incrementalClassificationKernel
Description
The incrementalClassificationKernel
function creates an
incrementalClassificationKernel
model object, which represents a binary Gaussian kernel
classification model for incremental learning. The kernel model maps data in a lowdimensional
space into a highdimensional space, then fits a linear model in the highdimensional space.
Supported linear models include support vector machine (SVM) and logistic
regression.
Unlike other Statistics and Machine Learning Toolbox™ model objects, incrementalClassificationKernel
can be called directly. Also,
you can specify learning options, such as performance metrics configurations and the objective
solver, before fitting the model to data. After you create an incrementalClassificationKernel
object, it is prepared for incremental learning.
incrementalClassificationKernel
is best suited for incremental learning. For a traditional
approach to training a kernel model for binary classification (such as creating a model by
fitting it to data, performing crossvalidation, tuning hyperparameters, and so on), see
fitckernel
. For
multiclass incremental learning, see incrementalClassificationECOC
and incrementalClassificationNaiveBayes
.
Creation
You can create an incrementalClassificationKernel
model object in several ways:
Call the function directly — Configure incremental learning options, or specify learnerspecific options, by calling
incrementalClassificationKernel
directly. This approach is best when you do not have data yet or you want to start incremental learning immediately.Convert a traditionally trained model — To initialize a model for incremental learning using the model parameters and hyperparameters of a trained model object, you can convert the traditionally trained model (
ClassificationKernel
) to anincrementalClassificationKernel
model object by passing it to theincrementalLearner
function.Convert a template object — You can convert a template object (
templateKernel
) to anincrementalClassificationKernel
model object by passing it to theincrementalLearner
function.Call an incremental learning function —
fit
,updateMetrics
, andupdateMetricsAndFit
accept a configuredincrementalClassificationKernel
model object and data as input, and return anincrementalClassificationKernel
model object updated with information learned from the input model and data.
Description
returns a default incremental learning model object for binary Gaussian kernel
classification, Mdl
= incrementalClassificationKernel()Mdl
. Properties of a default model contain
placeholders for unknown model parameters. You must train a default model before you can
track its performance or generate predictions from it.
sets properties and additional
options using namevalue arguments. For example,
Mdl
= incrementalClassificationKernel(Name=Value
)incrementalClassificationKernel(Solver="sgd",LearnRateSchedule="constant")
specifies to use the stochastic gradient descent (SGD) solver with a constant learning
rate.
Input 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.
Example: Metrics="logit",MetricsWarmupPeriod=100
sets the model
performance metric to the logistic loss and the metrics warmup period to
100
.
RandomStream
— Random number stream
global stream (default)  random stream object
Random number stream for reproducibility of data transformation, specified as a random stream object. For details, see Random Feature Expansion.
Use RandomStream
to reproduce the random basis functions used by
incrementalClassificationKernel
to transform the predictor data to a
highdimensional space. For details, see Managing the Global Stream Using RandStream
and Creating and Controlling a Random Number Stream.
Example: RandomStream=RandStream("mlfg6331_64")
Standardize
— Flag to standardize predictor data
"auto"
(default)  false
 true
Since R2023b
Flag to standardize the predictor data, specified as a value in this table.
Value  Description 

"auto"  incrementalClassificationKernel determines whether the predictor
variables need to be standardized. See Standardize Data. 
true  The software standardizes the predictor data. For more details, see Standardize Data. 
false  The software does not standardize the predictor data. 
Example: Standardize=true
Data Types: logical
 char
 string
BatchSize
— Minibatch size
10
(default)  positive integer
Minibatch size, specified as a positive integer. At each learning cycle during
training, incrementalClassificationKernel
uses BatchSize
observations to compute the subgradient.
The number of observations in the last minibatch (last learning cycle in each
function call of fit
or
updateMetricsAndFit
) can be smaller than BatchSize
.
For example, if you supply 25 observations to fit
or
updateMetricsAndFit
, the function uses 10 observations for the
first two learning cycles and 5 observations for the last learning cycle.
Example: BatchSize=5
Data Types: single
 double
Lambda
— Ridge (L2) regularization term strength
1e5
(default)  nonnegative scalar
Ridge (L2) regularization term strength, specified as a nonnegative scalar.
Example: Lambda=0.01
Data Types: single
 double
LearnRate
— Initial learning rate
"auto"
(default)  positive scalar
Initial learning rate, specified as "auto"
or a positive
scalar.
The learning rate controls the optimization step size by scaling the objective
subgradient. LearnRate
specifies an initial value for the learning
rate, and LearnRateSchedule
determines the learning rate for subsequent learning cycles.
When you specify "auto"
:
The initial learning rate is
0.7
.If
EstimationPeriod
>0
,fit
andupdateMetricsAndFit
change the rate to1/sqrt(1+max(sum(X.^2,2)))
at the end ofEstimationPeriod
.
Example: LearnRate=0.001
Data Types: single
 double
 char
 string
LearnRateSchedule
— Learning rate schedule
"decaying"
(default)  "constant"
Learning rate schedule, specified as a value in this table, where LearnRate
specifies the initial
learning rate ɣ_{0}.
Value  Description 

"constant"  The learning rate is ɣ_{0} for all learning cycles. 
"decaying"  The learning rate at learning cycle t is $${\gamma}_{t}=\frac{{\gamma}_{0}}{{\left(1+\lambda {\gamma}_{0}t\right)}^{c}}.$$

Example: LearnRateSchedule="constant"
Data Types: char
 string
Shuffle
— Flag for shuffling observations
true
or 1
(default)  false
or 0
Flag for shuffling the observations at each iteration, specified as logical
1
(true
) or 0
(false
).
Value  Description 

logical 1 (true )  The software shuffles the observations in an incoming chunk of
data before the fit function fits the model. This
action reduces bias induced by the sampling scheme. 
logical 0 (false )  The software processes the data in the order received. 
Example: Shuffle=false
Data Types: logical
Metrics
— Model performance metrics to track during incremental learning
"classiferror"
(default)  string vector  function handle  cell vector  structure array  "binodeviance"
 "exponential"
 "hinge"
 "logit"
 "quadratic"
Model performance metrics to track during incremental learning, specified as a
builtin loss function name, string vector of names, function handle
(@metricName
), structure array of function handles, or cell
vector of names, function handles, or structure arrays.
When Mdl
is warm (see IsWarm
), updateMetrics
and updateMetricsAndFit
track performance metrics in the Metrics
property of
Mdl
.
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"  Classification error 
"exponential"  Exponential loss 
"hinge"  Hinge loss 
"logit"  Logistic loss 
"quadratic"  Quadratic loss 
For more details on the builtin loss functions, see loss
.
Example: Metrics=["classiferror","hinge"]
To specify a custom function that returns a performance metric, use function handle notation. The function must have this form:
metric = customMetric(C,S)
The output argument
metric
is an nby1
numeric vector, where each element is the loss of the corresponding observation in the data processed by the incremental learning functions during a learning cycle.You specify the function name (
customMetric
).C
is an nby2
logical matrix with rows indicating the class to which the corresponding observation belongs. 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 nby2
numeric matrix of predicted classification scores.S
is similar to thescore
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
To specify multiple custom metrics and assign a custom name to each, use a structure array. To specify a combination of builtin and custom metrics, use a cell vector.
Example: Metrics=struct(Metric1=@customMetric1,Metric2=@customMetric2)
Example: Metrics={@customMetric1,@customMetric2,"logit",struct(Metric3=@customMetric3)}
updateMetrics
and updateMetricsAndFit
store specified metrics in a table in the Metrics
property. The
data type of Metrics
determines the row names of the
table.
Metrics Value Data Type  Description of Metrics Property Row Name  Example 

String or character vector  Name of corresponding builtin metric  Row name for "classiferror" is
"ClassificationError" 
Structure array  Field name  Row name for struct(Metric1=@customMetric1) is
"Metric1" 
Function handle to function stored in a program file  Name of function  Row name for @customMetric is
"customMetric" 
Anonymous function  CustomMetric_ , where
is metric
in
Metrics  Row name for @(C,S)customMetric(C,S)... is
CustomMetric_1 
For more details on performance metrics options, see Performance Metrics.
Data Types: char
 string
 struct
 cell
 function_handle
Properties
You can set most properties by using namevalue argument syntax when you call
incrementalClassificationKernel
directly. You can set some properties when you call
incrementalLearner
to convert a traditionally trained model object or
model template object. You cannot set the properties FittedLoss
,
NumTrainingObservations
, SolverOptions
, and
IsWarm
.
Classification Model Parameters
ClassNames
— Unique class labels
categorical array  character array  string array  logical vector  numeric vector  cell array of character vectors
This property is readonly.
Unique class labels used in training the model, specified as a categorical,
character, or string array, a logical or numeric vector, or a cell array of character
vectors. ClassNames
and the response data must have the same data
type. (The software treats string arrays as cell arrays of character
vectors.)
The default ClassNames
value depends on how you create the model:
If you convert a traditionally trained model to create
Mdl
,ClassNames
is specified by the corresponding property of the traditionally trained model.Otherwise, incremental fitting functions infer
ClassNames
during training.
Data Types: single
 double
 logical
 char
 string
 cell
 categorical
FittedLoss
— Loss function used to fit linear model
'hinge'
 'logit'
This property is readonly.
Loss function used to fit the linear model, specified as 'hinge'
or 'logit'
.
Value  Algorithm  Loss Function  Learner Value 

'hinge'  Support vector machine  Hinge: $$\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,1yf\left(x\right)\right]$$  'svm' 
'logit'  Logistic regression  Deviance (logistic): $$\ell \left[y,f\left(x\right)\right]=\mathrm{log}\left\{1+\mathrm{exp}\left[yf\left(x\right)\right]\right\}$$  'logistic' 
KernelScale
— Kernel scale parameter
"auto"
 positive scalar
This property is readonly.
Kernel scale parameter, specified as "auto"
or a positive
scalar. incrementalClassificationKernel
stores the KernelScale
value as a numeric scalar. The software obtains a random basis for feature expansion
by using the kernel scale parameter. For details, see Random Feature Expansion.
If you specify "auto"
when creating the model object, the
software selects an appropriate kernel scale parameter using a heuristic procedure.
This procedure uses subsampling, so estimates can vary from one call to another.
Therefore, to reproduce results, set a random number seed by using rng
before training.
The default KernelScale
value depends on how you create the model:
If you convert a traditionally trained model object or template model object to create
Mdl
,KernelScale
is specified by the corresponding property of the object.Otherwise, the default value is
1
.
Data Types: char
 string
 single
 double
Learner
— Linear classification model type
"svm"
 "logistic"
This property is readonly.
Linear classification model type, specified as "svm"
or
"logistic"
. incrementalClassificationKernel
stores the
Learner
value as a character vector.
In the following table, $$f\left(x\right)=T(x)\beta +b.$$
x is an observation (row vector) from p predictor variables.
$$T(\xb7)$$ is a transformation of an observation (row vector) for feature expansion. T(x) maps x in $${\mathbb{R}}^{p}$$ to a highdimensional space ($${\mathbb{R}}^{m}$$).
β is a vector of coefficients.
b is the scalar bias.
Value  Algorithm  Loss Function  FittedLoss Value 

"svm"  Support vector machine  Hinge loss: $$\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,1yf\left(x\right)\right]$$  'hinge' 
"logistic"  Logistic regression  Deviance (logistic loss): $$\ell \left[y,f\left(x\right)\right]=\mathrm{log}\left\{1+\mathrm{exp}\left[yf\left(x\right)\right]\right\}$$  'logit' 
The default Learner
value depends on how you create the model:
If you convert a traditionally trained model object or template model object to create
Mdl
,Learner
is specified by the corresponding property of the object.Otherwise, the default value is
"svm"
.
Data Types: char
 string
NumExpansionDimensions
— Number of dimensions of expanded space
"auto"
 positive integer
This property is readonly.
Number of dimensions of the expanded space, specified as "auto"
or a positive integer. incrementalClassificationKernel
stores the
NumExpansionDimensions
value as a numeric scalar.
For "auto"
, the software selects the number of dimensions using
2.^ceil(min(log2(p)+5,15))
, where p
is the
number of predictors. For details, see Random Feature Expansion.
The default NumExpansionDimensions
value depends on how you
create the model:
If you convert a traditionally trained model object or template model object to create
Mdl
,NumExpansionDimensions
is specified by the corresponding property of the object.Otherwise, the default value is
"auto"
.
Data Types: char
 string
 single
 double
NumPredictors
— Number of predictor variables
nonnegative numeric scalar
This property is readonly.
Number of predictor variables, specified as a nonnegative numeric scalar.
The default NumPredictors
value depends on how you create the model:
If you convert a traditionally trained model to create
Mdl
,NumPredictors
is specified by the corresponding property of the traditionally trained model.If you create
Mdl
by callingincrementalClassificationKernel
directly, you can specifyNumPredictors
by using namevalue argument syntax.Otherwise, the default value is
0
, and incremental fitting functions inferNumPredictors
from the predictor data during training.
Data Types: double
NumTrainingObservations
— Number of observations fit to incremental model
0
(default)  nonnegative numeric scalar
This property is readonly.
Number of observations fit to the incremental model Mdl
, specified as a nonnegative numeric scalar. NumTrainingObservations
increases when you pass Mdl
and training data to fit
or updateMetricsAndFit
.
Note
If you convert a traditionally trained model to create Mdl
, incrementalClassificationKernel
does not add the number of observations fit to the traditionally trained model to NumTrainingObservations
.
Data Types: double
Prior
— Prior class probabilities
"empirical"
 "uniform"
 numeric vector
This property is readonly.
Prior class probabilities, specified as "empirical"
,
"uniform"
, or a numeric vector. incrementalClassificationKernel
stores the Prior
value as a numeric vector.
Value  Description 

"empirical"  Incremental learning functions infer prior class probabilities from the
observed class relative frequencies in the response data during incremental
training (after the estimation period
EstimationPeriod ). 
"uniform"  For each class, the prior probability is 1/2. 
numeric vector  Custom, normalized prior probabilities. The order of the elements of
Prior corresponds to the elements of the
ClassNames property. 
The default Prior
value depends on how you create the model:
If you convert a traditionally trained model to create
Mdl
,Prior
is specified by the corresponding property of the traditionally trained model.Otherwise, the default value is
"empirical"
.
Data Types: single
 double
 char
 string
ScoreTransform
— Score transformation function
character vector  string scalar  function handle
This property is readonly.
Score transformation function describing how incremental learning functions
transform raw response values, specified as a character vector, string scalar, or
function handle. incrementalClassificationKernel
stores the
ScoreTransform
value as a character vector or function
handle.
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 
For a MATLAB^{®} function or a function that you define, enter its function handle; for
example, ScoreTransform=@function
, where:
function
accepts an nby2
matrix (the original scores) and returns a matrix of the same size (the transformed scores). The column order corresponds to the class order in theClassNames
property.n is the number of observations, and row j of the matrix contains the class scores of observation j.
The default ScoreTransform
value depends on how you create
the model:
If you convert a traditionally trained model to create
Mdl
,ScoreTransform
is specified by the corresponding property of the traditionally trained model.Otherwise, the default value is
"none"
(whenLearner
is"svm"
) or"logit"
(whenLearner
is"logistic"
).
Data Types: char
 string
 function_handle
Training Parameters
EstimationPeriod
— Number of observations processed to estimate hyperparameters
nonnegative integer
This property is readonly.
Number of observations processed by the incremental model to estimate hyperparameters before training or tracking performance metrics, specified as a nonnegative integer.
Note
If
Mdl
is prepared for incremental learning (all hyperparameters required for training are specified),incrementalClassificationKernel
forcesEstimationPeriod
to0
.If
Mdl
is not prepared for incremental learning,incrementalClassificationKernel
setsEstimationPeriod
to1000
.
For more details, see Estimation Period.
Data Types: single
 double
Mu
— Predictor means
numeric vector  []
Since R2023b
This property is readonly.
Predictor means, specified as a numeric vector.
If Mu
is an empty array []
and you specify Standardize=true
, incremental fitting functions set Mu
to the predictor variable means estimated during the estimation period specified by EstimationPeriod
.
You cannot specify Mu
directly.
Data Types: single
 double
Sigma
— Predictor standard deviations
numeric vector  []
Since R2023b
This property is readonly.
Predictor standard deviations, specified as a numeric vector.
If Sigma
is an empty array []
and you specify Standardize=true
, incremental fitting functions set Sigma
to the predictor variable standard deviations estimated during the estimation period specified by EstimationPeriod
.
You cannot specify Sigma
directly.
Data Types: single
 double
Solver
— Objective function minimization technique
"scaleinvariant"
 "sgd"
 "asgd"
This property is readonly.
Objective function minimization technique, specified as "scaleinvariant"
, "sgd"
, or "asgd"
. incrementalClassificationKernel
stores the Solver
value as a character vector.
Value  Description  Notes 

"scaleinvariant"  Adaptive scaleinvariant solver for incremental learning [1] 

"sgd"  Stochastic gradient descent (SGD) [2][3] 

"asgd"  Average stochastic gradient descent (ASGD) [4] 

The default Solver
value depends on how you create the model:
If you convert a traditionally trained model to create
Mdl
, theSolver
namevalue argument of theincrementalLearner
function sets this property. The default value of the argument is"scaleinvariant"
.Otherwise, the default value is
"scaleinvariant"
.
Data Types: char
 string
SolverOptions
— Objective solver configurations
structure array
This property is readonly.
Objective solver configurations, specified as a structure array. The fields of
SolverOptions
depend on Solver
.
For the SGD and ASGD solvers, the structure array includes the
Solver
,BatchSize
,Lambda
,LearnRate
, andLearnRateSchedule
fields.For the adaptive scaleinvariant solver, the structure array includes the
Solver
andShuffle
fields.
You can specify the field values using the corresponding namevalue arguments when you
create the model object by calling incrementalClassificationKernel
directly, or when you
convert a traditionally trained model using the incrementalLearner
function.
Data Types: struct
Performance Metrics Parameters
IsWarm
— Flag indicating whether model tracks performance metrics
false
or 0
 true
or 1
This property is readonly.
Flag indicating whether the incremental model tracks performance metrics, specified as
logical 0
(false
) or 1
(true
).
The incremental model Mdl
is warm
(IsWarm
becomes true
) after incremental
fitting functions fit (EstimationPeriod
+
MetricsWarmupPeriod
) observations to the incremental
model.
Value  Description 

true or 1  The incremental model Mdl is warm. Consequently,
updateMetrics and
updateMetricsAndFit track performance metrics
in the Metrics property of
Mdl . 
false or 0  updateMetrics and
updateMetricsAndFit do not track performance
metrics. 
Data Types: logical
Metrics
— Model performance metrics
table
This property is readonly.
Model performance metrics updated during incremental learning by
updateMetrics
and updateMetricsAndFit
,
specified as a table with two columns and m rows, where
m is the number of metrics specified by the Metrics
namevalue
argument.
The columns of Metrics
are labeled Cumulative
and Window
.
Cumulative
: Elementj
is the model performance, as measured by metricj
, from the time the model became warm (IsWarm
is1
).Window
: Elementj
is the model performance, as measured by metricj
, evaluated over all observations within the window specified by theMetricsWindowSize
property. The software updatesWindow
after it processesMetricsWindowSize
observations.
Rows are labeled by the specified metrics. For details, see the
Metrics
namevalue argument of
incrementalLearner
or incrementalClassificationKernel
.
Data Types: table
MetricsWarmupPeriod
— Number of observations fit before tracking performance metrics
nonnegative integer
This property is readonly.
Number of observations the incremental model must be fit to before it tracks performance metrics in its Metrics
property, specified as a nonnegative integer.
The default MetricsWarmupPeriod
value depends on how you create
the model:
If you convert a traditionally trained model to create
Mdl
, theMetricsWarmupPeriod
namevalue argument of theincrementalLearner
function sets this property. The default value of the argument is0
.Otherwise, the default value is
1000
.
For more details, see Performance Metrics.
Data Types: single
 double
MetricsWindowSize
— Number of observations to use to compute window performance metrics
positive integer
This property is readonly.
Number of observations to use to compute window performance metrics, specified as a positive integer.
The default MetricsWindowSize
value depends on how you create the model:
If you convert a traditionally trained model to create
Mdl
, theMetricsWindowSize
namevalue argument of theincrementalLearner
function sets this property. The default value of the argument is200
.Otherwise, the default value is
200
.
For more details on performance metrics options, see Performance Metrics.
Data Types: single
 double
Object Functions
fit  Train kernel model for incremental learning 
updateMetrics  Update performance metrics in kernel incremental learning model given new data 
updateMetricsAndFit  Update performance metrics in kernel incremental learning model given new data and train model 
loss  Loss of kernel incremental learning model on batch of data 
predict  Predict responses for new observations from kernel incremental learning model 
perObservationLoss  Per observation classification error of model for incremental learning 
reset  Reset incremental classification model 
Examples
Create Incremental Learner Without Any Prior Information
Create an incremental kernel model without any prior information. Track the model performance on streaming data, and fit the model to the data.
Create a default incremental kernel SVM model for binary classification.
Mdl = incrementalClassificationKernel()
Mdl = incrementalClassificationKernel IsWarm: 0 Metrics: [1x2 table] ClassNames: [1x0 double] ScoreTransform: 'none' NumExpansionDimensions: 0 KernelScale: 1
Mdl
is an incrementalClassificationKernel
model object. All its properties are readonly.
Mdl
must be fit to data before you can use it to perform any other operations.
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.
Responses can be one of five classes: Sitting, Standing, Walking, Running, or Dancing. Dichotomize the response by identifying whether the subject is moving (actid
> 2).
Y = Y > 2;
Fit the incremental model to the training data by using the updateMetricsAndFit
function. Simulate a data stream by processing chunks of 50 observations at a time. At each iteration:
Process 50 observations.
Overwrite the previous incremental model with a new one fitted to the incoming observations.
Store the cumulative metrics, window metrics, and number of training observations to see how they evolve during incremental learning.
% Preallocation numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); ce = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); numtrainobs = zeros(nchunk,1); % Incremental learning for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j1) + 1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; Mdl = updateMetricsAndFit(Mdl,X(idx,:),Y(idx)); ce{j,:} = Mdl.Metrics{"ClassificationError",:}; numtrainobs(j) = Mdl.NumTrainingObservations; end
Mdl
is an incrementalClassificationKernel
model object trained on all the data in the stream. During incremental learning and after the model is warmed up, updateMetricsAndFit
checks the performance of the model on the incoming observations, and then fits the model to those observations.
Plot a trace plot of the number of training observations and the performance metrics on separate tiles.
t = tiledlayout(2,1); nexttile plot(numtrainobs) xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"") xlim([0 nchunk]) ylabel("Number of Training Observations") nexttile plot(ce.Variables) xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"") xlim([0 nchunk]) ylabel("Classification Error") legend(ce.Properties.VariableNames,Location="best") xlabel(t,"Iteration")
The plot suggests that updateMetricsAndFit
does the following:
Fit the model during all incremental learning iterations
Compute the performance metrics after the metrics warmup period only.
Compute the cumulative metrics during each iteration.
Compute the window metrics after processing 200 observations (4 iterations).
Configure Incremental Learning Options
Prepare an incremental kernel SVM learner by specifying a metrics warmup period and a metrics window size. Train the model by using SGD, and adjust the SGD batch size, learning rate, and regularization parameter.
Load the human activity data set. Randomly shuffle the data.
load humanactivity n = numel(actid); rng("default") % For reproducibility idx = randsample(n,n); X = feat(idx,:); Y = actid(idx);
For details on the data set, enter Description
at the command line.
Responses can be one of five classes: Sitting, Standing, Walking, Running, or Dancing. Dichotomize the response by identifying whether the subject is moving (actid
> 2).
Y = Y > 2;
Create an incremental kernel model for binary classification. Configure the model as follows:
Specify the SGD solver.
Assume that a ridge regularization parameter value of 0.001, SGD batch size of 20, and learning rate of 0.002 work well for the problem.
Specify a metrics warmup period of 5000 observations.
Specify a metrics window size of 500 observations.
Track the classification and hinge error metrics to measure the performance of the model.
Mdl = incrementalClassificationKernel( ... Solver="sgd",Lambda=0.001,BatchSize=20,LearnRate=0.002, ... MetricsWarmupPeriod=5000,MetricsWindowSize=500, ... Metrics=["classiferror","hinge"])
Mdl = incrementalClassificationKernel IsWarm: 0 Metrics: [2x2 table] ClassNames: [1x0 double] ScoreTransform: 'none' NumExpansionDimensions: 0 KernelScale: 1
Mdl
is an incrementalClassificationKernel
model object configured for incremental learning.
Fit the incremental model to the rest of the data by using the updateMetricsAndFit
function. At each iteration:
Simulate a data stream by processing a chunk of 50 observations. Note that the chunk size is different from the SGD batch size.
Overwrite the previous incremental model with a new one fitted to the incoming observations.
Store the cumulative metrics, window metrics, and number of training observations to see how they evolve during incremental learning.
% Preallocation numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); ce = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); hinge = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); numtrainobs = zeros(nchunk,1); % Incremental fitting for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j1) + 1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; Mdl = updateMetricsAndFit(Mdl,X(idx,:),Y(idx)); ce{j,:} = Mdl.Metrics{"ClassificationError",:}; hinge{j,:} = Mdl.Metrics{"HingeLoss",:}; numtrainobs(j) = Mdl.NumTrainingObservations; end
Mdl
is an incrementalClassificationKernel
model object trained on all the data in the stream. During incremental learning and after the model is warmed up, updateMetricsAndFit
checks the performance of the model on the incoming observations, and then fits the model to those observations.
Plot a trace plot of the number of training observations and the performance metrics on separate tiles.
t = tiledlayout(3,1); nexttile plot(numtrainobs) xlim([0 nchunk]) ylabel(["Number of","Training Observations"]) xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"") nexttile plot(ce.Variables) xlim([0 nchunk]) ylabel("Classification Error") xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"") legend(ce.Properties.VariableNames,Location="best") nexttile plot(hinge.Variables) xlim([0 nchunk]) ylabel("Hinge Loss") xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"") legend(hinge.Properties.VariableNames,Location="best") xlabel(t,"Iteration")
The plot suggests that updateMetricsAndFit
does the following:
Fit the model during all incremental learning iterations.
Compute the performance metrics after the metrics warmup period only.
Compute the cumulative metrics during each iteration.
Compute the window metrics after processing 500 observations (10 iterations).
Convert Traditionally Trained Model to Incremental Learner
Train a kernel model for binary classification by using fitckernel
, convert it to an incremental learner, track its performance, and fit it to streaming data. Carry over training options from traditional to incremental learning.
Load and Preprocess Data
Load the human activity data set. Randomly shuffle the data.
load humanactivity rng(1) % For reproducibility n = numel(actid); idx = randsample(n,n); X = feat(idx,:); Y = actid(idx);
For details on the data set, enter Description
at the command line.
Responses can be one of five classes: Sitting, Standing, Walking, Running, or Dancing. Dichotomize the response by identifying whether the subject is moving (actid
> 2).
Y = Y > 2;
Suppose that the data collected when the subject was stationary (Y
= false
) has double the quality than when the subject was moving. Create a weight variable that attributes 2 to observations collected from a stationary subject, and 1 to a moving subject.
W = ones(n,1) + ~Y;
Train Kernel Model for Binary Classification
Fit a kernel model for binary classification to a random sample of half the data.
idxtt = randsample([true false],n,true); Mdl = fitckernel(X(idxtt,:),Y(idxtt),Weights=W(idxtt))
Mdl = ClassificationKernel ResponseName: 'Y' ClassNames: [0 1] Learner: 'svm' NumExpansionDimensions: 2048 KernelScale: 1 Lambda: 8.2967e05 BoxConstraint: 1
Mdl
is a ClassificationKernel
model object representing a traditionally trained kernel model for binary classification.
Convert Trained Model
Convert the traditionally trained classification model to a model for incremental learning.
IncrementalMdl = incrementalLearner(Mdl)
IncrementalMdl = incrementalClassificationKernel IsWarm: 1 Metrics: [1x2 table] ClassNames: [0 1] ScoreTransform: 'none' NumExpansionDimensions: 2048 KernelScale: 1
IncrementalMdl
is an incrementalClassificationKernel
model object configured for incremental learning.
Separately Track Performance Metrics and Fit Model
Perform incremental learning on the rest of the data by using the updateMetrics
and fit
functions. Simulate a data stream by processing 50 observations at a time. At each iteration:
Call
updateMetrics
to update the cumulative and window classification error of the model given the incoming chunk of observations. Overwrite the previous incremental model to update theMetrics
property. Note that the function does not fit the model to the chunk of data—the chunk is "new" data for the model. Specify the observation weights.Call
fit
to fit the model to the incoming chunk of observations. Overwrite the previous incremental model to update the model parameters. Specify the observation weights.Store the classification error and number of training observations.
% Preallocation idxil = ~idxtt; nil = sum(idxil); numObsPerChunk = 50; nchunk = floor(nil/numObsPerChunk); ce = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); numtrainobs = zeros(nchunk,1); Xil = X(idxil,:); Yil = Y(idxil); Wil = W(idxil); % Incremental fitting for j = 1:nchunk ibegin = min(nil,numObsPerChunk*(j1) + 1); iend = min(nil,numObsPerChunk*j); idx = ibegin:iend; IncrementalMdl = updateMetrics(IncrementalMdl,Xil(idx,:),Yil(idx), ... Weights=Wil(idx)); ce{j,:} = IncrementalMdl.Metrics{"ClassificationError",:}; IncrementalMdl = fit(IncrementalMdl,Xil(idx,:),Yil(idx), ... Weights=Wil(idx)); numtrainobs(j) = IncrementalMdl.NumTrainingObservations; end
IncrementalMdl
is an incrementalClassificationKernel
model object trained on all the data in the stream.
Alternatively, you can use updateMetricsAndFit
to update performance metrics of the model given a new chunk of data, and then fit the model to the data.
Plot a trace plot of the number of training observations and the performance metrics.
t = tiledlayout(2,1); nexttile plot(numtrainobs) xlim([0 nchunk]) ylabel("Number of Training Observations") nexttile plot(ce.Variables) xlim([0 nchunk]) legend(ce.Properties.VariableNames) ylabel("Classification Error") xlabel(t,"Iteration")
The cumulative loss is stable and decreases gradually, whereas the window loss jumps.
More About
Incremental Learning
Incremental learning, or online learning, is a branch of machine learning concerned with processing incoming data from a data stream, possibly given little to no knowledge of the distribution of the predictor variables, aspects of the prediction or objective function (including tuning parameter values), or whether the observations are labeled. Incremental learning differs from traditional machine learning, where enough labeled data is available to fit to a model, perform crossvalidation to tune hyperparameters, and infer the predictor distribution.
Given incoming observations, an incremental learning model processes data in any of the following ways, but usually in this order:
Predict labels.
Measure the predictive performance.
Check for structural breaks or drift in the model.
Fit the model to the incoming observations.
For more details, see Incremental Learning Overview.
Adaptive ScaleInvariant Solver for Incremental Learning
The adaptive scaleinvariant solver for incremental learning, introduced in [1], is a gradientdescentbased objective solver for training linear predictive models. The solver is hyperparameter free, insensitive to differences in predictor variable scales, and does not require prior knowledge of the distribution of the predictor variables. These characteristics make it well suited to incremental learning.
The standard SGD and ASGD solvers are sensitive to differing scales among the predictor variables, resulting in models that can perform poorly. To achieve better accuracy using SGD and ASGD, you can standardize the predictor data, and tune the regularization and learning rate parameters. For traditional machine learning, enough data is available to enable hyperparameter tuning by crossvalidation and predictor standardization. However, for incremental learning, enough data might not be available (for example, observations might be available only one at a time) and the distribution of the predictors might be unknown. These characteristics make parameter tuning and predictor standardization difficult or impossible to do during incremental learning.
The incremental fitting functions fit
and
updateMetricsAndFit
use the more aggressive ScInOL2 version of the
algorithm.
Random Feature Expansion
Random feature expansion, such as Random Kitchen Sinks[5] or Fastfood[6], is a scheme to approximate Gaussian kernels of the kernel classification algorithm to use for big data in a computationally efficient way. Random feature expansion is more practical for big data applications that have large training sets, but can also be applied to smaller data sets that fit in memory.
The kernel classification algorithm searches for an optimal hyperplane that separates the data into two classes after mapping features into a highdimensional space. Nonlinear features that are not linearly separable in a lowdimensional space can be separable in the expanded highdimensional space. All the calculations for hyperplane classification use only dot products. You can obtain a nonlinear classification model by replacing the dot product x_{1}x_{2}' with the nonlinear kernel function $$G({x}_{1},{x}_{2})=\langle \phi ({x}_{1}),\phi ({x}_{2})\rangle $$, where x_{i} is the ith observation (row vector) and φ(x_{i}) is a transformation that maps x_{i} to a highdimensional space (called the “kernel trick”). However, evaluating G(x_{1},x_{2}) (Gram matrix) for each pair of observations is computationally expensive for a large data set (large n).
The random feature expansion scheme finds a random transformation so that its dot product approximates the Gaussian kernel. That is,
$$G({x}_{1},{x}_{2})=\langle \phi ({x}_{1}),\phi ({x}_{2})\rangle \approx T({x}_{1})T({x}_{2})\text{'},$$
where T(x) maps x in $${\mathbb{R}}^{p}$$ to a highdimensional space ($${\mathbb{R}}^{m}$$). The Random Kitchen Sinks scheme uses the random transformation
$$T(x)={m}^{1/2}\mathrm{exp}\left(iZx\text{'}\right)\text{'},$$
where $$Z\in {\mathbb{R}}^{m\times p}$$ is a sample drawn from $$N\left(0,{\sigma}^{2}\right)$$ and σ is a kernel scale. This scheme requires O(mp) computation and storage.
The Fastfood scheme introduces another random
basis V instead of Z using Hadamard matrices combined
with Gaussian scaling matrices. This random basis reduces the computation cost to O(mlog
p) and reduces storage to O(m).
incrementalClassificationKernel
uses the
Fastfood scheme for random feature expansion, and uses linear classification to train a Gaussian
kernel classification model. You can specify values for m and
σ using the NumExpansionDimensions
and
KernelScale
namevalue arguments, respectively, when you create a
traditionally trained model using fitckernel
or when
you callincrementalClassificationKernel
directly to create the model
object.
Algorithms
Estimation Period
During the estimation period, the incremental fitting functions fit
and updateMetricsAndFit
use the first incoming
EstimationPeriod
observations to estimate (tune) hyperparameters
required for incremental training. Estimation occurs only when
EstimationPeriod
is positive. This table describes the hyperparameter
and when they are estimated, or tuned.
Hyperparameter  Model Property  Usage  Conditions 

Predictor means and standard deviations  Mu and Sigma  Standardize predictor data  The hyperparameters are estimated when both of these conditions apply:

Learning rate  LearnRate field of SolverOptions  Adjust the solver step size  The hyperparameter is estimated when both of these conditions apply:

Kernel scale parameter  KernelScale  Set a kernel scale parameter value for random feature expansion  The hyperparameter is estimated when you set the
KernelScale namevalue argument to
"auto" . 
During the estimation period, fit
does not fit the model, and updateMetricsAndFit
does not fit the model or update the performance metrics. At the end of the estimation period, the functions update the properties that store the hyperparameters.
Standardize Data
If incremental learning functions are configured to standardize predictor variables,
they do so using the means and standard deviations stored in the Mu
and
Sigma
properties, respectively, of the incremental learning model
Mdl
.
When you set
Standardize=true
and specify a positive estimation period (seeEstimationPeriod
), andMdl.Mu
andMdl.Sigma
are empty, incremental fitting functions estimate means and standard deviations using the estimation period observations.When you set
Standardize="auto"
(the default), the following conditions apply:If you create
incrementalClassificationKernel
by converting a traditionally trained binary kernel model (ClassificationKernel
), and theMu
andSigma
properties of the traditionally trained model are empty arrays[]
, incremental learning functions do not standardize predictor variables. If theMu
andSigma
properties of the traditionally trained model are nonempty, incremental learning functions standardize the predictor variables using the specified means and standard deviations. Incremental fitting functions do not estimate new means and standard deviations, regardless of the length of the estimation period.If you do not convert a traditionally trained model, incremental learning functions standardize the predictor data only when you specify an SGD solver (see
Solver
) and a positive estimation period (seeEstimationPeriod
).
When incremental fitting functions estimate predictor means and standard deviations, the functions compute weighted means and weighted standard deviations using the estimation period observations. Specifically, the functions standardize predictor j (x_{j}) using
$${x}_{j}^{\ast}=\frac{{x}_{j}{\mu}_{j}^{\ast}}{{\sigma}_{j}^{\ast}}.$$
x_{j} is predictor j, and x_{jk} is observation k of predictor j in the estimation period.
$${\mu}_{j}^{\ast}=\frac{1}{{\displaystyle \sum _{k}{w}_{k}^{\ast}}}{\displaystyle \sum _{k}{w}_{k}^{\ast}{x}_{jk}}.$$
$${\left({\sigma}_{j}^{\ast}\right)}^{2}=\frac{1}{{\displaystyle \sum _{k}{w}_{k}^{\ast}}}{\displaystyle \sum _{k}{w}_{k}^{\ast}{\left({x}_{jk}{\mu}_{j}^{\ast}\right)}^{2}}.$$
$${w}_{j}^{\ast}=\frac{{w}_{j}}{{\displaystyle \sum _{\forall j\in \text{Class}k}{w}_{j}}}{p}_{k},$$
p_{k} is the prior probability of class k (
Prior
property of the incremental model).w_{j} is observation weight j.
Performance Metrics
The
updateMetrics
andupdateMetricsAndFit
functions are incremental learning functions that track model performance metrics (Metrics
) from new data only when the incremental model is warm (IsWarm
property istrue
). An incremental model becomes warm afterfit
orupdateMetricsAndFit
fits the incremental model toMetricsWarmupPeriod
observations, which is the metrics warmup period.If
EstimationPeriod
> 0, thefit
andupdateMetricsAndFit
functions estimate hyperparameters before fitting the model to data. Therefore, the functions must process an additionalEstimationPeriod
observations before the model starts the metrics warmup period.The
Metrics
property of the incremental model stores two forms of each performance metric as variables (columns) of a table,Cumulative
andWindow
, with individual metrics in rows. When the incremental model is warm,updateMetrics
andupdateMetricsAndFit
update the metrics at the following frequencies:Cumulative
— The functions compute cumulative metrics since the start of model performance tracking. The functions update metrics every time you call the functions and base the calculation on the entire supplied data set.Window
— The functions compute metrics based on all observations within a window determined byMetricsWindowSize
, which also determines the frequency at which the software updatesWindow
metrics. For example, ifMetricsWindowSize
is 20, the functions compute metrics based on the last 20 observations in the supplied data (X((end – 20 + 1):end,:)
andY((end – 20 + 1):end)
).Incremental functions that track performance metrics within a window use the following process:
Store a buffer of length
MetricsWindowSize
for each specified metric, and store a buffer of observation weights.Populate elements of the metrics buffer with the model performance based on batches of incoming observations, and store corresponding observation weights in the weights buffer.
When the buffer is full, overwrite the
Window
field of theMetrics
property with the weighted average performance in the metrics window. If the buffer overfills when the function processes a batch of observations, the latest incomingMetricsWindowSize
observations enter the buffer, and the earliest observations are removed from the buffer. For example, supposeMetricsWindowSize
is 20, the metrics buffer has 10 values from a previously processed batch, and 15 values are incoming. To compose the length 20 window, the functions use the measurements from the 15 incoming observations and the latest 5 measurements from the previous batch.
The software omits an observation with a
NaN
score when computing theCumulative
andWindow
performance metric values.
References
[1] Kempka, Michał, Wojciech Kotłowski, and Manfred K. Warmuth. "Adaptive ScaleInvariant Online Algorithms for Learning Linear Models." Preprint, submitted February 10, 2019. https://arxiv.org/abs/1902.07528.
[2] Langford, J., L. Li, and T. Zhang. “Sparse Online Learning Via Truncated Gradient.” J. Mach. Learn. Res., Vol. 10, 2009, pp. 777–801.
[3] ShalevShwartz, S., Y. Singer, and N. Srebro. “Pegasos: Primal Estimated SubGradient Solver for SVM.” Proceedings of the 24th International Conference on Machine Learning, ICML ’07, 2007, pp. 807–814.
[4] Xu, Wei. “Towards Optimal One Pass Large Scale Learning with Averaged Stochastic Gradient Descent.” CoRR, abs/1107.2490, 2011.
[5] Rahimi, A., and B. Recht. “Random Features for LargeScale Kernel Machines.” Advances in Neural Information Processing Systems. Vol. 20, 2008, pp. 1177–1184.
[6] Le, Q., T. Sarlós, and A. Smola. “Fastfood — Approximating Kernel Expansions in Loglinear Time.” Proceedings of the 30th International Conference on Machine Learning. Vol. 28, No. 3, 2013, pp. 244–252.
[7] Huang, P. S., H. Avron, T. N. Sainath, V. Sindhwani, and B. Ramabhadran. “Kernel methods match Deep Neural Networks on TIMIT.” 2014 IEEE International Conference on Acoustics, Speech and Signal Processing. 2014, pp. 205–209.
Version History
Introduced in R2022aR2023b: Kernel models support standardization of predictors
incrementalClassificationKernel
supports the standardization of numeric predictors. That is, you can specify the Standardize
value as true
to center and scale each numeric predictor variable by the corresponding column mean and standard deviation. The software does not standardize the categorical predictors.
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