# Documentation

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# loss

Class: ClassificationKNN

Loss of k-nearest neighbor classifier

## Syntax

• L = loss(mdl,tbl,ResponseVarName)
• L = loss(mdl,tbl,Y)
• L = loss(mdl,X,Y)
• L = loss(___,Name,Value)

## Description

L = loss(mdl,tbl,ResponseVarName) returns a scalar representing how well mdl classifies the data in tbl, when tbl.ResponseVarName contains the true classifications.

When computing the loss, loss normalizes the class probabilities in tbl.ResponseVarNames to the class probabilities used for training, stored in the Prior property of mdl.

L = loss(mdl,tbl,Y) returns a scalar representing how well mdl classifies the data in tbl, when Y contains the true classifications.

When computing the loss, loss normalizes the class probabilities in Y to the class probabilities used for training, stored in the Prior property of mdl.

L = loss(mdl,X,Y) returns a scalar representing how well mdl classifies the data in X, when Y contains the true classifications.

When computing the loss, loss normalizes the class probabilities in Y to the class probabilities used for training, stored in the Prior property of mdl.

L = loss(___,Name,Value) returns the loss with additional options specified by one or more Name,Value pair arguments, using any of the previous syntaxes.

## Input Arguments

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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.

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. Optionally, tbl can contain one additional column for the response variable. Multi-column 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 this method 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, then you must do so as a character vector. For example, if the response variable is stored as tbl.response, then specify it as 'response'. Otherwise, the software treats all columns of tbl, including tbl.response, as predictors.

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

Matrix of predictor values. Each column of X represents one variable, and each row represents one observation.

A categorical array, cell array of character vectors, character array, logical vector, or a numeric vector with the same number of rows as X. Each row of Y represents the classification of the corresponding row of X.

### 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.

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Loss function, specified as the comma-separated pair consisting of 'LossFun' and a built-in, loss-function name or function handle.

• The following lists available loss functions. Specify one using its corresponding character vector.

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 k-nearest neighbor models return posterior probabilities as classification scores by default (see predict).

• 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(mdl.ClassNames)). 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 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. 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.

Data Types: char | function_handle

Observation weights, specified as the comma-separated pair consisting of 'Weights' and a numeric vector or the name of a variable in tbl.

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

If you specify Weights as the name of a variable in tbl, you must do so as a character vector, such as 'W'. For example, if the weights are stored as tbl.w, then specify it as 'w'. Otherwise, the software treats all columns of tbl, including tbl.w, as predictors.

loss normalizes the weights so that observation weights in each class sum to the prior probability of that class. When you supply Weights, loss computes weighted classification loss.

Data Types: single | double

## Output Arguments

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Classification loss, returned as a scalar value. The meaning of the error depends on the values in weights and lossfun.

## Definitions

### Classification Loss

Classification loss functions measure the predictive inaccuracy of classification models. When comparing the same type of loss among many models, lower loss indicates a better predictive model.

Suppose that:

• 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 to the average loss.

• For algorithms that support multiclass classification (that is, K ≥ 3):

• yj* is a vector of K – 1 zeros, and a 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,

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

The supported loss functions are:

• Binomial deviance, specified using 'LossFun','binodeviance'. Its equation is

$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.$

• Exponential loss, specified using 'LossFun','exponential'. Its equation is

$L=\sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right).$

• Classification error, specified using 'LossFun','classiferror'. It is the weighted fraction of misclassified observations, with equation

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

${\stackrel{^}{y}}_{j}$ is the class label corresponding to the class with the maximal posterior probability. I{x} is the indicator function.

• Hinge loss, specified using 'LossFun','hinge'. Its equation is

$L=\sum _{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.$

• Logit loss, specified using 'LossFun','logit'. Its equation is

$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right).$

• Minimal cost, specified using 'LossFun','mincost'. 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 the input model stores in the property Cost.

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

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

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

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

### 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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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 loss of the classifier for a mean observation classified 'versicolor'.

X = mean(meas);
Y = {'versicolor'};
L = loss(mdl,X,Y)
L =

0

The classifier has no doubt that 'versicolor' is the correct classification (all five nearest neighbors classify as 'versicolor').