loss

Classification error

Syntax

L = loss(ens,tbl,ResponseVarName)
L = loss(ens,tbl,Y)
L = loss(ens,X,Y)
L = loss(___,Name,Value)


Description

L = loss(ens,tbl,ResponseVarName) returns the classification error for ensemble ens computed using table of predictors tbl and true class labels tbl.ResponseVarName.

L = loss(ens,tbl,Y) returns the classification error for ensemble ens computed using table of predictors tbl and true class labels Y.

L = loss(ens,X,Y) returns the classification error for ensemble ens computed using matrix of predictors X and true class labels Y.

L = loss(___,Name,Value) computes classification error with additional options specified by one or more Name,Value pair arguments, using any of the previous syntaxes.

When computing the loss, loss normalizes the class probabilities in ResponseVarName or Y to the class probabilities used for training, stored in the Prior property of ens.

Note

If the predictor data X or the predictor variables in tbl contain any missing values, the loss function can return NaN. For more details, see loss can return NaN for predictor data with missing values.

Input Arguments

ens	Classification ensemble created with fitcensemble, or a compact classification ensemble created with compact.
tbl	Sample data, specified as a table. Each row of tbl corresponds to one observation, and each column corresponds to one predictor variable. tbl must contain all of the predictors used to train the model. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed. If you trained ens using sample data contained in a table, then the input data for this method must also be in a table.
ResponseVarName	Response variable name, specified as the name of a variable in tbl. You must specify ResponseVarName as a character vector or string scalar. For example, if the response variable Y is stored as tbl.Y, then specify it as 'Y'. Otherwise, the software treats all columns of tbl, including Y, as predictors when training the model.
X	Matrix of data to classify. Each row of X represents one observation, and each column represents one predictor. X must have the same number of columns as the data used to train ens. X should have the same number of rows as the number of elements in Y. If you trained ens using sample data contained in a matrix, then the input data for this method must also be in a matrix.
Y	Class labels of observations in tbl or X. Y should be of the same type as the classification used to train ens, and its number of elements should equal the number of rows of tbl or X.

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.

learners	Indices of weak learners in the ensemble ranging from 1 to ens.NumTrained. loss uses only these learners for calculating loss. Default: 1:NumTrained
Lossfun	Loss function, specified as the comma-separated pair consisting of 'LossFun' and 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. Bagged and subspace ensembles return posterior probabilities by default (ens.Method is 'Bag' or 'Subspace'). If the ensemble method is 'AdaBoostM1', 'AdaBoostM2', GentleBoost, or 'LogitBoost', then, to use posterior probabilities as classification scores, you must specify the double-logit score transform by entering ens.ScoreTransform = 'doublelogit'; For all other ensemble methods, the software does not support posterior probabilities as classification scores. Specify your own function using function handle notation. Suppose that n be the number of observations in X and K be the number of distinct classes (numel(ens.ClassNames), ens 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 ens.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 ens.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. For more details on loss functions, see Classification Loss. Default: 'classiferror'
mode	Meaning of the output L: 'ensemble' — L is a scalar value, the loss for the entire ensemble. 'individual' — L is a vector with one element per trained learner. 'cumulative' — L is a vector in which element J is obtained by using learners 1:J from the input list of learners. Default: 'ensemble'
UseObsForLearner	A logical matrix of size N-by-T, where: N is the number of rows of X. T is the number of weak learners in ens. When UseObsForLearner(i,j) is true, learner j is used in predicting the class of row i of X. Default: true(N,T)
UseParallel	Indication to perform inference in parallel, specified as false (compute serially) or true (compute in parallel). Parallel computation requires Parallel Computing Toolbox™. Parallel inference can be faster than serial inference, especially for large datasets. Parallel computation is supported only for tree learners. Default: false
weights	Vector of observation weights, with nonnegative entries. The length of weights must equal the number of rows in X. When you specify weights, loss normalizes the weights so that observation weights in each class sum to the prior probability of that class. Default: ones(size(X,1),1)

Output Arguments

L	Classification loss, by default the fraction of misclassified data. L can be a vector, and can mean different things, depending on the name-value pair settings.

Examples

Estimate Classification Error

Load Fisher's iris data set.

load fisheriris

Train a classification ensemble of 100 decision trees using AdaBoostM2. Specify tree stumps as the weak learners.

t = templateTree('MaxNumSplits',1);
ens = fitcensemble(meas,species,'Method','AdaBoostM2','Learners',t);

Estimate the classification error of the model using the training observations.

L = loss(ens,meas,species)

L = 0.0333

Alternatively, if ens is not compact, then you can estimate the training-sample classification error by passing ens to resubLoss.

Assess Performance of Ensemble of Boosted Trees

Create an ensemble of boosted trees and inspect the importance of each predictor. Using test data, assess the classification accuracy of the ensemble.

Load the arrhythmia data set. Determine the class representations in the data.

load arrhythmia
Y = categorical(Y);
tabulate(Y)

  Value    Count   Percent
      1      245     54.20%
      2       44      9.73%
      3       15      3.32%
      4       15      3.32%
      5       13      2.88%
      6       25      5.53%
      7        3      0.66%
      8        2      0.44%
      9        9      1.99%
     10       50     11.06%
     14        4      0.88%
     15        5      1.11%
     16       22      4.87%

The data set contains 16 classes, but not all classes are represented (for example, class 13). Most observations are classified as not having arrhythmia (class 1). The data set is highly discrete with imbalanced classes.

Combine all observations with arrhythmia (classes 2 through 15) into one class. Remove those observations with an unknown arrhythmia status (class 16) from the data set.

idx = (Y ~= "16");
Y = Y(idx);
X = X(idx,:);
Y(Y ~= "1") = "WithArrhythmia";
Y(Y == "1") = "NoArrhythmia";
Y = removecats(Y);

Create a partition that evenly splits the data into training and test sets.

rng("default") % For reproducibility
cvp = cvpartition(Y,"Holdout",0.5);
idxTrain = training(cvp);
idxTest = test(cvp);

cvp is a cross-validation partition object that specifies the training and test sets.

Train an ensemble of 100 boosted classification trees using AdaBoostM1. Specify to use tree stumps as the weak learners. Also, because the data set contains missing values, specify to use surrogate splits.

t = templateTree("MaxNumSplits",1,"Surrogate","on");
numTrees = 100;
mdl = fitcensemble(X(idxTrain,:),Y(idxTrain),"Method","AdaBoostM1", ...
    "NumLearningCycles",numTrees,"Learners",t);

mdl is a trained ClassificationEnsemble model.

Inspect the importance measure for each predictor.

predImportance = predictorImportance(mdl);
bar(predImportance)
title("Predictor Importance")
xlabel("Predictor")
ylabel("Importance Measure")

Identify the top ten predictors in terms of their importance.

[~,idxSort] = sort(predImportance,"descend");
idx10 = idxSort(1:10)

idx10 = 1×10

   228   233   238    93    15   224    91   177   260   277

Classify the test set observations. View the results using a confusion matrix. Blue values indicate correct classifications, and red values indicate misclassified observations.

predictedValues = predict(mdl,X(idxTest,:));
confusionchart(Y(idxTest),predictedValues)

Compute the accuracy of the model on the test data.

error = loss(mdl,X(idxTest,:),Y(idxTest), ...
    "LossFun","classiferror");
accuracy = 1 - error

accuracy = 0.7731

accuracy estimates the fraction of correctly classified observations.

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).

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

Usage notes and limitations:

• You cannot use UseParallel with tall arrays.

For more information, see Tall Arrays.

Automatic Parallel Support

Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

To run in parallel, set the UseParallel name-value argument to true in the call to this function.

For more general information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).

You cannot use UseParallel with tall or GPU arrays.

GPU Arrays

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

Usage notes and limitations:

• The loss function does not support ensembles trained using decision tree learners with surrogate splits.

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

Version History

R2022a: loss 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 loss function returns a different value compared to previous releases.

The loss function uses the prior probabilities stored in the Prior property to normalize the observation weights of the input data. 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 Prior 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 can 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 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: loss can return NaN for predictor data with missing values

The loss function no longer omits an observation with a NaN score when computing the weighted average classification loss. Therefore, loss can now return NaN when the predictor data X or the predictor variables in tbl contain any missing values. In most cases, if the test set observations do not contain missing predictors, the loss function does not return NaN.

This change improves the automatic selection of a classification model when you use fitcauto. Before this change, the software might select a model (expected to best classify new data) with few non-NaN predictors.

If loss in your code returns NaN, you can update your code to avoid this result. Remove or replace the missing values by using rmmissing or fillmissing, respectively.

The following table shows the classification models for which the loss object function might return NaN. For more details, see the Compatibility Considerations for each loss function.

Model Type	Full or Compact Model Object	loss Object Function
Discriminant analysis classification model	ClassificationDiscriminant, CompactClassificationDiscriminant	loss
Ensemble of learners for classification	ClassificationEnsemble, CompactClassificationEnsemble	loss
Gaussian kernel classification model	ClassificationKernel	loss
k-nearest neighbor classification model	ClassificationKNN	loss
Linear classification model	ClassificationLinear	loss
Neural network classification model	ClassificationNeuralNetwork, CompactClassificationNeuralNetwork	loss
Support vector machine (SVM) classification model	ClassificationSVM, CompactClassificationSVM	loss