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 modelClassificationNaiveBayes
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 dataSample 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 nametbl
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 dataPredictor 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 labelsClass 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'classiferror'
(default) | 'binodeviance'
| 'exponential'
| 'hinge'
| 'logit'
| 'mincost'
| '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' | 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)
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 weightsones(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 lossClassification 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_{j} is the observed class label. The software codes it as –1 or 1, indicating the negative or positive class, respectively.
f(X_{j}) is the raw 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
the ClassNames
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. 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 Function | Value of LossFun | Equation |
---|---|---|
Binomial deviance | 'binodeviance' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}}.$$ |
Exponential loss | 'exponential' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right)}.$$ |
Classification error | 'classiferror' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}I\left\{{\widehat{y}}_{j}\ne {y}_{j}\right\}.$$ The classification error is the weighted fraction of misclassified observations where $${\widehat{y}}_{j}$$ is the class label corresponding to the class with the maximal posterior probability. I{x} is the indicator function. |
Hinge loss | 'hinge' | $$L={\displaystyle \sum}_{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.$$ |
Logit loss | 'logit' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right)}.$$ |
Minimal cost | 'mincost' | The software computes the weighted minimal cost using this procedure for observations j = 1,...,n.
The weighted, average, minimum cost loss is $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{c}_{j}}.$$ |
Quadratic loss | 'quadratic' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{\left(1-{m}_{j}\right)}^{2}}.$$ |
This figure compares the loss functions (except 'mincost'
) for one
observation over m. Some functions are normalized to pass through [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.
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
A modified version of this example exists on your system. Do you want to open this version instead?
You clicked a link that corresponds to this MATLAB command:
Run the command by entering it in the MATLAB Command Window. Web browsers do not support MATLAB commands.
Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select: .
Select web siteYou can also select a web site from the following list:
Select the China site (in Chinese or English) for best site performance. Other MathWorks country sites are not optimized for visits from your location.