# loss

Regression error for regression tree model

## Syntax

``L = loss(tree,Tbl,ResponseVarName)``
``L = loss(tree,Tbl,Y)``
``L = loss(tree,X,Y)``
``L = loss(___,Name=Value)``
``````[L,SE,Nleaf,BestLevel] = loss(___)``````

## Description

example

````L = loss(tree,Tbl,ResponseVarName)` returns the mean squared error (MSE) `L` for the trained regression tree model `tree` using the predictor data in table `Tbl` and the true responses in `Tbl.ResponseVarName`. The interpretation of `L` depends on the loss function (`LossFun`) and weighting scheme (`Weights`).```
````L = loss(tree,Tbl,Y)` uses the predictor data in table `Tbl` and the true responses in `Y`.```
````L = loss(tree,X,Y)` uses the predictor data in matrix `X` and the true responses in `Y`.```
````L = loss(___,Name=Value)` 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.```

example

``````[L,SE,Nleaf,BestLevel] = loss(___)``` also returns the standard error of the loss, number of leaf nodes in the trees of the pruning sequence, and best pruning level as defined in the `TreeSize` name-value argument.```

## Examples

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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 ```

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");```

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

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Regression tree model, specified as a `RegressionTree` model object trained with `fitrtree`, or a `CompactRegressionTree` model object created with `compact`.

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 trained `tree` using sample data contained in a table, then the input data for `loss` must also be in a table.

Data Types: `table`

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`

Response data, specified as a numeric column vector with the same number of rows as `Tbl` or `X`. Each entry in `Y` is the response to the data in the corresponding row of `Tbl` or `X`.

Data Types: `single` | `double`

Predictor data, specified as a numeric matrix. Each column of `X` represents one variable, and each row represents one observation.

`X` must have the same number of columns as the data used to train `tree`. `X` must have the same number of rows as the number of rows in `Y`.

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.

Example: `L = loss(tree,X,Y,Subtrees="all")` specifies to prune all subtrees.

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.

The returned value of `fun(Y,Yfit,W)` must be a scalar.

Example: `LossFun="mse"`

Example: `LossFun=@Lossfun`

Data Types: `char` | `string` | `function_handle`

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`

Tree size, specified as one of the following:

• `"se"` — The `loss` function returns `BestLevel` corresponding to the smallest tree whose MSE is within one standard error of the minimum MSE.

• `"min"` — The `loss` function returns `BestLevel` corresponding to the minimal MSE tree.

Example: `TreeSize="min"`

Data Types: `char` | `string`

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`, you must do so as 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

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Regression error, returned as a numeric vector that has the same length as `Subtrees`. The error for each tree is the mean squared error, weighted with `Weights`. If you specify `LossFun`, then `L` reflects the loss calculated with `LossFun`.

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

Number of leaf nodes in the pruned subtrees, returned as a numeric vector that has the same length as `Subtrees`. Leaf nodes are terminal nodes, which give responses, not splits.

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`.

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### Mean Squared Error

The mean squared error m of the predictions f(Xn) with weight vector w is

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

## Version History

Introduced in R2011a