kfoldLoss

Classification loss for cross-validated classification model

Syntax

L = kfoldLoss(CVMdl)

L = kfoldLoss(CVMdl,Name,Value)

Description

L = kfoldLoss(CVMdl) returns the classification loss obtained by the cross-validated classification model CVMdl. For every fold, kfoldLoss computes the classification loss for validation-fold observations using a classifier trained on training-fold observations. CVMdl.X and CVMdl.Y contain both sets of observations.

L = kfoldLoss(CVMdl,Name,Value) returns the classification loss with additional options specified by one or more name-value arguments. For example, you can specify a custom loss function.

Examples

Estimate Cross-Validated Classification Error

Load the ionosphere data set.

load ionosphere

Grow a classification tree.

tree = fitctree(X,Y);

Cross-validate the classification tree using 10-fold cross-validation.

cvtree = crossval(tree);

Estimate the cross-validated classification error.

L = kfoldLoss(cvtree)

L = 
0.1083

Estimate Cross-Validated Classification Error

Load the ionosphere data set.

load ionosphere

Train a classification ensemble of 100 decision trees using AdaBoostM1. Specify tree stumps as the weak learners.

t = templateTree('MaxNumSplits',1);
ens = fitcensemble(X,Y,'Method','AdaBoostM1','Learners',t);

Cross-validate the ensemble using 10-fold cross-validation.

cvens = crossval(ens);

Estimate the cross-validated classification error.

L = kfoldLoss(cvens)

L = 
0.0655

Find Optimal Number of Trees for GAM Using kfoldLoss

Train a cross-validated generalized additive model (GAM) with 10 folds. Then, use kfoldLoss to compute cumulative cross-validation classification errors (misclassification rate in decimal). Use the errors to determine the optimal number of trees per predictor (linear term for predictor) and the optimal number of trees per interaction term.

Alternatively, you can find optimal values of fitcgam name-value arguments by using the OptimizeHyperparameters name-value argument. For an example, see Optimize GAM Using OptimizeHyperparameters.

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

load ionosphere

Create a cross-validated GAM by using the default cross-validation option. Specify the 'CrossVal' name-value argument as 'on'. Specify to include all available interaction terms whose p-values are not greater than 0.05.

rng('default') % For reproducibility
CVMdl = fitcgam(X,Y,'CrossVal','on','Interactions','all','MaxPValue',0.05);

If you specify 'Mode' as 'cumulative' for kfoldLoss, then the function returns cumulative errors, which are the average errors across all folds obtained using the same number of trees for each fold. Display the number of trees for each fold.

CVMdl.NumTrainedPerFold 

ans = struct with fields:
      PredictorTrees: [65 64 59 61 60 66 65 62 64 61]
    InteractionTrees: [1 2 2 2 2 1 2 2 2 2]

kfoldLoss can compute cumulative errors using up to 59 predictor trees and one interaction tree.

Plot the cumulative, 10-fold cross-validated, classification error (misclassification rate in decimal). Specify 'IncludeInteractions' as false to exclude interaction terms from the computation.

L_noInteractions = kfoldLoss(CVMdl,'Mode','cumulative','IncludeInteractions',false);
figure
plot(0:min(CVMdl.NumTrainedPerFold.PredictorTrees),L_noInteractions)

The first element of L_noInteractions is the average error over all folds obtained using only the intercept (constant) term. The (J+1)th element of L_noInteractions is the average error obtained using the intercept term and the first J predictor trees per linear term. Plotting the cumulative loss allows you to monitor how the error changes as the number of predictor trees in GAM increases.

Find the minimum error and the number of predictor trees used to achieve the minimum error.

[M,I] = min(L_noInteractions)

M = 
0.0655

I = 
23

The GAM achieves the minimum error when it includes 22 predictor trees.

Compute the cumulative classification error using both linear terms and interaction terms.

L = kfoldLoss(CVMdl,'Mode','cumulative')

L = 2×1

    0.0712
    0.0712

The first element of L is the average error over all folds obtained using the intercept (constant) term and all predictor trees per linear term. The second element of L is the average error obtained using the intercept term, all predictor trees per linear term, and one interaction tree per interaction term. The error does not decrease when interaction terms are added.

If you are satisfied with the error when the number of predictor trees is 22, you can create a predictive model by training the univariate GAM again and specifying 'NumTreesPerPredictor',22 without cross-validation.

Input Arguments

CVMdl — Cross-validated partitioned classifier

ClassificationPartitionedModel object | ClassificationPartitionedEnsemble object | ClassificationPartitionedGAM object

Cross-validated partitioned classifier, specified as a ClassificationPartitionedModel, ClassificationPartitionedEnsemble, or ClassificationPartitionedGAM object. You can create the object in two ways:

• Pass a trained classification model listed in the following table to its crossval object function.

• Train a classification model using a function listed in the following table and specify one of the cross-validation name-value arguments for the function.

Classification Model	Function
ClassificationDiscriminant	fitcdiscr
ClassificationEnsemble	fitcensemble
ClassificationGAM	fitcgam
ClassificationKNN	fitcknn
ClassificationNaiveBayes	fitcnb
ClassificationNeuralNetwork	fitcnet
ClassificationSVM	fitcsvm
ClassificationTree	fitctree

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 2 3 5]) specifies to use the first, second, third, and fifth folds to compute the classification loss, but to exclude the fourth fold.

Folds — Fold indices to use

1:CVMdl.KFold (default) | positive integer vector

Fold indices to use, specified as a positive integer vector. The elements of Folds must be within the range from 1 to CVMdl.KFold.

The software uses only the folds specified in Folds.

Example: 'Folds',[1 4 10]

Data Types: single | double

IncludeInteractions — Flag to include interaction terms

true | false

Flag to include interaction terms of the model, specified as true or false. This argument is valid only for a generalized additive model (GAM). That is, you can specify this argument only when CVMdl is ClassificationPartitionedGAM.

The default value is true if the models in CVMdl (CVMdl.Trained) contain interaction terms. The value must be false if the models do not contain interaction terms.

Data Types: logical

LossFun — Loss function

'binodeviance' | 'classifcost' | 'classiferror' | 'crossentropy' | 'exponential' | 'hinge' | 'logit' | 'mincost' | 'quadratic' | function handle

Loss function, specified as a built-in loss function name or a function handle. The default loss function depends on the model type of CVMdl.

• The default value is 'classiferror' if the model type is an ensemble or support vector machine classifier.

• The default value is 'mincost' if the model type is a discriminant analysis, k-nearest neighbor, naive Bayes, neural network, or tree classifier.

• If the model type is a generalized additive model classifier, the default value is 'mincost' if the ScoreTransform property of the input model object (CVMdl.ScoreTransform) is 'logit'; otherwise, the default value is 'classiferror'.

'classiferror' and 'mincost' are equivalent when you use the default cost matrix. See Algorithms for more information.

• This table lists the available loss functions. Specify one using its corresponding character vector or string scalar.

  Value	Description
  'binodeviance'	Binomial deviance
  'classifcost'	Observed misclassification cost
  'classiferror'	Misclassified rate in decimal
  'crossentropy'	Cross-entropy loss (for neural networks only)
  '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. The predict and kfoldPredict functions of discriminant analysis, generalized additive model, k-nearest neighbor, naive Bayes, neural network, and tree classifiers return such scores by default.

  • For ensemble models that use 'Bag' or 'Subspace' methods, classification scores are posterior probabilities by default. For ensemble models that use 'AdaBoostM1', 'AdaBoostM2', GentleBoost, or 'LogitBoost' methods, you can use posterior probabilities as classification scores by specifying the double-logit score transform. For example, enter:

    CVMdl.ScoreTransform = 'doublelogit';

    For all other ensemble methods, the software does not support posterior probabilities as classification scores.

  • For SVM models, you can specify to use posterior probabilities as classification scores by setting 'FitPosterior',true when you cross-validate the model using fitcsvm.

• Specify your own function using function handle notation.

  Suppose that n is the number of observations in the training data (CVMdl.NumObservations) and K is the number of classes (numel(CVMdl.ClassNames)). Your function must have the signature lossvalue = lossfun(C,S,W,Cost), where:

  • 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. The input S resembles the output argument score of kfoldPredict.

  • W is an n-by-1 numeric vector of observation weights. If you pass W, the software normalizes its elements 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','hinge'

Data Types: char | string | function_handle

Mode — Aggregation level for output

'average' (default) | 'individual' | 'cumulative'

Aggregation level for the output, specified as 'average', 'individual', or 'cumulative'.

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.
'cumulative'	Note If you want to specify this value, CVMdl must be a ClassificationPartitionedEnsemble object or ClassificationPartitionedGAM object. If CVMdl is ClassificationPartitionedEnsemble, then the output is a vector of length min(CVMdl.NumTrainedPerFold). Each element j is an average over all folds that the function obtains by using ensembles trained with weak learners 1:j. If CVMdl is ClassificationPartitionedGAM, then the output value depends on the IncludeInteractions value. If IncludeInteractions is false, then L is a (1 + min(NumTrainedPerFold.PredictorTrees))-by-1 numeric column vector. The first element of L is an average over all folds that is obtained only the intercept (constant) term. The (j + 1)th element of L is an average obtained using the intercept term and the first j predictor trees per linear term. If IncludeInteractions is true, then L is a (1 + min(NumTrainedPerFold.InteractionTrees))-by-1 numeric column vector. The first element of L is an average over all folds that is obtained using the intercept (constant) term and all predictor trees per linear term. The (j + 1)th element of L is an average obtained using the intercept term, all predictor trees per linear term, and the first j interaction trees per interaction term.

Example: 'Mode','individual'

Output Arguments

L — Classification loss

numeric scalar | numeric column vector

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

• If Mode is 'average', then L is the average classification loss over all folds.

• If Mode is 'individual', then L is a k-by-1 numeric column vector containing the classification loss for each fold, where k is the number of folds.

• If Mode is 'cumulative' and CVMdl is ClassificationPartitionedEnsemble, then L is a min(CVMdl.NumTrainedPerFold)-by-1 numeric column vector. Each element j is the average classification loss over all folds that the function obtains by using ensembles trained with weak learners 1:j.

• If Mode is 'cumulative' and CVMdl is ClassificationPartitionedGAM, then the output value depends on the IncludeInteractions value.

  • If IncludeInteractions is false, then L is a (1 + min(NumTrainedPerFold.PredictorTrees))-by-1 numeric column vector. The first element of L is the average classification loss over all folds that is obtained using only the intercept (constant) term. The (j + 1)th element of L is the average loss obtained using the intercept term and the first j predictor trees per linear term.

  • If IncludeInteractions is true, then L is a (1 + min(NumTrainedPerFold.InteractionTrees))-by-1 numeric column vector. The first element of L is the average classification loss over all folds that is obtained using the intercept (constant) term and all predictor trees per linear term. The (j + 1)th element of L is the average loss obtained using the intercept term, all predictor trees per linear term, and the first j interaction trees per interaction term.

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_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.

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

  • y_j^* is a vector of K – 1 zeros, with 1 in the position corresponding to the true, observed class y_j. For example, if the true class of the second observation is the third class and K = 4, then y₂^* = [0 0 1 0]′. The order of the classes corresponds to the order in the ClassNames property of the input model.

  • f(X_j) 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.

  • m_j = y_j^*′f(X_j). Therefore, m_j is the scalar classification score that the model predicts for the true, observed class.

• 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).

Algorithms

kfoldLoss computes the classification loss as described in the corresponding loss object function. For a model-specific description, see the appropriate loss function reference page in the following table.

Model Type	loss Function
Discriminant analysis classifier	loss
Ensemble classifier	loss
Generalized additive model classifier	loss
k-nearest neighbor classifier	loss
Naive Bayes classifier	loss
Neural network classifier	loss
Support vector machine classifier	loss
Binary decision tree for multiclass classification	loss

Extended Capabilities

GPU Arrays

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

Usage notes and limitations:

• This function fully supports GPU arrays for the following cross-validated model objects:

  • Ensemble classifier trained with fitcensemble

  • k-nearest neighbor classifier trained with fitcknn

  • Support vector machine classifier trained with fitcsvm

  • Binary decision tree for multiclass classification trained with fitctree

  • Neural network for classification trained with fitcnet

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

Version History

Introduced in R2011a

R2024b: Specify GPU arrays for neural network models (requires Parallel Computing Toolbox)

kfoldLoss fully supports GPU arrays for ClassificationPartitionedModel models trained using fitcnet.

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 cross-validated SVM and ensemble classifiers with a nondefault cost matrix

If you specify a nondefault cost matrix when you cross-validate the input model object for an SVM or ensemble classification model, 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 cross-validated SVM and ensemble model objects with a nondefault cost matrix, so the function can return a different value.

For details about the property value change, see Cost property stores the user-specified cost matrix (cross-validated SVM classifier) or Cost property stores the user-specified cost matrix (cross-validated ensemble classifier).

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.

R2022a: Default LossFun value has changed for cross-validated GAM and neural network classifiers

Starting in R2022a, the default value of the LossFun name-value argument has changed for both a generalized additive model (GAM) and a neural network model, so that the kfoldLoss function uses the "mincost" option (minimal expected misclassification cost) as the default when a cross-validated classification object uses posterior probabilities for classification scores.

• If the model type of the input model object CVMdl is a GAM classifier, the default value is "mincost" if the ScoreTransform property of CVMdl (CVMdl.ScoreTransform) is 'logit'; otherwise, the default value is "classiferror".

• If the model type of CVMdl is a neural network model classifier, the default value is "mincost".

In previous releases, the default value was 'classiferror'.

You do not need to make any changes to your code if you use the default cost matrix (whose element value is 0 for correct classification and 1 for incorrect classification). The "mincost" option is equivalent to the "classiferror" option for the default cost matrix.