# resubLoss

Class: ClassificationKNN

Loss of k-nearest neighbor classifier by resubstitution

## Syntax

L = resubLoss(mdl)
L = resubLoss(mdl,Name,Value)

## Description

L = resubLoss(mdl) returns the resubstitution loss, meaning the loss computed for the data that fitcknn used to create mdl.

L = resubLoss(mdl,Name,Value) returns loss statistics with additional options specified by one or more Name,Value pair arguments.

## Input Arguments

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### mdl — Classifier modelclassifier model object

k-nearest neighbor classifier model, returned as a classifier model object.

Note that using the 'CrossVal', 'KFold', 'Holdout', 'Leaveout', or 'CVPartition' options results in a model of class ClassificationPartitionedModel. You cannot use a partitioned tree for prediction, so this kind of tree does not have a predict method.

Otherwise, mdl is of class ClassificationKNN, and you can use the predict method to make predictions.

### Name-Value Pair 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 single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

 'lossfun ' Function handle or string representing a loss function. Built-in loss functions are: 'binodeviance' — See Loss Functions.'classiferror' — Fraction of misclassified observations. See Loss Functions.'exponential' — See Loss Functions.'hinge' — See Loss Functions.'mincost' — Smallest misclassification cost as given by the mdl.Cost matrix. You can write your own loss function using the syntax described in Loss Functions. Default: 'mincost'

## Output Arguments

 L Classification error, a scalar. The meaning of the error depends on the values in weights and lossfun. See Classification Error.

## Definitions

### Classification Error

The default classification error is the fraction of data X that mdl misclassifies, where Y represents the true classifications.

The weighted classification error is the sum of weight i times the Boolean value that is 1 when mdl misclassifies the ith row of X, divided by the sum of the weights.

### Loss Functions

The built-in loss functions are:

• 'binodeviance' — For binary classification, assume the classes yn are -1 and 1. With weight vector w normalized to have sum 1, and predictions of row n of data X as f(Xn), the binomial deviance is

$\sum {w}_{n}\mathrm{log}\left(1+\mathrm{exp}\left(-2{y}_{n}f\left({X}_{n}\right)\right)\right).$

• 'exponential' — With the same definitions as for 'binodeviance', the exponential loss is

$\sum {w}_{n}\mathrm{exp}\left(-{y}_{n}f\left({X}_{n}\right)\right).$

• 'classiferror' — Predict the label with the largest posterior probability. The loss is then the fraction of misclassified observations.

• 'hinge' — Classification error measure that has the form

$L=\frac{\sum _{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{y}_{j}\prime f\left({X}_{j}\right)\right\}}{\sum _{j=1}^{n}{w}_{j}},$

where:

• wj is weight j.

• For binary classification, yj = 1 for the positive class and -1 for the negative class. For problems where the number of classes K > 3, yj is a vector of 0s, but with a 1 in the position corresponding to the true class, e.g., if the second observation is in the third class and K = 4, then y2 = [0 0 1 0]′.

• $f\left({X}_{j}\right)$ is, for binary classification, the posterior probability or, for K > 3, a vector of posterior probabilities for each class given observation j.

• 'mincost' — Predict the label with the smallest expected misclassification cost, with expectation taken over the posterior probability, and cost as given by the Cost property of the classifier (a matrix). The loss is then the true misclassification cost averaged over the observations.

To write your own loss function, create a function file in this form:

function loss = lossfun(C,S,W,COST)
• N is the number of rows of X.

• K is the number of classes in the classifier, represented in the ClassNames property.

• C is an N-by-K logical matrix, with one true per row for the true class. The index for each class is its position in the ClassNames property.

• S is an N-by-K numeric matrix. S is a matrix of posterior probabilities for classes with one row per observation, similar to the posterior output from predict.

• W is a numeric vector with N elements, the observation weights. If you pass W, the elements are normalized to sum to the prior probabilities in the respective classes.

• COST is a K-by-K numeric matrix of misclassification costs. For example, you can use COST = ones(K) - eye(K), which means a cost of 0 for correct classification, and 1 for misclassification.

• The output loss should be a scalar.

Pass the function handle @lossfun as the value of the LossFun name-value pair.

### True Misclassification Cost

There are two costs associated with KNN classification: the true misclassification cost per class, and the expected misclassification cost per observation.

You can set the true misclassification cost per class in the Cost name-value pair when you run fitcknn. Cost(i,j) is the cost of classifying an observation into class j if its true class is i. By default, Cost(i,j)=1 if i~=j, and Cost(i,j)=0 if i=j. In other words, the cost is 0 for correct classification, and 1 for incorrect classification.

### Expected Cost

There are two costs associated with KNN classification: the true misclassification cost per class, and the expected misclassification cost per observation. The third output of predict is the expected misclassification cost per observation.

Suppose you have Nobs observations that you want to classify with a trained classifier mdl. Suppose you have K classes. You place the observations into a matrix Xnew with one observation per row. The command

[label,score,cost] = predict(mdl,Xnew)

returns, among other outputs, a cost matrix of size Nobs-by-K. Each row of the cost matrix contains the expected (average) cost of classifying the observation into each of the K classes. cost(n,k) is

$\sum _{i=1}^{K}\stackrel{^}{P}\left(i|Xnew\left(n\right)\right)C\left(k|i\right),$

where

• K is the number of classes.

• $\stackrel{^}{P}\left(i|Xnew\left(n\right)\right)$ is the posterior probability of class i for observation Xnew(n).

• $C\left(k|i\right)$ is the true misclassification cost of classifying an observation as k when its true class is i.

## Examples

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### Loss Calculation

Construct a k-nearest neighbor classifier for the Fisher iris data, where k = 5.

Construct a classifier for 5-nearest neighbors.

mdl = fitcknn(meas,species,'NumNeighbors',5);

Examine the resubstitution loss of the classifier.

L = resubLoss(mdl)
L =

0.0333

The classifier predicts incorrect classifications for 1/30 of its training data.