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loss

Loss of k-nearest neighbor classifier

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. If tbl contains the response variable used to train mdl, then you do not need to specify ResponseVarName.

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

The meaning of the classification loss (L) depends on the loss function and weighting scheme, but, in general, better classifiers yield smaller classification loss values. For more details, see Classification Loss.

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, the loss function normalizes the class probabilities in Y to the class probabilities used for training, which are stored in the Prior property of mdl.

example

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, the loss function normalizes the class probabilities in Y to the class probabilities used for training, which are stored in the Prior property of mdl.

L = loss(___,Name,Value) specifies options using one or more name-value pair arguments in addition to the input arguments in previous syntaxes. For example, you can specify the loss function and the classification weights.

Examples

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

X = mean(meas);
Y = {'versicolor'};
L = loss(mdl,X,Y)
L = 0

All five nearest neighbors classify as 'versicolor'.

Input Arguments

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

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. 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 train 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.

You must specify ResponseVarName as a character vector or string scalar. 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, character, or string 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.

Data Types: char | string

Predictor data, specified as a numeric matrix. Each row of X represents one observation, and each column represents one variable.

Data Types: single | double

Class labels, specified as a categorical, character, or string array, logical or numeric vector, or cell array of character vectors. Each row of Y represents the classification of the corresponding row of X.

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

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 quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: loss(mdl,tbl,'response','LossFun','exponential','Weights','w') returns the weighted exponential loss of mdl classifying the data in tbl. Here, tbl.response is the response variable, and tbl.w is the weight variable.

Loss function, specified as the comma-separated pair consisting of 'LossFun' and 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

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 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, the name must be 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.

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 the weighted classification loss.

Example: 'Weights','w'

Data Types: single | double | char | string

Algorithms

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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, 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 predict is the expected misclassification cost per observation.

Suppose you have Nobs observations that you want to classify with a trained classifier mdl, and 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 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

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

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).

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