# fit

Train linear model for incremental learning

## Description

The `fit`

function fits a configured incremental learning model for linear regression (`incrementalRegressionLinear`

object) or linear binary classification (`incrementalClassificationLinear`

object) to streaming data. To additionally track performance metrics using the data as it arrives, use `updateMetricsAndFit`

instead.

To fit or cross-validate a regression or classification model to an entire batch of data at once, see the other machine learning models in Regression or Classification.

## Examples

### Incrementally Train Model

Create a default incremental linear SVM model for binary classification. Specify an estimation period of 5000 observations and the SGD solver.

Mdl = incrementalClassificationLinear('EstimationPeriod',5000,'Solver','sgd')

Mdl = incrementalClassificationLinear IsWarm: 0 Metrics: [1x2 table] ClassNames: [1x0 double] ScoreTransform: 'none' Beta: [0x1 double] Bias: 0 Learner: 'svm' Properties, Methods

`Mdl`

is an `incrementalClassificationLinear`

model. All its properties are read-only.

`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, in chunks of 50 observations at a time, by using the `fit`

function. At each iteration:

Simulate a data stream by processing 50 observations.

Overwrite the previous incremental model with a new one fitted to the incoming observations.

Store ${\beta}_{1}$, the number of training observations, and the prior probability of whether the subject moved (

`Y`

=`true`

) to see how they evolve during incremental training.

% Preallocation numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); beta1 = zeros(nchunk,1); numtrainobs = zeros(nchunk,1); priormoved = zeros(nchunk,1); % Incremental fitting for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j-1) + 1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; Mdl = fit(Mdl,X(idx,:),Y(idx)); beta1(j) = Mdl.Beta(1); numtrainobs(j) = Mdl.NumTrainingObservations; priormoved(j) = Mdl.Prior(Mdl.ClassNames == true); end

`Mdl`

is an `incrementalClassificationLinear`

model object trained on all the data in the stream.

To see how the parameters evolve during incremental learning, plot them on separate tiles.

tiledlayout(2,2) nexttile plot(beta1) ylabel('\beta_1') xline(Mdl.EstimationPeriod/numObsPerChunk,'r-.') xlabel('Iteration') axis tight nexttile plot(numtrainobs) ylabel('Number of Training Observations') xline(Mdl.EstimationPeriod/numObsPerChunk,'r-.') xlabel('Iteration') axis tight nexttile plot(priormoved) ylabel('\pi(Subject Is Moving)') xline(Mdl.EstimationPeriod/numObsPerChunk,'r-.') xlabel('Iteration') axis tight

The plot suggests that `fit`

does not fit the model to the data or update the parameters until after the estimation period.

### Specify Orientation of Observations and Observation Weights

Train a linear model for binary classification by using `fitclinear`

, convert it to an incremental learner, track its performance, and fit it to streaming data. Orient the observations in columns, and specify observation weights.

**Load and Preprocess Data**

Load the human activity data set. Randomly shuffle the data. Orient the observations of the predictor data in columns.

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 not moving (`Y`

= `false`

) has double the quality than when the subject was moving. Create a weight variable that attributes 2 to observations collected from a still subject, and 1 to a moving subject.

W = ones(n,1) + ~Y;

**Train Linear Model for Binary Classification**

Fit a linear model for binary classification to a random sample of half the data.

idxtt = randsample([true false],n,true); TTMdl = fitclinear(X(:,idxtt),Y(idxtt),'ObservationsIn','columns', ... 'Weights',W(idxtt))

TTMdl = ClassificationLinear ResponseName: 'Y' ClassNames: [0 1] ScoreTransform: 'none' Beta: [60x1 double] Bias: -0.1107 Lambda: 8.2967e-05 Learner: 'svm' Properties, Methods

`TTMdl`

is a `ClassificationLinear`

model object representing a traditionally trained linear model for binary classification.

**Convert Trained Model**

Convert the traditionally trained classification model to a binary classification linear model for incremental learning.

IncrementalMdl = incrementalLearner(TTMdl)

IncrementalMdl = incrementalClassificationLinear IsWarm: 1 Metrics: [1x2 table] ClassNames: [0 1] ScoreTransform: 'none' Beta: [60x1 double] Bias: -0.1107 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. At each iteration:

Simulate a data stream by processing 50 observations at a time.

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 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 classification error and first estimated coefficient ${\beta}_{1}$.

% Preallocation idxil = ~idxtt; nil = sum(idxil); numObsPerChunk = 50; nchunk = floor(nil/numObsPerChunk); ce = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]); beta1 = [IncrementalMdl.Beta(1); zeros(nchunk,1)]; Xil = X(:,idxil); Yil = Y(idxil); Wil = W(idxil); % Incremental fitting for j = 1:nchunk ibegin = min(nil,numObsPerChunk*(j-1) + 1); iend = min(nil,numObsPerChunk*j); idx = ibegin:iend; IncrementalMdl = updateMetrics(IncrementalMdl,Xil(:,idx),Yil(idx), ... 'ObservationsIn','columns','Weights',Wil(idx)); ce{j,:} = IncrementalMdl.Metrics{"ClassificationError",:}; IncrementalMdl = fit(IncrementalMdl,Xil(:,idx),Yil(idx),'ObservationsIn','columns', ... 'Weights',Wil(idx)); beta1(j + 1) = IncrementalMdl.Beta(1); end

`IncrementalMdl`

is an `incrementalClassificationLinear`

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 ${\beta}_{1}$.

t = tiledlayout(2,1); nexttile h = plot(ce.Variables); xlim([0 nchunk]) ylabel('Classification Error') legend(h,ce.Properties.VariableNames) nexttile plot(beta1) ylabel('\beta_1') xlim([0 nchunk]) xlabel(t,'Iteration')

The cumulative loss is stable and gradually decreases, whereas the window loss jumps.

${\beta}_{1}$ changes gradually, then levels off, as `fit`

processes more chunks.

### Perform Conditional Training

Incrementally train a linear regression model only when its performance degrades.

Load and shuffle the 2015 NYC housing data set. For more details on the data, see NYC Open Data.

load NYCHousing2015 rng(1) % For reproducibility n = size(NYCHousing2015,1); shuffidx = randsample(n,n); NYCHousing2015 = NYCHousing2015(shuffidx,:);

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.

idxnum = varfun(@isnumeric,NYCHousing2015,'OutputFormat','uniform'); X = [dumvarmat NYCHousing2015{:,idxnum}];

Configure a linear regression model for incremental learning so that it does not have an estimation or metrics warm-up period. Specify a metrics window size of 1000. Fit the configured model to the first 100 observations.

Mdl = incrementalRegressionLinear('EstimationPeriod',0, ... 'MetricsWarmupPeriod',0,'MetricsWindowSize',1000); numObsPerChunk = 100; Mdl = fit(Mdl,X(1:numObsPerChunk,:),Y(1:numObsPerChunk));

`Mdl`

is an `incrementalRegressionLinear`

model object.

Perform incremental learning, with conditional fitting, by following this procedure for each iteration:

Simulate a data stream by processing a chunk of 100 observations at a time.

Update the model performance by computing the epsilon insensitive loss, within a 200 observation window.

Fit the model to the chunk of data only when the loss more than doubles from the minimum loss experienced.

When tracking performance and fitting, overwrite the previous incremental model.

Store the epsilon insensitive loss and ${\beta}_{313}$ to see how the loss and coefficient evolve during training.

Track when

`fit`

trains the model.

% Preallocation n = numel(Y) - numObsPerChunk; nchunk = floor(n/numObsPerChunk); beta313 = zeros(nchunk,1); ei = array2table(nan(nchunk,2),'VariableNames',["Cumulative" "Window"]); trained = false(nchunk,1); % Incremental fitting for j = 2:nchunk ibegin = min(n,numObsPerChunk*(j-1) + 1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; Mdl = updateMetrics(Mdl,X(idx,:),Y(idx)); ei{j,:} = Mdl.Metrics{"EpsilonInsensitiveLoss",:}; minei = min(ei{:,2}); pdiffloss = (ei{j,2} - minei)/minei*100; if pdiffloss > 100 Mdl = fit(Mdl,X(idx,:),Y(idx)); trained(j) = true; end beta313(j) = Mdl.Beta(end); end

`Mdl`

is an `incrementalRegressionLinear`

model object trained on all the data in the stream.

To see how the model performance and ${\beta}_{313}$ evolve during training, plot them on separate tiles.

t = tiledlayout(2,1); nexttile plot(beta313) hold on plot(find(trained),beta313(trained),'r.') xlim([0 nchunk]) ylabel('\beta_{313}') xline(Mdl.EstimationPeriod/numObsPerChunk,'r-.') legend('\beta_{313}','Training occurs','Location','southeast') hold off nexttile plot(ei.Variables) xlim([0 nchunk]) ylabel('Epsilon Insensitive Loss') xline(Mdl.EstimationPeriod/numObsPerChunk,'r-.') legend(ei.Properties.VariableNames) xlabel(t,'Iteration')

The trace plot of ${\beta}_{313}$ shows periods of constant values, during which the loss did not double from the minimum experienced.

## Input Arguments

`Mdl`

— Incremental learning model

`incrementalClassificationLinear`

model object | `incrementalRegressionLinear`

model object

Incremental learning model to fit to streaming data, specified as an `incrementalClassificationLinear`

or `incrementalRegressionLinear`

model object. You can create
`Mdl`

directly or by converting a supported, traditionally trained
machine learning model using the `incrementalLearner`

function. For
more details, see the corresponding reference page.

`X`

— Chunk of predictor data

floating-point matrix

Chunk of predictor data to which the model is fit, specified as a floating-point
matrix of *n* observations and `Mdl.NumPredictors`

predictor variables. The value of the `ObservationsIn`

name-value
argument determines the orientation of the variables and observations. The default
`ObservationsIn`

value is `"rows"`

, which indicates that
observations in the predictor data are oriented along the rows of
`X`

.

The length of the observation labels `Y`

and the number of
observations in `X`

must be equal;
`Y(`

is the label of observation
* j*)

*j*(row or column) in

`X`

.**Note**

If

`Mdl.NumPredictors`

= 0,`fit`

infers the number of predictors from`X`

, and sets the corresponding property of the output model. Otherwise, if the number of predictor variables in the streaming data changes from`Mdl.NumPredictors`

,`fit`

issues an error.`fit`

supports only floating-point input predictor data. If your input data includes categorical data, you must prepare an encoded version of the categorical data. Use`dummyvar`

to convert each categorical variable to a numeric matrix of dummy variables. Then, concatenate all dummy variable matrices and any other numeric predictors. For more details, see Dummy Variables.

**Data Types: **`single`

| `double`

`Y`

— Chunk of responses (labels)

categorical array | character array | string array | logical vector | floating-point vector | cell array of character vectors

Chunk of responses (labels) to which the model is fit, specified as a categorical, character, or string array, logical or floating-point vector, or cell array of character vectors for classification problems; or a floating-point vector for regression problems.

The length of the observation labels `Y`

and the number of observations in `X`

must be equal; `Y(`

is the label of observation * j*)

*j*(row or column) in

`X`

.For classification problems:

`fit`

supports binary classification only.When the

`ClassNames`

property of the input model`Mdl`

is nonempty, the following conditions apply:If

`Y`

contains a label that is not a member of`Mdl.ClassNames`

,`fit`

issues an error.The data type of

`Y`

and`Mdl.ClassNames`

must be the same.

**Data Types: **`char`

| `string`

| `cell`

| `categorical`

| `logical`

| `single`

| `double`

**Note**

If an observation (predictor or label) or weight contains at least one missing (

`NaN`

) value,`fit`

ignores the observation. Consequently,`fit`

uses fewer than*n*observations to create an updated model, where*n*is the number of observations in`X`

.The chunk size

*n*and the stochastic gradient descent (SGD) hyperparameter mini-batch size (`Mdl.BatchSize`

) can be different values, and*n*does not have to be an exact multiple of the mini-batch size. If*n*<`Mdl.BatchSize`

,`fit`

uses the*n*available observations when it applies SGD. If*n*>`Mdl.BatchSize`

, the function updates the model with a mini-batch of the specified size multiple times, and then uses the rest of the observations for the last mini-batch. The number of observations for the last mini-batch can be smaller than`Mdl.BatchSize`

.

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value 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: **`'ObservationsIn','columns','Weights',W`

specifies that the
columns of the predictor matrix correspond to observations, and the vector
`W`

contains observation weights to apply during incremental
learning.

`ObservationsIn`

— Predictor data observation dimension

`'rows'`

(default) | `'columns'`

Predictor data observation dimension, specified as the comma-separated pair consisting of `'ObservationsIn'`

and `'columns'`

or `'rows'`

.

**Data Types: **`char`

| `string`

`Weights`

— Chunk of observation weights

floating-point vector of positive values

Chunk of observation weights, specified as the comma-separated pair consisting of `'Weights'`

and a floating-point vector of positive values. `fit`

weighs the observations in `X`

with the corresponding values in `Weights`

. The size of `Weights`

must equal *n*, which is the number of observations in `X`

.

By default, `Weights`

is `ones(`

.* n*,1)

For more details, including normalization schemes, see Observation Weights.

**Data Types: **`double`

| `single`

## Output Arguments

`Mdl`

— Updated incremental learning model

`incrementalClassificationLinear`

model object | `incrementalRegressionLinear`

model object

Updated incremental learning model, returned as an incremental learning model object of the same data type as the input model `Mdl`

, either `incrementalClassificationLinear`

or `incrementalRegressionLinear`

.

If `Mdl.EstimationPeriod`

> 0, the incremental fitting functions
`updateMetricsAndFit`

and `fit`

estimate
hyperparameters using the first `Mdl.EstimationPeriod`

observations
passed to either function; they do not train the input model to that data. However, if
an incoming chunk of *n* observations is greater than or equal to the
number of observations remaining in the estimation period *m*,
`fit`

estimates hyperparameters using the first
*n* – *m* observations, and fits the input model to
the remaining *m* observations. Consequently, the software updates the
`Beta`

and `Bias`

properties, hyperparameter
properties, and recordkeeping properties such as
`NumTrainingObservations`

.

For classification problems, if the `ClassNames`

property of the input model `Mdl`

is an empty array, `fit`

sets the `ClassNames`

property of the output model `Mdl`

to `unique(Y)`

.

## Tips

Unlike traditional training, incremental learning might not have a separate test (holdout) set. Therefore, to treat each incoming chunk of data as a test set, pass the incremental model and each incoming chunk to

`updateMetrics`

before training the model on the same data.

## Algorithms

### Observation Weights

For classification problems, if the prior class probability distribution is known (in other words, the prior distribution is not empirical), `fit`

normalizes observation weights to sum to the prior class probabilities in the respective classes. This action implies that observation weights are the respective prior class probabilities by default.

For regression problems or if the prior class probability distribution is empirical, the software normalizes the specified observation weights to sum to 1 each time you call `fit`

.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Use

`saveLearnerForCoder`

,`loadLearnerForCoder`

, and`codegen`

(MATLAB Coder) to generate code for the`fit`

function. Save a trained model by using`saveLearnerForCoder`

. Define an entry-point function that loads the saved model by using`loadLearnerForCoder`

and calls the`fit`

function. Then use`codegen`

to generate code for the entry-point function.To generate single-precision C/C++ code for

`fit`

, specify the name-value argument`"DataType","single"`

when you call the`loadLearnerForCoder`

function.This table contains notes about the arguments of

`fit`

. Arguments not included in this table are fully supported.Argument Notes and Limitations `Mdl`

For usage notes and limitations of the model object, see

`incrementalClassificationLinear`

or`incrementalRegressionLinear`

.`X`

Batch-to-batch, the number of observations can be a variable size, but must equal the number of observations in

`Y`

.The number of predictor variables must equal to

`Mdl.NumPredictors`

.`X`

must be`single`

or`double`

.

`Y`

Batch-to-batch, the number of observations can be a variable size, but must equal the number of observations in

`X`

.For classification problems, all labels in

`Y`

must be represented in`Mdl.ClassNames`

.`Y`

and`Mdl.ClassNames`

must have the same data type.

The following restrictions apply:

If you configure

`Mdl`

to shuffle data (`Mdl.Shuffle`

is`true`

, or`Mdl.Solver`

is`'sgd'`

or`'asgd'`

), the`fit`

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^{®}. Therefore, the fitted coefficients computed in MATLAB and the generated code might not be equal.Use a homogeneous data type for all floating-point input arguments and object properties, specifically, either

`single`

or`double`

.

For more information, see Introduction to Code Generation.

## Version History

**Introduced in R2020b**

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