# 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.

ValueDescription
'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)

'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.

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

expand all

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

t = templateTree('MaxNumSplits',1);

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.

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.

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;
"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.