loss
Loss of naive Bayes incremental learning classification model on batch of data
Description
loss
returns the classification loss of a configured
naive Bayes classification model for incremental learning (incrementalClassificationNaiveBayes
object).
To measure model performance on a data stream and store the results in the output model,
call updateMetrics
or updateMetricsAndFit
.
Examples
Measure Model Performance During Incremental Learning
Three different ways to measure performance of an incremental model on streaming data exist:
Cumulative metrics measure the performance since the start of incremental learning.
Window metrics measure the performance on a specified window of observations. The metrics are updated every time the model processes the specified window.
The
loss
function measures the performance on a specified batch of data only.
Load the human activity data set. Randomly shuffle the data.
load humanactivity n = numel(actid); rng(1) % For reproducibility idx = randsample(n,n); X = feat(idx,:); Y = actid(idx);
For details on the data set, enter Description
at the command line.
Create a naive Bayes classification model for incremental learning. Specify the class names and a metrics window size of 1000 observations. Configure the model for loss
by fitting it to the first 10 observations.
Mdl = incrementalClassificationNaiveBayes('ClassNames',unique(Y),'MetricsWindowSize',1000); initobs = 10; Mdl = fit(Mdl,X(1:initobs,:),Y(1:initobs)); canComputeLoss = (size(Mdl.DistributionParameters,2) == Mdl.NumPredictors) + ... (size(Mdl.DistributionParameters,1) > 1) > 1
canComputeLoss = logical
1
Mdl
is an incrementalClassificationNaiveBayes
model. All its properties are readonly.
Simulate a data stream, and perform the following actions on each incoming chunk of 500 observations:
Call
updateMetrics
to measure the cumulative performance and the performance within a window of observations. Overwrite the previous incremental model with a new one to track performance metrics.Call
loss
to measure the model performance on the incoming chunk.Call
fit
to fit the model to the incoming chunk. Overwrite the previous incremental model with a new one fitted to the incoming observations.Store all performance metrics to see how they evolve during incremental learning.
% Preallocation numObsPerChunk = 500; nchunk = floor((n  initobs)/numObsPerChunk); mc = array2table(zeros(nchunk,3),'VariableNames',["Cumulative" "Window" "Chunk"]); % Incremental learning for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j1) + 1 + initobs); iend = min(n,numObsPerChunk*j + initobs); idx = ibegin:iend; Mdl = updateMetrics(Mdl,X(idx,:),Y(idx)); mc{j,["Cumulative" "Window"]} = Mdl.Metrics{"MinimalCost",:}; mc{j,"Chunk"} = loss(Mdl,X(idx,:),Y(idx)); Mdl = fit(Mdl,X(idx,:),Y(idx)); end
Now, Mdl
is an incrementalClassificationNaiveBayes
model object trained on all the data in the stream. During incremental learning and after the model is warmed up, updateMetrics
checks the performance of the model on the incoming observations, and then the fit
function fits the model to those observations. loss
is agnostic of the metrics warmup period, so it measures the minimal cost for every chunk.
To see how the performance metrics evolve during training, plot them.
figure plot(mc.Variables) xlim([0 nchunk]) ylim([0 0.1]) ylabel('Minimal Cost') xline(Mdl.MetricsWarmupPeriod/numObsPerChunk + 1,'r.') legend(mc.Properties.VariableNames) xlabel('Iteration')
The yellow line represents the minimal cost on each incoming chunk of data. After the metrics warmup period, Mdl
tracks the cumulative and window metrics.
Compute Custom Loss on Incoming Chunks of Data
Fit a naive Bayes classification model for incremental learning to streaming data, and compute the multiclass cross entropy loss on the incoming chunks of data.
Load the human activity data set. Randomly shuffle the data.
load humanactivity n = numel(actid); rng(1); % For reproducibility idx = randsample(n,n); X = feat(idx,:); Y = actid(idx);
For details on the data set, enter Description
at the command line.
Create a naive Bayes classification model for incremental learning. Configure the model as follows:
Specify the class names.
Specify a metrics warmup period of 1000 observations.
Specify a metrics window size of 2000 observations.
Track the multiclass cross entropy loss to measure the performance of the model. Create an anonymous function that measures the multiclass cross entropy loss of each new observation, and include a tolerance for numerical stability. Create a structure array containing the name
CrossEntropy
and its corresponding function handle.Compute the classification loss by fitting the model to the first 10 observations.
tolerance = 1e10; crossentropy = @(z,zfit,w,cost)log(max(zfit(z),tolerance)); ce = struct("CrossEntropy",crossentropy); Mdl = incrementalClassificationNaiveBayes('ClassNames',unique(Y),'MetricsWarmupPeriod',1000, ... 'MetricsWindowSize',2000,'Metrics',ce); initobs = 10; Mdl = fit(Mdl,X(1:initobs,:),Y(1:initobs));
Mdl
is an incrementalClassificationNaiveBayes
model object configured for incremental learning.
Perform incremental learning. At each iteration:
Simulate a data stream by processing a chunk of 100 observations.
Call
updateMetrics
to compute cumulative and window metrics on the incoming chunk of data. Overwrite the previous incremental model with a new one fitted to overwrite the previous metrics.Call
loss
to compute the cross entropy on the incoming chunk of data. Whereas the cumulative and window metrics require that custom losses return the loss for each observation,loss
requires the loss for the entire chunk. Compute the mean of the losses within a chunk.Call
fit
to fit the incremental model to the incoming chunk of data.Store the cumulative, window, and chunk metrics to see how they evolve during incremental learning.
% Preallocation numObsPerChunk = 100; nchunk = floor((n  initobs)/numObsPerChunk); tanloss = array2table(zeros(nchunk,3),'VariableNames',["Cumulative" "Window" "Chunk"]); % Incremental fitting for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j1) + 1 + initobs); iend = min(n,numObsPerChunk*j + initobs); idx = ibegin:iend; Mdl = updateMetrics(Mdl,X(idx,:),Y(idx)); tanloss{j,1:2} = Mdl.Metrics{"CrossEntropy",:}; tanloss{j,3} = loss(Mdl,X(idx,:),Y(idx),'LossFun',@(z,zfit,w,cost)mean(crossentropy(z,zfit,w,cost))); Mdl = fit(Mdl,X(idx,:),Y(idx)); end
Mdl
is an incrementalClassificationNaiveBayes
model object trained on all the data in the stream. During incremental learning and after the model is warmed up, updateMetrics
checks the performance of the model on the incoming observations, and then the fit
function fits the model to those observations.
Plot the performance metrics to see how they evolve during incremental learning.
figure h = plot(tanloss.Variables); xlim([0 nchunk]) ylabel('Cross Entropy') xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'r.') xlabel('Iteration') legend(h,tanloss.Properties.VariableNames)
The plot suggests the following:
updateMetrics
computes the performance metrics after the metrics warmup period only.updateMetrics
computes the cumulative metrics during each iteration.updateMetrics
computes the window metrics after processing 2000 observations (20 iterations).Because
Mdl
is configured to predict observations from the beginning of incremental learning,loss
can compute the cross entropy on each incoming chunk of data.
Input Arguments
Mdl
— Naive Bayes classification model for incremental learning
incrementalClassificationNaiveBayes
model object
Naive Bayes classification model for incremental learning, specified as an incrementalClassificationNaiveBayes
model object. You can create Mdl
directly or by converting a supported, traditionally trained machine learning model using the incrementalLearner
function. For more details, see the corresponding reference page.
You must configure Mdl
to compute its loss on a batch of observations.
If
Mdl
is a converted, traditionally trained model, you can compute its loss without any modifications.Otherwise, you must fit the input model
Mdl
to data that contains all expected classes. That is,Mdl.DistributionParameters
must be a cell matrix withMdl.NumPredictors
columns and at least one row, where each row corresponds to each class name inMdl.ClassNames
.
X
— Batch of predictor data
floatingpoint matrix
Batch of predictor data with which to compute the loss, specified as an
nbyMdl.NumPredictors
floatingpoint
matrix.
The length of the observation labels Y
and the number of
observations in X
must be equal;
Y(
is the label of observation
j (row) in j
)X
.
Data Types: single
 double
Y
— Batch of labels
categorical array  character array  string array  logical vector  floatingpoint vector  cell array of character vectors
Batch of labels with which to compute the loss, specified as a categorical, character, or string array; logical or floatingpoint vector; or cell array of character vectors.
The length of the observation labels Y
and the number of
observations in X
must be equal;
Y(
is the label of observation
j (row) in j
)X
.
If Y
contains a label that is not a member of
Mdl.ClassNames
, loss
issues an error.
The data type of Y
and Mdl.ClassNames
must be the
same.
Data Types: char
 string
 cell
 categorical
 logical
 single
 double
NameValue Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Namevalue 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: 'LossFun','classiferror','Weights',W
specifies returning the misclassification error rate, and the observation weights W
.
Cost
— Cost of misclassifying an observation
Mdl.Cost
(default)  square matrix  structure array
Cost of misclassifying an observation, specified as a value in the table, where
c is the number of classes in Mdl.ClassNames
.
The specified value overrides the value of Mdl.Cost
.
Value  Description 

cbyc numeric matrix 

Structure array  A structure array having two fields:

Example: Cost=struct('ClassNames',Mdl.ClassNames,'ClassificationCosts',[0 2; 1 0])
Data Types: single
 double
 struct
LossFun
— Loss function
'mincost'
(default)  string vector  function handle  cell vector  structure array  ...
Loss function, specified as a builtin loss function name or function handle.
The following table lists the builtin loss function names. You can specify more than one by using a string vector.
Name  Description 

"binodeviance"  Binomial deviance 
"classiferror"  Misclassification error rate 
"exponential"  Exponential 
"hinge"  Hinge 
"logit"  Logistic 
"mincost"  Minimal expected misclassification cost 
"quadratic"  Quadratic 
For more details, see Classification Loss.
To specify a custom loss function, use function handle notation. The function must have this form:
lossval = lossfcn(C,S,W,Cost)
The output argument
lossval
is an nby1 floatingpoint vector, where n is the number of observations inX
. The value inlossval(
is the classification loss of observationj
)
.j
You specify the function name (
).lossfcn
C
is an nbyK logical matrix with rows indicating the class to which the corresponding observation belongs.K
is the number of distinct classes (numel(Mdl.ClassNames)
, and the column order corresponds to the class order in theClassNames
property. CreateC
by settingC(
=p
,q
)1
, if observation
is in classp
, for each observation in the specified data. Set the other element in rowq
top
0
.S
is an nbyK numeric matrix of predicted classification scores.S
is similar to thePosterior
output ofpredict
, where rows correspond to observations in the data and the column order corresponds to the class order in theClassNames
property.S(
is the classification score of observationp
,q
)
being classified in classp
.q
W
is an nby1 numeric vector of observation weights.Cost
is a KbyK numeric matrix of misclassification costs.
Example: 'LossFun',"classiferror"
Example: 'LossFun',@
lossfcn
Data Types: char
 string
 function_handle
Prior
— Prior class probabilities
Mdl.Prior
(default)  numeric vector
Prior class probabilities, specified as a value in this numeric vector. Prior
has the same length as the number of classes in Mdl.ClassNames
, and the order of the elements corresponds to the class order in Mdl.ClassNames
. loss
normalizes the vector so that the sum of the result is 1.
The specified value overrides the value of Mdl.Prior
.
Data Types: single
 double
ScoreTransform
— Score transformation function
Mdl.ScoreTransform
(default)  string scalar  character vector
Score transformation function describing how incremental learning functions transform raw response values, specified as a character vector, string scalar, or function handle. The specified value overrides the value of Mdl.ScoreTransform
.
This table describes the available builtin functions for score transformation.
Value  Description 

"doublelogit"  1/(1 + e^{–2x}) 
"invlogit"  log(x / (1 – x)) 
"ismax"  Sets the score for the class with the largest score to 1, and sets the scores for all other classes to 0 
"logit"  1/(1 + e^{–x}) 
"none" or "identity"  x (no transformation) 
"sign"  –1 for x < 0 0 for x = 0 1 for x > 0 
"symmetric"  2x – 1 
"symmetricismax"  Sets the score for the class with the largest score to 1, and sets the scores for all other classes to –1 
"symmetriclogit"  2/(1 + e^{–x}) – 1 
Data Types: char
 string
Weights
— Chunk of observation weights
floatingpoint vector of positive values
Chunk of observation weights, specified as a floatingpoint vector of positive values.
loss
weighs the observations in X
with the corresponding values in Weights
. The size of
Weights
must equal n, the number of
observations in X
.
By default, Weights
is ones(
.n
,1)
For more details, including normalization schemes, see Observation Weights.
Data Types: double
 single
Output Arguments
L
— Classification loss
numeric scalar
Classification loss, returned as a numeric scalar. L
is a measure of model quality. Its interpretation depends on the loss function and weighting scheme.
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 positiveclass 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
namevalue 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 userspecified 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. 
Crossentropy loss  'crossentropy' 
The weighted crossentropy 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).
Algorithms
Observation Weights
For each conditional predictor distribution, loss
computes the weighted average and standard deviation.
If the prior class probability distribution is known (in other words, the prior distribution is not empirical), loss
normalizes observation weights to sum to the prior class probabilities in the respective classes. This action implies that the default observation weights are the respective prior class probabilities.
If the prior class probability distribution is empirical, the software normalizes the specified observation weights to sum to 1 each time you call loss
.
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