# loss

Classification error

## Syntax

## Description

returns a scalar representing how well `L`

= loss(`tree`

,`TBL`

,`ResponseVarName`

)`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`

.

returns the loss with additional options specified by one or more
`L`

= loss(___,`Name,Value`

)`Name,Value`

pair arguments, using any of the previous
syntaxes. For example, you can specify the loss function or observation
weights.

## Input Arguments

`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 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 this method
must also be in a table.

**Data Types: **`table`

`X`

— Data to classify

numeric matrix

`ResponseVarName`

— Response variable name

name 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
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.ResponseVarName`

, as predictors.

The response variable must be a categorical, character, or string 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.

**Data Types: **`char`

| `string`

`Y`

— Class labels

categorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

Class labels, specified as a categorical, character, or string 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: **`categorical`

| `char`

| `string`

| `logical`

| `single`

| `double`

| `cell`

### Name-Value 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 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 table lists the available loss functions. Specify one using its corresponding character vector or string scalar.

Value Description `'binodeviance'`

Binomial deviance `'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`

be the number of observations in`X`

and`K`

be the number of distinct classes (`numel(tree.ClassNames)`

). Your function must have this signaturewhere:`lossvalue =`

(C,S,W,Cost)`lossfun`

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.

**Data Types: **`char`

| `string`

| `function_handle`

`Weights`

— Observation weights

`ones(size(X,1),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 vector or
string scalar. 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`

| `char`

| `string`

`Name,Value`

arguments associated with pruning subtrees:

`Subtrees`

— Pruning level

0 (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 `loss`

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

.

`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'`

**Data Types: **`single`

| `double`

| `char`

| `string`

`TreeSize`

— Tree size

`'se'`

(default) | `'min'`

Tree size, specified as the comma-separated pair consisting of
`'TreeSize'`

and one of the following
values:

`'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

`L`

— Classification loss

vector 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 loss

vector of scalar values

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

.

`NLeaf`

— Number of leaf nodes

vector of integer values

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

.

`bestlevel`

— Best pruning level

scalar 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.

## Examples

### 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 = `*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')

## More About

### 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:

*y*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_{j}`ClassNames`

property), respectively.*f*(*X*) is the positive-class classification score for observation (row)_{j}*j*of the predictor data*X*.*m*=_{j}*y*_{j}*f*(*X*) is the classification score for classifying observation_{j}*j*into the class corresponding to*y*. Positive values of_{j}*m*indicate correct classification and do not contribute much to the average loss. Negative values of_{j}*m*indicate incorrect classification and contribute significantly to the average loss._{j}

For algorithms that support multiclass classification (that is,

*K*≥ 3):*y*is a vector of_{j}^{*}*K*– 1 zeros, with 1 in the position corresponding to the true, observed class*y*. For example, if the true class of the second observation is the third class and_{j}*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*(*X*) is the length_{j}*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.*m*=_{j}*y*_{j}^{*}′*f*(*X*). Therefore,_{j}*m*is the scalar classification score that the model predicts for the true, observed class._{j}

The weight for observation

*j*is*w*. 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,_{j}$$\sum _{j=1}^{n}{w}_{j}}=1.$$

Given this scenario, the following table describes the supported loss
functions that you can specify by using the `'LossFun'`

name-value pair
argument.

Loss Function | Value of `LossFun` | Equation |
---|---|---|

Binomial deviance | `'binodeviance'` | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}}.$$ |

Misclassified rate in decimal | `'classiferror'` | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}I\left\{{\widehat{y}}_{j}\ne {y}_{j}\right\}.$$ $${\widehat{y}}_{j}$$ is the class label corresponding to the class with the
maximal score. |

Cross-entropy loss | `'crossentropy'` |
The weighted cross-entropy loss is $$L=-{\displaystyle \sum _{j=1}^{n}\frac{{\tilde{w}}_{j}\mathrm{log}({m}_{j})}{Kn}},$$ where the weights $${\tilde{w}}_{j}$$ are normalized to sum to |

Exponential loss | `'exponential'` | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right)}.$$ |

Hinge loss | `'hinge'` | $$L={\displaystyle \sum}_{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.$$ |

Logit loss | `'logit'` | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right)}.$$ |

Minimal expected misclassification cost | `'mincost'` |
The software computes
the weighted minimal expected classification cost using this procedure
for observations Estimate the expected misclassification cost of classifying the observation *X*into the class_{j}*k*:$${\gamma}_{jk}={\left(f{\left({X}_{j}\right)}^{\prime}C\right)}_{k}.$$ *f*(*X*) is the column vector of class posterior probabilities for binary and multiclass classification for the observation_{j}*X*._{j}*C*is the cost matrix stored in the`Cost` property of the model.For observation *j*, predict the class label corresponding to the minimal expected misclassification cost:$${\widehat{y}}_{j}=\underset{k=1,\mathrm{...},K}{\text{argmin}}{\gamma}_{jk}.$$ Using *C*, identify the cost incurred (*c*) for making the prediction._{j}
The weighted average of the minimal expected misclassification cost loss is $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{c}_{j}}.$$ If you use the default cost matrix (whose element
value is 0 for correct classification and 1 for incorrect
classification), then the |

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

This figure compares the loss functions (except `'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(`

is*n*,*k*)

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

where:

*K*is the number of classes.$$\widehat{P}\left(i|X(n)\right)$$ 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*.

### 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 = `*10×3*
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.

## Extended Capabilities

### Tall Arrays

Calculate with arrays that have more rows than fit in memory.

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:

`loss`

executes on a GPU in these cases only:The input argument

`X`

is a`gpuArray`

.The input argument

`tbl`

contains`gpuArray`

predictor variables.The input argument

`mdl`

was fitted with GPU array input arguments.

If the classification tree model was trained with surrogate splits, these limitations apply:

You cannot specify the input argument

`X`

as a`gpuArray`

.You cannot specify the input argument

`tbl`

as a table containing`gpuArray`

elements.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

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