This property is read-only.

Predictor values, returned as a real matrix or table. Each column of X represents one variable (predictor), and each row represents one observation.

Data Types: double | table

Y — Class labels

categorical array | cell array of character vectors | character array | logical vector | numeric vector

This property is read-only.

Class labels corresponding to the observations in X, returned as a categorical array, cell array of character vectors, character array, logical vector, or a numeric vector. Each row of Y represents the classification of the corresponding row of X.

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

Object Functions

compact	Reduce size of classification tree model
compareHoldout	Compare accuracies of two classification models using new data
crossval	Cross-validate machine learning model
cvloss	Classification error by cross-validation for classification tree model
edge	Classification edge for classification tree model
gather	Gather properties of Statistics and Machine Learning Toolbox object from GPU
lime	Local interpretable model-agnostic explanations (LIME)
loss	Classification loss for classification tree model
margin	Classification margins for classification tree model
nodeVariableRange	Retrieve variable range of decision tree node
partialDependence	Compute partial dependence
plotPartialDependence	Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
predict	Predict labels using classification tree model
predictorImportance	Estimates of predictor importance for classification tree
prune	Produce sequence of classification subtrees by pruning classification tree
resubEdge	Resubstitution classification edge for classification tree model
resubLoss	Resubstitution classification loss for classification tree model
resubMargin	Resubstitution classification margins for classification tree model
resubPredict	Classify observations in classification tree by resubstitution
shapley	Shapley values
surrogateAssociation	Mean predictive measure of association for surrogate splits in classification tree
testckfold	Compare accuracies of two classification models by repeated cross-validation
view	View classification tree

Examples

Grow a Classification Tree

Grow a classification tree using the ionosphere data set.

load ionosphere
tc = fitctree(X,Y)

tc = 
  ClassificationTree
             ResponseName: 'Y'
    CategoricalPredictors: []
               ClassNames: {'b'  'g'}
           ScoreTransform: 'none'
          NumObservations: 351

Control Tree Depth

You can control the depth of the trees using the MaxNumSplits, MinLeafSize, or MinParentSize name-value pair parameters. fitctree grows deep decision trees by default. You can grow shallower trees to reduce model complexity or computation time.

Load the ionosphere data set.

load ionosphere

The default values of the tree depth controllers for growing classification trees are:

• n - 1 for MaxNumSplits. n is the training sample size.

• 1 for MinLeafSize.

• 10 for MinParentSize.

These default values tend to grow deep trees for large training sample sizes.

Train a classification tree using the default values for tree depth control. Cross-validate the model by using 10-fold cross-validation.

rng(1); % For reproducibility
MdlDefault = fitctree(X,Y,'CrossVal','on');

Draw a histogram of the number of imposed splits on the trees. Also, view one of the trees.

numBranches = @(x)sum(x.IsBranch);
mdlDefaultNumSplits = cellfun(numBranches, MdlDefault.Trained);

figure;
histogram(mdlDefaultNumSplits)

view(MdlDefault.Trained{1},'Mode','graph')

The average number of splits is around 15.

Suppose that you want a classification tree that is not as complex (deep) as the ones trained using the default number of splits. Train another classification tree, but set the maximum number of splits at 7, which is about half the mean number of splits from the default classification tree. Cross-validate the model by using 10-fold cross-validation.

Mdl7 = fitctree(X,Y,'MaxNumSplits',7,'CrossVal','on');
view(Mdl7.Trained{1},'Mode','graph')

Compare the cross-validation classification errors of the models.

classErrorDefault = kfoldLoss(MdlDefault)

classErrorDefault = 
0.1168

classError7 = kfoldLoss(Mdl7)

classError7 = 
0.1311

Mdl7 is much less complex and performs only slightly worse than MdlDefault.

More About

Impurity and Node Error

A decision tree splits nodes based on either impurity or node error.

Impurity means one of several things, depending on your choice of the SplitCriterion name-value argument:

• Gini's Diversity Index (gdi) — The Gini index of a node is

  $$1-{\displaystyle \sum _i{p}^2(i)},$$

  where the sum is over the classes i at the node, and p(i) is the observed fraction of classes with class i that reach the node. A node with just one class (a pure node) has Gini index 0; otherwise, the Gini index is positive. So the Gini index is a measure of node impurity.

• Deviance ("deviance") — With p(i) defined the same as for the Gini index, the deviance of a node is

  $$-{\displaystyle \sum _{i}p(i){\log }_{2}p(i)}.$$

  A pure node has deviance 0; otherwise, the deviance is positive.

• Twoing rule ("twoing") — Twoing is not a purity measure of a node, but is a different measure for deciding how to split a node. Let L(i) denote the fraction of members of class i in the left child node after a split, and R(i) denote the fraction of members of class i in the right child node after a split. Choose the split criterion to maximize

  $$P(L)P(R){\left({\displaystyle \sum _i\left|L(i)-R(i)\right|}\right)}^2,$$

  where P(L) and P(R) are the fractions of observations that split to the left and right, respectively. If the expression is large, the split made each child node purer. Similarly, if the expression is small, the split made each child node similar to each other and, therefore, similar to the parent node. The split did not increase node purity.

• Node error — The node error is the fraction of misclassified classes at a node. If j is the class with the largest number of training samples at a node, the node error is

  1 – p(j).

References

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

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The predict and update functions support code generation.

• To integrate the prediction of a classification tree model into Simulink^®, you can use the ClassificationTree Predict block in the Statistics and Machine Learning Toolbox™ library or a MATLAB^® Function block with the predict function.

• When you train a classification tree using fitctree, the following restrictions apply.

  • The value of the 'ScoreTransform' name-value pair argument cannot be an anonymous function. For fixed-point code generation, the 'ScoreTransform' value cannot be 'invlogit'.

  • You cannot use surrogate splits; that is, the value of the 'Surrogate' name-value pair argument must be 'off'.

  • For fixed-point code generation and code generation with a coder configurer, the following additional restrictions apply.

    • Categorical predictors (logical, categorical, char, string, or cell) are not supported. You cannot use the CategoricalPredictors name-value argument. To include categorical predictors in a model, preprocess them by using dummyvar before fitting the model.

    • Class labels with the categorical data type are not supported. Both the class label value in the training data (Tbl or Y) and the value of the ClassNames name-value argument cannot be an array with the categorical data type.

For more information, see Introduction to Code Generation.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• The following object functions fully support GPU arrays:

  • compact

  • crossval

  • cvloss

  • gather

  • nodeVariableRange

  • predictorImportance

  • resubEdge

  • resubLoss

  • resubMargin

  • resubPredict

  • surrogateAssociation

• The following object functions offer limited support for GPU arrays:

  • compareHoldout

  • edge

  • loss

  • margin

  • partialDependence

  • plotPartialDependence

  • predict

  • prune

  • view

• The object functions execute on a GPU if at least one of the following applies:

  • The model was fitted with GPU arrays.

  • The predictor data that you pass to the object function is a GPU array.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2011a