# loss

Regression error

## Syntax

## Description

returns the mean squared error between the predictions of `L`

= loss(`tree`

,`Tbl`

,`ResponseVarName`

)`tree`

to the data in `Tbl`

, compared to the true responses
`Tbl.ResponseVarName`

.

computes the error in prediction with additional options specified by one or more
`L`

= loss(___,`Name,Value`

)`Name,Value`

pair arguments, using any of the previous
syntaxes.

## Input Arguments

`tree`

— Trained regression tree

`RegressionTree`

object | `CompactRegressionTree`

object

Trained regression tree, specified as a `RegressionTree`

object
constructed by `fitrtree`

or a `CompactRegressionTree`

object
constructed by `compact`

.

`Tbl`

— sample data

table

Sample data, specified as a table. Each row of `Tbl`

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

must contain all of the predictors used
to train `tree`

. Optionally, `Tbl`

can
contain additional columns for the response variable and observation
weights. Multicolumn variables and cell arrays other than cell arrays of
character vectors are not allowed.

If `Tbl`

contains the response variable used to train
`tree`

, then you do not need to specify
`ResponseVarName`

or `Y`

.

If you trained `tree`

using sample data contained in a
`table`

, then the input data for this method must also
be in a table.

**Data Types: **`table`

`X`

— Predictor values

numeric matrix

`ResponseVarName`

— Response variable name

name of a variable in `Tbl`

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

. If `Tbl`

contains the
response variable used to train `tree`

, then you do not
need to specify `ResponseVarName`

.

If you specify `ResponseVarName`

, then you must do so
as a character vector or string scalar. For example, if the response
variable is stored as `Tbl.Response`

, then specify it as
`'Response'`

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

, including
`Tbl.ResponseVarName`

, as predictors.

**Data Types: **`char`

| `string`

`Y`

— Response data

numeric column vector

Response data, specified 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: **`single`

| `double`

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

`LossFun`

— Loss function

`'mse'`

(default) | function handle

Loss function, specified as the comma-separated pair consisting of
`'LossFun'`

and a function handle for loss, or
`'mse'`

representing mean-squared error. If you
pass a function handle `fun`

, `loss`

calls `fun`

as:

fun(Y,Yfit,W)

`Y`

is the vector of observed responses.`Yfit`

is the vector of predicted responses.`W`

is the observation weights. If you pass`W`

, the elements are normalized to sum to`1`

.

All the vectors have the same number of rows as
`Y`

.

**Example: **`'LossFun','mse'`

**Data Types: **`function_handle`

| `char`

| `string`

`Subtrees`

— Pruning level

0 (default) | vector of nonnegative integers | `'all'`

Pruning level, specified as the comma-separated pair consisting
of `'Subtrees'`

and a vector of nonnegative integers
in ascending order or `'all'`

.

If you specify a vector, then all elements must be at least `0`

and
at most `max(tree.PruneList)`

. `0`

indicates
the full, unpruned tree and `max(tree.PruneList)`

indicates
the completely pruned tree (i.e., just the root node).

If you specify `'all'`

, then `loss`

operates
on all subtrees (i.e., the entire pruning sequence). This specification
is equivalent to using `0:max(tree.PruneList)`

.

`loss`

prunes `tree`

to
each level indicated in `Subtrees`

, and then estimates
the corresponding output arguments. The size of `Subtrees`

determines
the size of some output arguments.

To invoke `Subtrees`

, the properties `PruneList`

and `PruneAlpha`

of `tree`

must
be nonempty. In other words, grow `tree`

by setting `'Prune','on'`

,
or by pruning `tree`

using `prune`

.

**Example: **`'Subtrees','all'`

**Data Types: **`single`

| `double`

| `char`

| `string`

`TreeSize`

— Tree size

`'se'`

(default) | `'min'`

Tree size, specified as the comma-separated pair consisting of
`'TreeSize'`

and one of the following:

`'se'`

—`loss`

returns`bestlevel`

that corresponds to the smallest tree whose mean squared error (MSE) is within one standard error of the minimum MSE.`'min'`

—`loss`

returns`bestlevel`

that corresponds to the minimal MSE tree.

**Example: **`'TreeSize','min'`

`Weights`

— Observation weights

`ones(size(X,1),1)`

(default) | vector of scalar values | name of a variable in `Tbl`

Observation 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 variable name.
For example, if weights vector `W`

is stored as
`Tbl.W`

, then specify `Weights`

as
`'W'`

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

, including `W`

, as
predictors when training the model.

**Data Types: **`single`

| `double`

| `char`

| `string`

## Output Arguments

`L`

— Classification error

vector of scalar values

Classification error, returned as a vector the length of
`Subtrees`

. The error for each tree is the mean squared
error, weighted with `Weights`

. If you include
`LossFun`

, `L`

reflects the loss
calculated with `LossFun`

.

`se`

— Standard error of loss

vector of scalar values

Standard error of loss, returned as a vector the length of
`Subtrees`

.

`NLeaf`

— Number of leaf nodes

vector of integer values

Number of leaves (terminal nodes) in the pruned subtrees, returned as a
vector the length of `Subtrees`

.

`bestlevel`

— Best pruning level

scalar value

Best pruning level as defined in the `TreeSize`

name-value pair, returned as a scalar whose value depends on
`TreeSize`

:

`TreeSize`

=`'se'`

—`loss`

returns the highest pruning level with loss within one standard deviation of the minimum (`L`

+`se`

, where`L`

and`se`

relate to the smallest value in`Subtrees`

).`TreeSize`

=`'min'`

—`loss`

returns the element of`Subtrees`

with smallest loss, usually the smallest element of`Subtrees`

.

## Examples

### Compute the In-Sample MSE

Load the `carsmall`

data set. Consider `Displacement`

, `Horsepower`

, and `Weight`

as predictors of the response `MPG`

.

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

Grow a regression tree using all observations.

tree = fitrtree(X,MPG);

Estimate the in-sample MSE.

L = loss(tree,X,MPG)

L = 4.8952

### Find the Pruning Level Yielding the Optimal In-sample Loss

Load the `carsmall`

data set. Consider `Displacement`

, `Horsepower`

, and `Weight`

as predictors of the response `MPG`

.

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

Grow a regression tree using all observations.

Mdl = fitrtree(X,MPG);

View the regression tree.

view(Mdl,'Mode','graph');

Find the best pruning level that yields the optimal in-sample loss.

[L,se,NLeaf,bestLevel] = loss(Mdl,X,MPG,'Subtrees','all'); bestLevel

bestLevel = 1

The best pruning level is level 1.

Prune the tree to level 1.

pruneMdl = prune(Mdl,'Level',bestLevel); view(pruneMdl,'Mode','graph');

### Examine the MSE for Each Subtree

Unpruned decision trees tend to overfit. One way to balance model complexity and out-of-sample performance is to prune a tree (or restrict its growth) so that in-sample and out-of-sample performance are satisfactory.

Load the `carsmall`

data set. Consider `Displacement`

, `Horsepower`

, and `Weight`

as predictors of the response `MPG`

.

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

Partition the data into training (50%) and validation (50%) sets.

n = size(X,1); rng(1) % For reproducibility idxTrn = false(n,1); idxTrn(randsample(n,round(0.5*n))) = true; % Training set logical indices idxVal = idxTrn == false; % Validation set logical indices

Grow a regression tree using the training set.

Mdl = fitrtree(X(idxTrn,:),Y(idxTrn));

View the regression tree.

view(Mdl,'Mode','graph');

The regression tree has seven pruning levels. Level 0 is the full, unpruned tree (as displayed). Level 7 is just the root node (i.e., no splits).

Examine the training sample MSE for each subtree (or pruning level) excluding the highest level.

```
m = max(Mdl.PruneList) - 1;
trnLoss = resubLoss(Mdl,'SubTrees',0:m)
```

`trnLoss = `*7×1*
5.9789
6.2768
6.8316
7.5209
8.3951
10.7452
14.8445

The MSE for the full, unpruned tree is about 6 units.

The MSE for the tree pruned to level 1 is about 6.3 units.

The MSE for the tree pruned to level 6 (i.e., a stump) is about 14.8 units.

Examine the validation sample MSE at each level excluding the highest level.

`valLoss = loss(Mdl,X(idxVal,:),Y(idxVal),'SubTrees',0:m)`

`valLoss = `*7×1*
32.1205
31.5035
32.0541
30.8183
26.3535
30.0137
38.4695

The MSE for the full, unpruned tree (level 0) is about 32.1 units.

The MSE for the tree pruned to level 4 is about 26.4 units.

The MSE for the tree pruned to level 5 is about 30.0 units.

The MSE for the tree pruned to level 6 (i.e., a stump) is about 38.5 units.

To balance model complexity and out-of-sample performance, consider pruning `Mdl`

to level 4.

pruneMdl = prune(Mdl,'Level',4); view(pruneMdl,'Mode','graph')

## More About

### Mean Squared Error

The mean squared error *m* of the predictions
*f*(*X _{n}*) with weight
vector

*w*is

$$m=\frac{{\displaystyle \sum {w}_{n}{\left(f\left({X}_{n}\right)-{Y}_{n}\right)}^{2}}}{{\displaystyle \sum {w}_{n}}}.$$

## Extended Capabilities

### Tall Arrays

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

Usage notes and limitations:

Only one output is supported.

You can use models trained on either in-memory or tall data with this function.

For more information, see Tall Arrays.

### GPU Arrays

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

Usage notes and limitations:

The

`loss`

function does not support decision tree models trained with surrogate splits.

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

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