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# loss

Classification loss for Gaussian kernel classification model

## Syntax

L = loss(Mdl,X,Y)
L = loss(Mdl,X,Y,Name,Value)

## Description

example

L = loss(Mdl,X,Y) returns the classification loss for the binary Gaussian kernel classification model Mdl using the predictor data in X and the corresponding class labels in Y.

example

L = loss(Mdl,X,Y,Name,Value) uses additional options specified by one or more name-value pair arguments. For example, you can specify a classification loss function and observation weights. Then, loss returns the weighted classification loss using the specified loss function.

## Examples

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Load the ionosphere data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b') or good ('g').

Partition the data set into training and test sets. Specify a 15% holdout sample for the test set.

rng('default') % For reproducibility
Partition = cvpartition(Y,'Holdout',0.15);
trainingInds = training(Partition); % Indices for the training set
testInds = test(Partition); % Indices for the test set

Train a binary kernel classification model using the training set.

Mdl = fitckernel(X(trainingInds,:),Y(trainingInds));

Estimate the training-set classification error and the test-set classification error.

ceTrain = loss(Mdl,X(trainingInds,:),Y(trainingInds))
ceTrain = 0.0067
ceTest = loss(Mdl,X(testInds,:),Y(testInds))
ceTest = 0.1140

Load the ionosphere data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b') or good ('g').

Partition the data set into training and test sets. Specify a 15% holdout sample for the test set.

rng('default') % For reproducibility
Partition = cvpartition(Y,'Holdout',0.15);
trainingInds = training(Partition); % Indices for the training set
testInds = test(Partition); % Indices for the test set

Train a binary kernel classification model using the training set.

Mdl = fitckernel(X(trainingInds,:),Y(trainingInds));

Create an anonymous function that measures linear loss, that is,

$L=\frac{\sum _{j}-{w}_{j}{y}_{j}{f}_{j}}{\sum _{j}{w}_{j}}.$

${w}_{j}$ is the weight for observation j, ${y}_{j}$ is response j (-1 for the negative class, and 1 otherwise), and ${f}_{j}$ is the raw classification score of observation j.

linearloss = @(C,S,W,Cost)sum(-W.*sum(S.*C,2))/sum(W);

Custom loss functions must be written in a particular form. For rules on writing a custom loss function, see the 'LossFun' name-value pair argument.

Estimate the training-set classification loss and the test-set classification loss using the linear loss function.

ceTrain = loss(Mdl,X(trainingInds,:),Y(trainingInds),'LossFun',linearloss)
ceTrain = -1.0851
ceTest = loss(Mdl,X(testInds,:),Y(testInds),'LossFun',linearloss)
ceTest = -0.7821

## Input Arguments

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Binary kernel classification model, specified as a ClassificationKernel model object. You can create a ClassificationKernel model object using fitckernel.

Predictor data, specified as an n-by-p numeric matrix, where n is the number of observations and p is the number of predictors used to train Mdl.

The length of Y and the number of observations in X must be equal.

Data Types: single | double

Class labels, specified as a categorical, character, or string array, logical or numeric vector, or cell array of character vectors.

• The data type of Y must be the same as the data type of Mdl.ClassNames. (The software treats string arrays as cell arrays of character vectors.)

• The distinct classes in Y must be a subset of Mdl.ClassNames.

• If Y is a character array, then each element must correspond to one row of the array.

• The length of Y and the number of observations in X must be equal.

Data Types: categorical | char | string | logical | single | double | 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 quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: L = loss(Mdl,X,Y,'LossFun','quadratic','Weights',weights) returns the weighted classification loss using the quadratic loss function.

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 value.

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)

'mincost' is appropriate for classification scores that are posterior probabilities. For kernel classification models, logistic regression learners return posterior probabilities as classification scores by default, but SVM learners do not (see predict).

• Specify your own function by using function handle notation.

Let n be the number of observations in X and K be the number of distinct classes (numel(Mdl.ClassNames), where Mdl is the input model). Your function must have this signature:

lossvalue = lossfun(C,S,W,Cost)

• 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 Mdl.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 Mdl.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.

Example: 'LossFun',@lossfun

Data Types: char | string | function_handle

Observation weights, specified as the comma-separated pair consisting of 'Weights' and a positive numeric vector of length n, where n is the number of observations in X. If you supply weights, loss computes the weighted classification loss.

The default value is ones(n,1).

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

Data Types: double | single

## Output Arguments

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Classification loss, returned as a numeric scalar. The interpretation of L depends on Weights and LossFun.

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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.

Suppose the following:

• L is the weighted average classification loss.

• n is the sample size.

• 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 the transformed observation (row) j of the predictor data X using feature expansion.

• 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.

• 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 that they sum to 1. Therefore,

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

This table describes the supported loss functions that you can specify by using the 'LossFun' name-value pair argument.

Loss FunctionValue of LossFunEquation
Binomial deviance'binodeviance'$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.$
Exponential loss'exponential'$L=\sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right).$
Classification error'classiferror'

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

The classification error is the weighted fraction of misclassified observations where ${\stackrel{^}{y}}_{j}$ is the class label corresponding to the class with the maximal posterior probability. I{x} is the indicator function.

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 cost'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. C is the cost matrix that the input model stores in the Cost property.

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

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