incrementalRegressionLinear
Description
incrementalRegressionLinear
creates an incrementalRegressionLinear
model object, which represents an incremental linear model for regression problems. Supported learners include support vector machine (SVM) and least squares.
Unlike other Statistics and Machine Learning Toolbox™ model objects, incrementalRegressionLinear
can be called directly. Also, you can specify learning options, such as performance metrics configurations, parameter values, and the objective solver, before fitting the model to data. After you create an incrementalRegressionLinear
object, it is prepared for incremental learning.
incrementalRegressionLinear
is best suited for incremental learning. For a traditional approach to training an SVM or linear regression model (such as creating a model by fitting it to data, performing crossvalidation, tuning hyperparameters, and so on), see fitrsvm
or fitrlinear
.
Creation
You can create an incrementalRegressionLinear
model object in several ways:
Call the function directly — Configure incremental learning options, or specify initial values for linear model parameters and hyperparameters, by calling
incrementalRegressionLinear
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 an linear regression model for incremental learning using the model coefficients and hyperparameters of a trained model object, you can convert the traditionally trained model to an
incrementalRegressionLinear
model object by passing it to theincrementalLearner
function. This table contains links to the appropriate reference pages.Convertible Model Object Conversion Function RegressionSVM
orCompactRegressionSVM
incrementalLearner
RegressionLinear
incrementalLearner
Call an incremental learning function —
fit
,updateMetrics
, andupdateMetricsAndFit
accept a configuredincrementalRegressionLinear
model object and data as input, and return anincrementalRegressionLinear
model object updated with information learned from the input model and data.
Description
returns
a default incremental model object for linear regression, Mdl
= incrementalRegressionLinear()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. Enclose each name in quotes. For example,
Mdl
= incrementalRegressionLinear(Name
,Value
)incrementalRegressionLinear('Beta',[0.1
0.3],'Bias',1,'MetricsWarmupPeriod',100)
sets the vector of linear model
coefficients β to [0.1 0.3]
, the bias
β_{0} to 1
, and the metrics
warmup period to 100
.
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.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: 'Standardize',true
standardizes the predictor data using the predictor means and standard deviations estimated during the estimation period.
Metrics
— Model performance metrics to track during incremental learning
"epsiloninsensitive"
 "mse"
 string vector  function handle  cell vector  structure array
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 and which learners,
specified in Learner
, support them. You can specify
more than one loss function by using a string vector.
Name  Description  Learner Supporting Metric 

"epsiloninsensitive"  Epsilon insensitive loss  'svm' 
"mse"  Weighted mean squared error  'svm' and 'leastsquares' 
For more details on the builtin loss functions, see loss
.
Example: 'Metrics',["epsiloninsensitive" "mse"]
To specify a custom function that returns a performance metric, use function handle notation. The function must have this form:
metric = customMetric(Y,YFit)
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
).Y
is a length n numeric vector of observed responses, where n is the sample size.YFit
is a length n numeric vector of corresponding predicted responses.
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 'mse'
struct('Metric3',@customMetric3)}
updateMetrics
and updateMetricsAndFit
store specified metrics in a table in the property Metrics
. 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 "epsiloninsensitive" is "EpsilonInsensitiveLoss" 
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 @(Y,YFit)customMetric(Y,YFit)... is CustomMetric_1 
By default:
Metrics
is"epsiloninsensitive"
ifLearner
is'svm'
.Metrics
is"mse"
ifLearner
is'leastsquares'
.
For more details on performance metrics options, see Performance Metrics.
Data Types: char
 string
 struct
 cell
 function_handle
Standardize
— Flag to standardize predictor data
'auto'
(default)  false
 true
Flag to standardize the predictor data, specified as a value in this table.
Value  Description 

'auto'  incrementalRegressionLinear 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
Shuffle
— Flag for shuffling observations
true
(default)  false
Flag for shuffling the observations at each iteration, specified as a value in this table.
Value  Description 

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. 
false  The software processes the data in the order received. 
This option is valid only when Solver
is
'scaleinvariant'
. When Solver
is
'sgd'
or 'asgd'
, the software always
shuffles the observations in an incoming chunk of data before processing the
data.
Example: 'Shuffle',false
Data Types: logical
Properties
You can set most properties by using namevalue argument syntax only when you call
incrementalRegressionLinear
. You can set some properties when you call
incrementalLearner
to convert a traditionally trained model. You cannot
set the properties FittedLoss
,
NumTrainingObservations
, Mu
,
Sigma
, SolverOptions
, and
IsWarm
.
Regression Model Parameters
Beta
— Linear model coefficients β
numeric vector
This property is readonly.
Linear model coefficients β, specified as a NumPredictors
by1 numeric vector.
Incremental fitting functions estimate Beta
during training. The
default initial Beta
value depends on how you create the model:
If you convert a traditionally trained model to create
Mdl
, the initial value is specified by the corresponding property of the traditionally trained model.Otherwise, the initial value is
zeros(NumPredictors,1)
.
Data Types: single
 double
Bias
— Model intercept β_{0}
numeric scalar
This property is readonly.
Model intercept β_{0}, or bias term, specified as a numeric scalar.
Incremental fitting functions estimate Bias
during training. The
default initial Bias
value depends on how you create the model:
If you convert a traditionally trained model to create
Mdl
, the initial value is specified by the corresponding property of the traditionally trained model.Otherwise, the initial value is
0
.
Data Types: single
 double
Epsilon
— Half of the width of epsilon insensitive band
'auto'
 nonnegative scalar
This property is readonly.
Half of the width of the epsilon insensitive band, specified as
'auto'
or a nonnegative scalar. incrementalRegressionLinear
stores the Epsilon
value as a numeric scalar.
If you specify 'auto'
when you call
incrementalRegressionLinear
, incremental fitting functions estimate
Epsilon
during the estimation period, specified by EstimationPeriod
, using
this procedure:
If
iqr(Y)
≠ 0,Epsilon
isiqr(Y)/13.49
, whereY
is the estimation period response data.If
iqr(Y)
= 0 or before you fitMdl
to data,Epsilon
is0.1
.
The default Epsilon
value depends on how you create the model:
If you convert a traditionally trained SVM regression model (
Learner
is'svm'
),Epsilon
is specified by the corresponding property of the traditionally trained model.Otherwise, the default value is
'auto'
.
If Learner
is 'leastsquares'
, you cannot
set Epsilon
and its value is NaN
.
Data Types: single
 double
FittedLoss
— Loss function used to fit linear model
'epsiloninsensitive'
 'mse'
This property is readonly.
Loss function used to fit the linear model, specified as
'epsiloninsensitive'
or 'mse'
.
Value  Algorithm  Loss Function  Learner Value 

'epsiloninsensitive'  Support vector machine regression  Epsilon insensitive: $$\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,\leftyf\left(x\right)\right\epsilon \right]$$  'svm' 
'mse'  Linear regression through ordinary least squares  Mean squared error (MSE): $$\ell \left[y,f\left(x\right)\right]=\frac{1}{2}{\left[yf\left(x\right)\right]}^{2}$$  'leastsquares' 
Learner
— Linear regression model type
'svm'
 'leastsquares'
This property is readonly.
Linear regression model type, specified as 'svm'
or
'leastsquares'
. incrementalRegressionLinear
stores the
Learner
value as a character vector.
In the following table, $$f\left(x\right)=x\beta +b.$$
β is
Beta
.x is an observation from p predictor variables.
β_{0} is
Bias
.
Value  Algorithm  Loss Function  FittedLoss Value 

'svm'  Support vector machine regression  Epsilon insensitive: $$\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,\leftyf\left(x\right)\right\epsilon \right]$$  'epsiloninsensitive' 
'leastsquares'  Linear regression through ordinary least squares  Mean squared error (MSE): $$\ell \left[y,f\left(x\right)\right]=\frac{1}{2}{\left[yf\left(x\right)\right]}^{2}$$  'mse' 
The default Learner
value depends on how you create the model:
If you convert a traditionally trained model to create
Mdl
:Learner
is'svm'
when the traditionally trained model isRegressionSVM
orCompactRegressionSVM
.Learner
is specified by the corresponding property of the traditionally trained model when the traditionally trained model isRegressionLinear
.
Otherwise, the default value is
'svm'
.
Data Types: char
 string
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 callingincrementalRegressionLinear
directly, you can specifyNumPredictors
by using namevalue argument syntax. If you do not specify the value, then the default value is0
, 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
, incrementalRegressionLinear
does not add the number of observations fit to the traditionally trained model to NumTrainingObservations
.
Data Types: double
ResponseTransform
— Response transformation function
'none'
 function handle
This property is readonly.
Response transformation function, specified as 'none'
or a
function handle. incrementalRegressionLinear
stores the
ResponseTransform
value as a character vector or function
handle.
ResponseTransform
describes how incremental learning
functions transform raw response values.
For a MATLAB^{®} function or a function that you define, enter its function handle; for
example, 'ResponseTransform',@function
, where
function
accepts an
nby1 vector (the original responses) and returns a vector of the
same length (the transformed responses).
The default ResponseTransform
value depends on how you create
the model:
If you convert a traditionally trained model to create
Mdl
,ResponseTransform
is specified by the corresponding property of the traditionally trained model.Otherwise, the default value is
"none"
.
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),incrementalRegressionLinear
forcesEstimationPeriod
to0
.If
Mdl
is not prepared for incremental learning,incrementalRegressionLinear
setsEstimationPeriod
to1000
.
For more details, see Estimation Period.
Data Types: single
 double
FitBias
— Linear model intercept inclusion flag
true
 false
This property is readonly.
Linear model intercept inclusion flag, specified as true
or
false
.
Value  Description 

true  incrementalRegressionLinear includes the bias term
β_{0} in the linear model, which
incremental fitting functions fit to data. 
false  incrementalRegressionLinear sets
β_{0} = 0. 
If Bias
≠ 0, FitBias
must be
true
. In other words, incrementalRegressionLinear
does not
support an equality constraint on β_{0}.
The default FitBias
value depends on how you create the model:
If you convert a traditionally trained linear regression model (
RegressionLinear
) to createMdl
,FitBias
is specified by theFitBias
value of theModelParameters
property of the traditionally trained model.Otherwise, the default value is
true
.
Data Types: logical
Mu
— Predictor means
vector of numeric values  []
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
vector of numeric values  []
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'
. incrementalRegressionLinear
stores the
Solver
value as a character vector.
Value  Description  Notes 

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

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

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

The default Solver
value depends on how you create the model:
If you create
Mdl
by callingincrementalRegressionLinear
directly, the default value is'scaleinvariant'
.If you convert a traditionally trained linear regression model (
RegressionLinear
) to createMdl
, and the traditionally trained model'sRegularization
property is'ridge (L2)'
andModelParameters.Solver
is'sgd'
or'asgd'
,Solver
is specified by theSolver
value of theModelParameters
property of the traditionally trained model.Otherwise, the
Solver
namevalue argument of theincrementalLearner
function sets this property. The default value of the argument 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
are properties specific to the specified solver
Solver
.
Data Types: struct
SGD and ASGD Solver Parameters
BatchSize
— Minibatch size
positive integer
This property is readonly.
Minibatch size, specified as a positive integer. At each learning cycle during
training, incrementalRegressionLinear
uses BatchSize
observations to compute the subgradient.
The number of observations for 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.
The default BatchSize
value depends on how you create the model:
If you create
Mdl
by callingincrementalRegressionLinear
directly, the default value is10
.If you convert a traditionally trained linear regression model (
RegressionLinear
) to createMdl
, and the traditionally trained model'sRegularization
property is'ridge (L2)'
andModelParameters.Solver
is'sgd'
or'asgd'
,BatchSize
is specified by theBatchSize
value of theModelParameters
property of the traditionally trained model.Otherwise, the
BatchSize
namevalue argument of theincrementalLearner
function sets this property. The default value of the argument is10
.
Data Types: single
 double
Lambda
— Ridge (L2) regularization term strength
nonnegative scalar
This property is readonly.
Ridge (L2) regularization term strength, specified as a nonnegative scalar.
The default Lambda
value depends on how you create the model:
If you create
Mdl
by callingincrementalRegressionLinear
directly, the default value is1e5
.If you convert a traditionally trained linear regression model (
RegressionLinear
) to createMdl
, and the traditionally trained model'sRegularization
property is'ridge (L2)'
andModelParameters.Solver
is'sgd'
or'asgd'
,Lambda
is specified by the corresponding property of the traditionally trained model.Otherwise, the
Lambda
namevalue argument of theincrementalLearner
function sets this property. The default value of the argument is1e5
.
Data Types: double
 single
LearnRate
— Initial learning rate
'auto'
 positive scalar
This property is readonly.
Initial learning rate, specified as 'auto'
or a positive
scalar. incrementalRegressionLinear
stores the LearnRate
value
as 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,obsDim)))
at the end ofEstimationPeriod
. When the observations are the columns of the predictor dataX
collected during the estimation period, theobsDim
value is1
; otherwise, the value is2
.
The default LearnRate
value depends on how you create the model:
If you create
Mdl
by callingincrementalRegressionLinear
directly, the default value is'auto'
.If you convert a traditionally trained linear regression model (
RegressionLinear
) to createMdl
, and the traditionally trained model'sRegularization
property is'ridge (L2)'
andModelParameters.Solver
is'sgd'
or'asgd'
,LearnRate
is specified by theLearnRate
value of theModelParameters
property of the traditionally trained model.Otherwise, the
LearnRate
namevalue argument of theincrementalLearner
function sets this property. The default value of the argument is'auto'
.
Example: 'LearnRate',0.001
Data Types: single
 double
 char
 string
LearnRateSchedule
— Learning rate schedule
'decaying'
 'constant'
This property is readonly.
Learning rate schedule, specified as a value in this table, where LearnRate
specifies
the initial learning rate ɣ_{0}.
incrementalRegressionLinear
stores the LearnRateSchedule
value as a character vector.
Value  Description 

'constant'  The learning rate is ɣ_{0} for all learning cycles. 
'decaying'  The learning rate at learning cycle t is $${\gamma}_{t}=\frac{{\gamma}_{0}}{{\left(1+\lambda {\gamma}_{0}t\right)}^{c}}.$$

The default LearnRateSchedule
value depends on how you
create the model:
If you convert a traditionally trained model to create
Mdl
, theLearnRateSchedule
namevalue argument of theincrementalLearner
function sets this property. The default value of the argument is'decaying'
.Otherwise, the default value is
'decaying'
.
Data Types: char
 string
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 incrementalRegressionLinear
.
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 linear model for incremental learning 
updateMetricsAndFit  Update performance metrics in linear incremental learning model given new data and train model 
updateMetrics  Update performance metrics in linear incremental learning model given new data 
loss  Loss of linear incremental learning model on batch of data 
predict  Predict responses for new observations from linear incremental learning model 
perObservationLoss  Per observation regression error of model for incremental learning 
reset  Reset incremental regression model 
Examples
Create Incremental Learner Without Any Prior Information
Create a default incremental linear model for regression.
Mdl = incrementalRegressionLinear()
Mdl = incrementalRegressionLinear IsWarm: 0 Metrics: [1x2 table] ResponseTransform: 'none' Beta: [0x1 double] Bias: 0 Learner: 'svm'
Mdl.EstimationPeriod
ans = 1000
Mdl
is an incrementalRegressionLinear
model object. All its properties are readonly.
Mdl
must be fit to data before you can use it to perform any other operations. The software sets the estimation period to 1000 because half the width of the epsilon insensitive band Epsilon
is unknown. You can set Epsilon
to a positive floatingpoint scalar by using the Epsilon
namevalue argument. This action results in a default estimation period of 0.
Load the robot arm data set.
load robotarm
For details on the data set, enter Description
at the command line.
Fit the incremental model to the training data by using the updateMetricsAndFit
function. To simulate a data stream fit the model in 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 ${\beta}_{1}$, the cumulative metrics, and the window metrics to see how they evolve during incremental learning.
% Preallocation n = numel(ytrain); numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); ei = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]); beta1 = zeros(nchunk+1,1); % Incremental fitting rng("default"); % For reproducibility for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j1) + 1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; Mdl = updateMetricsAndFit(Mdl,Xtrain(idx,:),ytrain(idx)); ei{j,:} = Mdl.Metrics{"EpsilonInsensitiveLoss",:}; beta1(j + 1) = Mdl.Beta(1); end
Mdl
is an incrementalRegressionLinear
model object trained on all the data in the stream. While updateMetricsAndFit
processes the first 1000 observations, it stores the response values to estimate Epsilon
; the function does not fit the coefficients until after this estimation period. 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.
To see how the performance metrics and ${\beta}_{1}$ evolve during training, plot them on separate tiles.
t = tiledlayout(2,1); nexttile plot(beta1) ylabel('\beta_1') xlim([0 nchunk]) xline(Mdl.EstimationPeriod/numObsPerChunk,'r.') nexttile h = plot(ei.Variables); xlim([0 nchunk]) ylabel('Epsilon Insensitive Loss') xline(Mdl.EstimationPeriod/numObsPerChunk,'r.') xline((Mdl.EstimationPeriod + Mdl.MetricsWarmupPeriod)/numObsPerChunk,'g.') legend(h,ei.Properties.VariableNames) xlabel(t,'Iteration')
The plot suggests that updateMetricsAndFit
does the following:
After the estimation period (first 20 iterations), fit ${\beta}_{1}$ 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 (4 iterations).
Configure Incremental Learning Options
Prepare an incremental regression learner by specifying a metrics warmup period, during which the updateMetricsAndFit
function only fits the model. Specify a metrics window size of 500 observations. Train the model by using SGD, and adjust the SGD batch size, learning rate, and regularization parameter.
Load the robot arm data set.
load robotarm
n = numel(ytrain);
For details on the data set, enter Description
at the command line.
Create an incremental linear model for regression. Configure the model as follows:
Specify the SGD solver.
Assume that these settings work well for the problem: a ridge regularization parameter value of 0.001, SGD batch size of 20, learning rate of 0.002, and half the width of the epsilon insensitive band for SVM of 0.05.
Specify that the incremental fitting functions process the raw (unstandardized) predictor data.
Specify a metrics warmup period of 1000 observations.
Specify a metrics window size of 500 observations.
Track the epsilon insensitive loss, MSE, and mean absolute error (MAE) to measure the performance of the model. The software supports epsilon insensitive loss and MSE. Create an anonymous function that measures the absolute error of each new observation. Create a structure array containing the name
MeanAbsoluteError
and its corresponding function.
maefcn = @(z,zfit)abs(z  zfit); maemetric = struct("MeanAbsoluteError",maefcn); Mdl = incrementalRegressionLinear('Epsilon',0.05, ... 'Solver','sgd','Lambda',0.001,'BatchSize',20,'LearnRate',0.002, ... 'Standardize',false, ... 'MetricsWarmupPeriod',1000,'MetricsWindowSize',500, ... 'Metrics',{'epsiloninsensitive' 'mse' maemetric})
Mdl = incrementalRegressionLinear IsWarm: 0 Metrics: [3x2 table] ResponseTransform: 'none' Beta: [0x1 double] Bias: 0 Learner: 'svm'
Mdl
is an incrementalRegressionLinear
model object configured for incremental learning without an estimation period.
Fit the incremental model to 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 SGD batch size.
Overwrite the previous incremental model with a new one fitted to the incoming observations.
Store the estimated coefficient ${\beta}_{10}$, the cumulative metrics, and the window metrics to see how they evolve during incremental learning.
% Preallocation numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); ei = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]); mse = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]); mae = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]); beta10 = zeros(nchunk+1,1); % Incremental fitting rng("default"); % For reproducibility for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j1) + 1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; Mdl = updateMetricsAndFit(Mdl,Xtrain(idx,:),ytrain(idx)); ei{j,:} = Mdl.Metrics{"EpsilonInsensitiveLoss",:}; mse{j,:} = Mdl.Metrics{"MeanSquaredError",:}; mae{j,:} = Mdl.Metrics{"MeanAbsoluteError",:}; beta10(j + 1) = Mdl.Beta(10); end
Mdl
is an incrementalRegressionLinear
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.
To see how the performance metrics and ${\beta}_{10}$ evolve during training, plot them on separate tiles.
tiledlayout(2,2) nexttile plot(beta10) ylabel('\beta_{10}') xlim([0 nchunk]) xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'g.') xlabel('Iteration') nexttile h = plot(ei.Variables); xlim([0 nchunk]) ylabel('Epsilon Insensitive Loss') xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'g.') legend(h,ei.Properties.VariableNames) xlabel('Iteration') nexttile h = plot(mse.Variables); xlim([0 nchunk]) ylabel('MSE') xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'g.') legend(h,mse.Properties.VariableNames) xlabel('Iteration') nexttile h = plot(mae.Variables); xlim([0 nchunk]) ylabel('MAE') xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'g.') legend(h,mae.Properties.VariableNames) xlabel('Iteration')
The plot suggests that updateMetricsAndFit
does the following:
Fit ${\beta}_{10}$ 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 linear regression model by using fitrlinear
, 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 2015 NYC housing data set, and shuffle the data. For more details on the data, see NYC Open Data.
load NYCHousing2015 rng(1); % For reproducibility n = size(NYCHousing2015,1); idxshuff = randsample(n,n); NYCHousing2015 = NYCHousing2015(idxshuff,:);
Suppose that the data collected from Manhattan (BOROUGH
= 1
) was collected using a new method that doubles its quality. Create a weight variable that attributes 2
to observations collected from Manhattan, and 1
to all other observations.
NYCHousing2015.W = ones(n,1) + (NYCHousing2015.BOROUGH == 1);
Extract the response variable SALEPRICE
from the table. For numerical stability, scale SALEPRICE
by 1e6
.
Y = NYCHousing2015.SALEPRICE/1e6; NYCHousing2015.SALEPRICE = [];
Create dummy variable matrices from the categorical predictors.
catvars = ["BOROUGH" "BUILDINGCLASSCATEGORY" "NEIGHBORHOOD"]; dumvarstbl = varfun(@(x)dummyvar(categorical(x)),NYCHousing2015, ... 'InputVariables',catvars); dumvarmat = table2array(dumvarstbl); NYCHousing2015(:,catvars) = [];
Treat all other numeric variables in the table as linear predictors of sales price. Concatenate the matrix of dummy variables to the rest of the predictor data. Transpose the resulting predictor matrix.
idxnum = varfun(@isnumeric,NYCHousing2015,'OutputFormat','uniform'); X = [dumvarmat NYCHousing2015{:,idxnum}]';
Train Linear Regression Model
Fit a linear regression model to a random sample of half the data.
idxtt = randsample([true false],n,true); TTMdl = fitrlinear(X(:,idxtt),Y(idxtt),'ObservationsIn','columns', ... 'Weights',NYCHousing2015.W(idxtt))
TTMdl = RegressionLinear ResponseName: 'Y' ResponseTransform: 'none' Beta: [313x1 double] Bias: 0.1116 Lambda: 2.1977e05 Learner: 'svm'
TTMdl
is a RegressionLinear
model object representing a traditionally trained linear regression model.
Convert Trained Model
Convert the traditionally trained linear regression model to a linear regression model for incremental learning.
IncrementalMdl = incrementalLearner(TTMdl)
IncrementalMdl = incrementalRegressionLinear IsWarm: 1 Metrics: [1x2 table] ResponseTransform: 'none' Beta: [313x1 double] Bias: 0.1116 Learner: 'svm'
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 500 observations at a time. At each iteration:
Call
updateMetrics
to update the cumulative and window epsilon insensitive loss of the model given the incoming chunk of observations. Overwrite the previous incremental model to update the losses in 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 that the observations are oriented in columns, and 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 that the observations are oriented in columns, and specify the observation weights.Store the losses and last estimated coefficient ${\beta}_{313}$.
% Preallocation idxil = ~idxtt; nil = sum(idxil); numObsPerChunk = 500; nchunk = floor(nil/numObsPerChunk); ei = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]); beta313 = [IncrementalMdl.Beta(end); zeros(nchunk,1)]; Xil = X(:,idxil); Yil = Y(idxil); Wil = NYCHousing2015.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), ... 'ObservationsIn','columns','Weights',Wil(idx)); ei{j,:} = IncrementalMdl.Metrics{"EpsilonInsensitiveLoss",:}; IncrementalMdl = fit(IncrementalMdl,Xil(:,idx),Yil(idx),'ObservationsIn','columns', ... 'Weights',Wil(idx)); beta313(j + 1) = IncrementalMdl.Beta(end); end
IncrementalMdl
is an incrementalRegressionLinear
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 performance metrics and estimated coefficient ${\beta}_{313}$.
t = tiledlayout(2,1); nexttile h = plot(ei.Variables); xlim([0 nchunk]) ylabel('Epsilon Insensitive Loss') legend(h,ei.Properties.VariableNames) nexttile plot(beta313) ylabel('\beta_{313}') xlim([0 nchunk]) xlabel(t,'Iteration')
The cumulative loss gradually changes with each iteration (chunk of 500 observations), whereas the window loss jumps. Because the metrics window is 200 by default, updateMetrics
measures the performance based on the latest 200 observations in each 500 observation chunk.
${\beta}_{313}$ changes abruptly, then levels off as fit
processes chunks of observations.
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 for regression fit
and updateMetricsAndFit
use the more conservative ScInOL1 version of the algorithm.
Tips
After creating a model, you can generate C/C++ code that performs incremental learning on a data stream. Generating C/C++ code requires MATLAB Coder™. For details, see Introduction to Code Generation.
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
hyperparameters and when they are estimated, or tuned.
Hyperparameter  Model Property  Usage  Conditions 

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

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

Half the width of the epsilon insensitive band  Epsilon
 Control the number of support vectors  The hyperparameter is estimated when both of these conditions apply:

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 of the incremental learning model Mdl
.
When you set
'Standardize',true
and 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
incrementalRegressionLinear
by converting a traditionally trained SVM regression model (CompactRegressionSVM
orRegressionSVM
), and theMu
andSigma
properties of the model being converted are empty arrays[]
, incremental learning functions do not standardize predictor variables. If theMu
andSigma
properties of the model being converted 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 create
incrementalRegressionLinear
by converting a linear regression model (RegressionLinear
), incremental learning functions does not standardize the data 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.
$${\mu}_{j}^{\ast}=\frac{1}{{\displaystyle \sum _{k}{w}_{k}}}{\displaystyle \sum _{k}{w}_{k}{x}_{jk}}.$$
$${\left({\sigma}_{j}^{\ast}\right)}^{2}=\frac{1}{{\displaystyle \sum _{k}{w}_{k}}}{\displaystyle \sum _{k}{w}_{k}{\left({x}_{jk}{\mu}_{j}^{\ast}\right)}^{2}}.$$
w_{j} is observation weight j.
Performance Metrics
The
updateMetrics
andupdateMetricsAndFit
functions track model performance metrics ('Metrics'
) from new data when the incremental model is warm (IsWarm property). An incremental model becomes warm afterfit
orupdateMetricsAndFit
fit the incremental model to MetricsWarmupPeriod observations, which is the metrics warmup period.If EstimationPeriod > 0, the functions estimate hyperparameters before fitting the model to data. Therefore, the functions must process an additional
EstimationPeriod
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 by the MetricsWindowSize namevalue pair argument.MetricsWindowSize
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 filled, overwrite
Mdl.Metrics.Window
with the weighted average performance in the metrics window. If the buffer is overfilled 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
prediction 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.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
All object functions of an
incrementalRegressionLinear
model object support code generation.If you configure
Mdl
to shuffle data (see Solver and Shuffle), thefit
function randomly shuffles each incoming batch of observations before it fits the model to the batch. The order of the shuffled observations might not match the order generated by MATLAB.When you generate code that loads or creates an
incrementalRegressionLinear
model object, theNumPredictors
property must reflect the number of predictor variables.
For more information, see Introduction to Code Generation.
Version History
Introduced in R2020b
See Also
Functions
fit
updateMetrics
updateMetricsAndFit
predict
incrementalLearner (RegressionLinear)
incrementalLearner (RegressionSVM)
Objects
Topics
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