# RegressionTree

Regression tree

## Description

A decision tree with binary splits for regression. An object of class `RegressionTree`

can predict responses for new data with the
`predict`

method. The object contains the
data used for training, so can compute resubstitution predictions using `resubPredict`

.

## Creation

Create a `RegressionTree`

object by using `fitrtree`

.

## 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 `NaN`

s, then the corresponding
`Xbinned`

values are `NaN`

s.
`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`

`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.

`NodeError`

— Mean squared error for each node

`n`

-element vector

This property is read-only.

Mean squared error for each node in `tree`

, returned as an `n`

-element vector, where `n`

is the number of nodes in the tree.

**Data Types: **`double`

`NodeMean`

— Mean observation values for each node

`n`

-element vector

This property is read-only.

Mean observation values for each node in `tree`

, returned as an
`n`

-element vector, where `n`

is the number of
nodes in the tree. Every element in `NodeMean`

is the average of the
true `Y`

values over all observations in the node.

**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.

**Data Types: **`double`

`NodeRisk`

— Risk of each node

`n`

-element vector

This property is read-only.

Risk of each node in `tree`

, returned as an `n`

-element vector, where `n`

is the number of nodes in the tree. The risk for each node is the node error weighted by the node probability.

**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`

`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`

`ResponseTransform`

— Function for transforming responses

`'none'`

(default) | function handle

Function for transforming the raw response values (mean squared error), specified as a function handle or `'none'`

. The default `'none'`

means no transformation; equivalently, `'none'`

means `@(x)x`

. A function handle must accept a matrix of response values and return a matrix of the same size.

Add or change a `ResponseTransform`

function using dot notation:

tree.ResponseTransform = @function

**Data Types: **`char`

| `function_handle`

`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`

`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`

— Response data

numeric column vector

This property is read-only.

Response data, returned as a numeric column vector with the same number of
rows as `X`

. Each entry in `Y`

is the
response to the data in the corresponding row of
`X`

.

**Data Types: **`double`

## Object Functions

`compact` | Reduce size of regression tree model |

`crossval` | Cross-validate machine learning model |

`cvloss` | Regression error by cross-validation for regression tree model |

`gather` | Gather properties of Statistics and Machine Learning Toolbox object from GPU |

`lime` | Local interpretable model-agnostic explanations (LIME) |

`loss` | Regression error for regression 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 responses using regression tree model |

`predictorImportance` | Estimates of predictor importance for regression tree |

`prune` | Produce sequence of regression subtrees by pruning regression tree |

`resubLoss` | Resubstitution loss for regression tree model |

`resubPredict` | Predict response of regression tree by resubstitution |

`shapley` | Shapley values |

`surrogateAssociation` | Mean predictive measure of association for surrogate splits in regression tree |

`view` | View regression tree |

## Examples

### Construct Regression Tree

Load the sample data.

`load carsmall`

Construct a regression tree using the sample data. The response variable is miles per gallon, MPG.

tree = fitrtree([Weight, Cylinders],MPG,... 'CategoricalPredictors',2,'MinParentSize',20,... 'PredictorNames',{'W','C'})

tree = RegressionTree PredictorNames: {'W' 'C'} ResponseName: 'Y' CategoricalPredictors: 2 ResponseTransform: 'none' NumObservations: 94

Predict the mileage of 4,000-pound cars with 4, 6, and 8 cylinders.

MPG4Kpred = predict(tree,[4000 4; 4000 6; 4000 8])

`MPG4Kpred = `*3×1*
19.2778
19.2778
14.3889

## 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:

To integrate the prediction of a regression tree model into Simulink

^{®}, you can use the RegressionTree Predict block in the Statistics and Machine Learning Toolbox™ library or a MATLAB^{®}Function block with the`predict`

function.When you train a regression tree model by using

`fitrtree`

, the following restrictions apply.The value of the

`ResponseTransform`

name-value argument cannot be an anonymous function. For fixed-point code generation, the value must be`'none'`

(default).You cannot use surrogate splits; that is, the value of the

`Surrogate`

name-value argument must be`'off'`

.Fixed-point code generation and code generation with a coder configurer do not support categorical predictors (

`logical`

,`categorical`

,`char`

,`string`

, or`cell`

). You cannot use the`CategoricalPredictors`

name-value argument. To include categorical predictors in a model, preprocess them by using`dummyvar`

before fitting the model.

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:

The following object functions offer limited support for GPU arrays:

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.

The response 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**

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