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Classification loss for naive Bayes classifier

returns the Classification Loss, a scalar representing how well the trained naive
Bayes classifier `L`

= loss(`Mdl`

,`tbl`

,`ResponseVarName`

)`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`

.

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.`L`

= loss(___,`Name,Value`

)

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.

`Mdl`

— Naive Bayes classification model`ClassificationNaiveBayes`

model object | `CompactClassificationNaiveBayes`

model objectNaive Bayes classification model, specified as a `ClassificationNaiveBayes`

model object or `CompactClassificationNaiveBayes`

model object returned by `fitcnb`

or `compact`

,
respectively.

`tbl`

— Sample datatable

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.

`ResponseVarName`

— Response variable namename of a variable in

`tbl`

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`

`X`

— Predictor datanumeric matrix

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`

`Y`

— Class labelscategorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

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`

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`

.

`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`

.`'LossFun'`

— Loss function`'mincost'`

(default) | `'binodeviance'`

| `'classiferror'`

| `'exponential'`

| `'hinge'`

| `'logit'`

| `'quadratic'`

| function handleLoss 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.

Value Description `'binodeviance'`

Binomial deviance `'classiferror'`

Misclassified rate in decimal `'exponential'`

Exponential loss `'hinge'`

Hinge loss `'logit'`

Logistic loss `'mincost'`

Minimal expected misclassification cost (for classification scores that are posterior probabilities) `'quadratic'`

Quadratic loss `'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 signaturewhere:`lossvalue =`

(C,S,W,Cost)`lossfun`

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`

`'Weights'`

— Observation weights`ones(size(X,1),1)`

(default) | numeric vector | name of a variable in `tbl`

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`

`L`

— Classification lossscalar

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.

*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:

*y*is the observed class label. The software codes it as –1 or 1, indicating the negative or positive class (or the first or second class in the_{j}`ClassNames`

property), respectively.*f*(*X*) is the positive-class classification score for observation (row)_{j}*j*of the predictor data*X*.*m*=_{j}*y*_{j}*f*(*X*) is the classification score for classifying observation_{j}*j*into the class corresponding to*y*. Positive values of_{j}*m*indicate correct classification and do not contribute much to the average loss. Negative values of_{j}*m*indicate incorrect classification and contribute significantly to the average loss._{j}

For algorithms that support multiclass classification (that is,

*K*≥ 3):*y*is a vector of_{j}^{*}*K*– 1 zeros, with 1 in the position corresponding to the true, observed class*y*. For example, if the true class of the second observation is the third class and_{j}*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*(*X*) is the length_{j}*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.*m*=_{j}*y*_{j}^{*}′*f*(*X*). Therefore,_{j}*m*is the scalar classification score that the model predicts for the true, observed class._{j}

The weight for observation

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

Given this scenario, the following table describes the supported loss
functions that you can specify by using the `'LossFun'`

name-value pair
argument.

Loss Function | Value of `LossFun` | Equation |
---|---|---|

Binomial deviance | `'binodeviance'` | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}}.$$ |

Misclassified rate in decimal | `'classiferror'` | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}I\left\{{\widehat{y}}_{j}\ne {y}_{j}\right\}.$$ $${\widehat{y}}_{j}$$ is the class label corresponding to the class with the
maximal score. |

Cross-entropy loss | `'crossentropy'` |
The weighted cross-entropy loss is $$L=-{\displaystyle \sum _{j=1}^{n}\frac{{\tilde{w}}_{j}\mathrm{log}({m}_{j})}{Kn}},$$ where the weights $${\tilde{w}}_{j}$$ are normalized to sum to |

Exponential loss | `'exponential'` | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right)}.$$ |

Hinge loss | `'hinge'` | $$L={\displaystyle \sum}_{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.$$ |

Logit loss | `'logit'` | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right)}.$$ |

Minimal expected misclassification cost | `'mincost'` |
The software computes
the weighted minimal expected classification cost using this procedure
for observations Estimate the expected misclassification cost of classifying the observation *X*into the class_{j}*k*:$${\gamma}_{jk}={\left(f{\left({X}_{j}\right)}^{\prime}C\right)}_{k}.$$ *f*(*X*) is the column vector of class posterior probabilities for binary and multiclass classification for the observation_{j}*X*._{j}*C*is the cost matrix stored in the`Cost` property of the model.For observation *j*, predict the class label corresponding to the minimal expected misclassification cost:$${\widehat{y}}_{j}=\underset{k=1,\mathrm{...},K}{\text{argmin}}{\gamma}_{jk}.$$ Using *C*, identify the cost incurred (*c*) for making the prediction._{j}
The weighted average of the minimal expected misclassification cost loss is $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{c}_{j}}.$$ If you use the default cost matrix (whose element
value is 0 for correct classification and 1 for incorrect
classification), then the |

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

This figure compares the loss functions (except `'crossentropy'`

and
`'mincost'`

) over the score *m* for one observation.
Some functions are normalized to pass through the point (0,1).

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

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

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 (*x*_{1},...,*x _{P}*)
is

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

where:

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

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

Calculate with arrays that have more rows than fit in memory.

This function fully supports tall arrays. You can use models trained on either in-memory or tall data with this function.

For more information, see Tall Arrays.

`ClassificationNaiveBayes`

| `CompactClassificationNaiveBayes`

| `fitcnb`

| `predict`

| `resubLoss`

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