# fit

Train kernel model for incremental learning

## Description

The `fit`

function fits a configured incremental learning model
for kernel regression (`incrementalRegressionKernel`

object) or binary kernel classification (`incrementalClassificationKernel`

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

instead.

To fit or cross-validate a kernel regression or classification model to an entire batch of
data at once, see `fitrkernel`

or
`fitckernel`

,
respectively.

## Examples

### Incrementally Train Model

Configure incremental learning options for an `incrementalClassificationKernel`

model object when you call the `incrementalClassificationKernel`

function. Fit the model to incoming observations.

Create an incremental kernel model for binary classification. Specify an estimation period of 5000 observations and the stochastic gradient descent (SGD) solver.

`Mdl = incrementalClassificationKernel(EstimationPeriod=5000,Solver="sgd")`

Mdl = incrementalClassificationKernel IsWarm: 0 Metrics: [1x2 table] ClassNames: [1x0 double] ScoreTransform: 'none' NumExpansionDimensions: 0 KernelScale: 1 Properties, Methods

`Mdl`

is an `incrementalClassificationKernel`

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 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); 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)); numtrainobs(j) = Mdl.NumTrainingObservations; priormoved(j) = Mdl.Prior(Mdl.ClassNames == true); end

`Mdl`

is an `incrementalClassificationKernel`

model object trained on all the data in the stream.

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

t = tiledlayout(2,1); nexttile plot(numtrainobs) xlim([0 nchunk]) ylabel("Number of Training Observations") xline(Mdl.EstimationPeriod/numObsPerChunk,"-.") nexttile plot(priormoved) xlim([0 nchunk]) ylabel("\pi(Subject Is Moving)") xline(Mdl.EstimationPeriod/numObsPerChunk,"-.") xlabel(t,"Iteration")

The plot suggests that `fit`

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

### Specify Observation Weights

Train a kernel model for binary classification by using `fitckernel`

, and convert it to an incremental learner by using `incrementalLearner`

. Track the model performance and fit the model to streaming data. Specify the observation weights when you call `fitckernel`

and incremental learning functions.

**Load and Preprocess Data**

Load the human activity data set. Randomly shuffle the data.

load humanactivity rng(1) % For reproducibility n = numel(actid); idx = randsample(n,n); X = feat(idx,:); Y = actid(idx);

For details on the data set, enter `Description`

at the command line.

Responses can be one of five classes: Sitting, Standing, Walking, Running, or Dancing. Dichotomize the response by identifying whether the subject is moving (`actid`

> 2).

Y = Y > 2;

Suppose that the data collected when the subject was 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 stationary subject, and 1 to a moving subject.

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

**Train Kernel Model for Binary Classification**

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

idxtt = randsample([true false],n,true); TTMdl = fitckernel(X(idxtt,:),Y(idxtt),Weights=W(idxtt))

TTMdl = ClassificationKernel ResponseName: 'Y' ClassNames: [0 1] Learner: 'svm' NumExpansionDimensions: 2048 KernelScale: 1 Lambda: 8.2967e-05 BoxConstraint: 1 Properties, Methods

`TTMdl`

is a `ClassificationKernel`

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

**Convert Trained Model**

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

IncrementalMdl = incrementalLearner(TTMdl)

IncrementalMdl = incrementalClassificationKernel IsWarm: 1 Metrics: [1x2 table] ClassNames: [0 1] ScoreTransform: 'none' NumExpansionDimensions: 2048 KernelScale: 1 Properties, Methods

`IncrementalMdl`

is an `incrementalClassificationKernel`

model. All its properties are read-only.

**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 the observation weights.Call

`fit`

to fit the model to the incoming chunk of observations. Overwrite the previous incremental model to update the model parameters. Specify the observation weights.Store the classification error and number of training observations.

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

`IncrementalMdl`

is an `incrementalClassificationKernel`

model object trained on all the data in the stream.

Alternatively, you can use `updateMetricsAndFit`

to update performance metrics of the model given a new chunk of data, and then fit the model to the data.

Plot a trace plot of the number of training observations and the performance metrics.

t = tiledlayout(2,1); nexttile plot(numtrainobs) xlim([0 nchunk]) ylabel("Number of Training Observations") nexttile plot(ce.Variables) xlim([0 nchunk]) legend(ce.Properties.VariableNames) ylabel("Classification Error") xlabel(t,"Iteration")

The plot suggests that the `fit`

function fits the model during all incremental learning iterations. The cumulative loss is stable and gradually decreases, whereas the window loss jumps.

### Perform Conditional Training

Incrementally train a kernel 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 = [];

To reduce computational cost for this example, remove the `NEIGHBORHOOD`

column, which contains a categorical variable with 254 categories.

NYCHousing2015.NEIGHBORHOOD = [];

Create dummy variable matrices from the other categorical predictors.

catvars = ["BOROUGH","BUILDINGCLASSCATEGORY"]; dumvarstbl = varfun(@(x)dummyvar(categorical(x)),NYCHousing2015, ... InputVariables=catvars); dumvarmat = table2array(dumvarstbl); NYCHousing2015(:,catvars) = [];

Treat all other numeric variables in the table as 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 kernel 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. Prepare the model for `updateMetrics`

by fitting it to the first 100 observations.

```
Mdl = incrementalRegressionKernel(EstimationPeriod=0, ...
MetricsWarmupPeriod=0,MetricsWindowSize=1000);
initobs = 100;
Mdl = fit(Mdl,X(1:initobs,:),Y(1:initobs));
```

`Mdl`

is an `incrementalRegressionKernel`

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 number of training observations to see how they evolve during training.

Track when

`fit`

trains the model.

% Preallocation numObsPerChunk = 100; nchunk = floor((n - initobs)/numObsPerChunk); ei = array2table(nan(nchunk,2),VariableNames=["Cumulative","Window"]); numtrainobs = zeros(nchunk,1); trained = false(nchunk,1); % Incremental fitting for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j-1) + 1 + initobs); iend = min(n,numObsPerChunk*j + initobs); 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 numtrainobs(j) = Mdl.NumTrainingObservations; end

`Mdl`

is an `incrementalRegressionKernel`

model object trained on all the data in the stream.

To see how the number of training observations and model performance evolve during training, plot them on separate tiles.

t = tiledlayout(2,1); nexttile plot(numtrainobs) hold on plot(find(trained),numtrainobs(trained),"r.") xlim([0 nchunk]) ylabel("Number of Training Observations") legend("Number of Training Observations","Training occurs",Location="best") hold off nexttile plot(ei.Variables) xlim([0 nchunk]) ylabel("Epsilon Insensitive Loss") legend(ei.Properties.VariableNames) xlabel(t,"Iteration")

The trace plot of the number of training observations shows periods of constant values, during which the loss does not double from the minimum experienced.

## Input Arguments

`Mdl`

— Incremental learning model

`incrementalClassificationKernel`

model object | `incrementalRegressionKernel`

model object

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

or `incrementalRegressionKernel`

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, specified as a floating-point matrix of *n*
observations and `Mdl.NumPredictors`

predictor variables.

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) 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), specified as a categorical, character, or string array, a logical or floating-point vector, or a 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) 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`

`weights`

— Chunk of observation weights

floating-point vector of positive values

Chunk of observation weights, specified as 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*, 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`

**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.SolverOptions.BatchSize`

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

uses the`BatchSize`

observations when it applies SGD for each learning cycle. The number of observations in the last mini-batch for the last learning cycle can be less than or equal to`Mdl.SolverOptions.BatchSize`

.

## Output Arguments

`Mdl`

— Updated incremental learning model

`incrementalClassificationKernel`

model object | `incrementalRegressionKernel`

model object

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

, either `incrementalClassificationKernel`

or `incrementalRegressionKernel`

.

If `Mdl.EstimationPeriod`

> 0,
`fit`

estimates hyperparameters using the first
`Mdl.EstimationPeriod`

observations passed to it; the function does not
train the input model using 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 model
parameters, 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`

.

## Version History

**Introduced in R2022a**

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