# loss

Find classification error for support vector machine (SVM) classifier

## Syntax

## Description

returns the classification error (see Classification Loss), a
scalar representing how well the trained support vector machine (SVM) classifier
(`L`

= loss(`SVMModel`

,`TBL`

,`ResponseVarName`

)`SVMModel`

) classifies the predictor data in table
`TBL`

compared to the true class labels in
`TBL.ResponseVarName`

.

`loss`

normalizes the class probabilities in
`TBL.ResponseVarName`

to the prior class probabilities that
`fitcsvm`

used for training, stored
in the `Prior`

property of `SVMModel`

.

The classification loss (`L`

) is a generalization or
resubstitution quality measure. Its interpretation depends on the loss function
and weighting scheme, but, in general, better classifiers yield smaller
classification loss values.

specifies options using one or more name-value pair arguments in addition to the
input arguments in previous syntaxes. For example, you can specify the loss
function and the classification weights.`L`

= loss(___,`Name,Value`

)

## Examples

### Determine Test Sample Classification Error of SVM Classifiers

Load the `ionosphere`

data set.

load ionosphere rng(1); % For reproducibility

Train an SVM classifier. Specify a 15% holdout sample for testing, standardize the data, and specify that `'g'`

is the positive class.

CVSVMModel = fitcsvm(X,Y,'Holdout',0.15,'ClassNames',{'b','g'},... 'Standardize',true); CompactSVMModel = CVSVMModel.Trained{1}; % Extract the trained, compact classifier testInds = test(CVSVMModel.Partition); % Extract the test indices XTest = X(testInds,:); YTest = Y(testInds,:);

`CVSVMModel`

is a `ClassificationPartitionedModel`

classifier. It contains the property `Trained`

, which is a 1-by-1 cell array holding a `CompactClassificationSVM`

classifier that the software trained using the training set.

Determine how well the algorithm generalizes by estimating the test sample classification error.

L = loss(CompactSVMModel,XTest,YTest)

L = 0.0787

The SVM classifier misclassifies approximately 8% of the test sample.

### Determine Test Sample Hinge Loss of SVM Classifiers

Load the `ionosphere`

data set.

load ionosphere rng(1); % For reproducibility

Train an SVM classifier. Specify a 15% holdout sample for testing, standardize the data, and specify that `'g'`

is the positive class.

CVSVMModel = fitcsvm(X,Y,'Holdout',0.15,'ClassNames',{'b','g'},... 'Standardize',true); CompactSVMModel = CVSVMModel.Trained{1}; % Extract the trained, compact classifier testInds = test(CVSVMModel.Partition); % Extract the test indices XTest = X(testInds,:); YTest = Y(testInds,:);

`CVSVMModel`

is a `ClassificationPartitionedModel`

classifier. It contains the property `Trained`

, which is a 1-by-1 cell array holding a `CompactClassificationSVM`

classifier that the software trained using the training set.

Determine how well the algorithm generalizes by estimating the test sample hinge loss.

L = loss(CompactSVMModel,XTest,YTest,'LossFun','hinge')

L = 0.2998

The hinge loss is approximately 0.3. Classifiers with hinge losses close to 0 are preferred.

## Input Arguments

`SVMModel`

— SVM classification model

`ClassificationSVM`

model object | `CompactClassificationSVM`

model object

SVM classification model, specified as a `ClassificationSVM`

model object or `CompactClassificationSVM`

model object returned by `fitcsvm`

or `compact`

,
respectively.

`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 of the
predictors used to train `SVMModel`

.
Multicolumn variables and cell arrays other than cell arrays of
character vectors are not allowed.

If `TBL`

contains the response variable
used to train `SVMModel`

, then you do not need
to specify `ResponseVarName`

or `Y`

.

If you trained `SVMModel`

using sample data contained in a table, then the
input data for `loss`

must also be in
a table.

If you set `'Standardize',true`

in `fitcsvm`

when training `SVMModel`

, then the software
standardizes the columns of the predictor data using the
corresponding means in `SVMModel.Mu`

and the
standard deviations in `SVMModel.Sigma`

.

**Data Types: **`table`

`ResponseVarName`

— Response variable name

name of variable in `TBL`

Response variable name, specified as the name of a variable in
`TBL`

.

You must specify `ResponseVarName`

as a character vector
or string scalar. For example, if the response variable `Y`

is stored as `TBL.Y`

, then specify
`ResponseVarName`

as `'Y'`

.
Otherwise, the software treats all columns of `TBL`

,
including `Y`

, as predictors when training the
model.

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`

`X`

— Predictor data

numeric matrix

Predictor data, specified as a numeric matrix.

Each row of `X`

corresponds to one observation (also known as an instance
or example), and each column corresponds to one variable (also known as a feature). The
variables in the columns of `X`

must be the same as the variables
that trained the `SVMModel`

classifier.

The length of `Y`

and the number of rows in `X`

must be
equal.

If you set `'Standardize',true`

in `fitcsvm`

to train `SVMModel`

, then the software
standardizes the columns of `X`

using the corresponding means in
`SVMModel.Mu`

and the standard deviations in
`SVMModel.Sigma`

.

**Data Types: **`double`

| `single`

`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, logical or numeric
vector, or cell array of character vectors. `Y`

must be the same as the data type of
`SVMModel.ClassNames`

. (The software treats string arrays as cell arrays of character
vectors.)

The length of `Y`

must equal the number of rows in `TBL`

or the number of rows in `X`

.

### 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`

.

**Example:**

`loss(SVMModel,TBL,Y,'Weights',W)`

weighs the
observations in each row of `TBL`

using the corresponding weight in
each row of the variable `W`

in
`TBL`

.`LossFun`

— Loss function

`'classiferror'`

(default) | `'binodeviance'`

| `'exponential'`

| `'hinge'`

| `'logit'`

| `'mincost'`

| `'quadratic'`

| function handle

Loss function, specified as the comma-separated pair consisting of
`'LossFun'`

and a built-in loss function name or a
function handle.

This 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. You can specify to use posterior probabilities as classification scores for SVM models by setting`'FitPosterior',true`

when you cross-validate the model using`fitcsvm`

.Specify your own function by using function handle notation.

Suppose that

`n`

is the number of observations in`X`

, and`K`

is the number of distinct classes (`numel(SVMModel.ClassNames)`

) used to create the input model (`SVMModel`

). 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 the class to which the corresponding observation belongs. The column order corresponds to the class order in`SVMModel.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, similar to the output of`predict`

. The column order corresponds to the class order in`SVMModel.ClassNames`

.`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.

Specify your function using

`'LossFun',@`

.`lossfun`

For more details on loss functions, see Classification Loss.

**Example: **`'LossFun','binodeviance'`

**Data Types: **`char`

| `string`

| `function_handle`

`Weights`

— Observation weights

`ones(size(X,1),1)`

(default) | numeric vector | name of variable in `TBL`

Observation weights, specified as the comma-separated pair consisting
of `'Weights'`

and a numeric vector or the name of a
variable in `TBL`

. The software weighs the
observations in each row of `X`

or
`TBL`

with the corresponding weight in
`Weights`

.

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.

If you do not specify your own loss function, then the software
normalizes `Weights`

to sum up to the value of the
prior probability in the respective class.

**Example: **`'Weights','W'`

**Data Types: **`single`

| `double`

| `char`

| `string`

## 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).

### Classification Score

The SVM *classification score* for
classifying observation *x* is the signed distance
from *x* to the decision boundary ranging from -∞
to +∞. A positive score for a class indicates that *x* is
predicted to be in that class. A negative score indicates otherwise.

The positive class classification score $$f(x)$$ is the trained SVM classification function. $$f(x)$$ is also the numerical predicted response for *x*, or the
score for predicting *x* into the positive class.

$$f(x)={\displaystyle \sum _{j=1}^{n}{\alpha}_{j}}{y}_{j}G({x}_{j},x)+b,$$

where $$({\alpha}_{1},\mathrm{...},{\alpha}_{n},b)$$ are the estimated SVM parameters, $$G({x}_{j},x)$$ is the dot product in the predictor space between *x* and
the support vectors, and the sum includes the training set observations. The negative class
classification score for *x*, or the score for predicting
*x* into the negative class, is
–*f*(*x*).

If *G*(*x _{j}*,

*x*) =

*x*′

_{j}*x*(the linear kernel), then the score function reduces to

$$f\left(x\right)=\left(x/s\right)\prime \beta +b.$$

*s* is
the kernel scale and *β* is the vector of fitted
linear coefficients.

For more details, see Understanding Support Vector Machines.

## References

[1] Hastie, T., R. Tibshirani, and J. Friedman. *The Elements of
Statistical Learning*, second edition. Springer, New York,
2008.

## Extended Capabilities

### Tall Arrays

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

This function fully supports tall arrays. For more information, see Tall Arrays.

### GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

## See Also

`ClassificationSVM`

| `CompactClassificationSVM`

| `fitcsvm`

| `predict`

**Introduced in R2014a**

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