kfoldLoss

Classification loss for cross-validated kernel classification model

Syntax

loss = kfoldLoss(CVMdl)

loss = kfoldLoss(CVMdl,Name,Value)

Description

loss = kfoldLoss(CVMdl) returns the classification loss obtained by the cross-validated, binary kernel model (ClassificationPartitionedKernel) CVMdl. For every fold, kfoldLoss computes the classification loss for validation-fold observations using a model trained on training-fold observations.

By default, kfoldLoss returns the classification error.

loss = kfoldLoss(CVMdl,Name,Value) returns the classification loss with additional options specified by one or more name-value pair arguments. For example, specify the classification loss function, number of folds, or aggregation level.

Examples

Estimate k-Fold Cross-Validation Classification Error

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

load ionosphere

Cross-validate a binary kernel classification model using the data.

CVMdl = fitckernel(X,Y,'Crossval','on')

CVMdl = 
  ClassificationPartitionedKernel
    CrossValidatedModel: 'Kernel'
           ResponseName: 'Y'
        NumObservations: 351
                  KFold: 10
              Partition: [1x1 cvpartition]
             ClassNames: {'b'  'g'}
         ScoreTransform: 'none'

CVMdl is a ClassificationPartitionedKernel model. By default, the software implements 10-fold cross-validation. To specify a different number of folds, use the 'KFold' name-value pair argument instead of 'Crossval'.

Estimate the cross-validated classification loss. By default, the software computes the classification error.

loss = kfoldLoss(CVMdl)

loss = 
0.0940

Alternatively, you can obtain the per-fold classification errors by specifying the name-value pair 'Mode','individual' in kfoldLoss.

Specify Custom Classification Loss

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

load ionosphere

Cross-validate a binary kernel classification model using the data.

CVMdl = fitckernel(X,Y,'Crossval','on')

CVMdl = 
  ClassificationPartitionedKernel
    CrossValidatedModel: 'Kernel'
           ResponseName: 'Y'
        NumObservations: 351
                  KFold: 10
              Partition: [1x1 cvpartition]
             ClassNames: {'b'  'g'}
         ScoreTransform: 'none'

CVMdl is a ClassificationPartitionedKernel model. By default, the software implements 10-fold cross-validation. To specify a different number of folds, use the 'KFold' name-value pair argument instead of 'Crossval'.

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 the 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 cross-validated classification loss using the linear loss function.

loss = kfoldLoss(CVMdl,'LossFun',linearloss)

loss = 
-0.7792

Input Arguments

CVMdl — Cross-validated, binary kernel classification model

ClassificationPartitionedKernel model object

Cross-validated, binary kernel classification model, specified as a ClassificationPartitionedKernel model object. You can create a ClassificationPartitionedKernel model by using fitckernel and specifying any one of the cross-validation name-value pair arguments.

To obtain estimates, kfoldLoss applies the same data used to cross-validate the kernel classification model (X and Y).

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: kfoldLoss(CVMdl,'Folds',[1 3 5]) specifies to use only the first, third, and fifth folds to calculate the classification loss.

Folds — Fold indices for prediction

1:CVMdl.KFold (default) | numeric vector of positive integers

Fold indices for prediction, specified as the comma-separated pair consisting of 'Folds' and a numeric vector of positive integers. The elements of Folds must be within the range from 1 to CVMdl.KFold.

The software uses only the folds specified in Folds for prediction.

Example: 'Folds',[1 4 10]

Data Types: single | double

LossFun — Loss function

'classiferror' (default) | 'binodeviance' | 'classifcost' | '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 value.

  Value	Description
  "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. For kernel classification models, logistic regression learners return posterior probabilities as classification scores by default, but SVM learners do not (see kfoldPredict).

• Specify your own function by using function handle notation.

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

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

  • 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 CVMdl.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 CVMdl.ClassNames. S is a matrix of classification scores, similar to the output of kfoldPredict.

  • 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

Mode — Aggregation level for output

'average' (default) | 'individual'

Aggregation level for the output, specified as the comma-separated pair consisting of 'Mode' and 'average' or 'individual'.

This table describes the values.

Value	Description
'average'	The output is a scalar average over all folds.
'individual'	The output is a vector of length k containing one value per fold, where k is the number of folds.

Example: 'Mode','individual'

Output Arguments

loss — Classification loss

numeric scalar | numeric column vector

Classification loss, returned as a numeric scalar or numeric column vector.

If Mode is 'average', then loss is the average classification loss over all folds. Otherwise, loss is a k-by-1 numeric column vector containing the classification loss for each fold, where k is the number of folds.

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.

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

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

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

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

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

Loss Function	Value of LossFun	Equation
Binomial deviance	"binodeviance"	$$L={\displaystyle \sum _{j=1}^n{w}_j\log \left\{1+\exp \left[-2m_j\right]\right\}}.$$
Observed misclassification cost	"classifcost"	$$L={\displaystyle \sum _{j=1}^n{w}_j}{c}_{y_j{\widehat{y}}_j},$$ where $${\widehat{y}}_j$$ is the class label corresponding to the class with the maximal score, and $$c_{y_j{\widehat{y}}_j}$$ is the user-specified cost of classifying an observation into class $${\widehat{y}}_j$$ when its true class is yj.
Misclassified rate in decimal	"classiferror"	$$L={\displaystyle \sum _{j=1}^n{w}_j}I\left\{{\widehat{y}}_j\ne y_j\right\},$$ where I{·} is the indicator function.
Cross-entropy loss	"crossentropy"	"crossentropy" is appropriate only for neural network models. The weighted cross-entropy loss is $$L=-{\displaystyle \sum _{j=1}^n\frac{{\tilde{w}}_j\log (m_j)}{Kn}},$$ where the weights $${\tilde{w}}_j$$ are normalized to sum to n instead of 1.
Exponential loss	"exponential"	$$L={\displaystyle \sum _{j=1}^n{w}_j\exp \left(-m_j\right)}.$$
Hinge loss	"hinge"	$$L={\displaystyle \sum }_{j=1}^n{w}_j\max \left\{0,1-m_j\right\}.$$
Logit loss	"logit"	$$L={\displaystyle \sum _{j=1}^n{w}_j\log \left(1+\exp \left(-m_j\right)\right)}.$$
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. 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. 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 (cj) for making the prediction. The weighted average of the minimal expected misclassification cost loss is $$L={\displaystyle \sum _{j=1}^n{w}_j{c}_j}.$$
Quadratic loss	"quadratic"	$$L={\displaystyle \sum _{j=1}^n{w}_j{\left(1-m_j\right)}^2}.$$

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).

Version History

Introduced in R2018b

R2023b: Observations with missing predictor values are used in resubstitution and cross-validation computations

Starting in R2023b, the following classification model object functions use observations with missing predictor values as part of resubstitution ("resub") and cross-validation ("kfold") computations for classification edges, losses, margins, and predictions.

Model Type	Model Objects	Object Functions
Discriminant analysis classification model	ClassificationDiscriminant	resubEdge, resubLoss, resubMargin, resubPredict
	ClassificationPartitionedModel	kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
Ensemble of discriminant analysis learners for classification	ClassificationEnsemble	resubEdge, resubLoss, resubMargin, resubPredict
	ClassificationPartitionedEnsemble	kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
Gaussian kernel classification model	ClassificationPartitionedKernel	kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
	ClassificationPartitionedKernelECOC	kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
Linear classification model	ClassificationPartitionedLinear	kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
	ClassificationPartitionedLinearECOC	kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
Neural network classification model	ClassificationNeuralNetwork	resubEdge, resubLoss, resubMargin, resubPredict
	ClassificationPartitionedModel	kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
Support vector machine (SVM) classification model	ClassificationSVM	resubEdge, resubLoss, resubMargin, resubPredict
	ClassificationPartitionedModel	kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict

In previous releases, the software omitted observations with missing predictor values from the resubstitution and cross-validation computations.

R2022a: kfoldLoss returns a different value for a model with a nondefault cost matrix

If you specify a nondefault cost matrix when you train the input model object, the kfoldLoss function returns a different value compared to previous releases.

The kfoldLoss function uses the observation weights stored in the W property. Also, the function uses the cost matrix stored in the Cost property if you specify the LossFun name-value argument as "classifcost" or "mincost". The way the function uses the W and Cost property values has not changed. However, the property values stored in the input model object have changed for a model with a nondefault cost matrix, so the function might return a different value.

For details about the property value change, see Cost property stores the user-specified cost matrix.

If you want the software to handle the cost matrix, prior probabilities, and observation weights in the same way as in previous releases, adjust the prior probabilities and observation weights for the nondefault cost matrix, as described in Adjust Prior Probabilities and Observation Weights for Misclassification Cost Matrix. Then, when you train a classification model, specify the adjusted prior probabilities and observation weights by using the Prior and Weights name-value arguments, respectively, and use the default cost matrix.