# loss

Classification error

## 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

````L = loss(tree,TBL,ResponseVarName)` returns a scalar representing how well `tree` classifies the data in `TBL`, when `TBL.ResponseVarName` contains the true classifications.When computing the loss, `loss` normalizes the class probabilities in `Y` to the class probabilities used for training, stored in the `Prior` property of `tree`.```
````L = loss(tree,TBL,Y)` returns a scalar representing how well `tree` classifies the data in `TBL`, when `Y` contains the true classifications.```
````L = loss(tree,X,Y)` returns a scalar representing how well `tree` classifies the data in `X`, when `Y` contains the true classifications.```
````L = loss(___,Name,Value)` returns the loss with additional options specified by one or more `Name,Value` pair arguments, using any of the previous syntaxes. For example, you can specify the loss function or observation weights.```
``````[L,se,NLeaf,bestlevel] = loss(___)``` also returns the vector of standard errors of the classification errors (`se`), the vector of numbers of leaf nodes in the trees of the pruning sequence (`NLeaf`), and the best pruning level as defined in the `TreeSize` name-value pair (`bestlevel`).```
 Note:   `loss` returns `se` and further outputs only when the `LossFun` name-value pair is the default `'classiferror'`.

## Input Arguments

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### `tree` — Trained classification tree`ClassificationTree` model object | `CompactClassificationTree` model object

Trained classification tree, specified as a `ClassificationTree` or `CompactClassificationTree` model object. That is, `tree` is a trained classification model returned by `fitctree` or `compact`.

### `TBL` — Sample datatable

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`. Multi-column 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 this method must also be in a table.

Data Types: `table`

### `X` — Data to classifynumeric matrix

Data to classify, specified as a numeric matrix. Each row of `X` represents one observation, and each column represents one predictor. `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 elements in `Y`.

Data Types: `single` | `double`

### `ResponseVarName` — Response variable namename 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. 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.

The response variable must be a categorical or character array, logical or numeric vector, or cell array of character vectors. If the response variable is a character array, then each element must correspond to one row of the array.

### `Y` — Class labelscategorical array | character array | logical vector | vector of numeric values | cell array of character vectors

Class labels, specified as a categorical or character array, a logical or numeric vector, or a cell array of character vectors. `Y` must be of the same type as the classification used to train `tree`, and its number of elements must equal the number of rows of `X`.

Data Types: `single` | `double` | `categorical` | `char` | `logical` | `cell`

### Name-Value Pair Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside single quotes (`' '`). You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

### `'LossFun'` — Loss function`'mincost'` (default) | `'binodeviance'` | `'classiferror'` | `'exponential'` | `'hinge'` | `'logit'` | `'quadratic'` | function handle

Loss function, specified as the comma-separated pair consisting of `'LossFun'` and a built-in, loss-function name or function handle.

• The following lists available loss functions. Specify one using its corresponding character vector.

ValueDescription
`'binodeviance'`Binomial deviance
`'classiferror'`Classification error
`'exponential'`Exponential
`'hinge'`Hinge
`'logit'`Logistic
`'mincost'`Minimal expected misclassification cost (for classification scores that are posterior probabilities)
`'quadratic'`Quadratic

`'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` be the number of observations in `X` and `K` be the number of distinct classes (`numel(tree.ClassNames)`). Your function must have this signature

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

• The output argument `lossvalue` is a scalar.

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

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

Construct `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 them 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.

Specify your function using `'LossFun',@lossfun`.

For more details on loss functions, see Classification Loss.

### `'Weights'` — Observation weights`ones(size(X,1))` (default) | name of a variable in `TBL` | numeric vector of positive values

Observation weights, specified as the comma-separated pair consisting of `'Weights'` and 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 vectors. For example, if the weights are stored as `TBL.W`, then specify it as `'W'`. Otherwise, the software treats all columns of `TBL`, including `TBL.W`, as predictors.

`loss` normalizes the weights so that observation weights in each class sum to the prior probability of that class. When you supply `Weights`, `loss` computes weighted classification loss.

Data Types: `single` | `double`

`Name,Value` arguments associated with pruning subtrees:

### `'Subtrees'` — Pruning level0 (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 `CompactClassificationTree.loss` operates on all subtrees (i.e., the entire pruning sequence). This specification is equivalent to using `0:max(tree.PruneList)`.

`CompactClassificationTree.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'`

### `'TreeSize'` — Tree size`'se'` (default) | `'min'`

Tree size, specified as the comma-separated pair consisting of `'TreeSize'` and one of the following character vectors:

• `'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`).

• `'min'``loss` returns the element of `Subtrees` with smallest loss, usually the smallest element of `Subtrees`.

## Output Arguments

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### `L` — Classification lossvector of scalar values

Classification loss, returned as a vector the length of `Subtrees`. The meaning of the error depends on the values in `Weights` and `LossFun`.

### `se` — Standard error of lossvector of scalar values

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

### `NLeaf` — Number of leaf nodesvector of integer values

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

### `bestlevel` — Best pruning levelscalar 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`.

By default, `bestlevel` is the pruning level that gives loss within one standard deviation of minimal loss.

## Definitions

### Classification Loss

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

Suppose that:

• 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, respectively.

• f(Xj) is the raw 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 to the average loss.

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

• yj* is a vector of K – 1 zeros, and a 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 y*2 = [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. The software also normalizes the prior probabilities so they sum to 1. Therefore,

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

The supported loss functions are:

• Binomial deviance, specified using `'LossFun','binodeviance'`. Its equation is

`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.$`
• Exponential loss, specified using `'LossFun','exponential'`. Its equation is

`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right).$`
• Classification error, specified using `'LossFun','classiferror'`. It is the weighted fraction of misclassified observations, with equation

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

${\stackrel{^}{y}}_{j}$ is the class label corresponding to the class with the maximal posterior probability. I{x} is the indicator function.

• Hinge loss, specified using `'LossFun','hinge'`. Its equation is

`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.$`
• Logit loss, specified using `'LossFun','logit'`. Its equation is

`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right).$`
• Minimal cost, specified using `'LossFun','mincost'`. The software computes the weighted minimal cost using this procedure for observations j = 1,...,n:

1. Estimate the 1-by-K vector of expected classification costs for observation j

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

f(Xj) is the column vector of class posterior probabilities for binary and multiclass classification. C is the cost matrix the input model stores in the property `Cost`.

2. For observation j, predict the class label corresponding to the minimum, expected classification cost:

`${\stackrel{^}{y}}_{j}=\underset{j=1,...,K}{\mathrm{min}}\left({\gamma }_{j}\right).$`
3. Using C, identify the cost incurred (cj) for making the prediction.

The weighted, average, minimum cost loss is

`$L=\sum _{j=1}^{n}{w}_{j}{c}_{j}.$`
• Quadratic loss, specified using `'LossFun','quadratic'`. Its equation is

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

This figure compares some of the loss functions for one observation over m (some functions are normalized to pass through [0,1]).

### True Misclassification Cost

There are two costs associated with classification: the true misclassification cost per class, and the expected misclassification cost per observation.

You can set the true misclassification cost per class in the `Cost` name-value pair when you create the classifier using the `fitctree` method. `Cost(i,j)` is the cost of classifying an observation into class `j` if 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

There are two costs associated with classification: the true misclassification cost per class, and the expected misclassification cost per observation.

Suppose you have `Nobs` observations that you want to classify with a trained classifier. Suppose you have `K` classes. You place the observations into a matrix `Xnew` 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

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

where

• K is the number of classes.

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

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

### Score (tree)

For trees, the score of a classification of a leaf node is the posterior probability of the classification at that node. The posterior probability of the classification at a node is the number of training sequences that lead to that node with the classification, divided by the number of training sequences that lead to that node.

For example, consider classifying a predictor `X` as `true` when `X` < `0.15` or `X` > `0.95`, and `X` is false otherwise.

Generate 100 random points and classify them:

```rng(0,'twister') % for reproducibility X = rand(100,1); Y = (abs(X - .55) > .4); tree = fitctree(X,Y); view(tree,'Mode','Graph') ```

Prune the tree:

```tree1 = prune(tree,'Level',1); view(tree1,'Mode','Graph') ```

The pruned tree correctly classifies observations that are less than 0.15 as `true`. It also correctly classifies observations from .15 to .94 as `false`. However, it incorrectly classifies observations that are greater than .94 as `false`. Therefore, the score for observations that are greater than .15 should be about .05/.85=.06 for `true`, and about .8/.85=.94 for `false`.

Compute the prediction scores for the first 10 rows of `X`:

```[~,score] = predict(tree1,X(1:10)); [score X(1:10,:)] ```
```ans = 0.9059 0.0941 0.8147 0.9059 0.0941 0.9058 0 1.0000 0.1270 0.9059 0.0941 0.9134 0.9059 0.0941 0.6324 0 1.0000 0.0975 0.9059 0.0941 0.2785 0.9059 0.0941 0.5469 0.9059 0.0941 0.9575 0.9059 0.0941 0.9649 ```

Indeed, every value of `X` (the right-most column) that is less than 0.15 has associated scores (the left and center columns) of `0` and `1`, while the other values of `X` have associated scores of `0.91` and `0.09`. The difference (score `0.09` instead of the expected `.06`) is due to a statistical fluctuation: there are `8` observations in `X` in the range `(.95,1)` instead of the expected `5` observations.

## Examples

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### Compute the In-sample Classification Error

Compute the resubstituted classification error for the `ionosphere` data set.

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

### Examine the Classification Error 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 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 = 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 = 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') ```