loss
Classification loss for naive Bayes classifier
Description
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
)
Examples
Determine Test Sample Classification Loss of Naive Bayes Classifier
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
.
Determine Test Sample Logit Loss of Naive Bayes Classifier
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
Mdl
— Naive Bayes classification model
ClassificationNaiveBayes
model object | CompactClassificationNaiveBayes
model object
Naive Bayes classification model, specified as a ClassificationNaiveBayes
model object or CompactClassificationNaiveBayes
model object returned by fitcnb
or compact
,
respectively.
tbl
— Sample data
table
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 name
name 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 data
numeric 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 labels
categorical 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
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
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
.
LossFun
— Loss function
'mincost'
(default) | 'binodeviance'
| 'classifcost'
| 'classiferror'
| 'exponential'
| 'hinge'
| 'logit'
| 'quadratic'
| function handle
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.
Value Description 'binodeviance'
Binomial deviance 'classifcost'
Observed misclassification cost '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 (seepredict
).Specify your own function using function handle notation.
Suppose that
n
is the number of observations inX
andK
is the number of distinct classes (numel(Mdl.ClassNames)
, whereMdl
is the input model). Your function must have this signaturewhere:lossvalue =
lossfun
(C,S,W,Cost)The output argument
lossvalue
is a scalar.You specify the function name (
lossfun
).C
is ann
-by-K
logical matrix with rows indicating the class to which the corresponding observation belongs. The column order corresponds to the class order inMdl.ClassNames
.Create
C
by settingC(p,q) = 1
if observationp
is in classq
, for each row. Set all other elements of rowp
to0
.S
is ann
-by-K
numeric matrix of classification scores. The column order corresponds to the class order inMdl.ClassNames
.S
is a matrix of classification scores, similar to the output ofpredict
.W
is ann
-by-1 numeric vector of observation weights. If you passW
, the software normalizes the weights to sum to1
.Cost
is aK
-by-K
numeric matrix of misclassification costs. For example,Cost = ones(K) - eye(K)
specifies a cost of0
for correct classification and1
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
Output Arguments
L
— Classification loss
scalar
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.
More About
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:
y_{j} 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
ClassNames
property), respectively.f(X_{j}) is the positive-class classification score for observation (row) j of the predictor data X.
m_{j} = y_{j}f(X_{j}) is the classification score for classifying observation j into the class corresponding to y_{j}. Positive values of m_{j} indicate correct classification and do not contribute much to the average loss. Negative values of m_{j} indicate incorrect classification and contribute significantly to the average loss.
For algorithms that support multiclass classification (that is, K ≥ 3):
y_{j}^{*} is a vector of K – 1 zeros, with 1 in the position corresponding to the true, observed class y_{j}. 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 theClassNames
property of the input model.f(X_{j}) 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.m_{j} = y_{j}^{*}′f(X_{j}). Therefore, m_{j} is the scalar classification score that the model predicts for the true, observed class.
The weight for observation j is w_{j}. The software normalizes the observation weights so that they sum to the corresponding prior class probability stored in the
Prior
property. 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 argument.
Loss Function | Value of LossFun | Equation |
---|---|---|
Binomial deviance | 'binodeviance' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}}.$$ |
Observed misclassification cost | 'classifcost' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}{c}_{{y}_{j}{\widehat{y}}_{j}},$$ where $${\widehat{y}}_{j}$$ is the class label corresponding to the class with the maximal score, and $${c}_{{y}_{j}{\widehat{y}}_{j}}$$ is the user-specified cost of classifying an observation into class $${\widehat{y}}_{j}$$ when its true class is y_{j}. |
Misclassified rate in decimal | 'classiferror' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}I\left\{{\widehat{y}}_{j}\ne {y}_{j}\right\},$$ where I{·} is the indicator function. |
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 n instead of 1. |
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 j = 1,...,n.
The weighted average of the minimal expected misclassification cost loss is $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{c}_{j}}.$$ |
Quadratic loss | 'quadratic' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{\left(1-{m}_{j}\right)}^{2}}.$$ |
If you use the default cost matrix (whose element value is 0 for correct classification
and 1 for incorrect classification), then the loss values for
'classifcost'
, 'classiferror'
, and
'mincost'
are identical. For a model with a nondefault cost matrix,
the 'classifcost'
loss is equivalent to the 'mincost'
loss most of the time. These losses can be different if prediction into the class with
maximal posterior probability is different from prediction into the class with minimal
expected cost. Note that 'mincost'
is appropriate only if classification
scores are posterior probabilities.
This figure compares the loss functions (except 'classifcost'
,
'crossentropy'
, and 'mincost'
) over the score
m for one observation. Some functions are normalized to pass through
the point (0,1).
Misclassification Cost
A misclassification cost is the relative severity of a classifier labeling an observation into the wrong class.
Two types of misclassification cost exist: true and expected. Let K be the number of classes.
True misclassification cost — A K-by-K matrix, where element (i,j) indicates the cost of classifying 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 ifi
≠j
, andMdl.Cost(i,j)
= 0 ifi
=j
. In other words, the cost is0
for correct classification and1
for any incorrect classification.Expected misclassification cost — A K-dimensional vector, where element k is the weighted average 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 into the class 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 (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 $$P({X}_{1},\mathrm{...},{X}_{P})={\displaystyle \sum _{k=1}^{K}P}({X}_{1},\mathrm{...},{X}_{P}|y=k)\pi (Y=k).$$
Prior Probability
The prior probability of a class is the assumed relative frequency with which observations from that class occur in a population.
Extended Capabilities
Tall Arrays
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.
Version History
See Also
ClassificationNaiveBayes
| CompactClassificationNaiveBayes
| predict
| fitcnb
| resubLoss
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