# kfoldLoss

Classification loss for cross-validated classification model

## Description

example

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.

example

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

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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

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

t = templateTree('MaxNumSplits',1);

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

cvens = crossval(ens);

Estimate the cross-validated classification error.

L = kfoldLoss(cvens)
L = 0.0655

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').

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

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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.

### 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: 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.

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

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

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, generalized additive model, neural network, or support vector machine classifier.

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

'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.

ValueDescription
'binodeviance'Binomial deviance
'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)

'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.

For more details on loss functions, see Classification Loss.

Example: 'LossFun','hinge'

Data Types: char | string | function_handle

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

ValueDescription
'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

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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.

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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.

Consider the following scenario.

• L is the weighted average classification loss.

• n is the sample size.

• For binary classification:

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

• 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 significantly to the average loss.

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

• yj* is a vector of K – 1 zeros, with 1 in the position corresponding to the true, observed class yj. 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(Xj) 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.

• mj = yj*f(Xj). Therefore, mj is the scalar classification score that the model predicts for the true, observed class.

• 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 they sum to 1. Therefore,

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

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

Loss FunctionValue of LossFunEquation
Binomial deviance'binodeviance'$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.$
Misclassified rate in decimal'classiferror'

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

${\stackrel{^}{y}}_{j}$ is the class label corresponding to the class with the maximal score. I{·} is the indicator function.

Cross-entropy loss'crossentropy'

'crossentropy' is appropriate only for neural network models.

The weighted cross-entropy loss is

$L=-\sum _{j=1}^{n}\frac{{\stackrel{˜}{w}}_{j}\mathrm{log}\left({m}_{j}\right)}{Kn},$

where the weights ${\stackrel{˜}{w}}_{j}$ are normalized to sum to n instead of 1.

Exponential loss'exponential'$L=\sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right).$
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 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.

1. 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 binary and multiclass classification for the observation Xj. C is the cost matrix stored in the Cost property of the model.

2. For observation j, predict the class label corresponding to the minimal expected misclassification cost:

${\stackrel{^}{y}}_{j}=\underset{k=1,...,K}{\text{argmin}}{\gamma }_{jk}.$

3. Using C, identify the cost incurred (cj) for making the prediction.

The weighted average of the minimal expected misclassification cost loss is

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

If you use the default cost matrix (whose element value is 0 for correct classification and 1 for incorrect classification), then the 'mincost' loss is equivalent to the 'classiferror' loss.

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

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).

## 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 Typeloss Function
Discriminant analysis classifierloss
Ensemble classifierloss
k-nearest neighbor classifierloss
Naive Bayes classifierloss
Neural network classifierloss
Support vector machine classifierloss
Binary decision tree for multiclass classificationloss