Linear regression model for incremental learning
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
.
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 SVM or linear regression model object, you can convert the traditionally trained model to an incrementalRegressionLinear
model object by passing it to the incrementalLearner
function. This table contains links to the appropriate reference pages.
Convertible Model Object  Conversion Function 

RegressionSVM or CompactRegressionSVM  incrementalLearner 
RegressionLinear  incrementalLearner 
Call an incremental learning function — fit
, updateMetrics
, and updateMetricsAndFit
accept a configured incrementalRegressionLinear
model object and data as input, and return an incrementalRegressionLinear
model object updated with information learned from the input model and data.
returns a default incremental linear model for 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 or additional options using namevalue pair 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
.
Specify optional
commaseparated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.
'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 the commaseparated pair consisting of 'Metrics'
and 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 lossfunction names and which learners, specified in Mdl.Learner
, support them. You can specify more than one loss function by using a string vector.
Name  Description  Learners 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 select 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 @customeMetric2 '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"
if Mdl.Learner
is 'svm'
.
Metrics
is "mse"
if Mdl.Learner
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 the commaseparated pair consisting of 'Standardize'
and 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
You can set most properties by using namevalue pair 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
.
Beta
— Linear model coefficients βThis property is readonly.
Linear model coefficients β, specified as a NumPredictors
by1 numeric vector.
If you convert a traditionally trained model to create Mdl
, Beta
is specified by the value of the Beta
property of the traditionally trained model. Otherwise, by default, Beta
is zeros(NumPredictors,1)
.
Data Types: single
 double
Bias
— Model intercept β_{0}This property is readonly.
Model intercept β_{0}, or bias term, specified as a numeric scalar.
If you convert a traditionally trained model to create Mdl
, Bias
is specified by the value of the Bias
property of the traditionally trained model. Otherwise, by default, Bias
is 0
.
Data Types: single
 double
Epsilon
— Half of the width of epsilon insensitive band'auto'
 nonnegative scalarThis property is readonly.
Half of the width of the epsilon insensitiveband, specified as 'auto'
or a nonnegative 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
is iqr(Y)/13.49
, where Y
is the estimation period response data.
If iqr(Y)
= 0 or before you fit Mdl
to data, Epsilon
is 0.1
.
If you convert a traditionally trained SVM regression model to create Mdl
(Learner
is 'svm'
), Epsilon
is specified by the value of the Epsilon
property of the traditionally trained model.
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'leastsquares'
 'svm'
This property is readonly.
Linear regression model type, specified as 'leastsquares'
or 'svm'
.
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 

'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' 
'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' 
If you convert a traditionally trained model to create Mdl
, Leaner
is the learner of the traditionally trained model.
If the traditionally trained model is CompactRegressionSVM
or RegressionSVM
, Learner
is 'svm'
.
If the traditionally trained model is RegressionLinear
, Learner
is the value of the Learner
property of the traditionally trained model.
NumPredictors
— Number of predictor variables0
(default)  nonnegative numeric scalarThis property is readonly.
Number of predictor variables, specified as a nonnegative numeric scalar.
If you convert a traditionally trained model to create Mdl
, NumPredictors
is specified by the congruent property of the traditionally trained model. Otherwise, incremental fitting functions infer NumPredictors
from the predictor data during training.
Data Types: uint32
NumTrainingObservations
— Number of observations fit to incremental model0
(default)  nonnegative numeric scalarThis 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: uint64
ResponseTransform
— Response transformation function'none'
 function handleThis property is readonly.
Response transformation function, specified as 'none'
or a 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).
If you convert a traditionally trained model to create Mdl
, ResponseTransform
is specified by the congruent property of the traditionally trained model.
Otherwise, ResponseTransform
is 'none'
.
Data Types: char
 function_handle
EstimationPeriod
— Number of observations processed to estimate hyperparametersThis 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
forces 'EstimationPeriod'
to 0
.
If Mdl
is not prepared for incremental learning, incrementalRegressionLinear
sets 'EstimationPeriod'
to 1000
.
For more details, see Estimation Period.
Data Types: single
 double
FitBias
— Linear model intercept inclusion flagtrue
 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}.
If you convert a traditionally trained linear regression model (RegressionLinear
) to create Mdl
, FitBias
is specified by the value of the ModelParameters.FitBias
property of the traditionally trained model.
Data Types: logical
Mu
— Predictor means[]
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[]
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'
(default)  'sgd'
 'asgd'
This property is readonly.
Objective function minimization technique, specified as a value in this table.
Value  Description  Notes 

'scaleinvariant'  Adaptive scaleinvariant solver for incremental learning [1]  This algorithm is parameter free and can adapt to differences in predictor scales. Try this algorithm before using SGD or ASGD. 
'sgd'  Stochastic gradient descent (SGD) [3][2]  To train effectively with SGD, standardize the data and specify adequate values for hyperparameters using options listed in SGD and ASGD Solver Parameters. 
'asgd'  Average stochastic gradient descent (ASGD) [4]  To train effectively with ASGD, standardize the data and specify adequate values for hyperparameters using options listed in SGD and ASGD Solver Parameters. 
If you convert a traditionally trained linear regression model (RegressionLinear
) to create Mdl
, whose ModelParameters.Solver
property is 'sgd'
or 'asgd'
, Solver
is specified by the ModelParameters.Solver
property of the traditionally trained model.
Data Types: char
 string
SolverOptions
— Objective solver configurationsThis 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
BatchSize
— Minibatch sizeThis property is readonly.
Minibatch size, specified as a positive integer. At each iteration during training, incrementalRegressionLinear
uses min(BatchSize,numObs)
observations to compute the subgradient, where numObs
is the number of observations in the training data passed to fit
or updateMetricsAndFit
.
If you convert a traditionally trained linear regression model (RegressionLinear
) to create Mdl
, whose ModelParameters.Solver
property is 'sgd'
or 'asgd'
, BatchSize
is specified by the ModelParameters.BatchSize
property of the traditionally trained model. Otherwise, the default is 10
.
Data Types: single
 double
Lambda
— Ridge (L2) regularization term strengthThis property is readonly.
Ridge (L2) regularization term strength, specified as a nonnegative scalar.
If you convert a traditionally trained linear model for ridge regression (RegressionLinear
object with the Regularization
property equal to 'ridge (L2)'
) to create Mdl
, Lambda
is specified by the value of the Lambda
property of the traditionally trained model. Otherwise, the default is 1e5
.
Data Types: double
 single
LearnRate
— Learning rate'auto'
 positive scalarThis property is readonly.
Learning rate, specified as 'auto'
or a positive scalar. LearnRate
controls the optimization step size by scaling the objective subgradient.
When you specify 'auto'
:
If EstimationPeriod
is 0
, the initial learning rate is 0.7
.
If EstimationPeriod
>
0
, the initial learning rate is 1/sqrt(1+max(sum(X.^2,obsDim)))
, where obsDim
is 1
if the observations compose the columns of the predictor data, and 2
otherwise. fit
and updateMetricsAndFit
set the value when you pass the model and training data to either.
If you convert a traditionally trained linear regression model (RegressionLinear
) to create Mdl
, whose ModelParameters.Solver
property is 'sgd'
or 'asgd'
, LearnRate
is specified by the ModelParameters.LearnRate
property of the traditionally trained model.
The LearnRateSchedule
property determines the learning rate for subsequent learning cycles.
Example: 'LearnRate',0.001
Data Types: single
 double
 char
 string
LearnRateSchedule
— Learning rate schedule'decaying'
(default)  'constant'
This property is readonly.
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 $${\gamma}_{t}=\frac{{\gamma}_{0}}{{\left(1+\lambda {\gamma}_{0}t\right)}^{c}}.$$

If you convert a traditionally trained linear regression model (RegressionLinear
) to create Mdl
, whose ModelParameters.Solver
property is 'sgd'
or 'asgd'
, LearnRate
is 'decaying'
.
Data Types: char
 string
Shuffle
— Flag for shuffling observations in batchtrue
(default)  false
This property is readonly.
Flag for shuffling the observations in the batch at each learning cycle, specified as a value in this table.
Value  Description 

true  The software shuffles observations in each incoming batch of data before processing the set. This action reduces bias induced by the sampling scheme. 
false  The software processes the data in the order received. 
Data Types: logical
IsWarm
— Flag indicating whether model tracks performance metricsfalse
 true
This property is readonly.
Flag indicating whether the incremental model tracks performance metrics, specified as false
or true
. The incremental model Mdl
is warm (IsWarm
becomes true
) after incremental fitting functions fit MetricsWarmupPeriod
observations to the incremental model (that is, EstimationPeriod
+ MetricsWarmupPeriod
observations).
Value  Description 

true  The incremental model Mdl is warm. Consequently, updateMetrics and updateMetricsAndFit track performance metrics in the Metrics property of Mdl . 
false  updateMetrics and updateMetricsAndFit do not track performance metrics. 
Data Types: logical
Metrics
— Model performance metricsThis 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 pair argument.
The columns of Metrics
are labeled Cumulative
and Window
.
Cumulative
: Element j
is the model performance, as measured by metric j
, from the time the model became warm (IsWarm is 1
).
Window
: Element j
is the model performance, as measured by metric j
, evaluated over all observations within the window specified by the MetricsWindowSize
property. The software updates Window
after it processes MetricsWindowSize
observations.
Rows are labeled by the specified metrics. For details, see 'Metrics'
.
Data Types: table
MetricsWarmupPeriod
— Number of observations fit before tracking performance metrics1000
(default)  nonnegative integerThis 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.
For more details, see IsWarm and Performance Metrics.
Data Types: single
 double
MetricsWindowSize
— Number of observations to use to compute window performance metrics200
(default)  positive integerThis property is readonly.
Number of observations to use to compute window performance metrics, specified as a positive integer.
For more details on performance metrics options, see Performance Metrics.
Data Types: single
 double
fit  Train incremental learning model 
updateMetricsAndFit  Update incremental learning model performance metrics on new data, then train model 
updateMetrics  Update incremental learning model performance metrics on new data 
loss  Loss of incremental learning model on batch of data 
predict  Predict responses for new observations from incremental learning model 
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' Properties, Methods
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 floating point scalar by using the 'Epsilon'
namevalue pair 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 bt 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 observation.
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); % Incremental fitting 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
IncrementalMdl
is an incrementalRegressionLinear
model object trained on all the data in the stream. While updateMetricsAndFit
processes the first 1000 observations, it stores a buffer 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 observation, and then fits the model to that observation.
To see how the performance metrics and ${\beta}_{1}$ evolved during training, plot them on separate subplots.
figure; subplot(2,1,1) plot(beta1) ylabel('\beta_1') xlim([0 nchunk]); xline(Mdl.EstimationPeriod/numObsPerChunk,'r.'); subplot(2,1,2) 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('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 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).
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 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 insenstivie band for SVM of 0.05 work well for the problem.
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' Properties, Methods
Mdl
is an incrementalRegressionLinear
model object configured for incremental learning without an estimation period.
Fit the incremental model to the data by using updateMetricsAndfit
function. At each iteration:
Simulate a data stream by processing a chunk of 50 observations. Note that chunk size is different from SGD batch size.
Overwrite the previous incremental model with a new one fitted to the incoming observation.
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); % Incremental fitting 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
IncrementalMdl
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 observation, and then fits the model to that observation.
To see how the performance metrics and ${\beta}_{10}$ evolved during training, plot them on separate subplots.
figure; subplot(2,2,1) plot(beta10) ylabel('\beta_{10}') xlim([0 nchunk]); xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'r.'); xlabel('Iteration') subplot(2,2,2) h = plot(ei.Variables); xlim([0 nchunk]); ylabel('Epsilon Insensitive Loss') xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'r.'); legend(h,ei.Properties.VariableNames) xlabel('Iteration') subplot(2,2,3) h = plot(mse.Variables); xlim([0 nchunk]); ylabel('MSE') xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'r.'); legend(h,mse.Properties.VariableNames) xlabel('Iteration') subplot(2,2,4) h = plot(mae.Variables); xlim([0 nchunk]); ylabel('MAE') xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'r.'); legend(h,mae.Properties.VariableNames) xlabel('Iteration')
The plot suggests that updateMetricsAndFit
does the following:
Fit ${\beta}_{10}$ during all incremental learning iterations.
Compute 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).
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: [313×1 double] Bias: 0.1116 Lambda: 2.1977e05 Learner: 'svm' Properties, Methods
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: [1×2 table] ResponseTransform: 'none' Beta: [313×1 double] Bias: 0.1116 Learner: 'svm' Properties, Methods
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 the Metrics
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 ${\beta}_{313}$.
figure; subplot(2,1,1) h = plot(ei.Variables); xlim([0 nchunk]); ylabel('Epsilon Insensitive Loss') legend(h,ei.Properties.VariableNames) subplot(2,1,2) plot(beta313) ylabel('\beta_{313}') xlim([0 nchunk]); xlabel('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.
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.
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 can require tuning. 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.
During the estimation period, incremental fitting functions fit
and updateMetricsAndFit
use the first incoming EstimationPeriod
observations to estimate (tune) hyperparameters required for incremental training. This table describes the hyperparameters and when they are estimated or tuned. Estimation occurs only when EstimationPeriod
is positive.
Hyperparameter  Model Property  Use  Hyperparameters Estimated 

Predictor means and standard deviations 
 Standardize predictor data  When both these conditions apply:

Learning rate  LearnRate  Adjust solver step size  When both these conditions apply:

Half the width of the epsilon insensitive band  Epsilon  Control number of support vectors  When both these conditions apply:

The functions fit only the last estimation period observation to the incremental model, and they do not use any of the observations to track the performance of the model. At the end of the estimation period, the functions update the properties that store the hyperparameters.
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 (see EstimationPeriod), and Mdl.Mu
and Mdl.Sigma
are empty, incremental fitting functions estimate means and standard deviations using the estimation period observations.
If you set 'Standardize','auto'
(the default), the following conditions apply.
If you create incrementalRegressionLinear
by converting a traditionally trained SVM regression model (CompactRegressionSVM
or RegressionSVM
), and the Mu
and Sigma
properties of the model being converted are empty arrays []
, incremental learning functions do not standardize predictor variables. If the Mu
and Sigma
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 (see EstimationPeriod).
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.
The updateMetrics
and updateMetricsAndFit
functions track model performance metrics ('Metrics'
) from new data when the incremental model is warm (IsWarm property). An incremental model is warm after fit
or updateMetricsAndFit
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
and Window
, with individual metrics in rows. When the incremental model is warm, updateMetrics
and updateMetricsAndFit
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 updates Window
metrics. For example, if MetricsWindowSize
is 20, the functions compute metrics based on the last 20 observations in the supplied data (X((end – 20 + 1):end,:)
and Y((end – 20 + 1):end)
).
Incremental functions that track performance metrics within a window use the following process:
For each specified metric, store a buffer of length MetricsWindowSize
and a buffer of observation weights.
Populate elements of the metrics buffer with the model performance based on batches of incoming observations, and store corresponding observations 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 incoming MetricsWindowSize
observations enter the buffer, and the earliest observations are removed from the buffer. For example, suppose MetricsWindowSize
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.
[1] Kempka, Michał, Wojciech Kotłowski, and Manfred K. Warmuth. "Adaptive ScaleInvariant Online Algorithms for Learning Linear Models." CoRR (February 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.
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