# resubLoss

Loss of k-nearest neighbor classifier by resubstitution

## Description

example

L = resubLoss(mdl) returns the classification loss by resubstitution, which is the loss computed for the data used by fitcknn to create mdl.

The classification loss (L) is a numeric scalar, whose interpretation depends on the loss function and the observation weights in mdl.

L = resubLoss(mdl,'LossFun',lossfun) returns the resubstitution loss for the loss function lossfun, specified as a name-value pair argument.

## Examples

collapse all

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

Load the Fisher iris data set.

Create a classifier for five 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.

## Input Arguments

collapse all

k-nearest neighbor classifier model, specified as a ClassificationKNN object.

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

• The following table lists the available loss functions.

ValueDescription
'binodeviance'Binomial deviance
'classiferror'Classification error
'exponential'Exponential
'hinge'Hinge
'logit'Logistic
'mincost'Minimal expected misclassification cost (for classification scores that are posterior probabilities)

'mincost' is appropriate for classification scores that are posterior probabilities. By default, k-nearest neighbor models return posterior probabilities as classification scores (see predict).

• You can specify a function handle for a custom loss function using @ (for example, @lossfun). Let n be the number of observations in X and K be the number of distinct classes (numel(mdl.ClassNames)). Your custom loss function must have this form:

function lossvalue = lossfun(C,S,W,Cost)

• 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. 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 mdl.ClassNames. The argument 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.

• The output argument lossvalue is a scalar.

For more details on loss functions, see Classification Loss.

Data Types: char | string | function_handle

## Algorithms

collapse all

### 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, respectively.

• f(Xj) is the raw 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 y*2 = [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. The software also normalizes the prior probabilities so they sum to 1. 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 pair argument.

Loss FunctionValue of LossFunEquation
Binomial deviance'binodeviance'$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.$
Exponential loss'exponential'$L=\sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right).$
Classification error'classiferror'

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

It is the weighted fraction of misclassified observations where ${\stackrel{^}{y}}_{j}$ is the class label corresponding to the class with the maximal posterior probability. I{x} is the indicator function.

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 cost'mincost'

Minimal cost. The software computes the weighted minimal cost using this procedure for observations j = 1,...,n.

1. Estimate the 1-by-K vector of expected classification costs for observation j:

${\gamma }_{j}=f{\left({X}_{j}\right)}^{\prime }C.$

f(Xj) is the column vector of class posterior probabilities for binary and multiclass classification. C is the cost matrix that the input model stores in the Cost property.

2. For observation j, predict the class label corresponding to the minimum expected classification cost:

${\stackrel{^}{y}}_{j}=\underset{j=1,...,K}{\mathrm{min}}\left({\gamma }_{j}\right).$

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

The weighted, average, minimum cost loss is

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

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

This figure compares the loss functions (except 'mincost') for one observation over m. Some functions are normalized to pass through [0,1].

### True Misclassification Cost

Two costs are 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 by using the 'Cost' name-value pair argument when you run fitcknn. The value 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

Two costs are associated with KNN classification: the true misclassification cost per class and the expected misclassification cost per observation. The third output of resubPredict is the expected misclassification cost per observation.

Suppose you have Nobs observations that you classified with a trained classifier mdl, and you have K classes. The command

[label,score,cost] = resubPredict(mdl)

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

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

where

• K is the number of classes.

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

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