# resubLoss

Resubstitution classification loss for classification tree model

## Syntax

``L = resubLoss(tree)``
``L = resubLoss(tree,Name=Value)``
``[L,SE,Nleaf,BestLevel] = resubLoss(___)``

## Description

example

````L = resubLoss(tree)` returns the classification loss `L` by resubstitution for the trained classification tree model `tree` using the training data stored in `tree.X` and the corresponding true class labels stored in `tree.Y`. By default, `resubLoss` uses the loss computed for the data used by `fitctree` to create `tree`.The classification loss (`L`) is a resubstitution quality measure. Its interpretation depends on the loss function (`LossFun`), but in general, better classifiers yield smaller classification loss values. The default `LossFun` value is `"mincost"` (minimal expected misclassification cost).```

example

````L = resubLoss(tree,Name=Value)` specifies additional options using one or more name-value arguments. For example, you can specify the loss function, pruning level, and the tree size that `resubLoss` uses to calculate the classification loss.```
````[L,SE,Nleaf,BestLevel] = resubLoss(___)` also returns the standard errors of the classification errors, the number of leaf nodes in the trees of the pruning sequence, and the best pruning level as defined in the `TreeSize` name-value argument. By default, `BestLevel` is the pruning level that gives loss within one standard deviation of the minimal loss.```

## Examples

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Compute the resubstitution classification error for the `ionosphere` data.

```load ionosphere tree = fitctree(X,Y); L = resubLoss(tree)```
```L = 0.0114 ```

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 Fisher's iris data set. Partition the data into training (50%) and validation (50%) sets.

```load fisheriris n = size(meas,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 classification tree using the training set.

`Mdl = fitctree(meas(idxTrn,:),species(idxTrn));`

View the classification tree.

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

The classification tree has four pruning levels. Level 0 is the full, unpruned tree (as displayed). Level 3 is just the root node (i.e., no splits).

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

```m = max(Mdl.PruneList) - 1; trnLoss = resubLoss(Mdl,'Subtrees',0:m)```
```trnLoss = 3×1 0.0267 0.0533 0.3067 ```
• The full, unpruned tree misclassifies about 2.7% of the training observations.

• The tree pruned to level 1 misclassifies about 5.3% of the training observations.

• The tree pruned to level 2 (i.e., a stump) misclassifies about 30.6% of the training observations.

Examine the validation sample classification error at each level excluding the highest level.

`valLoss = loss(Mdl,meas(idxVal,:),species(idxVal),'Subtrees',0:m)`
```valLoss = 3×1 0.0369 0.0237 0.3067 ```
• The full, unpruned tree misclassifies about 3.7% of the validation observations.

• The tree pruned to level 1 misclassifies about 2.4% of the validation observations.

• The tree pruned to level 2 (i.e., a stump) misclassifies about 30.7% of the validation observations.

To balance model complexity and out-of-sample performance, consider pruning `Mdl` to level 1.

```pruneMdl = prune(Mdl,'Level',1); view(pruneMdl,'Mode','graph')```

## Input Arguments

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Classification tree model, specified as a `ClassificationTree` model object trained with `fitctree`.

### 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 = resubLoss(tree,Subtrees="all")` specifies to use all subtrees when computing the resubstitution classification loss for `tree`.

Loss function, specified as a built-in loss function name or a function handle.

The following table describes the values for the built-in loss functions.

ValueDescription
`"binodeviance"`Binomial deviance
`"classifcost"`Observed misclassification cost
`"classiferror"`Misclassified rate in decimal
`"exponential"`Exponential loss
`"hinge"`Hinge loss
`"logit"`Logistic loss
`"mincost"`Minimal expected misclassification cost (for classification scores that are posterior probabilities)
`"quadratic"`Quadratic loss

`"mincost"` is appropriate for classification scores that are posterior probabilities. Classification trees return posterior probabilities as classification scores by default (see `predict`).

Specify your own function using function handle notation. Suppose that n is the number of observations in `X`, and K is the number of distinct classes (`numel(tree.ClassNames)`). Your function must have the signature

``lossvalue = lossfun(C,S,W,Cost)``
where:

• The output argument `lossvalue` is a scalar.

• You specify the function name (`lossfun`).

• `C` is an n-by-K logical matrix with rows indicating the class to which the corresponding observation belongs. The column order corresponds to the class order in `tree.ClassNames`.

Create `C` by setting `C(p,q) = 1`, if observation `p` is in class `q`, for each row. Set all other elements of row `p` to `0`.

• `S` is an n-by-K numeric matrix of classification scores. The column order corresponds to the class order in `tree.ClassNames`. `S` is a matrix of classification scores, similar to the output of `predict`.

• `W` is an n-by-1 numeric vector of observation weights. If you pass `W`, the software normalizes the weights to sum to `1`.

• `Cost` is a K-by-K numeric matrix of misclassification costs. For example, ```Cost = ones(K) - eye(K)``` specifies a cost of `0` for correct classification and `1` for misclassification.

For more details on loss functions, see Classification Loss.

Example: `LossFun="binodeviance"`

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 `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 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 `fitctree`, or by pruning `tree` using `prune`.

Example: `Subtrees="all"`

Data Types: `single` | `double` | `char` | `string`

Tree size, specified as one of these values:

• `"se"``resubLoss` returns the best pruning level (`BestLevel`), which corresponds to the highest pruning level with the loss within one standard deviation of the minimum (`L`+`se`, where `L` and `se` relate to the smallest value in `Subtrees`).

• `"min"``resubLoss` returns the best pruning level, which corresponds to the element of `Subtrees` with the smallest loss. This element is usually the smallest element of `Subtrees`.

Example: `TreeSize="min"`

Data Types: `char` | `string`

## Output Arguments

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Classification loss, returned as a vector of scalar values that has the same length as `Subtrees`. The meaning of the error depends on the loss function (`LossFun`).

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

Number of leaf nodes in the pruned subtrees, returned as a vector of integer values 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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### Classification Loss

Classification loss functions measure the predictive inaccuracy of classification models. When you compare the same type of loss among many models, a lower loss indicates a better predictive model.

Consider the following scenario.

• L is the weighted average classification loss.

• n is the sample size.

• For binary classification:

• yj is the observed class label. The software codes it as –1 or 1, indicating the negative or positive class (or the first or second class in the `ClassNames` property), respectively.

• f(Xj) is the positive-class classification score for observation (row) j of the predictor data X.

• mj = yjf(Xj) is the classification score for classifying observation j into the class corresponding to yj. Positive values of mj indicate correct classification and do not contribute much to the average loss. Negative values of mj indicate incorrect classification and contribute significantly to the average loss.

• For algorithms that support multiclass classification (that is, K ≥ 3):

• yj* is a vector of K – 1 zeros, with 1 in the position corresponding to the true, observed class yj. For example, if the true class of the second observation is the third class and K = 4, then y2* = [`0 0 1 0`]′. The order of the classes corresponds to the order in the `ClassNames` property of the input model.

• f(Xj) is the length K vector of class scores for observation j of the predictor data X. The order of the scores corresponds to the order of the classes in the `ClassNames` property of the input model.

• mj = yj*f(Xj). Therefore, mj is the scalar classification score that the model predicts for the true, observed class.

• The weight for observation j is wj. The software normalizes the observation weights so that they sum to the corresponding prior class probability stored in the `Prior` property. Therefore,

`$\sum _{j=1}^{n}{w}_{j}=1.$`

Given this scenario, the following table describes the supported loss functions that you can specify by using the `LossFun` name-value argument.

Loss FunctionValue of `LossFun`Equation
Binomial deviance`"binodeviance"`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.$
Observed misclassification cost`"classifcost"`

$L=\sum _{j=1}^{n}{w}_{j}{c}_{{y}_{j}{\stackrel{^}{y}}_{j}},$

where ${\stackrel{^}{y}}_{j}$ is the class label corresponding to the class with the maximal score, and ${c}_{{y}_{j}{\stackrel{^}{y}}_{j}}$ is the user-specified cost of classifying an observation into class ${\stackrel{^}{y}}_{j}$ when its true class is yj.

Misclassified rate in decimal`"classiferror"`

$L=\sum _{j=1}^{n}{w}_{j}I\left\{{\stackrel{^}{y}}_{j}\ne {y}_{j}\right\},$

where I{·} is the indicator function.

Cross-entropy loss`"crossentropy"`

`"crossentropy"` is appropriate only for neural network models.

The weighted cross-entropy loss is

`$L=-\sum _{j=1}^{n}\frac{{\stackrel{˜}{w}}_{j}\mathrm{log}\left({m}_{j}\right)}{Kn},$`

where the weights ${\stackrel{˜}{w}}_{j}$ are normalized to sum to n instead of 1.

Exponential loss`"exponential"`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right).$
Hinge loss`"hinge"`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.$
Logit loss`"logit"`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right).$
Minimal expected misclassification cost`"mincost"`

`"mincost"` is appropriate only if classification scores are posterior probabilities.

The software computes the weighted minimal expected classification cost using this procedure for observations j = 1,...,n.

1. Estimate the expected misclassification cost of classifying the observation Xj into the class k:

`${\gamma }_{jk}={\left(f{\left({X}_{j}\right)}^{\prime }C\right)}_{k}.$`

f(Xj) is the column vector of class posterior probabilities for the observation Xj. C is the cost matrix stored in the `Cost` property of the model.

2. For observation j, predict the class label corresponding to the minimal expected misclassification cost:

`${\stackrel{^}{y}}_{j}=\underset{k=1,...,K}{\text{argmin}}{\gamma }_{jk}.$`

3. Using C, identify the cost incurred (cj) for making the prediction.

The weighted average of the minimal expected misclassification cost loss is

`$L=\sum _{j=1}^{n}{w}_{j}{c}_{j}.$`

Quadratic loss`"quadratic"`$L=\sum _{j=1}^{n}{w}_{j}{\left(1-{m}_{j}\right)}^{2}.$

If you use the default cost matrix (whose element value is 0 for correct classification and 1 for incorrect classification), then the loss values for `"classifcost"`, `"classiferror"`, and `"mincost"` are identical. For a model with a nondefault cost matrix, the `"classifcost"` loss is equivalent to the `"mincost"` loss most of the time. These losses can be different if prediction into the class with maximal posterior probability is different from prediction into the class with minimal expected cost. Note that `"mincost"` is appropriate only if classification scores are posterior probabilities.

This figure compares the loss functions (except `"classifcost"`, `"crossentropy"`, and `"mincost"`) over the score m for one observation. Some functions are normalized to pass through the point (0,1).

### True Misclassification Cost

The true misclassification cost is the cost of classifying an observation into an incorrect class.

You can set the true misclassification cost per class by using the `Cost` name-value argument when you create the classifier. `Cost(i,j)` is the cost of classifying an observation into class `j` when its true class is `i`. By default, `Cost(i,j)=1` if `i~=j`, and `Cost(i,j)=0` if `i=j`. In other words, the cost is `0` for correct classification and `1` for incorrect classification.

### Expected Misclassification Cost

The expected misclassification cost per observation is an averaged cost of classifying the observation into each class.

Suppose you have `Nobs` observations that you want to classify with a trained classifier, and you have `K` classes. You place the observations into a matrix `X` with one observation per row.

The expected cost matrix `CE` has size `Nobs`-by-`K`. Each row of `CE` contains the expected (average) cost of classifying the observation into each of the `K` classes. `CE(n,k)` is

`$\sum _{i=1}^{K}\stackrel{^}{P}\left(i|X\left(n\right)\right)C\left(k|i\right),$`

where:

• K is the number of classes.

• $\stackrel{^}{P}\left(i|X\left(n\right)\right)$ is the posterior probability of class i for observation X(n).

• $C\left(k|i\right)$ is the true misclassification cost of classifying an observation as k when its true class is i.

## Version History

Introduced in R2011a