kfoldLoss

Classification loss for cross-validated linear classification model

Syntax

L = kfoldLoss(CVMdl)

L = kfoldLoss(CVMdl,Name=Value)

Description

L = kfoldLoss(CVMdl) returns the cross-validated classification losses obtained by the cross-validated, binary, linear classification model CVMdl. That is, for every fold, kfoldLoss estimates the classification loss for observations that it holds out when it trains using all other observations.

L contains a classification loss for each regularization strength in the linear classification models that compose CVMdl.

L = kfoldLoss(CVMdl,Name=Value) uses additional options specified by one or more name-value arguments. For example, indicate which folds to use for the loss calculation, or specify the classification loss function.

Examples

Estimate k-Fold Cross-Validation Classification Error

Load the NLP data set.

load nlpdata

X is a sparse matrix of predictor data, and Y is a categorical vector of class labels. There are more than two classes in the data.

The models should identify whether the word counts in a web page are from the Statistics and Machine Learning Toolbox™ documentation. So, identify the labels that correspond to the Statistics and Machine Learning Toolbox™ documentation web pages.

Ystats = Y == 'stats';

Cross-validate a binary, linear classification model that can identify whether the word counts in a documentation web page are from the Statistics and Machine Learning Toolbox™ documentation.

rng(1); % For reproducibility 
CVMdl = fitclinear(X,Ystats,'CrossVal','on');

CVMdl is a ClassificationPartitionedLinear model. By default, the software implements 10-fold cross validation. You can alter the number of folds using the 'KFold' name-value pair argument.

Estimate the average of the out-of-fold, classification error rates.

ce = kfoldLoss(CVMdl)

ce = 
7.6017e-04

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

Specify Custom Classification Loss

Load the NLP data set. Preprocess the data as in Estimate k-Fold Cross-Validation Classification Error, and transpose the predictor data.

load nlpdata
Ystats = Y == 'stats';
X = X';

Cross-validate a binary, linear classification model using 5-fold cross-validation. Optimize the objective function using SpaRSA. Specify that the predictor observations correspond to columns.

rng(1) % For reproducibility 
CVMdl = fitclinear(X,Ystats,'Solver','sparsa','KFold',5, ...
    'ObservationsIn','columns');
CMdl = CVMdl.Trained{1};

CVMdl is a ClassificationPartitionedLinear model. It contains the property Trained, which is a 5-by-1 cell array holding a ClassificationLinear models that the software trained using the training set of each fold.

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. 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. Because the function does not use classification cost, use ~ to have kfoldLoss ignore its position.

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

Estimate the average cross-validated classification loss using the linear loss function. Also, obtain the loss for each fold.

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

ce = 
-8.0982

ceFold = kfoldLoss(CVMdl,'LossFun',linearloss,'Mode','individual')

ceFold = 5×1

   -8.3165
   -8.7633
   -7.4342
   -8.0423
   -7.9347

Find Good Lasso Penalty Using k-fold Classification Loss

To determine a good lasso-penalty strength for a linear classification model that uses a logistic regression learner, compare test-sample classification error rates.

Load the NLP data set. Preprocess the data as in Specify Custom Classification Loss.

load nlpdata
Ystats = Y == 'stats';
X = X';

Create a set of 11 logarithmically-spaced regularization strengths from $$10^{-6}$$ through $$10^{0.5}$$.

Lambda = logspace(-6,-0.5,11);

Cross-validate binary, linear classification models using 5-fold cross-validation, and that use each of the regularization strengths. Optimize the objective function using SpaRSA. Lower the tolerance on the gradient of the objective function to 1e-8.

rng(10); % For reproducibility
CVMdl = fitclinear(X,Ystats,'ObservationsIn','columns',...
    'KFold',5,'Learner','logistic','Solver','sparsa',...
    'Regularization','lasso','Lambda',Lambda,'GradientTolerance',1e-8)

CVMdl = 
  ClassificationPartitionedLinear
    CrossValidatedModel: 'Linear'
           ResponseName: 'Y'
        NumObservations: 31572
                  KFold: 5
              Partition: [1×1 cvpartition]
             ClassNames: [0 1]
         ScoreTransform: 'none'


  Properties, Methods

Extract a trained linear classification model.

Mdl1 = CVMdl.Trained{1}

Mdl1 = 
  ClassificationLinear
      ResponseName: 'Y'
        ClassNames: [0 1]
    ScoreTransform: 'logit'
              Beta: [34023×11 double]
              Bias: [-13.5813 -13.5813 -13.5813 -13.5813 -13.5813 -7.0880 -5.4309 -4.7753 -3.5022 -3.2701 -2.9796]
            Lambda: [1.0000e-06 3.5481e-06 1.2589e-05 4.4668e-05 1.5849e-04 5.6234e-04 0.0020 0.0071 0.0251 0.0891 0.3162]
           Learner: 'logistic'


  Properties, Methods

Mdl1 is a ClassificationLinear model object. Because Lambda is a sequence of regularization strengths, you can think of Mdl as 11 models, one for each regularization strength in Lambda.

Estimate the cross-validated classification error.

ce = kfoldLoss(CVMdl);

Because there are 11 regularization strengths, ce is a 1-by-11 vector of classification error rates.

Higher values of Lambda lead to predictor variable sparsity, which is a good quality of a classifier. For each regularization strength, train a linear classification model using the entire data set and the same options as when you cross-validated the models. Determine the number of nonzero coefficients per model.

Mdl = fitclinear(X,Ystats,'ObservationsIn','columns',...
    'Learner','logistic','Solver','sparsa','Regularization','lasso',...
    'Lambda',Lambda,'GradientTolerance',1e-8);
numNZCoeff = sum(Mdl.Beta~=0);

In the same figure, plot the cross-validated, classification error rates and frequency of nonzero coefficients for each regularization strength. Plot all variables on the log scale.

figure;
[h,hL1,hL2] = plotyy(log10(Lambda),log10(ce),...
    log10(Lambda),log10(numNZCoeff)); 
hL1.Marker = 'o';
hL2.Marker = 'o';
ylabel(h(1),'log_{10} classification error')
ylabel(h(2),'log_{10} nonzero-coefficient frequency')
xlabel('log_{10} Lambda')
title('Test-Sample Statistics')
hold off

Choose the indexes of the regularization strength that balances predictor variable sparsity and low classification error. In this case, a value between $$10^{-4}$$ to $$10^{-1}$$ should suffice.

idxFinal = 7;

Select the model from Mdl with the chosen regularization strength.

MdlFinal = selectModels(Mdl,idxFinal);

MdlFinal is a ClassificationLinear model containing one regularization strength. To estimate labels for new observations, pass MdlFinal and the new data to predict.

Input Arguments

CVMdl — Cross-validated, binary, linear classification model

ClassificationPartitionedLinear model object

Cross-validated, binary, linear classification model, specified as a ClassificationPartitionedLinear model object. You can create a ClassificationPartitionedLinear model using fitclinear and specifying any one of the cross-validation, name-value pair arguments, for example, CrossVal.

To obtain estimates, kfoldLoss applies the same data used to cross-validate the linear 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 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 for classification-score prediction

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

Fold indices to use for classification-score prediction, specified as a numeric vector of positive integers. The elements of Folds must range from 1 through CVMdl.KFold.

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 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
  "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 linear classification models, logistic regression learners return posterior probabilities as classification scores by default, but SVM learners do not (see predict).

• Specify your own function using function handle notation.

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

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

  where:

  • 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 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 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.

Example: LossFun="classifcost"

Data Types: char | string | function_handle

Mode — Loss aggregation level

"average" (default) | "individual"

Loss aggregation level, specified as "average" or "individual".

Value	Description
"average"	Returns losses averaged over all folds
"individual"	Returns losses for each fold

Example: Mode="individual"

Output Arguments

L — Cross-validated classification losses

numeric scalar | numeric vector | numeric matrix

Cross-validated classification losses, returned as a numeric scalar, vector, or matrix. The interpretation of L depends on LossFun.

Let R be the number of regularizations strengths is the cross-validated models (stored in numel(CVMdl.Trained{1}.Lambda)) and F be the number of folds (stored in CVMdl.KFold).

• If Mode is 'average', then L is a 1-by-R vector. L(j) is the average classification loss over all folds of the cross-validated model that uses regularization strength j.

• Otherwise, L is an F-by-R matrix. L(i,j) is the classification loss for fold i of the cross-validated model that uses regularization strength j.

To estimate L, kfoldLoss uses the data that created CVMdl (see X and Y).

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"	$$L={\displaystyle \sum }_{j=1}^n{w}_j\max \left\{0,1-m_j\right\}.$$
Logistic 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).

Extended Capabilities

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).

Version History

Introduced in R2016a

R2024a: Specify GPU arrays (requires Parallel Computing Toolbox)

kfoldLoss fully supports GPU arrays.

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.