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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)
`'quadratic'`Quadratic

`'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.

Specify your function using `'LossFun',@lossfun`.

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

`$L=\sum _{j=1}^{n}{w}_{j}{c}_{j}.$`
• Quadratic loss, specified using `'LossFun','quadratic'`. Its equation is

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

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.

`load fisheriris`

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'`).