incrementalRegressionKernel

Kernel regression model for incremental learning

Description

The `incrementalRegressionKernel` function creates an `incrementalRegressionKernel` model object, which represents a binary Gaussian kernel regression model for incremental learning. The kernel model maps data in a low-dimensional space into a high-dimensional space, then fits a linear model in the high-dimensional space. Supported linear models include support vector machine (SVM) and least-squares regression.

Unlike other Statistics and Machine Learning Toolbox™ model objects, `incrementalRegressionKernel` can be called directly. Also, you can specify learning options, such as performance metrics configurations and the objective solver, before fitting the model to data. After you create an `incrementalRegressionKernel` object, it is prepared for incremental learning.

`incrementalRegressionKernel` is best suited for incremental learning. For a traditional approach to training a kernel regression model (such as creating a model by fitting it to data, performing cross-validation, tuning hyperparameters, and so on), see `fitrkernel`.

Creation

You can create an `incrementalRegressionKernel` model object in several ways:

• Call the function directly — Configure incremental learning options, or specify learner-specific options, by calling `incrementalRegressionKernel` directly. This approach is best when you do not have data yet or you want to start incremental learning immediately.

• Convert a traditionally trained model — To initialize a model for incremental learning using the model parameters and hyperparameters of a trained model object (`RegressionKernel`), you can convert the traditionally trained model to an `incrementalRegressionKernel` model object by passing it to the `incrementalLearner` function.

• Call an incremental learning function`fit`, `updateMetrics`, and `updateMetricsAndFit` accept a configured `incrementalRegressionKernel` model object and data as input, and return an `incrementalRegressionKernel` model object updated with information learned from the input model and data.

Syntax

``Mdl = incrementalRegressionKernel()``
``Mdl = incrementalRegressionKernel(Name=Value)``

Description

example

````Mdl = incrementalRegressionKernel()` returns a default incremental learning model object for binary Gaussian kernel regression, `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.```

example

````Mdl = incrementalRegressionKernel(Name=Value)` sets properties and additional options using name-value arguments. For example, `incrementalRegressionKernel(Solver="sgd",LearnRateSchedule="constant")` specifies to use the stochastic gradient descent (SGD) solver with a constant learning rate.```

Input Arguments

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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.

Example: `Metrics="mse",MetricsWarmupPeriod=100` sets the model performance metric to the weighted mean squared error and the metrics warm-up period to `100`.

Regression Options

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Random number stream for reproducibility of data transformation, specified as a random stream object. For details, see Random Feature Expansion.

Use `RandomStream` to reproduce the random basis functions used by `incrementalRegressionKernel` to transform the predictor data to a high-dimensional space. For details, see Managing the Global Stream Using RandStream and Creating and Controlling a Random Number Stream.

Example: `RandomStream=RandStream("mlfg6331_64")`

SGD and ASGD (Average SGD) Solver Options

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Mini-batch size, specified as a positive integer. At each learning cycle during training, `incrementalRegressionKernel` uses `BatchSize` observations to compute the subgradient.

The number of observations in the last mini-batch (last learning cycle in each function call of `fit` or `updateMetricsAndFit`) can be smaller than `BatchSize`. For example, if you supply 25 observations to `fit` or `updateMetricsAndFit`, the function uses 10 observations for the first two learning cycles and 5 observations for the last learning cycle.

Example: `BatchSize=5`

Data Types: `single` | `double`

Ridge (L2) regularization term strength, specified as a nonnegative scalar.

Example: `Lambda=0.01`

Data Types: `single` | `double`

Initial learning rate, specified as `"auto"` or a positive scalar.

The learning rate controls the optimization step size by scaling the objective subgradient. `LearnRate` specifies an initial value for the learning rate, and `LearnRateSchedule` determines the learning rate for subsequent learning cycles.

When you specify `"auto"`:

• The initial learning rate is `0.7`.

• If `EstimationPeriod` > `0`, `fit` and `updateMetricsAndFit` change the rate to `1/sqrt(1+max(sum(X.^2,2)))` at the end of `EstimationPeriod`.

Example: `LearnRate=0.001`

Data Types: `single` | `double` | `char` | `string`

Learning rate schedule, specified as a value in this table, where `LearnRate` specifies the initial learning rate ɣ0.

ValueDescription
`"constant"`The learning rate is ɣ0 for all learning cycles.
`"decaying"`

The learning rate at learning cycle t is

`${\gamma }_{t}=\frac{{\gamma }_{0}}{{\left(1+\lambda {\gamma }_{0}t\right)}^{c}}.$`

• λ is the value of `Lambda`.

• If `Solver` is `"sgd"`, c = `1`.

• If `Solver` is `"asgd"`:

• c = `2/3` if `Learner` is `"leastsquares"`.

• c = `3/4` if `Learner` is `"svm"` [4].

Example: `LearnRateSchedule="constant"`

Data Types: `char` | `string`

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Flag for shuffling the observations at each iteration, specified as logical `1` (`true`) or `0` (`false`).

ValueDescription
logical `1` (`true`)The software shuffles the observations in an incoming chunk of data before the `fit` function fits the model. This action reduces bias induced by the sampling scheme.
logical `0` (`false`)The software processes the data in the order received.

Example: `Shuffle=false`

Data Types: `logical`

Performance Metrics Options

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Model performance metrics to track during incremental learning, specified as a built-in 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 built-in loss function names and which learners, specified in `Learner`, support them. You can specify more than one loss function by using a string vector.

NameDescriptionLearner Supporting Metric
`"epsiloninsensitive"`Epsilon insensitive loss`"svm"`
`"mse"`Weighted mean squared error`"svm"` and `"leastsquares"`

For more details on the built-in 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 n-by-1 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 built-in 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 TypeDescription of `Metrics` Property Row NameExample
String or character vectorName of corresponding built-in metricRow name for `"epsiloninsensitive"` is `"EpsilonInsensitiveLoss"`
Structure arrayField nameRow name for `struct(Metric1=@customMetric1)` is `"Metric1"`
Function handle to function stored in a program fileName of functionRow name for `@customMetric` is `"customMetric"`
Anonymous function`CustomMetric_j`, where `j` is metric `j` in `Metrics`Row name for `@(Y,YFit)customMetric(Y,YFit)...` is `CustomMetric_1`

By default:

• `Metrics` is `"epsiloninsensitive"` if `Learner` is `"svm"`.

• `Metrics` is `"mse"` if `Learner` is `"leastsquares"`.

For more details on performance metrics options, see Performance Metrics.

Data Types: `char` | `string` | `struct` | `cell` | `function_handle`

Properties

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You can set most properties by using name-value argument syntax when you call `incrementalRegressionKernel` directly. You can set some properties when you call `incrementalLearner` to convert a traditionally trained model. You cannot set the properties `FittedLoss`, `NumTrainingObservations`, `SolverOptions`, and `IsWarm`.

Regression Model Parameters

Half of the width of the epsilon insensitive band, specified as `"auto"` or a nonnegative scalar. `incrementalRegressionKernel` stores the `Epsilon` value as a numeric scalar.

If you specify `"auto"` when you call `incrementalRegressionKernel`, 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`.

The default `Epsilon` value depends on how you create the model:

• If you convert a traditionally trained model whose `Learner` property 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`

Loss function used to fit the linear model, specified as `'epsiloninsensitive'` or `'mse'`.

ValueAlgorithmLoss Function`Learner` Value
`'epsiloninsensitive'`Support vector machine regressionEpsilon insensitive: $\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,|y-f\left(x\right)|-\epsilon \right]$`'svm'`
`'mse'`Linear regression through ordinary least squaresMean squared error (MSE): $\ell \left[y,f\left(x\right)\right]=\frac{1}{2}{\left[y-f\left(x\right)\right]}^{2}$`'leastsquares'`

Kernel scale parameter, specified as `"auto"` or a positive scalar. `incrementalRegressionKernel` stores the `KernelScale` value as a numeric scalar. The software obtains a random basis for feature expansion by using the kernel scale parameter. For details, see Random Feature Expansion.

If you specify `"auto"` when creating the model object, the software selects an appropriate kernel scale parameter using a heuristic procedure. This procedure uses subsampling, so estimates can vary from one call to another. Therefore, to reproduce results, set a random number seed by using `rng` before training.

The default `KernelScale` value depends on how you create the model:

• If you convert a traditionally trained model to create `Mdl`, `KernelScale` is specified by the corresponding property of the traditionally trained model.

• Otherwise, the default value is `1`.

Data Types: `char` | `string` | `single` | `double`

Linear regression model type, specified as `"svm"` or `"leastsquares"`. `incrementalRegressionKernel` stores the `Learner` value as a character vector.

In the following table, $f\left(x\right)=T\left(x\right)\beta +b.$

• x is an observation (row vector) from p predictor variables.

• $T\left(·\right)$ is a transformation of an observation (row vector) for feature expansion. T(x) maps x in ${ℝ}^{p}$ to a high-dimensional space (${ℝ}^{m}$).

• β is a vector of coefficients.

• b is the scalar bias.

ValueAlgorithmLoss Function`FittedLoss` Value
`"svm"`Support vector machine regressionEpsilon insensitive: $\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,|y-f\left(x\right)|-\epsilon \right]$`'epsiloninsensitive'`
`"leastsquares"`Linear regression through ordinary least squaresMean squared error (MSE): $\ell \left[y,f\left(x\right)\right]=\frac{1}{2}{\left[y-f\left(x\right)\right]}^{2}$`'mse'`

The default `Learner` value depends on how you create the model:

• If you convert a traditionally trained model to create `Mdl`, `Learner` is specified by the corresponding property of the traditionally trained model.

• Otherwise, the default value is `"svm"`.

Number of dimensions of the expanded space, specified as `"auto"` or a positive integer. `incrementalRegressionKernel` stores the `NumExpansionDimensions` value as a numeric scalar.

For `"auto"`, the software selects the number of dimensions using `2.^ceil(min(log2(p)+5,15))`, where `p` is the number of predictors. For details, see Random Feature Expansion.

The default `NumExpansionDimensions` value depends on how you create the model:

• If you convert a traditionally trained model to create `Mdl`, `NumExpansionDimensions` is specified by the corresponding property of the traditionally trained model.

• Otherwise, the default value is `"auto"`.

Data Types: `char` | `string` | `single` | `double`

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 calling `incrementalRegressionKernel` directly, you can specify `NumPredictors` by using name-value argument syntax. If you do not specify the value, then the default value is `0`, and incremental fitting functions infer `NumPredictors` from the predictor data during training.

Data Types: `double`

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`, `incrementalRegressionKernel` does not add the number of observations fit to the traditionally trained model to `NumTrainingObservations`.

Data Types: `double`

Response transformation function, specified as `"none"` or a function handle. `incrementalRegressionKernel` 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 n-by-1 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

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), `incrementalRegressionKernel` forces `EstimationPeriod` to `0`.

• If `Mdl` is not prepared for incremental learning, `incrementalRegressionKernel` sets `EstimationPeriod` to `1000`.

For more details, see Estimation Period.

Data Types: `single` | `double`

Objective function minimization technique, specified as `"scale-invariant"`, `"sgd"`, or `"asgd"`. `incrementalRegressionKernel` stores the `Solver` value as a character vector.

ValueDescriptionNotes
`"scale-invariant"`

• This algorithm is parameter free and can adapt to differences in predictor scales. Try this algorithm before using SGD or ASGD.

• To shuffle an incoming chunk of data before the `fit` function fits the model, set `Shuffle` to `true`.

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

• To train effectively with SGD, specify adequate values for hyperparameters using options listed in SGD and ASGD (Average SGD) Solver Options.

• The `fit` function always shuffles an incoming chunk of data before fitting the model.

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

• To train effectively with ASGD, specify adequate values for hyperparameters using options listed in SGD and ASGD (Average SGD) Solver Options.

• The `fit` function always shuffles an incoming chunk of data before fitting the model.

The default `Solver` value depends on how you create the model:

• If you convert a traditionally trained model to create `Mdl`, the `Solver` name-value argument of the `incrementalLearner` function sets this property. The default value of the argument is `"scale-invariant"`.

• Otherwise, the default value is `"scale-invariant"`.

Data Types: `char` | `string`

Objective solver configurations, specified as a structure array. The fields of `SolverOptions` depend on `Solver`.

You can specify the field values using the corresponding name-value arguments when you create the model object by calling `incrementalRegressionKernel` directly, or when you convert a traditionally trained model using the `incrementalLearner` function.

Data Types: `struct`

Performance Metrics Parameters

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.

ValueDescription
`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`

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` name-value 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 the `Metrics` name-value argument of `incrementalLearner` or `incrementalRegressionKernel`.

Data Types: `table`

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`, the `MetricsWarmupPeriod` name-value argument of the `incrementalLearner` function sets this property. The default value of the argument is `0`.

• Otherwise, the default value is `1000`.

For more details, see Performance Metrics.

Data Types: `single` | `double`

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`, the `MetricsWindowSize` name-value argument of the `incrementalLearner` function sets this property. The default value of the argument is `200`.

• Otherwise, the default value is `200`.

For more details on performance metrics options, see Performance Metrics.

Data Types: `single` | `double`

Object Functions

 `fit` Train kernel model for incremental learning `updateMetrics` Update performance metrics in kernel incremental learning model given new data `updateMetricsAndFit` Update performance metrics in kernel incremental learning model given new data and train model `loss` Loss of kernel incremental learning model on batch of data `predict` Predict responses for new observations from kernel incremental learning model `perObservationLoss` Per observation regression error of model for incremental learning `reset` Reset incremental regression model

Examples

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Create an incremental kernel model without any prior information. Track the model performance on streaming data, and fit the model to the data.

Create a default incremental kernel SVM model for regression.

`Mdl = incrementalRegressionKernel()`
```Mdl = incrementalRegressionKernel IsWarm: 0 Metrics: [1x2 table] ResponseTransform: 'none' NumExpansionDimensions: 0 KernelScale: 1 Properties, Methods ```
`Mdl.EstimationPeriod`
```ans = 1000 ```

`Mdl` is an `incrementalRegressionKernel` model object. All its properties are read-only.

`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` name-value 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 the cumulative metrics, window metrics, and number of training observations 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"]); numtrainobs = zeros(nchunk+1,1); % Incremental fitting for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j-1) + 1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; Mdl = updateMetricsAndFit(Mdl,Xtrain(idx,:),ytrain(idx)); ei{j,:} = Mdl.Metrics{"EpsilonInsensitiveLoss",:}; numtrainobs(j+1) = Mdl.NumTrainingObservations; end```

`Mdl` is an `incrementalRegressionKernel` 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 model 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.

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

```t = tiledlayout(2,1); nexttile plot(numtrainobs) xlim([0 nchunk]) ylabel("Number of Training Observations") xline(Mdl.EstimationPeriod/numObsPerChunk,"-.") xline((Mdl.EstimationPeriod + Mdl.MetricsWarmupPeriod)/numObsPerChunk,"--") nexttile plot(ei.Variables) xlim([0 nchunk]) ylabel("Epsilon Insensitive Loss") xline(Mdl.EstimationPeriod/numObsPerChunk,"-.") xline((Mdl.EstimationPeriod + Mdl.MetricsWarmupPeriod)/numObsPerChunk,"--") legend(ei.Properties.VariableNames,Location="best") xlabel(t,"Iteration")```

The plot suggests that `updateMetricsAndFit` does the following:

• After the estimation period (first 20 iterations), fit the model during all incremental learning iterations.

• Compute the performance metrics after the metrics warm-up period only.

• Compute the cumulative metrics during each iteration.

• Compute the window metrics after processing 200 observations (4 iterations).

Prepare an incremental regression learner by specifying a metrics warm-up period and a metrics window size. Train the model by using SGD, and adjust the SGD batch size, learning rate, and regularization parameter.

Load the robot arm data set.

```load robotarm n = numel(ytrain);```

For details on the data set, enter `Description` at the command line.

Create an incremental kernel 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 a metrics warm-up 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 = incrementalRegressionKernel(Solver="sgd", ... Lambda=0.001,BatchSize=20,LearnRate=0.002,Epsilon=0.05, ... MetricsWarmupPeriod=1000,MetricsWindowSize=500, ... Metrics={"epsiloninsensitive","mse",maemetric})```
```Mdl = incrementalRegressionKernel IsWarm: 0 Metrics: [3x2 table] ResponseTransform: 'none' NumExpansionDimensions: 0 KernelScale: 1 Properties, Methods ```

`Mdl` is an `incrementalRegressionKernel` 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 the SGD batch size.

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

• Store the cumulative metrics, window metrics, and number of training observations to see how they evolve during incremental learning.

```% Preallocation numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); ei = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); mse = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); mae = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); numtrainobs = zeros(nchunk,1); % Incremental fitting rng("default") % For reproducibility for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j-1) + 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",:}; numtrainobs(j) = Mdl.NumTrainingObservations; end```

`Mdl` is an `incrementalRegressionKernel` model object trained on all the data in the stream. During incremental learning and after the model is warmed up, `updateMetricsAndFit` checks the performance of the model on the incoming observations, and then fits the model to those observations.

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

```t = tiledlayout(4,1); nexttile plot(numtrainobs) xlim([0 nchunk]) ylabel(["Number of","Training Observations"]) xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--") nexttile plot(ei.Variables) xlim([0 nchunk]) ylabel(["Epsilon Insensitive","Loss"]) xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--") legend(ei.Properties.VariableNames) nexttile plot(mse.Variables) xlim([0 nchunk]) ylabel("MSE") xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--") legend(mse.Properties.VariableNames) nexttile plot(mae.Variables) xlim([0 nchunk]) ylabel("MAE") xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--") legend(mae.Properties.VariableNames) xlabel(t,"Iteration")```

The plot suggests that `updateMetricsAndFit` does the following:

• Fit the model during all incremental learning iterations.

• Compute the performance metrics after the metrics warm-up period only.

• Compute the cumulative metrics during each iteration.

• Compute the window metrics after processing 500 observations (10 iterations).

Train a kernel regression model by using `fitrkernel`, 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 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 = [];```

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}];```

Train Kernel Regression Model

Fit a kernel regression model to a random sample of half the data. Specify the observation weights.

```idxtt = randsample([true false],n,true); Mdl = fitrkernel(X(idxtt,:),Y(idxtt),Weights=NYCHousing2015.W(idxtt))```
```Mdl = RegressionKernel ResponseName: 'Y' Learner: 'svm' NumExpansionDimensions: 2048 KernelScale: 1 Lambda: 2.1977e-05 BoxConstraint: 1 Epsilon: 0.0547 Properties, Methods ```

`Mdl` is a `RegressionKernel` model object representing a traditionally trained kernel regression model.

Convert Trained Model

Convert the traditionally trained kernel regression model to a model for incremental learning.

`IncrementalMdl = incrementalLearner(Mdl)`
```IncrementalMdl = incrementalRegressionKernel IsWarm: 1 Metrics: [1x2 table] ResponseTransform: 'none' NumExpansionDimensions: 2048 KernelScale: 1 Properties, Methods ```

`IncrementalMdl` is an `incrementalRegressionKernel` model object configured for incremental learning.

Separately Track Performance Metrics and Fit Model

Perform incremental learning on the rest of the data by using the `updateMetrics` and `fit` functions. Simulate a data stream by processing 500 observations at a time. At each iteration:

1. 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 `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.

2. 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.

3. Store the losses and number of training observations.

```% Preallocation idxil = ~idxtt; nil = sum(idxil); numObsPerChunk = 500; nchunk = floor(nil/numObsPerChunk); ei = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); numtrainobs = zeros(nchunk,1); Xil = X(idxil,:); Yil = Y(idxil); Wil = NYCHousing2015.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)); ei{j,:} = IncrementalMdl.Metrics{"EpsilonInsensitiveLoss",:}; IncrementalMdl = fit(IncrementalMdl,Xil(idx,:),Yil(idx), ... Weights=Wil(idx)); numtrainobs(j) = IncrementalMdl.NumTrainingObservations; end```

`IncrementalMdl` is an `incrementalRegressionKernel` 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 on separate tiles.

```t = tiledlayout(2,1); nexttile plot(numtrainobs) xlim([0 nchunk]) ylabel("Number of Training Observations") nexttile plot(ei.Variables) xlim([0 nchunk]) ylabel("Epsilon Insensitive Loss") legend(ei.Properties.VariableNames) 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.

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References

[1] Kempka, Michał, Wojciech Kotłowski, and Manfred K. Warmuth. "Adaptive Scale-Invariant 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] Shalev-Shwartz, S., Y. Singer, and N. Srebro. “Pegasos: Primal Estimated Sub-Gradient Solver for SVM.” Proceedings of the 24th International Conference on Machine Learning, ICML ’07, 2007, pp. 807–814.

[4] Xu, Wei. “Towards Optimal One Pass Large Scale Learning with Averaged Stochastic Gradient Descent.” CoRR, abs/1107.2490, 2011.

[5] Rahimi, A., and B. Recht. “Random Features for Large-Scale Kernel Machines.” Advances in Neural Information Processing Systems. Vol. 20, 2008, pp. 1177–1184.

[6] Le, Q., T. Sarlós, and A. Smola. “Fastfood — Approximating Kernel Expansions in Loglinear Time.” Proceedings of the 30th International Conference on Machine Learning. Vol. 28, No. 3, 2013, pp. 244–252.

[7] Huang, P. S., H. Avron, T. N. Sainath, V. Sindhwani, and B. Ramabhadran. “Kernel methods match Deep Neural Networks on TIMIT.” 2014 IEEE International Conference on Acoustics, Speech and Signal Processing. 2014, pp. 205–209.

Version History

Introduced in R2022a