# incrementalClassificationKernel

Binary classification kernel model for incremental learning

## Description

The `incrementalClassificationKernel` function creates an `incrementalClassificationKernel` model object, which represents a binary Gaussian kernel classification 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 logistic regression.

Unlike other Statistics and Machine Learning Toolbox™ model objects, `incrementalClassificationKernel` 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 `incrementalClassificationKernel` object, it is prepared for incremental learning.

`incrementalClassificationKernel` is best suited for incremental learning. For a traditional approach to training a kernel model for binary classification (such as creating a model by fitting it to data, performing cross-validation, tuning hyperparameters, and so on), see `fitckernel`. For multiclass incremental learning, see `incrementalClassificationECOC` and `incrementalClassificationNaiveBayes`.

## Creation

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

• Call the function directly — Configure incremental learning options, or specify learner-specific options, by calling `incrementalClassificationKernel` 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, you can convert the traditionally trained model (`ClassificationKernel`) to an `incrementalClassificationKernel` model object by passing it to the `incrementalLearner` function.

• Convert a template object — You can convert a template object (`templateKernel`) to an `incrementalClassificationKernel` model object by passing it to the `incrementalLearner` function.

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

### Syntax

``Mdl = incrementalClassificationKernel()``
``Mdl = incrementalClassificationKernel(Name=Value)``

### Description

example

````Mdl = incrementalClassificationKernel()` returns a default incremental learning model object for binary Gaussian kernel classification, `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 = incrementalClassificationKernel(Name=Value)` sets properties and additional options using name-value arguments. For example, `incrementalClassificationKernel(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="logit",MetricsWarmupPeriod=100` sets the model performance metric to the logistic loss and the metrics warm-up period to `100`.

Classification 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 `incrementalClassificationKernel` 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, `incrementalClassificationKernel` 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"`, then c = 1.

• If `Solver` is `"asgd"`, then c = 0.75 [4].

Example: `LearnRateSchedule="constant"`

Data Types: `char` | `string`

Adaptive Scale-Invariant Solver Options

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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. You can specify more than one by using a string vector.

NameDescription
`"binodeviance"`Binomial deviance
`"classiferror"`Classification error
`"exponential"`Exponential loss
`"hinge"`Hinge loss
`"logit"`Logistic loss
`"quadratic"`Quadratic loss

For more details on the built-in loss functions, see `loss`.

Example: `Metrics=["classiferror","hinge"]`

To specify a custom function that returns a performance metric, use function handle notation. The function must have this form:

`metric = customMetric(C,S)`

• 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`).

• `C` is an n-by-`2` logical matrix with rows indicating the class to which the corresponding observation belongs. The column order corresponds to the class order in the `ClassNames` property. Create `C` by setting `C(p,q)` = `1`, if observation `p` is in class `q`, for each observation in the specified data. Set the other element in row `p` to `0`.

• `S` is an n-by-`2` numeric matrix of predicted classification scores. `S` is similar to the `score` output of `predict`, where rows correspond to observations in the data and the column order corresponds to the class order in the `ClassNames` property. `S(p,q)` is the classification score of observation `p` being classified in class `q`.

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,"logit",struct(Metric3=@customMetric3)}`

`updateMetrics` and `updateMetricsAndFit` store specified metrics in a table in the `Metrics` property. 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 `"classiferror"` is `"ClassificationError"`
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 `@(C,S)customMetric(C,S)...` is `CustomMetric_1`

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 `incrementalClassificationKernel` directly. You can set some properties when you call `incrementalLearner` to convert a traditionally trained model object or model template object. You cannot set the properties `FittedLoss`, `NumTrainingObservations`, `SolverOptions`, and `IsWarm`.

### Classification Model Parameters

This property is read-only.

Unique class labels used in training the model, specified as a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors. `ClassNames` and the response data must have the same data type. (The software treats string arrays as cell arrays of character vectors.)

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

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

• Otherwise, incremental fitting functions infer `ClassNames` during training.

Data Types: `single` | `double` | `logical` | `char` | `string` | `cell` | `categorical`

This property is read-only.

Loss function used to fit the linear model, specified as `'hinge'` or `'logit'`.

ValueAlgorithmLoss Function`Learner` Value
`'hinge'`Support vector machineHinge: $\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,1-yf\left(x\right)\right]$`'svm'`
`'logit'`Logistic regressionDeviance (logistic): $\ell \left[y,f\left(x\right)\right]=\mathrm{log}\left\{1+\mathrm{exp}\left[-yf\left(x\right)\right]\right\}$`'logistic'`

This property is read-only.

Kernel scale parameter, specified as `"auto"` or a positive scalar. `incrementalClassificationKernel` 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 object or template model object to create `Mdl`, `KernelScale` is specified by the corresponding property of the object.

• Otherwise, the default value is `1`.

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

This property is read-only.

Linear classification model type, specified as `"svm"` or `"logistic"`. `incrementalClassificationKernel` 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 machineHinge loss: $\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,1-yf\left(x\right)\right]$`'hinge'`
`"logistic"`Logistic regressionDeviance (logistic loss): $\ell \left[y,f\left(x\right)\right]=\mathrm{log}\left\{1+\mathrm{exp}\left[-yf\left(x\right)\right]\right\}$`'logit'`

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

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

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

Data Types: `char` | `string`

This property is read-only.

Number of dimensions of the expanded space, specified as `"auto"` or a positive integer. `incrementalClassificationKernel` 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 object or template model object to create `Mdl`, `NumExpansionDimensions` is specified by the corresponding property of the object.

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

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

This property is read-only.

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 `incrementalClassificationKernel` directly, you can specify `NumPredictors` by using name-value argument syntax.

• Otherwise, the default value is `0`, and incremental fitting functions infer `NumPredictors` from the predictor data during training.

Data Types: `double`

This property is read-only.

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

Data Types: `double`

This property is read-only.

Prior class probabilities, specified as `"empirical"`, `"uniform"`, or a numeric vector. `incrementalClassificationKernel` stores the `Prior` value as a numeric vector.

ValueDescription
`"empirical"`Incremental learning functions infer prior class probabilities from the observed class relative frequencies in the response data during incremental training (after the estimation period `EstimationPeriod`).
`"uniform"`For each class, the prior probability is 1/2.
numeric vectorCustom, normalized prior probabilities. The order of the elements of `Prior` corresponds to the elements of the `ClassNames` property.

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

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

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

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

This property is read-only.

Score transformation function describing how incremental learning functions transform raw response values, specified as a character vector, string scalar, or function handle. `incrementalClassificationKernel` stores the `ScoreTransform` value as a character vector or function handle.

This table describes the available built-in functions for score transformation.

ValueDescription
`"doublelogit"`1/(1 + e–2x)
`"invlogit"`log(x / (1 – x))
`"ismax"`Sets the score for the class with the largest score to 1, and sets the scores for all other classes to 0
`"logit"`1/(1 + ex)
`"none"` or `"identity"`x (no transformation)
`"sign"`–1 for x < 0
0 for x = 0
1 for x > 0
`"symmetric"`2x – 1
`"symmetricismax"`Sets the score for the class with the largest score to 1, and sets the scores for all other classes to –1
`"symmetriclogit"`2/(1 + ex) – 1

For a MATLAB® function or a function that you define, enter its function handle; for example, `ScoreTransform=@function`, where:

• `function` accepts an n-by-`2` matrix (the original scores) and returns a matrix of the same size (the transformed scores). The column order corresponds to the class order in the `ClassNames` property.

• n is the number of observations, and row j of the matrix contains the class scores of observation j.

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

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

• Otherwise, the default value is `"none"` (when `Learner` is `"svm"`) or `"logit"` (when `Learner` is `"logistic"`).

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

### Training Parameters

This property is read-only.

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

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

For more details, see Estimation Period.

Data Types: `single` | `double`

This property is read-only.

Objective function minimization technique, specified as `"scale-invariant"`, `"sgd"`, or `"asgd"`. `incrementalClassificationKernel` 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`

This property is read-only.

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 `incrementalClassificationKernel` directly, or when you convert a traditionally trained model using the `incrementalLearner` function.

Data Types: `struct`

### Performance Metrics Parameters

This property is read-only.

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`

This property is read-only.

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 `incrementalClassificationKernel`.

Data Types: `table`

This property is read-only.

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`

This property is read-only.

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 classification error of model for incremental learning `reset` Reset incremental classification 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 binary classification.

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

`Mdl` is an `incrementalClassificationKernel` model object. 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 by using the `updateMetricsAndFit` function. Simulate a data stream by processing 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 numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); ce = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); numtrainobs = zeros(nchunk,1); % Incremental learning for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j-1) + 1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; Mdl = updateMetricsAndFit(Mdl,X(idx,:),Y(idx)); ce{j,:} = Mdl.Metrics{"ClassificationError",:}; numtrainobs(j) = Mdl.NumTrainingObservations; end```

`Mdl` is an `incrementalClassificationKernel` 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(2,1); nexttile plot(numtrainobs) xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--") xlim([0 nchunk]) ylabel("Number of Training Observations") nexttile plot(ce.Variables) xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--") xlim([0 nchunk]) ylabel("Classification Error") legend(ce.Properties.VariableNames,Location="best") 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 200 observations (4 iterations).

Prepare an incremental kernel SVM 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 human activity data set. Randomly shuffle the data.

```load humanactivity n = numel(actid); rng("default") % 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;`

Create an incremental kernel model for binary classification. Configure the model as follows:

• Specify the SGD solver.

• Assume that a ridge regularization parameter value of 0.001, SGD batch size of 20, and learning rate of 0.002 work well for the problem.

• Specify a metrics warm-up period of 5000 observations.

• Specify a metrics window size of 500 observations.

• Track the classification and hinge error metrics to measure the performance of the model.

```Mdl = incrementalClassificationKernel( ... Solver="sgd",Lambda=0.001,BatchSize=20,LearnRate=0.002, ... MetricsWarmupPeriod=5000,MetricsWindowSize=500, ... Metrics=["classiferror","hinge"])```
```Mdl = incrementalClassificationKernel IsWarm: 0 Metrics: [2x2 table] ClassNames: [1x0 double] ScoreTransform: 'none' NumExpansionDimensions: 0 KernelScale: 1 Properties, Methods ```

`Mdl` is an `incrementalClassificationKernel` model object configured for incremental learning.

Fit the incremental model to the rest of 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); ce = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); hinge = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); numtrainobs = 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 = updateMetricsAndFit(Mdl,X(idx,:),Y(idx)); ce{j,:} = Mdl.Metrics{"ClassificationError",:}; hinge{j,:} = Mdl.Metrics{"HingeLoss",:}; numtrainobs(j) = Mdl.NumTrainingObservations; end```

`Mdl` is an `incrementalClassificationKernel` 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(3,1); nexttile plot(numtrainobs) xlim([0 nchunk]) ylabel(["Number of","Training Observations"]) xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--") nexttile plot(ce.Variables) xlim([0 nchunk]) ylabel("Classification Error") xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--") legend(ce.Properties.VariableNames,Location="best") nexttile plot(hinge.Variables) xlim([0 nchunk]) ylabel("Hinge Loss") xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--") legend(hinge.Properties.VariableNames,Location="best") 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 model for binary classification by using `fitckernel`, 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 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 stationary (`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); Mdl = fitckernel(X(idxtt,:),Y(idxtt),Weights=W(idxtt))```
```Mdl = ClassificationKernel ResponseName: 'Y' ClassNames: [0 1] Learner: 'svm' NumExpansionDimensions: 2048 KernelScale: 1 Lambda: 8.2967e-05 BoxConstraint: 1 Properties, Methods ```

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

`IncrementalMdl` is an `incrementalClassificationKernel` 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 50 observations at a time. At each iteration:

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

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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