# fitrtree

Binary decision tree for regression

## Syntax

• `tree = fitrtree(x,y)` example
• `tree = fitrtree(x,y,Name,Value)` example

## Description

example

````tree = fitrtree(x,y)` returns a regression tree based on the input variables (also known as predictors, features, or attributes) `x` and output (response) `y`. The returned tree is a binary tree where each branching node is split based on the values of a column of `x`.```

example

````tree = fitrtree(x,y,Name,Value)` fits a tree with additional options specified by one or more name-value pair arguments. For example, you can grow a cross-validated tree, hold out a fraction of data for validation, or specify observation weights.```

## Examples

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### Construct a Regression Tree

```load carsmall; ```

Construct a regression tree using the sample data.

```tree = fitrtree([Weight, Cylinders],MPG,... 'categoricalpredictors',2,'MinParentSize',20,... 'PredictorNames',{'W','C'}) ```
```tree = RegressionTree PredictorNames: {'W' 'C'} ResponseName: 'Y' ResponseTransform: 'none' CategoricalPredictors: 2 NumObservations: 94 ```

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

```mileage4K = predict(tree,[4000 4; 4000 6; 4000 8]) ```
```mileage4K = 19.2778 19.2778 14.3889 ```

### Control Regression Tree Depth

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

Load the `carsmall` data set. Consider `Displacement`, `Horsepower`, and `Weight` as predictors of the response `MPG`.

```load carsmall X = [Displacement Horsepower Weight]; ```

The default values of the tree-depth controllers for growing regression 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 regression tree using the default values for tree-depth control. Cross validate the model using 10-fold cross validation.

```rng(1); % For reproducibility MdlDefault = fitrtree(X,MPG,'CrossVal','on'); ```

Draw a histogram of the number of imposed on the trees. The number of imposed splits is one less than the number of leaves. 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 between 14 and 15.

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

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

Compare the cross validation MSEs of the models.

```mseDefault = kfoldLoss(MdlDefault) mse7 = kfoldLoss(Mdl7) ```
```mseDefault = 27.7277 mse7 = 28.3833 ```

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

## Input Arguments

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### `x` — Predictor valuesmatrix of scalar values

Predictor values, specified as a matrix of scalar values. Each column of `x` represents one variable, and each row represents one observation.

`fitrtree` considers `NaN` values in `x` as missing values. `fitrtree` does not use observations with all missing values for `x` in the fit. `fitrtree` uses observations with some missing values for `x` to find splits on variables for which these observations have valid values.

Data Types: `single` | `double`

### `y` — Response valuesvector of scalar values

Response values, specified as a vector of scalar values with the same number of rows as `x`. Each entry in `y` is the response to the data in the corresponding row of `x`.

`fitrtree` considers `NaN` values in `y` to be missing values. `fitrtree` does not use observations with missing values for `y` in the fit.

Data Types: `single` | `double`

### Name-Value Pair Arguments

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`.

Example: `'CrossVal','on','MinParentSize',30` specifies a cross-validated regression tree with a minimum of 30 observations per branch node.

### `'CategoricalPredictors'` — Categorical predictors listnumeric or logical vector | cell array of strings | character matrix | `'all'`

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

• A numeric vector with indices from `1` through `p`, where `p` is the number of columns of `x`.

• A logical vector of length `p`, where a `true` entry means that the corresponding column of `x` is a categorical variable.

• A cell array of strings, where each element in the array is the name of a predictor variable. The names must match entries in the `PredictorNames` property.

• A character matrix, where each row of the matrix is a name of a predictor variable. Pad the names with extra blanks so each row of the character matrix has the same length.

• `'all'`, meaning all predictors are categorical.

Data Types: `single` | `double` | `logical` | `char` | `cell`

### `'CrossVal'` — Cross-validation flag`'off'` (default) | `'on'`

Cross-validation flag, specified as the comma-separated pair consisting of `'CrossVal'` and either `'on'` or `'off'`.

If `'on'`, `fitrtree` 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 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` object

Partition for 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

Number of folds to use in a cross-validated tree, specified as the comma-separated pair consisting of `'KFold'` and a positive integer value.

If you use `'KFold'`, you cannot use any of the `'CVPartition'`, `'Holdout'`, or `'Leaveout'` name-value pair arguments.

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 either `'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 integer

Maximal number of decision splits (or branch nodes), specified as the comma-separated pair consisting of `'MaxNumSplits'` and a positive integer. `fitrtree` splits `MaxNumSplits` or fewer branch nodes. For more details on splitting behavior, see Algorithms.

Example: `'MaxNumSplits',5`

Data Types: `single` | `double`

### `'MergeLeaves'` — Leaf merge flag`'on'` (default) | `'off'`

Leaf merge flag, specified as the comma-separated pair consisting of `'MergeLeaves'` and either `'on'` or `'off'`.

If `MergeLeaves` is `'on'`, then `fitrtree` merges leaves that originate from the same parent node, and that give a sum of risk values greater or equal to the risk associated with the parent node. Otherwise, `fitrtree` does not merge leaves.

Example: `'MergeLeaves','off'`

### `'MinLeafSize'` — Minimum number of leaf node observations`1` (default) | positive integer value

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

Example: `'MinLeafSize',3`

Data Types: `single` | `double`

### `'MinParentSize'` — Minimum number of branch node observations`10` (default) | positive integer value

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

Example: `'MinParentSize',8`

Data Types: `single` | `double`

### `'NumVariablesToSample'` — Number of predictors to select at random for each split`'all'` (default) | positive integer value

Number 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`

### `'PredictorNames'` — Predictor variable names`{'x1','x2',...}` (default) | cell array of strings

Predictor variable names, specified as the comma-separated pair consisting of `'PredictorNames'` and a cell array of strings containing the names for the predictor variables, in the order in which they appear in `x`.

Data Types: `cell`

### `'Prune'` — Flag to estimate the 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 either `'on'` or `'off'`.

If `Prune` is `'on'`, then `fitrtree` grows the regression tree and estimates the optimal sequence of pruned subtrees, but does not prune the regression tree. Otherwise, `fitrtree` grows the regression tree without estimating the optimal sequence of pruned subtrees.

To prune a trained `RegressionTree` model, pass it to `prune`.

Example: `'Prune','off'`

### `'PruneCriterion'` — Pruning criterion`'mse'`

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

Example: `'PruneCriterion','mse'`

### `'QuadraticErrorTolerance'` — Quadratic error tolerance`1e-6` (default) | positive scalar value

Quadratic error tolerance per node, specified as the comma-separated pair consisting of `'QuadraticErrorTolerance'` and a positive scalar value. Splitting nodes stops when the quadratic error per node drops below `QuadraticErrorTolerance*QED`, where `QED` is the quadratic error for all data computed before the decision tree is grown.

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

### `'ResponseName'` — Response variable name`'Y'` (default) | string

Response variable name, specified as the comma-separated pair consisting of `'ResponseName'` and a string containing the name of the response variable in `y`.

Example: `'ResponseName','Response'`

Data Types: `char`

### `'ResponseTransform'` — Response transform function`'none'` (default) | function handle

Response transform function for transforming the raw response values, specified as the comma-separated pair consisting of `'ResponseTransform'` and either a function handle or `'none'`. The function handle should accept a matrix of response values and return a matrix of the same size. The default string `'none'` means `@(x)x`, or no transformation.

Add or change a `ResponseTransform` function using dot notation:

`tree.ResponseTransform = @function`

Data Types: `function_handle`

### `'SplitCriterion'` — Split criterion`'MSE'`

Split criterion, specified as the comma-separated pair consisting of `'SplitCriterion'` and `'MSE'`, meaning mean squared error.

Example: `'SplitCriterion','MSE'`

### `'Surrogate'` — Surrogate decision splits flag`'off'` | `'on'` | `'all'` | positive integer value

Surrogate decision splits flag, specified as the comma-separated pair consisting of `'Surrogate'` and one of `'on'`, `'off'`, `'all'`, or a positive integer value.

• When `'on'`, `fitrtree` finds at most 10 surrogate splits at each branch node.

• When set to a positive integer value, `fitrtree` finds at most the specified number of surrogate splits at each branch node.

• When set to `'all'`, `fitrtree` finds all surrogate splits at each branch node. The `'all'` setting can use considerable time and memory.

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.

Example: `'Surrogate','on'`

Data Types: `single` | `double`

### `'Weights'` — Observation weights`ones(size(X,1),1)` (default) | vector of scalar values

Observation weights, specified as the comma-separated pair consisting of `'Weights'` and a vector of scalar values. The length of `Weights` is the number of rows in `x`.

Data Types: `single` | `double`

## Output Arguments

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### `tree` — Regression treeregression tree object

Regression tree, returned as a regression tree object. Note that using the `'Crossval'`, `'KFold'`, `'Holdout'`, `'Leaveout'`, or `'CVPartition'` options results in a tree of class `RegressionPartitionedModel`. You cannot use a partitioned tree for prediction, so this kind of tree does not have a `predict` method.

Otherwise, `tree` is of class `RegressionTree`, and you can use the `predict` method to make predictions.

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### Tips

By default, `Prune` is `'on'`. However, this specification does not prune the regression tree. To prune a trained regression tree, pass the regression tree to `prune`.

### Algorithms

• 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`, `fitrtree` 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`, `fitrtree` follows this procedure:

1. Determine how many branch nodes in the current layer must be unsplit so that there are at most `MaxNumSplits` branch nodes.

2. Sort the branch nodes by their impurity gains.

3. Unsplit the number of least successful branches.

4. 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).

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

## References

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