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Fit binary classification decision tree for multiclass classification

`tree = fitctree(Tbl,ResponseVarName)`

`tree = fitctree(Tbl,formula)`

`tree = fitctree(Tbl,Y)`

`tree = fitctree(X,Y)`

`tree = fitctree(___,Name,Value)`

returns a fitted binary classification decision tree based on the input
variables (also known as predictors, features, or attributes) contained in the
table `tree`

= fitctree(`Tbl`

,`ResponseVarName`

)`Tbl`

and output (response or labels) contained in
`ResponseVarName`

. The returned binary tree splits
branching nodes based on the values of a column of
`Tbl`

.

fits a tree with additional options specified by one or more name-value pair
arguments, using any of the previous syntaxes. For example, you can specify the
algorithm used to find the best split on a categorical predictor, grow a
cross-validated tree, or hold out a fraction of the input data for
validation.`tree`

= fitctree(___,`Name,Value`

)

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

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 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 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) classError7 = kfoldLoss(Mdl7)

classErrorDefault = 0.1140 classError7 = 0.1254

`Mdl7`

is much less complex and performs only slightly worse than `MdlDefault`

.

This example shows how to optimize hyperparameters automatically using `fitctree`

. The example uses Fisher's iris data.

Load Fisher's iris data.

```
load fisheriris
```

Optimize the cross-validation loss of the classifier, using the data in `meas`

to predict the response in `species`

.

X = meas; Y = species; Mdl = fitctree(X,Y,'OptimizeHyperparameters','auto')

|======================================================================================| | Iter | Eval | Objective | Objective | BestSoFar | BestSoFar | MinLeafSize | | | result | | runtime | (observed) | (estim.) | | |======================================================================================| | 1 | Best | 0.066667 | 2.0052 | 0.066667 | 0.066667 | 31 | | 2 | Accept | 0.066667 | 0.76897 | 0.066667 | 0.066667 | 12 | | 3 | Best | 0.04 | 0.3351 | 0.04 | 0.040003 | 2 | | 4 | Accept | 0.66667 | 0.26359 | 0.04 | 0.15796 | 73 | | 5 | Accept | 0.04 | 0.54401 | 0.04 | 0.040009 | 2 | | 6 | Accept | 0.66667 | 0.20746 | 0.04 | 0.040012 | 75 | | 7 | Accept | 0.066667 | 0.3126 | 0.04 | 0.040012 | 19 | | 8 | Accept | 0.04 | 0.25334 | 0.04 | 0.040009 | 4 | | 9 | Best | 0.033333 | 0.22749 | 0.033333 | 0.033351 | 1 | | 10 | Accept | 0.066667 | 0.15892 | 0.033333 | 0.03335 | 7 | | 11 | Accept | 0.04 | 0.21844 | 0.033333 | 0.033349 | 3 | | 12 | Accept | 0.066667 | 0.24604 | 0.033333 | 0.033348 | 25 | | 13 | Accept | 0.046667 | 0.26025 | 0.033333 | 0.033347 | 5 | | 14 | Accept | 0.033333 | 0.46176 | 0.033333 | 0.03334 | 1 | | 15 | Accept | 0.033333 | 0.39522 | 0.033333 | 0.033337 | 1 | | 16 | Accept | 0.066667 | 0.4845 | 0.033333 | 0.033337 | 15 | | 17 | Accept | 0.033333 | 0.33524 | 0.033333 | 0.033336 | 1 | | 18 | Accept | 0.33333 | 0.25695 | 0.033333 | 0.033336 | 43 | | 19 | Accept | 0.066667 | 0.27693 | 0.033333 | 0.033336 | 9 | | 20 | Accept | 0.066667 | 0.30575 | 0.033333 | 0.033336 | 6 | |======================================================================================| | Iter | Eval | Objective | Objective | BestSoFar | BestSoFar | MinLeafSize | | | result | | runtime | (observed) | (estim.) | | |======================================================================================| | 21 | Accept | 0.066667 | 0.25222 | 0.033333 | 0.033336 | 10 | | 22 | Accept | 0.066667 | 0.22828 | 0.033333 | 0.033336 | 22 | | 23 | Accept | 0.066667 | 0.22946 | 0.033333 | 0.033336 | 35 | | 24 | Accept | 0.33333 | 0.43078 | 0.033333 | 0.034054 | 54 | | 25 | Accept | 0.04 | 0.4052 | 0.033333 | 0.034034 | 2 | | 26 | Accept | 0.04 | 0.2057 | 0.033333 | 0.034003 | 3 | | 27 | Accept | 0.04 | 0.18643 | 0.033333 | 0.033979 | 4 | | 28 | Accept | 0.066667 | 0.24762 | 0.033333 | 0.033947 | 17 | | 29 | Accept | 0.066667 | 0.29662 | 0.033333 | 0.03392 | 8 | | 30 | Accept | 0.066667 | 0.31063 | 0.033333 | 0.033898 | 13 | __________________________________________________________ Optimization completed. MaxObjectiveEvaluations of 30 reached. Total function evaluations: 30 Total elapsed time: 102.8604 seconds. Total objective function evaluation time: 11.1107 Best observed feasible point: MinLeafSize ___________ 1 Observed objective function value = 0.033333 Estimated objective function value = 0.033898 Function evaluation time = 0.22749 Best estimated feasible point (according to models): MinLeafSize ___________ 1 Estimated objective function value = 0.033898 Estimated function evaluation time = 0.3139 Mdl = ClassificationTree ResponseName: 'Y' CategoricalPredictors: [] ClassNames: {'setosa' 'versicolor' 'virginica'} ScoreTransform: 'none' NumObservations: 150 HyperparameterOptimizationResults: [1x1 BayesianOptimization]

Load the `census1994`

data set. Consider a model that predicts a person's salary category given their age, working class, education level, martial status, race, sex, capital gain and loss, and number of working hours per week.

load census1994 X = adultdata(:,{'age','workClass','education_num','marital_status','race',... 'sex','capital_gain','capital_loss','hours_per_week','salary'});

Display the number of categories represented in the categorical variables using `summary`

.

summary(X)

Variables: age: 32561x1 double Values: Min 17 Median 37 Max 90 workClass: 32561x1 categorical Values: Federal-gov 960 Local-gov 2093 Never-worked 7 Private 22696 Self-emp-inc 1116 Self-emp-not-inc 2541 State-gov 1298 Without-pay 14 NumMissing 1836 education_num: 32561x1 double Values: Min 1 Median 10 Max 16 marital_status: 32561x1 categorical Values: Divorced 4443 Married-AF-spouse 23 Married-civ-spouse 14976 Married-spouse-absent 418 Never-married 10683 Separated 1025 Widowed 993 race: 32561x1 categorical Values: Amer-Indian-Eskimo 311 Asian-Pac-Islander 1039 Black 3124 Other 271 White 27816 sex: 32561x1 categorical Values: Female 10771 Male 21790 capital_gain: 32561x1 double Values: Min 0 Median 0 Max 99999 capital_loss: 32561x1 double Values: Min 0 Median 0 Max 4356 hours_per_week: 32561x1 double Values: Min 1 Median 40 Max 99 salary: 32561x1 categorical Values: <=50K 24720 >50K 7841

Because there are few categories represented in the categorical variables compared to levels in the continuous variables, the standard CART, predictor-splitting algorithm prefers splitting a continuous predictor over the categorical variables.

Train a classification tree using the entire data set. To grow unbiased trees, specify usage of the curvature test for splitting predictors. Because there are missing observations in the data, specify usage of surrogate splits.

Mdl = fitctree(X,'salary','PredictorSelection','curvature',... 'Surrogate','on');

Estimate predictor importance values by summing changes in the risk due to splits on every predictor and dividing the sum by the number of branch nodes. Compare the estimates using a bar graph.

imp = predictorImportance(Mdl); figure; bar(imp); title('Predictor Importance Estimates'); ylabel('Estimates'); xlabel('Predictors'); h = gca; h.XTickLabel = Mdl.PredictorNames; h.XTickLabelRotation = 45; h.TickLabelInterpreter = 'none';

In this case, `capital_gain`

is the most important predictor, followed by `education_num`

.

`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. Optionally, `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 in `Tbl`

as
predictors, then specify the response variable using `ResponseVarName`

.

If `Tbl`

contains the response variable, and
you want to use only a subset of the remaining variables in `Tbl`

as
predictors, then specify a formula using `formula`

.

If `Tbl`

does not contain the response variable,
then specify a response variable using `Y`

. The
length of response variable and the number of rows of `Tbl`

must
be equal.

**Data Types: **`table`

`ResponseVarName`

— Response variable namename of variable in

`Tbl`

Response variable name, specified as the name of a variable
in `Tbl`

.

You must specify `ResponseVarName`

as a character
vector. For example, if the response variable `Y`

is
stored as `Tbl.Y`

, then specify it as `'Y'`

.
Otherwise, the software treats all columns of `Tbl`

,
including `Y`

, as predictors when training the model.

The response variable must be a categorical or character array,
logical or numeric vector, or cell array of character vectors. If `Y`

is
a character array, then each element must correspond to one row of
the array.

It is good practice to specify the order of the classes using
the `ClassNames`

name-value pair argument.

**Data Types: **`char`

`formula`

— Explanatory model of response and subset of predictor variablescharacter vector

Explanatory model of the response and a subset of the predictor
variables, specified as a character vector in the form of `'Y~X1+X2+X3'`

.
In this form, `Y`

represents the response variable,
and `X1`

, `X2`

, and `X3`

represent
the predictor variables. The variables must be variable names in `Tbl`

(`Tbl.Properties.VariableNames`

).

To specify a subset of variables in `Tbl`

as
predictors for training the model, use a formula. If you specify a
formula, then the software does not use any variables in `Tbl`

that
do not appear in `formula`

.

**Data Types: **`char`

`Y`

— Class labelsnumeric vector | categorical vector | logical vector | character array | cell array of character vectors

Class labels, specified as a numeric vector, categorical vector,
logical vector, character array, or cell array of character vectors.
Each row of `Y`

represents the classification of
the corresponding row of `X`

.

When fitting the tree, `fitctree`

considers `NaN`

, `''`

(empty
character vector), and `<undefined>`

values
in `Y`

to be missing values. `fitctree`

does
not use observations with missing values for `Y`

in
the fit.

For numeric `Y`

, consider fitting a regression
tree using `fitrtree`

instead.

**Data Types: **`single`

| `double`

| `char`

| `logical`

| `cell`

`X`

— Predictor datanumeric matrix

Predictor data, specified as a numeric matrix. Each row of `X`

corresponds
to one observation, and each column corresponds to one predictor variable.

`fitctree`

considers `NaN`

values
in `X`

as missing values. `fitctree`

does
not use observations with all missing values for `X`

in
the fit. `fitctree`

uses observations with some
missing values for `X`

to find splits on variables
for which these observations have valid values.

**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 single quotes (`' '`

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

.

`'CrossVal','on','MinLeafSize',40`

specifies a
cross-validated classification tree with a minimum of 40 observations per
leaf.You cannot use any cross-validation name-value pair along with `OptimizeHyperparameters`

.
You can modify the cross-validation for `OptimizeHyperparameters`

only
by using the `HyperparameterOptimizationOptions`

name-value
pair.

`'AlgorithmForCategorical'`

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

| `'PullLeft'`

| `'PCA'`

| `'OVAbyClass'`

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

and one of the
following values.

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

`'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. |

`fitctree`

automatically
selects the optimal subset of algorithms for each split using the known
number of classes and levels of a categorical predictor. For
*K* = 2 classes, `fitctree`

always performs
the exact search. To specify a particular algorithm, use the
`'AlgorithmForCategorical'`

name-value pair
argument.

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

`'CategoricalPredictors'`

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

`'all'`

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

and one of these values:

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

Vector of positive integers | An entry in the vector is the index value corresponding to the column of the
predictor data (`X` or `Tbl` ) that contains a
categorical variable. |

Logical 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. |

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`

), `fitctree`

assumes that a variable is categorical if it contains logical values, categorical values, or a
cell array of character vectors. If the predictor data is a matrix (`X`

),
`fitctree`

assumes all predictors are continuous. To identify any
categorical predictors when the data is a matrix, use the `'CategoricalPredictors'`

name-value pair
argument.

**Example: **`'CategoricalPredictors','all'`

**Data Types: **`single`

| `double`

| `logical`

| `char`

| `cell`

`'ClassNames'`

— Names of classes to use for trainingcategorical array | cell array of character vectors | character array | logical vector | numeric vector

Names of classes to use for training, specified as the comma-separated
pair consisting of `'ClassNames'`

and a categorical
or character array, logical or numeric vector, or cell array of character
vectors. `ClassNames`

must be the same data type
as `Y`

.

If `ClassNames`

is a character array, then
each element must correspond to one *row* of the
array.

Use `ClassNames`

to:

Order the classes during training.

Specify the order of any input or output argument dimension that corresponds to the class order. For example, use

`ClassNames`

to specify the order of the dimensions of`Cost`

or the column order of classification scores returned by`predict`

.Select a subset of classes for training. For example, suppose that the set of all distinct class names in

`Y`

is`{'a','b','c'}`

. To train the model using observations from classes`'a'`

and`'c'`

only, specify`'ClassNames',{'a','c'}`

.

The default is the set of all distinct class names in `Y`

.

**Example: **`'ClassNames',{'b','g'}`

**Data Types: **`categorical`

| `char`

| `logical`

| `single`

| `double`

| `cell`

`'Cost'`

— Cost of misclassificationsquare matrix | structure

Cost of misclassification of a point, specified as the comma-separated
pair consisting of `'Cost'`

and one of the
following:

Square matrix, where

`Cost(i,j)`

is the cost of classifying a point into class`j`

if its true class is`i`

(i.e., the rows correspond to the true class and the columns correspond to the predicted class). To specify the class order for the corresponding rows and columns of`Cost`

, also specify the`ClassNames`

name-value pair argument.Structure

`S`

having two fields:`S.ClassNames`

containing the group names as a variable of the same data type as`Y`

, and`S.ClassificationCosts`

containing the cost matrix.

The default is `Cost(i,j)=1`

if
`i~=j`

, and `Cost(i,j)=0`

if
`i=j`

.

**Data Types: **`single`

| `double`

| `struct`

`'MaxDepth'`

— Maximum tree depthpositive integer

Maximum tree depth, specified as a positive integer. Specify a value
for this parameter to return a tree with fewer levels that requires
fewer passes through the tall array to compute. Generally the algorithm
of `fitctree`

takes one pass through the data and an
additional pass for each tree level. There is no maximum tree depth by
default.

This option only applies when using `fitctree`

on tall arrays. See Extended Capabilities for more information.

`'MaxNumCategories'`

— Maximum category levels`10`

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

and a nonnegative
scalar value. `fitctree`

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

levels in the split
node. Otherwise, `fitctree`

finds the best
categorical split using one of the inexact algorithms.

Passing a small value can lead to loss of accuracy and passing a large value can increase computation time and memory overload.

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

`'MergeLeaves'`

— Leaf merge flag`'on'`

(default) | `'off'`

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

and `'on'`

or
`'off'`

.

If `MergeLeaves`

is `'on'`

, then
`fitctree`

:

Merges leaves that originate from the same parent node, and that yields a sum of risk values greater or equal to the risk associated with the parent node

Estimates the optimal sequence of pruned subtrees, but does not prune the classification tree

Otherwise, `fitctree`

does not merge
leaves.

**Example: **`'MergeLeaves','off'`

`'MinParentSize'`

— Minimum number of branch node observations`10`

(default) | positive integer valueMinimum number of branch node observations, 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`

,
`fitctree`

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

.

**Example: **`'MinParentSize',8`

**Data Types: **`single`

| `double`

`'PredictorNames'`

— Predictor variable namescell array of unique character vectors

Predictor variable names, specified as the comma-separated pair
consisting of `'PredictorNames'`

and a cell array
of unique character vectors. The functionality of `'PredictorNames'`

depends
on the way you supply the training data.

If you supply

`X`

and`Y`

, then you can use`'PredictorNames'`

to give the predictor variables in`X`

names.The order of the names in

`PredictorNames`

must correspond to the column order of`X`

. That is,`PredictorNames{1}`

is the name of`X(:,1)`

,`PredictorNames{2}`

is the name of`X(:,2)`

, and so on. Also,`size(X,2)`

and`numel(PredictorNames)`

must be equal.By default,

`PredictorNames`

is`{'x1','x2',...}`

.

If you supply

`Tbl`

, then you can use`'PredictorNames'`

to choose which predictor variables to use in training. That is,`fitctree`

uses the predictor variables in`PredictorNames`

and the response only in training.`PredictorNames`

must be a subset of`Tbl.Properties.VariableNames`

and cannot include the name of the response variable.By default,

`PredictorNames`

contains the names of all predictor variables.It good practice to specify the predictors for training using one of

`'PredictorNames'`

or`formula`

only.

**Example: **`'PredictorNames',{'SepalLength','SepalWidth','PedalLength','PedalWidth'}`

**Data Types: **`cell`

`'PredictorSelection'`

— Algorithm used to select the best split predictor`'allsplits'`

(default) | `'curvature'`

| `'interaction-curvature'`

Algorithm used to select the best split predictor at each node,
specified as the comma-separated pair consisting of
`'PredictorSelection'`

and a value in this
table.

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

`'allsplits'` |
Standard CART — Selects the split predictor that maximizes the split-criterion gain over all possible splits of all predictors [1]. |

`'curvature'` | Curvature test — Selects the split
predictor that minimizes the
p-value of chi-square tests of
independence between each predictor and the response
[4]. Training speed is similar to standard
CART. |

`'interaction-curvature'` | Interaction test — Chooses the
split predictor that minimizes the
p-value of chi-square tests of
independence between each predictor and the
response, and that minimizes the
p-value of a chi-square test of
independence between each pair of predictors and
response [3]. Training speed can be slower than standard
CART. |

For `'curvature'`

and
`'interaction-curvature'`

, if all tests yield
*p*-values greater than 0.05, then
`fitctree`

stops splitting nodes.

Standard CART tends to select split predictors containing many distinct values, e.g., continuous variables, over those containing few distinct values, e.g., categorical variables [4]. Consider specifying the curvature or interaction test if any of the following are true:

If there are predictors that have relatively fewer distinct values than other predictors, for example, if the predictor data set is heterogeneous.

If an analysis of predictor importance is your goal. For more on predictor importance estimation, see

`predictorImportance`

.

Trees grown using standard CART are not sensitive to predictor variable interactions. Also, such trees are less likely to identify important variables in the presence of many irrelevant predictors than the application of the interaction test. Therefore, to account for predictor interactions and identify importance variables in the presence of many irrelevant variables, specify the interaction test [3].

Prediction speed is unaffected by the value of

`'PredictorSelection'`

.

`fitctree`

selects split
predictors, see Node Splitting Rules.
**Example: **`'PredictorSelection','curvature'`

**Data Types: **`char`

`'Prior'`

— Prior probabilities`'empirical'`

(default) | `'uniform'`

| vector of scalar values | structurePrior probabilities for each class, specified as the comma-separated
pair consisting of `'Prior'`

and one of the
following.

A character vector:

`'empirical'`

determines class probabilities from class frequencies in`Y`

. If you pass observation weights,`fitctree`

uses the weights to compute the class probabilities.`'uniform'`

sets all class probabilities equal.

A vector (one scalar value for each class). To specify the class order for the corresponding elements of

`Prior`

, also specify the`ClassNames`

name-value pair argument.A structure

`S`

with two fields:`S.ClassNames`

containing the class names as a variable of the same type as`Y`

`S.ClassProbs`

containing a vector of corresponding probabilities

If you set values for both `weights`

and
`prior`

, the weights are renormalized to add up to
the value of the prior probability in the respective class.

**Example: **`'Prior','uniform'`

`'Prune'`

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

(default) | `'off'`

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
`fitctree`

grows the classification tree
without pruning it, but estimates the optimal sequence of pruned
subtrees. Otherwise, `fitctree`

grows the
classification tree without estimating the optimal sequence of pruned
subtrees.

To prune a trained `ClassificationTree`

model,
pass it to `prune`

.

**Example: **`'Prune','off'`

`'PruneCriterion'`

— Pruning criterion`'error'`

(default) | `'impurity'`

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

and `'error'`

or `'impurity'`

.

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

`'ResponseName'`

— Response variable name`'Y'`

(default) | character vectorResponse variable name, specified as the comma-separated pair
consisting of `'ResponseName'`

and a character vector
representing the name of the response variable.

This name-value pair is not valid when using the
`ResponseVarName`

or `formula`

input arguments.

**Example: **`'ResponseName','IrisType'`

`'ScoreTransform'`

— Score transformation`'none'`

(default) | character vector | function handleScore transformation, specified as the comma-separated pair consisting of
`'ScoreTransform'`

and either a character vector or a function
handle.

This table summarizes the available character vectors.

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

`'doublelogit'` | 1/(1 + e^{–2x}) |

`'invlogit'` | log(x / (1–x)) |

`'ismax'` | Set the score for the class with the largest score to `1` , and set the
scores for all other classes to `0` . |

`'logit'` | 1/(1 + e^{–x}) |

`'none'` or `'identity'` | x (no transformation) |

`'sign'` | –1 for x < 00 for x = 01 for x >
0 |

`'symmetric'` | 2x – 1 |

`'symmetricismax'` | Set the score for the class with the largest score to `1` ,
and set the scores for all other classes to `-1` . |

`'symmetriclogit'` | 2/(1 + e^{–x})
– 1 |

For a MATLAB^{®} function or a function you define, use its function handle. The function
handle must accept a matrix (the original scores) and return a matrix of the same size
(the transformed scores).

**Example: **`'ScoreTransform','logit'`

**Data Types: **`char`

| `function_handle`

`'Surrogate'`

— Surrogate decision splits flag`'off'`

(default) | `'on'`

| `'all'`

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

and `'on'`

,
`'off'`

, `'all'`

, or a positive
integer value.

When set to

`'on'`

,`fitctree`

finds at most 10 surrogate splits at each branch node.When set to

`'all'`

,`fitctree`

finds all surrogate splits at each branch node. The`'all'`

setting can use considerable time and memory.When set to a positive integer value,

`fitctree`

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. The setting also lets you compute measures of predictive association between predictors. For more details, see Node Splitting Rules.

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

**Data Types: **`single`

| `double`

| `char`

`'Weights'`

— Observation weights`ones(size(x,1),1)`

(default) | vector of scalar valuesObservation weights, specified as the comma-separated pair consisting
of `'Weights'`

and a vector of scalar values. The
software weights 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 in `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. For
example, if 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 when training the
model.

`fitctree`

normalizes the
weights in each class to add up to the value of the prior probability of
the class.

**Data Types: **`single`

| `double`

`'CrossVal'`

— Flag to grow cross-validated decision tree`'off'`

(default) | `'on'`

Flag to grow a cross-validated decision tree, specified as the
comma-separated pair consisting of `'CrossVal'`

and
`'on'`

or `'off'`

.

If `'on'`

, `fitctree`

grows a
cross-validated decision tree with 10 folds. You can override this
cross-validation setting using one of the `'KFold'`

,
`'Holdout'`

, `'Leaveout'`

, or
`'CVPartition'`

name-value pair arguments. You can
only use one of these four arguments at a time when creating a
cross-validated tree.

Alternatively, cross validate `tree`

later using
the `crossval`

method.

**Example: **`'CrossVal','on'`

`'CVPartition'`

— Partition for cross-validated tree`cvpartition`

objectPartition to use in a cross-validated tree, specified as the
comma-separated pair consisting of `'CVPartition'`

and
an object created using `cvpartition`

.

If you use `'CVPartition'`

, you cannot use any of the
`'KFold'`

, `'Holdout'`

, or
`'Leaveout'`

name-value pair arguments.

`'Holdout'`

— Fraction of data for holdout validation`0`

(default) | scalar value in the range `[0,1]`

Fraction of data used for holdout validation, specified as the
comma-separated pair consisting of `'Holdout'`

and a
scalar value in the range `[0,1]`

. Holdout validation
tests the specified fraction of the data, and uses the rest of the data
for training.

If you use `'Holdout'`

, you cannot use any of the
`'CVPartition'`

, `'KFold'`

, or
`'Leaveout'`

name-value pair arguments.

**Example: **`'Holdout',0.1`

**Data Types: **`single`

| `double`

`'KFold'`

— Number of folds`10`

(default) | positive integer value greater than 1Number of folds to use in a cross-validated classifier, specified
as the comma-separated pair consisting of `'KFold'`

and
a positive integer value greater than 1. If you specify, e.g., `'KFold',`

,
then the software:*k*

Randomly partitions the data into

*k*setsFor each set, reserves the set as validation data, and trains the model using the other

*k*– 1 setsStores the

compact, trained models in the cells of a`k`

-by-1 cell vector in the`k`

`Trained`

property of the cross-validated model.

To create a cross-validated model, you can use one of these
four options only: `CVPartition`

, `Holdout`

, `KFold`

,
or `Leaveout`

.

**Example: **`'KFold',8`

**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'`

. Specify
`'on'`

to use leave-one-out
cross-validation.

If you use `'Leaveout'`

, you cannot use any of the
`'CVPartition'`

, `'Holdout'`

, or
`'KFold'`

name-value pair arguments.

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

`'MaxNumSplits'`

— Maximal number of decision splits`size(X,1) - 1`

(default) | positive integerMaximal number of decision splits (or branch nodes), specified
as the comma-separated pair consisting of `'MaxNumSplits'`

and
a positive integer. `fitctree`

splits `MaxNumSplits`

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

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

**Data Types: **`single`

| `double`

`'MinLeafSize'`

— Minimum number of leaf node observations`1`

(default) | positive integer valueMinimum number of leaf node observations, 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`

, `fitctree`

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

.

**Example: **`'MinLeafSize',3`

**Data Types: **`single`

| `double`

`'NumVariablesToSample'`

— Number of predictors to select at random for each split`'all'`

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

and a positive integer
value. You can also specify `'all'`

to use all
available predictors.

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

**Data Types: **`single`

| `double`

`'SplitCriterion'`

— Split criterion`'gdi'`

(default) | `'twoing'`

| `'deviance'`

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

and `'gdi'`

(Gini's diversity index), `'twoing'`

for the twoing
rule, or `'deviance'`

for maximum deviance reduction
(also known as cross entropy).

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

`'OptimizeHyperparameters'`

— Parameters to optimize`'none'`

(default) | `'auto'`

| `'all'`

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

objectsParameters to optimize, specified as:

`'none'`

— Do not optimize.`'auto'`

— Use`{'MinLeafSize'}`

`'all'`

— Optimize all eligible parameters.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 `fitctree`

by varying the parameters. For
information about cross-validation loss (albeit in a different context),
see Classification Loss. To control the
cross-validation type and other aspects of the optimization, use the
`HyperparameterOptimizationOptions`

name-value
pair.

`OptimizeHyperparameters`

values override any
values you set using other name-value pairs. For example, setting `OptimizeHyperparameters`

to `'auto'`

causes
the `'auto'`

values to apply.

The eligible parameters for `fitctree`

are:

`MaxNumSplits`

—`fitctree`

searches among integers, by default log-scaled in the range`[1,max(2,NumObservations-1)]`

.`MinLeafSize`

—`fitctree`

searches among integers, by default log-scaled in the range`[1,max(2,floor(NumObservations/2))]`

.`SplitCriterion`

— For two classes,`fitctree`

searches among`'gdi'`

and`'deviance'`

. For three or more classes,`fitctree`

also searches among`'twoing'`

.`NumVariablesToSample`

—`fitctree`

does not optimize over this hyperparameter. If you pass`NumVariablesToSample`

as a parameter name,`fitctree`

simply uses the full number of predictors. However,`fitcensemble`

does optimize over this hyperparameter.

Set nondefault parameters by passing a vector of
`optimizableVariable`

objects that have nondefault
values. For example,

load fisheriris params = hyperparameters('fitctree',meas,species); params(1).Range = [1,30];

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 `HyperparameterOptimizationOptions`

name-value
pair, `Verbose`

field. To control the plots, set
the `HyperparameterOptimizationOptions`

name-value
pair, `ShowPlots`

field.

For an example, see Optimize Classification Tree.

**Example: **`'auto'`

**Data Types: **`char`

| `cell`

`'HyperparameterOptimizationOptions'`

— Options for optimizationstructure

Options for optimization, specified as a structure. Modifies
the effect of the `OptimizeHyperparameters`

name-value
pair. 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'`
`bayesopt ` `AcquisitionFunctionName` name-value pair, or Acquisition Function Types. | `'expected-improvement-per-second-plus'` |

`MaxObjectiveEvaluations` | Maximum number of objective function evaluations. | `30` for `'bayesopt'` or `'randomsearch'` , and the entire grid for `'gridsearch'` |

`MaxTime` | Time limit, specified as a positive real. The time limit is in seconds, as measured by | `Inf` |

`NumGridDivisions` | For `'gridsearch'` , the number of values in each dimension. Can be a vector of positive integers giving the number of values for each dimension, or a scalar that applies to all dimensions. Ignored for categorical variables. | `10` |

`ShowPlots` | Logical value indicating whether to show plots. If `true` , plots the best objective function value against 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` , 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
`bayesopt` `Verbose` name-value pair. | `1` |

`UseParallel` | Logical value indicating whether to run Bayesian optimization in parallel, which requires Parallel Computing Toolbox™. For details, see Parallel Bayesian Optimization. | `false` |

`Repartition` | Logical value indicating whether to repartition the cross-validation at every iteration. If
| `false` |

Use no more than one of the following three field names. | ||

`CVPartition` | A `cvpartition` object, as created by `cvpartition` | `Kfold` = `5` |

`Holdout` | A scalar in the range `(0,1)` representing the holdout fraction. | |

`Kfold` | An integer greater than 1. |

**Example: **`struct('MaxObjectiveEvaluations',60)`

**Data Types: **`struct`

`tree`

— Classification treeclassification tree object

Classification tree, returned as a classification tree object.

Using the `'CrossVal'`

, `'KFold'`

,
`'Holdout'`

, `'Leaveout'`

, or
`'CVPartition'`

options results in a tree of class
`ClassificationPartitionedModel`

.
You cannot use a partitioned tree for prediction, so this kind of tree does
not have a `predict`

method. Instead, use
`kfoldpredict`

to predict
responses for observations not used for training.

Otherwise, `tree`

is of class `ClassificationTree`

, and you can
use the `predict`

method to make
predictions.

The *curvature test* is
a statistical test assessing the null hypothesis that two variables
are unassociated.

The curvature test between predictor variable *x* and *y* is
conducted using this process.

If

*x*is continuous, then partition it into its quartiles. Create a nominal variable that bins observations according to which section of the partition they occupy. If there are missing values, then create an extra bin for them.For each level in the partitioned predictor

*j*= 1...*J*and class in the response*k*= 1,...,*K*, compute the weighted proportion of observations in class*k*$${\widehat{\pi}}_{jk}={\displaystyle \sum _{i=1}^{n}I\{{y}_{i}=k\}}{w}_{i}.$$

*w*is the weight of observation_{i}*i*, $$\sum {w}_{i}}=1$$,*I*is the indicator function, and*n*is the sample size. If all observations have the same weight, then $${\widehat{\pi}}_{jk}=\frac{{n}_{jk}}{n}$$, where*n*is the number of observations in level_{jk}*j*of the predictor that are in class*k*.Compute the test statistic

$${\widehat{\pi}}_{j+}={\displaystyle \sum _{k}{\widehat{\pi}}_{jk}}$$, that is, the marginal probability of observing the predictor at level$$t=n{\displaystyle \sum _{k=1}^{K}{\displaystyle \sum _{j=1}^{J}\frac{{\left({\widehat{\pi}}_{jk}-{\widehat{\pi}}_{j+}{\widehat{\pi}}_{+k}\right)}^{2}}{{\widehat{\pi}}_{j+}{\widehat{\pi}}_{+k}}}}$$

*j*. $${\widehat{\pi}}_{+k}={\displaystyle \sum _{j}{\widehat{\pi}}_{jk}}$$, that is the marginal probability of observing class*k*. If*n*is large enough, then*t*is distributed as a*χ*^{2}with (*K*– 1)(*J*– 1) degrees of freedom.If the

*p*-value for the test is less than 0.05, then reject the null hypothesis that there is no association between*x*and*y*.

When determining the best split predictor at each node, the standard CART algorithm prefers to select continuous predictors that have many levels. Sometimes, such a selection can be spurious and can also mask more important predictors that have fewer levels, such as categorical predictors.

The curvature test can be applied instead of standard CART to
determine the best split predictor at each node. In that case, the
best split predictor variable is the one that minimizes the significant *p*-values
(those less than 0.05) of curvature tests between each predictor and
the response variable. Such a selection is robust to the number of
levels in individual predictors.

If levels of a predictor are pure for a particular class, then
`fitctree`

merges those levels. Therefore, in step 3
of the algorithm, *J* can be less than the actual number of
levels in the predictor. For example, if *x* has 4 levels, and
all observations in bins 1 and 2 belong to class 1, then those levels are pure
for class 1. Consequently, `fitctree`

merges the
observations in bins 1 and 2, and *J* reduces to 3.

For more details on how the curvature test applies to growing classification trees, see Node Splitting Rules and [4].

`ClassificationTree`

splits
nodes based on either *impurity* or *node
error*.

Impurity means one of several things, depending on your choice
of the `SplitCriterion`

name-value pair 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$$-{\displaystyle \sum _{i}p(i){\mathrm{log}}_{2}p(i)}.$$

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 hence similar to the parent node, and so 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 is1 –

*p*(*j*).

The *interaction test* is a statistical test
that assesses the null hypothesis that there is no interaction between a pair of
predictor variables and the response variable.

The interaction test assessing the association between predictor variables
*x*_{1} and
*x*_{2} with respect to
*y* is conducted using this process.

If

*x*_{1}or*x*_{2}is continuous, then partition that variable into its quartiles. Create a nominal variable that bins observations according to which section of the partition they occupy. If there are missing values, then create an extra bin for them.Create the nominal variable

*z*with*J*=*J*_{1}*J*_{2}levels that assigns an index to observation*i*according to which levels of*x*_{1}and*x*_{2}it belongs. Remove any levels of*z*that do not correspond to any observations.Conduct a curvature test between

*z*and*y*.

When growing decision trees, if there are important interactions between pairs of predictors, but there are also many other less important predictors in the data, then standard CART tends to miss the important interactions. However, conducting curvature and interaction tests for predictor selection instead can improve detection of important interactions, which can yield more accurate decision trees.

For more details on how the interaction test applies to growing decision trees, see Curvature Test, Node Splitting Rules and [3].

The *predictive measure of association* is
a value that indicates the similarity between decision rules that
split observations. Among all possible decision splits that are compared
to the optimal split (found by growing the tree), the best surrogate decision
split yields the maximum predictive measure of association.
The second-best surrogate split has the second-largest predictive
measure of association.

Suppose *x _{j}* and

$${\lambda}_{jk}=\frac{\text{min}\left({P}_{L},{P}_{R}\right)-\left(1-{P}_{{L}_{j}{L}_{k}}-{P}_{{R}_{j}{R}_{k}}\right)}{\text{min}\left({P}_{L},{P}_{R}\right)}.$$

*P*is the proportion of observations in node_{L}*t*, such that*x*<_{j}*u*. The subscript*L*stands for the left child of node*t*.*P*is the proportion of observations in node_{R}*t*, such that*x*≥_{j}*u*. The subscript*R*stands for the right child of node*t*.$${P}_{{L}_{j}{L}_{k}}$$ is the proportion of observations at node

*t*, such that*x*<_{j}*u*and*x*<_{k}*v*.$${P}_{{R}_{j}{R}_{k}}$$ is the proportion of observations at node

*t*, such that*x*≥_{j}*u*and*x*≥_{k}*v*.Observations with missing values for

*x*or_{j}*x*do not contribute to the proportion calculations._{k}

*λ _{jk}* is a value
in (–∞,1]. If

A *surrogate decision split* is an alternative
to the optimal decision split at a given node in a decision tree.
The optimal split is found by growing the tree; the surrogate split
uses a similar or correlated predictor variable and split criterion.

When the value of the optimal split predictor for an observation is missing, the observation is sent to the left or right child node using the best surrogate predictor. When the value of the best surrogate split predictor for the observation is also missing, the observation is sent to the left or right child node using the second-best surrogate predictor, and so on. Candidate splits are sorted in descending order by their predictive measure of association.

By default,

`Prune`

is`'on'`

. However, this specification does not prune the classification tree. To prune a trained classification tree, pass the classification tree to`prune`

.After training a

`ClassificationTree`

model object by using`fitctree`

, use the function`predict`

and the trained model object to generate C code that predicts labels for new data. For details, see Code Generation.

`fitctree`

uses these processes to determine how to split
node *t*.

For standard CART (that is, if

`PredictorSelection`

is`'allpairs'`

) and for all predictors*x*,_{i}*i*= 1,...,*p*:`fitctree`

computes the weighted impurity of node*t*,*i*. For supported impurity measures, see_{t}`SplitCriterion`

.`fitctree`

estimates the probability that an observation is in node*t*using$$P\left(T\right)={\displaystyle \sum _{j\in T}{w}_{j}}.$$

*w*is the weight of observation_{j}*j*, and*T*is the set of all observation indices in node*t*. If you do not specify`Prior`

or`Weights`

, then*w*= 1/_{j}*n*, where*n*is the sample size.`fitctree`

sorts*x*in ascending order. Each element of the sorted predictor is a splitting candidate or cut point._{i}`fitctree`

stores any indices corresponding to missing values in the set*T*, which is the unsplit set._{U}`fitctree`

determines the best way to split node*t*using*x*by maximizing the impurity gain (Δ_{i}*I*) over all splitting candidates. That is, for all splitting candidates in*x*:_{i}`fitctree`

splits the observations in node*t*into left and right child nodes (*t*and_{L}*t*, respectively)._{R}`fitctree`

computes Δ*I*. Suppose that for a particular splitting candidate,*t*and_{L}*t*contain observation indices in the sets_{R}*T*and_{L}*T*, respectively._{R}If

*x*does not contain any missing values, then the impurity gain for the current splitting candidate is_{i}$$\Delta I=P\left(T\right){i}_{t}-P\left({T}_{L}\right){i}_{{t}_{L}}-P\left({T}_{R}\right){i}_{{t}_{R}}.$$

If

*x*contains missing values then, assuming that the observations are missing at random, the impurity gain is_{i}$$\Delta {I}_{U}=P\left(T-{T}_{U}\right){i}_{t}-P\left({T}_{L}\right){i}_{{t}_{L}}-P\left({T}_{R}\right){i}_{{t}_{R}}.$$

*T*–*T*is the set of all observation indices in node_{U}*t*that are not missing.If you use surrogate decision splits, then:

`fitctree`

computes the predictive measures of association between the decision split*x*<_{j}*u*and all possible decision splits*x*<_{k}*v*,*j*≠*k*.`fitctree`

sorts the possible alternative decision splits in descending order by their predictive measure of association with the optimal split. The surrogate split is the decision split yielding the largest measure.`fitctree`

decides the child node assignments for observations with a missing value for*x*using the surrogate split. If the surrogate predictor also contains a missing value, then_{i}`fitctree`

uses the decision split with the second largest measure, and so on, until there are no other surrogates. It is possible for`fitctree`

to split two different observations at node*t*using two different surrogate splits. For example, suppose the predictors*x*_{1}and*x*_{2}are the best and second best surrogates, respectively, for the predictor*x*,_{i}*i*∉ {1,2}, at node*t*. If observation*m*of predictor*x*is missing (i.e.,_{i}*x*is missing), but_{mi}*x*_{m1}is not missing, then*x*_{1}is the surrogate predictor for observation*x*. If observations_{mi}*x*_{(m + 1),i}and*x*(*m*+ 1),*1*are missing, but*x*_{(m + 1),2}is not missing, then*x*_{2}is the surrogate predictor for observation*m*+ 1.`fitctree`

uses the appropriate impurity gain formula. That is, if`fitctree`

fails to assign all missing observations in node*t*to children nodes using surrogate splits, then the impurity gain is Δ*I*. Otherwise,_{U}`fitctree`

uses Δ*I*for the impurity gain.

`fitctree`

chooses the candidate that yields the largest impurity gain.

`fitctree`

splits the predictor variable at the cut point that maximizes the impurity gain.For the curvature test (that is, if

`PredictorSelection`

is`'curvature'`

):`fitctree`

conducts curvature tests between each predictor and the response for observations in node*t*.If all

*p*-values are at least 0.05, then`fitctree`

does not split node*t*.If there is a minimal

*p*-value, then`fitctree`

chooses the corresponding predictor to split node*t*.If more than one

*p*-value is zero due to underflow, then`fitctree`

applies standard CART to the corresponding predictors to choose the split predictor.

If

`fitctree`

chooses a split predictor, then it uses standard CART to choose the cut point (see step 4 in the standard CART process).

For the interaction test (that is, if

`PredictorSelection`

is`'interaction-curvature'`

):For observations in node

*t*,`fitctree`

conducts curvature tests between each predictor and the response and interaction tests between each pair of predictors and the response.If all

*p*-values are at least 0.05, then`fitctree`

does not split node*t*.If there is a minimal

*p*-value and it is the result of a curvature test, then`fitctree`

chooses the corresponding predictor to split node*t*.If there is a minimal

*p*-value and it is the result of an interaction test, then`fitctree`

chooses the split predictor using standard CART on the corresponding pair of predictors.If more than one

*p*-value is zero due to underflow, then`fitctree`

applies standard CART to the corresponding predictors to choose the split predictor.

If

`fitctree`

chooses a split predictor, then it uses standard CART to choose the cut point (see step 4 in the standard CART process).

If

`MergeLeaves`

is`'on'`

and`PruneCriterion`

is`'error'`

(which are the default values for these name-value pair arguments), then the software applies pruning only to the leaves and by using classification error. This specification amounts to merging leaves that share the most popular class per leaf.To accommodate

`MaxNumSplits`

,`fitctree`

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`

,`fitctree`

follows this procedure:Determine how many branch nodes in the current layer must be unsplit so that there are at most

`MaxNumSplits`

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

Unsplit the number of least successful branches.

Return the decision tree grown so far.

This procedure produces maximally balanced trees.

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

There are

`MaxNumSplits`

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 is when all nodes are pure (i.e., all observations in the node have the same class).For values

`'curvature'`

or`'interaction-curvature'`

of`PredictorSelection`

, all tests yield*p*-values greater than 0.05.

`MaxNumSplits`

and`MinLeafSize`

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

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

, before`MaxNumSplits`

splits occur.

For dual-core systems and above, `fitctree`

parallelizes
training decision trees using Intel^{®} Threading Building Blocks (TBB). For details on Intel TBB, see https://software.intel.com/en-us/intel-tbb.

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

[2] 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.

[3] Loh, W.Y. “Regression Trees with Unbiased Variable
Selection and Interaction Detection.” *Statistica Sinica*,
Vol. 12, 2002, pp. 361–386.

[4] Loh, W.Y. and Y.S. Shih. “Split Selection Methods for
Classification Trees.” *Statistica Sinica*, Vol. 7, 1997,
pp. 815–840.

Calculate with arrays that have more rows than fit in memory.

Usage notes and limitations:

Supported syntaxes for tall arrays are:

`tree = fitctree(Tbl,Y)`

`tree = fitctree(X,Y)`

`tree = fitctree(___,Name,Value)`

Supported name-value pairs are:

`'AlgorithmForCategorical'`

`'CategoricalPredictors'`

`'ClassNames'`

`'MaxNumCategories'`

`'MaxNumSplits'`

`'MergeLeaves'`

`'MinLeafSize'`

`'MinParentSize'`

`'NumVariablesToSample'`

`'PredictorNames'`

`'ResponseName'`

`'ScoreTransform'`

`'SplitCriterion'`

`'Weights'`

There is an additional name-value pair specific to tall arrays:

`'MaxDepth'`

— A positive integer specifying the maximum depth of the output tree. Specify a value for this parameter to return a tree with fewer levels that requires fewer passes through the tall array to compute. Generally the algorithm of`fitctree`

takes one pass through the data and an additional pass for each tree level. There is no maximum tree depth by default.

For more information, see Tall Arrays (MATLAB).

`ClassificationPartitionedModel`

| `ClassificationTree`

| `kfoldpredict`

| `predict`

| `prune`

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