# loss

Classification loss for classification ensemble model

## Description

L = loss(ens,tbl,ResponseVarName) returns the Classification Loss L for the trained classification ensemble model ens using the predictor data in table tbl and the true class labels in tbl.ResponseVarName. The interpretation of L depends on the loss function (LossFun) and weighting scheme (Weights). In general, better classifiers yield smaller classification loss values. The default LossFun value is "classiferror" (misclassification rate in decimal).

L = loss(ens,tbl,Y) uses the predictor data in table tbl and the true class labels in Y.

example

L = loss(ens,X,Y) uses the predictor data in matrix X and the true class labels in Y.

example

L = loss(___,Name=Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in the previous syntaxes. For example, you can specify the indices of weak learners in the ensemble to use for calculating loss, specify a classification loss function, and perform computations in parallel.

Note

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

## Examples

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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.

## Input Arguments

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Full classification ensemble model, specified as a ClassificationEnsemble model object trained with fitcensemble, or a CompactClassificationEnsemble model object created with compact.

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 loss must also be in a table.

Data Types: table

Response variable name, specified as the name of a variable in tbl. If tbl contains the response variable used to train ens, then you do not need to specify ResponseVarName.

If you specify ResponseVarName, you must specify it 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.

The response variable must be a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors. If the response variable is a character array, then each element must correspond to one row of the array.

Data Types: char | string

Class labels, specified as a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors. Y must have the same data type as tbl or X. (The software treats string arrays as cell arrays of character vectors.)

Y must be of the same type as the classification used to train ens, and its number of elements must equal the number of rows of tbl or X.

Data Types: categorical | char | string | logical | single | double | cell

Predictor data, specified as a numeric matrix.

Each row of X corresponds to one observation, and each column corresponds to one variable. The variables in the columns of X must be the same as the variables used to train ens.

The number of rows in X must equal the number of rows in Y.

If you trained ens using sample data contained in a matrix, then the input data for loss must also be in a matrix.

Data Types: double | single

### 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: loss(ens,X,LossFun="exponential",UseParallel=true) specifies to use an exponential loss function, and to run in parallel.

Indices of weak learners in the ensemble to use in loss, specified as a vector of positive integers in the range [1:ens.NumTrained]. By default, all learners are used.

Example: Learners=[1 2 4]

Data Types: single | double

Loss function, specified as a built-in loss function name or a function handle.

The following table describes the values for the built-in loss functions.

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").

• To use posterior probabilities as classification scores when the ensemble method is "AdaBoostM1", "AdaBoostM2", "GentleBoost", or "LogitBoost", you must specify the double-logit score transform by entering the following:

ens.ScoreTransform = "doublelogit";

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

You can specify your own function using function handle notation. Suppose that n is the number of observations in X, and K is the number of distinct classes (numel(ens.ClassNames), where ens is the input model). 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 ens.ClassNames.

Create 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 the weights 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="binodeviance"

Example: LossFun=@Lossfun

Data Types: char | string | function_handle

Aggregation level for the output, specified as "ensemble", "individual", or "cumulative".

ValueDescription
"ensemble"The output is a scalar value, the loss for the entire ensemble.
"individual"The output is a vector with one element per trained learner.
"cumulative"The output is a vector in which element J is obtained by using learners 1:J from the input list of learners.

Example: Mode="individual"

Data Types: char | string

Option to use observations for learners, specified as 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 (default), learner j is used in predicting the class of row i of X.

Example: UseObsForLearner=logical([1 1; 0 1; 1 0])

Data Types: logical matrix

Flag to run in parallel, specified as a numeric or logical 1 (true) or 0 (false). If you specify UseParallel=true, the loss function executes for-loop iterations by using parfor. The loop runs in parallel when you have Parallel Computing Toolbox™.

Example: UseParallel=true

Data Types: logical

Observation weights, specified as a numeric vector or the name of a variable in tbl. If you supply weights, loss normalizes them so that the observation weights in each class sum to the prior probability of that class.

If you specify Weights as a numeric vector, the size of Weights must be equal to the number of observations in X or tbl. The software normalizes Weights to sum up to the value of the prior probability in the respective class.

If you specify Weights as the name of a variable in tbl, you must specify it as a character vector or string scalar. For example, if the weights are stored as tbl.W, then specify Weights as "W". Otherwise, the software treats all columns of tbl, including tbl.W, as predictors.

Example: Weights="W"

Data Types: single | double | char | string

## Output Arguments

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

• If Mode is "ensemble", then L is a scalar value, the loss for the entire ensemble.

• If Mode is "individual", then L is a vector with one element per trained learner.

• If Mode is "cumulative", then L is a vector in which element J is obtained by using learners 1:J from the input list of learners.

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

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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 stored in the Prior property. 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 argument.

Loss FunctionValue of LossFunEquation
Binomial deviance"binodeviance"$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.$
Observed misclassification cost"classifcost"

$L=\sum _{j=1}^{n}{w}_{j}{c}_{{y}_{j}{\stackrel{^}{y}}_{j}},$

where ${\stackrel{^}{y}}_{j}$ is the class label corresponding to the class with the maximal score, and ${c}_{{y}_{j}{\stackrel{^}{y}}_{j}}$ is the user-specified cost of classifying an observation into class ${\stackrel{^}{y}}_{j}$ when its true class is yj.

Misclassified rate in decimal"classiferror"

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

where I{·} is the indicator function.

Cross-entropy loss"crossentropy"

"crossentropy" is appropriate only for neural network models.

The weighted cross-entropy loss is

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

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 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

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

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

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).

## Version History

Introduced in R2011a

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