# loss

Regression error for regression tree model

## Syntax

## Description

specifies options using one or more name-value arguments in addition to any of the
input argument combinations in the previous syntaxes. For example, you can specify
the loss function and observation weights.`L`

= loss(___,`Name=Value`

)

## 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")
```

## Input Arguments

`tree`

— Trained regression tree

`RegressionTree`

model object | `CompactRegressionTree`

model object

Trained regression tree, specified as a `RegressionTree`

model object trained with `fitrtree`

, or a `CompactRegressionTree`

model object created with
`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. Optionally, `Tbl`

can contain additional
columns for the response variable and observation weights.
`Tbl`

must contain all the predictors used to train
`tree`

. 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 train `tree`

using sample data contained in a
table, then the input data for `loss`

must also be
in a table.

**Data Types: **`table`

`ResponseVarName`

— Response variable name

name of 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`

.

You must specify `ResponseVarName`

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.Response`

, as predictors.

**Data Types: **`char`

| `string`

`X`

— Predictor data

numeric matrix

### 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.*

**Example: **`L = loss(tree,X,Y,Subtrees="all")`

specifies to prune
all subtrees.

`LossFun`

— Loss function

`"mse"`

(default) | `function handle`

Loss function, specified as `"mse"`

(mean squared
error) or as a function handle. If you pass a function handle
`fun`

, `loss`

calls it as

fun(Y,Yfit,W)

where `Y`

, `Yfit`

, and
`W`

are numeric vectors of the same length.

`Y`

is the observed response.`Yfit`

is the predicted response.`W`

is the observation weights. If you pass`W`

, the elements are normalized to sum to 1.

The returned value of `fun(Y,Yfit,W)`

must be a
scalar.

**Example: **`LossFun="mse"`

**Example: **`LossFun=@`

`Lossfun`

**Data Types: **`char`

| `string`

| `function_handle`

`Subtrees`

— Pruning level

`0`

(default) | vector of nonnegative integers | `"all"`

Pruning level, specified as 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 (that is, just the root node).

If you specify `"all"`

, then `loss`

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

.

`loss`

prunes `tree`

to each level
specified by `Subtrees`

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

determines the size of some output
arguments.

For the function to invoke `Subtrees`

, the properties
`PruneList`

and `PruneAlpha`

of
`tree`

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

by setting `Prune="on"`

when you use
`fitrtree`

, 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 one of these values:

`"se"`

—`loss`

returns the best pruning level (`BestLevel`

), which corresponds to the smallest tree whose MSE is within one standard error of the minimum MSE.`"min"`

—`loss`

returns the best pruning level (`BestLevel`

), which corresponds to the minimal MSE tree.

**Example: **`TreeSize="min"`

**Data Types: **`char`

| `string`

`Weights`

— Observation weights

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

(default) | numeric vector of positive values | name of variable in `Tbl`

Observation weights, specified as a numeric vector of positive values
or the name of a variable in `Tbl`

.

If you specify `Weights`

as a numeric vector, then
the size of `Weights`

must be equal to the number of
rows in `X`

or `Tbl`

.

If you specify `Weights`

as the name of a variable
in `Tbl`

, then the name must be a character vector or
string scalar. For example, if the weights are stored as
`Tbl.W`

, then specify `Weights`

as `"W"`

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

, including `Tbl.W`

, as
predictors.

**Example: **`Weights="W"`

**Data Types: **`single`

| `double`

| `char`

| `string`

## Output Arguments

`SE`

— Standard error of loss

numeric vector

Standard error of loss, returned as a numeric vector that has the same
length as `Subtrees`

.

`Nleaf`

— Number of leaf nodes

numeric vector of nonnegative integers

Number of leaf nodes in the pruned subtrees, returned as a numeric vector
of nonnegative integers that has the same length as
`Subtrees`

. Leaf nodes are terminal nodes, which give
responses, not splits.

`BestLevel`

— Best pruning level

numeric scalar

Best pruning level, returned as a numeric scalar whose value depends on
`TreeSize`

:

When

`TreeSize`

is`"se"`

, the`loss`

function returns the highest pruning level whose loss is within one standard deviation of the minimum (`L`

+`se`

, where`L`

and`se`

relate to the smallest value in`Subtrees`

).When

`TreeSize`

is`"min"`

, the`loss`

function returns the element of`Subtrees`

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

.

## 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.

The
`loss`

function supports tall arrays with the following 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).

## Version History

**Introduced in R2011a**

## See Also

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