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loss

Classification loss for discriminant analysis classifier

Description

L = loss(Mdl,Tbl,ResponseVarName) returns the classification loss, a scalar representing how well the trained discriminant analysis classifier Mdl classifies the predictor data in table Tbl compared to the true class labels in Tbl.ResponseVarName.

The classification loss (L) is a generalization or resubstitution quality measure. Its interpretation depends on the loss function and weighting scheme, but, in general, better classifiers yield smaller classification loss values.

When computing L, loss normalizes the class probabilities in Tbl.ResponseVarName to the class probabilities used for training, stored in the Prior property of Mdl.

L = loss(Mdl,Tbl,Y) returns the classification loss for the classifier Mdl using the predictor data in table Tbl and the class labels in Y.

example

L = loss(Mdl,X,Y) returns the classification loss for the trained discriminant analysis classifier Mdl using the predictor data X and the corresponding class labels in Y.

L = loss(___,Name=Value) specifies additional 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 a classification loss function and the observation weights.

Note

If the predictor data X contains any missing values and LossFun is not set to "mincost" or "classiferror", the loss function might return NaN. For more information, see loss can return NaN for predictor data with missing values.

Examples

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Load Fisher's iris data set.

load fisheriris

Train a discriminant analysis model using all observations in the data.

Mdl = fitcdiscr(meas,species);

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

L = loss(Mdl,meas,species)
L = 0.0200

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

Input Arguments

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Trained discriminant analysis classifier, specified as a ClassificationDiscriminant model object trained with fitcdiscr, or a CompactClassificationDiscriminant model object created with compact.

Sample data used to train the model, specified as a table. Each row of Tbl corresponds to one observation, and each column corresponds to one predictor variable. Categorical predictor variables are not supported. Optionally, Tbl can contain additional columns for the response variable (which can be categorical) and observation weights. Tbl must contain all of the predictor variables used to train Mdl. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

If Tbl contains the response variable used to train Mdl, then you do not need to specify ResponseVarName or Y.

If you trained Mdl 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 Mdl, 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

Predictor data, specified as a numeric matrix. Each row of X corresponds to one observation, and each column corresponds to one predictor variable. Categorical predictor variables are not supported. The variables in the columns of X must be the same as the variables used to train Mdl. The number of rows in X must equal the number of rows in Y.

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

Data Types: single | double

Class labels, specified as a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors. Y must be of the same type as the classification used to train Mdl. (The software treats string arrays as cell arrays of character vectors.)

The length of Y must equal the number of rows in Tbl or X.

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

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: L = loss(Mdl,meas,species,LossFun="binodeviance")

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. Specify one using the 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)
"quadratic"Quadratic loss

"mincost" is appropriate for classification scores that are posterior probabilities. Discriminant analysis classifiers return posterior probabilities as classification scores by default (see predict).

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(Mdl.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 Mdl.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 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 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.

Example: LossFun="binodeviance"

Example: LossFun=@Lossfun

Data Types: char | string | function_handle

Observation weights, specified as a numeric vector or the name of a variable in Tbl. The software weighs the observations in each row of X or Tbl with the corresponding weight in Weights.

If you specify Weights as a numeric vector, then the size of Weights must be equal to the number of rows in X or Tbl.

If you specify Weights as the name of a variable in Tbl, you must do so 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.

If you do not specify your own loss function, then the software normalizes Weights to sum up to the value of the prior probability in the respective class.

Example: Weights="W"

Data Types: single | double | char | string

More About

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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,

    j=1nwj=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=j=1nwjlog{1+exp[2mj]}.
Observed misclassification cost"classifcost"

L=j=1nwjcyjy^j,

where y^j is the class label corresponding to the class with the maximal score, and cyjy^j is the user-specified cost of classifying an observation into class y^j when its true class is yj.

Misclassified rate in decimal"classiferror"

L=j=1nwjI{y^jyj},

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=j=1nw˜jlog(mj)Kn,

where the weights w˜j are normalized to sum to n instead of 1.

Exponential loss"exponential"L=j=1nwjexp(mj).
Hinge loss"hinge"L=j=1nwjmax{0,1mj}.
Logit loss"logit"L=j=1nwjlog(1+exp(mj)).
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:

    γjk=(f(Xj)C)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:

    y^j=argmink=1,...,Kγjk.

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

The weighted average of the minimal expected misclassification cost loss is

L=j=1nwjcj.

Quadratic loss"quadratic"L=j=1nwj(1mj)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).

Comparison of classification losses for different loss functions

Posterior Probability

The posterior probability that a point x belongs to class k is the product of the prior probability and the multivariate normal density. The density function of the multivariate normal with 1-by-d mean μk and d-by-d covariance Σk at a 1-by-d point x is

P(x|k)=1((2π)d|Σk|)1/2exp(12(xμk)Σk1(xμk)T),

where |Σk| is the determinant of Σk, and Σk1 is the inverse matrix.

Let P(k) represent the prior probability of class k. Then the posterior probability that an observation x is of class k is

P^(k|x)=P(x|k)P(k)P(x),

where P(x) is a normalization constant, the sum over k of P(x|k)P(k).

Prior Probability

The prior probability is one of three choices:

  • 'uniform' — The prior probability of class k is one over the total number of classes.

  • 'empirical' — The prior probability of class k is the number of training samples of class k divided by the total number of training samples.

  • Custom — The prior probability of class k is the kth element of the prior vector. See fitcdiscr.

After creating a classification model (Mdl) you can set the prior using dot notation:

Mdl.Prior = v;

where v is a vector of positive elements representing the frequency with which each element occurs. You do not need to retrain the classifier when you set a new prior.

Cost

The matrix of expected costs per observation is defined in Cost.

Extended Capabilities

Version History

Introduced in R2011b

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