# resubLoss

Regression error by resubstitution

## Syntax

``L = resubLoss(tree)``
``L = resubLoss(tree,Name=Value)``
``````[L,se] = resubLoss(___)``````
``````[L,se,NLeaf] = resubLoss(___)``````
``````[L,se,NLeaf,bestLevel] = resubLoss(___)``````

## Description

example

````L = resubLoss(tree)` returns the resubstitution loss, which is the loss computed for the data that `fitrtree` used to create `tree`.```

example

````L = resubLoss(tree,Name=Value)` returns the resubstitution loss with additional options specified by one or more name-value arguments.```
``````[L,se] = resubLoss(___)``` also returns the standard error of the classification error.```
``````[L,se,NLeaf] = resubLoss(___)``` also returns the numbers of leaf nodes.```
``````[L,se,NLeaf,bestLevel] = resubLoss(___)``` also returns the best pruning level. By default, `bestLevel` is the pruning level that gives loss within one standard deviation of minimal loss.```

## 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.

`Mdl = fitrtree(X,MPG);`

Compute the resubstitution MSE.

`resubLoss(Mdl)`
```ans = 4.8952 ```

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, specified as a `RegressionTree` object created using the `fitrtree` function.

### 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.

Example: `L = resubloss(tree,Subtrees="all")` prunes all subtrees.

Before R2021a, use commas to separate each name and value, and enclose `Name` in quotes.

Example: `L = resubloss(tree,"Subtrees","all")` prunes all subtrees.

Loss function, specified as a function handle or `"mse"` for mean squared error.

You can write your own loss function in the syntax described in Loss Functions.

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 (in other words, just the root node).

If you specify `"all"`, then `resubLoss` operates on all subtrees (in other words, the entire pruning sequence). This specification is equivalent to using `0:max(tree.PruneList)`.

`resubLoss` 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`

Tree size, specified as one of the following:

• `"se"` — The `resubloss` function 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`).

• `"min"` — The `resubloss` function returns the element of `Subtrees` with smallest loss, which is usually the smallest element of `Subtrees`.

Example: `TreeSize="min"`

## Output Arguments

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Regression loss, returned as a vector of the length of `Subtrees`.

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

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

Optimal pruning level, returned as a nonnegative numeric scalar whose value depends on `TreeSize`:

• When `TreeSize` is `"se"`, then `bestLevel` is 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`).

• When `TreeSize` is `"min"`, then `bestLevel` is the element of `Subtrees` with the smallest loss, usually the smallest element of `Subtrees`.

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### Loss Functions

The built-in loss function is `"mse"`, meaning mean squared error.

To write your own loss function, create a function file of the form

`function loss = lossfun(Y,Yfit,W)`
• `N` is the number of rows of `tree``.X`.

• `Y` is an `N`-element vector representing the observed response.

• `Yfit` is an `N`-element vector representing the predicted responses.

• `W` is an `N`-element vector representing the observation weights.

• The output `loss` should be a scalar.

Pass the function handle `@lossfun` as the value of the `LossFun` name-value argument.

## Version History

Introduced in R2011a