# loss

Classification loss for naive Bayes 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 the Classification Loss, a scalar representing how well the trained naive Bayes classifier `Mdl` classifies the predictor data in table `tbl` compared to the true class labels in `tbl.ResponseVarName`.`loss` normalizes the class probabilities in `tbl.ResponseVarName` to the prior class probabilities used by `fitcnb` for training, which are stored in the `Prior` property of `Mdl`.```
````L = loss(Mdl,tbl,Y)` returns the classification loss for the predictor data in table `tbl` and the true class labels in `Y`.```

example

````L = loss(Mdl,X,Y)` returns the classification loss based on the predictor data in matrix `X` compared to the true class labels in `Y`.```

example

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

## Examples

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Determine the test sample classification error (loss) of a naive Bayes classifier. When you compare the same type of loss among many models, a lower loss indicates a better predictive model.

Load the `fisheriris` data set. Create `X` as a numeric matrix that contains four petal measurements for 150 irises. Create `Y` as a cell array of character vectors that contains the corresponding iris species.

```load fisheriris X = meas; Y = species; rng('default') % for reproducibility```

Randomly partition observations into a training set and a test set with stratification, using the class information in `Y`. Specify a 30% holdout sample for testing.

`cv = cvpartition(Y,'HoldOut',0.30);`

Extract the training and test indices.

```trainInds = training(cv); testInds = test(cv);```

Specify the training and test data sets.

```XTrain = X(trainInds,:); YTrain = Y(trainInds); XTest = X(testInds,:); YTest = Y(testInds);```

Train a naive Bayes classifier using the predictors `XTrain` and class labels `YTrain`. A recommended practice is to specify the class names. `fitcnb` assumes that each predictor is conditionally and normally distributed.

`Mdl = fitcnb(XTrain,YTrain,'ClassNames',{'setosa','versicolor','virginica'})`
```Mdl = ClassificationNaiveBayes ResponseName: 'Y' CategoricalPredictors: [] ClassNames: {'setosa' 'versicolor' 'virginica'} ScoreTransform: 'none' NumObservations: 105 DistributionNames: {'normal' 'normal' 'normal' 'normal'} DistributionParameters: {3x4 cell} Properties, Methods ```

`Mdl` is a trained `ClassificationNaiveBayes` classifier.

Determine how well the algorithm generalizes by estimating the test sample classification error.

`L = loss(Mdl,XTest,YTest)`
```L = 0.0444 ```

The naive Bayes classifier misclassifies approximately 4% of the test sample.

You might decrease the classification error by specifying better predictor distributions when you train the classifier with `fitcnb`.

Load the `fisheriris` data set. Create `X` as a numeric matrix that contains four petal measurements for 150 irises. Create `Y` as a cell array of character vectors that contains the corresponding iris species.

```load fisheriris X = meas; Y = species; rng('default') % for reproducibility```

Randomly partition observations into a training set and a test set with stratification, using the class information in `Y`. Specify a 30% holdout sample for testing.

`cv = cvpartition(Y,'HoldOut',0.30);`

Extract the training and test indices.

```trainInds = training(cv); testInds = test(cv);```

Specify the training and test data sets.

```XTrain = X(trainInds,:); YTrain = Y(trainInds); XTest = X(testInds,:); YTest = Y(testInds);```

Train a naive Bayes classifier using the predictors `XTrain` and class labels `YTrain`. A recommended practice is to specify the class names. `fitcnb` assumes that each predictor is conditionally and normally distributed.

`Mdl = fitcnb(XTrain,YTrain,'ClassNames',{'setosa','versicolor','virginica'});`

`Mdl` is a trained `ClassificationNaiveBayes` classifier.

Determine how well the algorithm generalizes by estimating the test sample logit loss.

`L = loss(Mdl,XTest,YTest,'LossFun','logit')`
```L = 0.3359 ```

The logit loss is approximately 0.34.

## Input Arguments

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Naive Bayes classification model, specified as a `ClassificationNaiveBayes` model object or `CompactClassificationNaiveBayes` model object returned by `fitcnb` or `compact`, respectively.

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. `tbl` must contain all the predictors used to train `Mdl`. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed. Optionally, `tbl` can contain additional columns for the response variable and observation weights.

If you train `Mdl` using sample data contained in a table, then the input data for `loss` must also be in a table.

Response variable name, specified as the name of a variable in `tbl`.

You must specify `ResponseVarName` as a character vector or string scalar. For example, if the response variable `y` is stored as `tbl.y`, then specify it as `'y'`. Otherwise, the software treats all columns of `tbl`, including `y`, as predictors.

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

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` corresponds to one observation (also known as an instance or example), and each column corresponds to one variable (also known as a feature). The variables in the columns of `X` must be the same as the variables that trained the `Mdl` classifier.

The length of `Y` and the number of rows of `X` must be equal.

Data Types: `double` | `single`

Class labels, specified as a categorical, character, or string array, logical or numeric vector, or cell array of character vectors. `Y` must have the same data type as `Mdl.ClassNames`. (The software treats string arrays as cell arrays of character vectors.)

The length of `Y` must be equal to the number of rows of `tbl` or `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,Y,'Weights',W)` weighs the observations in each row of `tbl` using the corresponding weight in each row of the variable `W`.

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

• The following table lists the available loss functions. Specify one using its corresponding character vector or string scalar.

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. Naive Bayes models return posterior probabilities as classification scores by default (see `predict`).

• Specify your own function using function handle notation.

Suppose that `n` is the number of observations in `X` and `K` is the number of distinct classes (`numel(Mdl.ClassNames)`, where `Mdl` is the input model). Your function must have this signature

``lossvalue = lossfun(C,S,W,Cost)``
where:

• The output argument `lossvalue` is a scalar.

• You specify the function name (`lossfun`).

• `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`.

Create `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 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.

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

For more details on loss functions, see Classification Loss.

Data Types: `char` | `string` | `function_handle`

Observation weights, specified as a numeric vector or the name of a variable in `tbl`. The software weighs the observations in each row of `X` or `tbl` with the corresponding weights in `Weights`.

If you specify `Weights` as a numeric 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`, then 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.

If you do not specify a loss function, then the software normalizes `Weights` to add up to `1`.

Data Types: `double` | `char` | `string`

## Output Arguments

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Classification loss, returned as a scalar. `L` is a generalization or resubstitution quality measure. Its interpretation depends on the loss function and weighting scheme; in general, better classifiers yield smaller loss values.

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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 y2* = [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 `LossFun`Equation
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'`

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

The classification error 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'`

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 stored by the input model 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].

### Misclassification Cost

A misclassification cost is the relative severity of a classifier labeling an observation into the wrong class.

There are two types of misclassification costs: true and expected. Let K be the number of classes.

• True misclassification cost — A K-by-K matrix, where element (i,j) indicates the misclassification cost of predicting an observation into class j if its true class is i. The software stores the misclassification cost in the property `Mdl.Cost`, and uses it in computations. By default, `Mdl.Cost(i,j)` = 1 if `i``j`, and `Mdl.Cost(i,j)` = 0 if `i` = `j`. In other words, the cost is `0` for correct classification and `1` for any incorrect classification.

• Expected misclassification cost — A K-dimensional vector, where element k is the weighted average misclassification cost of classifying an observation into class k, weighted by the class posterior probabilities.

`${c}_{k}=\sum _{j=1}^{K}\stackrel{^}{P}\left(Y=j|{x}_{1},...,{x}_{P}\right)Cos{t}_{jk}.$`

In other words, the software classifies observations to the class corresponding with the lowest expected misclassification cost.

### Posterior Probability

The posterior probability is the probability that an observation belongs in a particular class, given the data.

For naive Bayes, the posterior probability that a classification is k for a given observation (x1,...,xP) is

`$\stackrel{^}{P}\left(Y=k|{x}_{1},..,{x}_{P}\right)=\frac{P\left({X}_{1},...,{X}_{P}|y=k\right)\pi \left(Y=k\right)}{P\left({X}_{1},...,{X}_{P}\right)},$`

where:

• $P\left({X}_{1},...,{X}_{P}|y=k\right)$ is the conditional joint density of the predictors given they are in class k. `Mdl.DistributionNames` stores the distribution names of the predictors.

• π(Y = k) is the class prior probability distribution. `Mdl.Prior` stores the prior distribution.

• $P\left({X}_{1},..,{X}_{P}\right)$ is the joint density of the predictors. The classes are discrete, so $P\left({X}_{1},...,{X}_{P}\right)=\sum _{k=1}^{K}P\left({X}_{1},...,{X}_{P}|y=k\right)\pi \left(Y=k\right).$

### Prior Probability

The prior probability of a class is the assumed relative frequency with which observations from that class occur in a population.