Documentation

# fitckernel

Fit Gaussian kernel classification model using random feature expansion

## Description

fitckernel trains or cross-validates a binary Gaussian kernel classification model for nonlinear classification. fitckernel is more practical for big data applications that have large training sets but can also be applied to smaller data sets that fit in memory.

fitckernel maps data in a low-dimensional space into a high-dimensional space, then fits a linear model in the high-dimensional space by minimizing the regularized objective function. Obtaining the linear model in the high-dimensional space is equivalent to applying the Gaussian kernel to the model in the low-dimensional space. Available linear classification models include regularized support vector machine (SVM) and logistic regression models.

To train a nonlinear SVM model for binary classification of in-memory data, see fitcsvm.

example

Mdl = fitckernel(X,Y) returns a binary Gaussian kernel classification model trained using the predictor data in X and the corresponding class labels in Y. The fitckernel function maps the predictors in a low-dimensional space into a high-dimensional space, then fits a binary SVM model to the transformed predictors and class labels. This linear model is equivalent to the Gaussian kernel classification model in the low-dimensional space.

example

Mdl = fitckernel(X,Y,Name,Value) returns a kernel classification model with additional options specified by one or more name-value pair arguments. For example, you can implement logistic regression, specify the number of dimensions of the expanded space, or specify to cross-validate.

example

[Mdl,FitInfo] = fitckernel(___) also returns the fit information in the structure array FitInfo using any of the input arguments in the previous syntaxes. You cannot request FitInfo for cross-validated models.

example

[Mdl,FitInfo,HyperparameterOptimizationResults] = fitckernel(___) also returns the hyperparameter optimization results HyperparameterOptimizationResults when you optimize hyperparameters by using the 'OptimizeHyperparameters' name-value pair argument.

## Examples

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Train a binary kernel classification model using SVM.

Load the ionosphere data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b') or good ('g').

[n,p] = size(X)
n = 351
p = 34
resp = unique(Y)
resp = 2x1 cell array
{'b'}
{'g'}

Train a binary kernel classification model that identifies whether the radar return is bad ('b') or good ('g'). Extract a fit summary to determine how well the optimization algorithm fits the model to the data.

rng('default') % For reproducibility
[Mdl,FitInfo] = fitckernel(X,Y)
Mdl =
ClassificationKernel
ResponseName: 'Y'
ClassNames: {'b'  'g'}
Learner: 'svm'
NumExpansionDimensions: 2048
KernelScale: 1
Lambda: 0.0028
BoxConstraint: 1

Properties, Methods

FitInfo = struct with fields:
Solver: 'LBFGS-fast'
LossFunction: 'hinge'
Lambda: 0.0028
BetaTolerance: 1.0000e-04
ObjectiveValue: 0.2604
RelativeChangeInBeta: 8.2512e-05
FitTime: 0.1617
History: []

Mdl is a ClassificationKernel model. To inspect the in-sample classification error, you can pass Mdl and the training data or new data to the loss function. Or, you can pass Mdl and new predictor data to the predict function to predict class labels for new observations. You can also pass Mdl and the training data to the resume function to continue training.

FitInfo is a structure array containing optimization information. Use FitInfo to determine whether optimization termination measurements are satisfactory.

For better accuracy, you can increase the maximum number of optimization iterations ('IterationLimit') and decrease the tolerance values ('BetaTolerance' and 'GradientTolerance') by using the name-value pair arguments. Doing so can improve measures like ObjectiveValue and RelativeChangeInBeta in FitInfo. You can also optimize model parameters by using the 'OptimizeHyperparameters' name-value pair argument.

Load the ionosphere data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b') or good ('g').

rng('default') % For reproducibility

Cross-validate a binary kernel classification model. By default, the software uses 10-fold cross-validation.

CVMdl = fitckernel(X,Y,'CrossVal','on')
CVMdl =
classreg.learning.partition.ClassificationPartitionedKernel
CrossValidatedModel: 'Kernel'
ResponseName: 'Y'
NumObservations: 351
KFold: 10
Partition: [1x1 cvpartition]
ClassNames: {'b'  'g'}
ScoreTransform: 'none'

Properties, Methods

numel(CVMdl.Trained)
ans = 10

CVMdl is a ClassificationPartitionedKernel model. Because fitckernel implements 10-fold cross-validation, CVMdl contains 10 ClassificationKernel models that the software trains on training-fold (in-fold) observations.

Estimate the cross-validated classification error.

kfoldLoss(CVMdl)
ans = 0.0940

The classification error rate is approximately 9%.

Optimize hyperparameters automatically using the 'OptimizeHyperparameters' name-value pair argument.

Load the ionosphere data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b') or good ('g').

Find hyperparameters that minimize five-fold cross-validation loss by using automatic hyperparameter optimization. Specify 'OptimizeHyperparameters' as 'auto' so that fitckernel finds optimal values of the 'KernelScale' and 'Lambda' name-value pair arguments. For reproducibility, set the random seed and use the 'expected-improvement-plus' acquisition function.

rng('default')
[Mdl,FitInfo,HyperparameterOptimizationResults] = fitckernel(X,Y,'OptimizeHyperparameters','auto',...
'HyperparameterOptimizationOptions',struct('AcquisitionFunctionName','expected-improvement-plus'))

|=====================================================================================================|
| Iter | Eval   | Objective   | Objective   | BestSoFar   | BestSoFar   |  KernelScale |       Lambda |
|      | result |             | runtime     | (observed)  | (estim.)    |              |              |
|=====================================================================================================|
|    1 | Best   |     0.35897 |      1.7313 |     0.35897 |     0.35897 |       64.836 |   4.4811e-06 |
|    2 | Accept |     0.35897 |      2.3447 |     0.35897 |     0.35897 |     0.036335 |     0.015885 |
|    3 | Accept |     0.39601 |      1.9929 |     0.35897 |     0.36053 |    0.0022147 |   6.8254e-06 |
|    4 | Accept |     0.35897 |     0.82793 |     0.35897 |     0.35898 |       5.1259 |      0.28097 |
|    5 | Accept |     0.35897 |      1.1595 |     0.35897 |     0.35897 |      0.24853 |      0.10828 |
|    6 | Accept |     0.35897 |      0.4125 |     0.35897 |     0.35897 |       885.09 |   0.00057316 |
|    7 | Best   |     0.10826 |      1.4876 |     0.10826 |     0.10833 |       8.0346 |    0.0048286 |
|    8 | Best   |    0.076923 |      1.3417 |    0.076923 |    0.076999 |       7.0902 |    0.0034068 |
|    9 | Accept |    0.091168 |      1.5055 |    0.076923 |    0.077059 |       9.1504 |    0.0020604 |
|   10 | Best   |    0.062678 |     0.82875 |    0.062678 |    0.062723 |       3.5487 |    0.0025912 |
|   11 | Accept |    0.062678 |      4.4023 |    0.062678 |    0.062741 |       2.3869 |     0.003321 |
|   12 | Accept |     0.41026 |      4.4349 |    0.062678 |    0.062536 |      0.14075 |    0.0022499 |
|   13 | Accept |    0.062678 |      2.1413 |    0.062678 |    0.062532 |       3.4215 |    0.0036803 |
|   14 | Accept |    0.062678 |      1.0142 |    0.062678 |    0.061956 |       3.2928 |    0.0030533 |
|   15 | Best   |     0.05698 |     0.74696 |     0.05698 |    0.057204 |       5.0598 |    0.0025499 |
|   16 | Accept |    0.062678 |     0.88619 |     0.05698 |    0.057186 |       5.3401 |    0.0015096 |
|   17 | Accept |     0.05698 |     0.72951 |     0.05698 |    0.057118 |        1.813 |    0.0069209 |
|   18 | Accept |    0.059829 |     0.90806 |     0.05698 |    0.057092 |       1.5122 |    0.0046637 |
|   19 | Accept |    0.059829 |      1.0088 |     0.05698 |     0.05718 |       1.9277 |    0.0056364 |
|   20 | Accept |    0.065527 |     0.70343 |     0.05698 |    0.057189 |       1.4064 |    0.0094306 |
|=====================================================================================================|
| Iter | Eval   | Objective   | Objective   | BestSoFar   | BestSoFar   |  KernelScale |       Lambda |
|      | result |             | runtime     | (observed)  | (estim.)    |              |              |
|=====================================================================================================|
|   21 | Accept |     0.05698 |     0.76405 |     0.05698 |    0.057033 |       5.1719 |    0.0023614 |
|   22 | Best   |    0.054131 |      1.7745 |    0.054131 |    0.054176 |       1.9618 |   6.5704e-05 |
|   23 | Best   |    0.042735 |      1.0991 |    0.042735 |    0.042763 |       1.9463 |   1.0169e-05 |
|   24 | Accept |    0.082621 |     0.80884 |    0.042735 |    0.042775 |       1.0661 |   1.3245e-05 |
|   25 | Accept |    0.054131 |      1.5182 |    0.042735 |    0.042789 |        3.288 |   2.0035e-05 |
|   26 | Accept |    0.062678 |      1.6483 |    0.042735 |    0.042769 |        2.657 |   3.0334e-06 |
|   27 | Accept |    0.059829 |      1.2015 |    0.042735 |    0.043054 |       2.0381 |   1.9791e-05 |
|   28 | Accept |    0.042735 |      2.0598 |    0.042735 |    0.042764 |       3.5043 |    0.0001237 |
|   29 | Accept |    0.054131 |      1.0533 |    0.042735 |    0.042764 |       1.3897 |   3.2288e-06 |
|   30 | Accept |    0.062678 |      1.3586 |    0.042735 |    0.042792 |       2.2414 |    0.0002259 |

__________________________________________________________
Optimization completed.
MaxObjectiveEvaluations of 30 reached.
Total function evaluations: 30
Total elapsed time: 86.9349 seconds.
Total objective function evaluation time: 43.8943

Best observed feasible point:
KernelScale      Lambda
___________    __________

1.9463       1.0169e-05

Observed objective function value = 0.042735
Estimated objective function value = 0.042792
Function evaluation time = 1.0991

Best estimated feasible point (according to models):
KernelScale     Lambda
___________    _________

3.5043       0.0001237

Estimated objective function value = 0.042792
Estimated function evaluation time = 1.2639
Mdl =
ClassificationKernel
ResponseName: 'Y'
ClassNames: {'b'  'g'}
Learner: 'svm'
NumExpansionDimensions: 2048
KernelScale: 3.5043
Lambda: 1.2370e-04
BoxConstraint: 23.0320

Properties, Methods

FitInfo = struct with fields:
Solver: 'LBFGS-fast'
LossFunction: 'hinge'
Lambda: 1.2370e-04
BetaTolerance: 1.0000e-04
ObjectiveValue: 0.0426
RelativeChangeInBeta: 8.9154e-05
FitTime: 0.2920
History: []

HyperparameterOptimizationResults =
BayesianOptimization with properties:

ObjectiveFcn: @createObjFcn/inMemoryObjFcn
VariableDescriptions: [4x1 optimizableVariable]
Options: [1x1 struct]
MinObjective: 0.0427
XAtMinObjective: [1x2 table]
MinEstimatedObjective: 0.0428
XAtMinEstimatedObjective: [1x2 table]
NumObjectiveEvaluations: 30
TotalElapsedTime: 86.9349
NextPoint: [1x2 table]
XTrace: [30x2 table]
ObjectiveTrace: [30x1 double]
ConstraintsTrace: []
UserDataTrace: {30x1 cell}
ObjectiveEvaluationTimeTrace: [30x1 double]
IterationTimeTrace: [30x1 double]
ErrorTrace: [30x1 double]
FeasibilityTrace: [30x1 logical]
FeasibilityProbabilityTrace: [30x1 double]
IndexOfMinimumTrace: [30x1 double]
ObjectiveMinimumTrace: [30x1 double]
EstimatedObjectiveMinimumTrace: [30x1 double]

For big data, the optimization procedure can take a long time. If the data set is too large to run the optimization procedure, you can try to optimize the parameters using only partial data. Use the datasample function and specify 'Replace','false' to sample data without replacement.

## Input Arguments

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Predictor data, specified as an n-by-p numeric matrix, where n is the number of observations and p is the number of predictors.

The length of Y and the number of observations in X must be equal.

Data Types: single | double

Class labels, specified as a categorical, character, or string array, logical or numeric vector, or cell array of character vectors.

• fitckernel supports only binary classification. Either Y must contain exactly two distinct classes, or you must specify two classes for training by using the ClassNames name-value pair argument.

• If Y is a character array, then each element must correspond to one row of the array.

• The length of Y and the number of observations in X must be equal.

• A good practice is to specify the class order by using the ClassNames name-value pair argument.

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

### Note

fitckernel removes missing observations, that is, observations with any of these characteristics:

• NaN, empty character vector (''), empty string (""), <missing>, and <undefined> elements in the response (Y)

• At least one NaN value in a predictor observation (row in X)

• NaN value or 0 weight ('Weights')

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: Mdl = fitckernel(X,Y,'Learner','logistic','NumExpansionDimensions',2^15,'KernelScale','auto') implements logistic regression after mapping the predictor data to the 2^15 dimensional space using feature expansion with a kernel scale parameter selected by a heuristic procedure.

### Note

You cannot use any cross-validation name-value pair argument along with the 'OptimizeHyperparameters' name-value pair argument. You can modify the cross-validation for 'OptimizeHyperparameters' only by using the 'HyperparameterOptimizationOptions' name-value pair argument.

#### Kernel Classification Options

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Linear classification model type, specified as the comma-separated pair consisting of 'Learner' and 'svm' or 'logistic'.

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 m coefficients.

• b is the scalar bias.

ValueAlgorithmResponse RangeLoss Function
'svm'Support vector machiney ∊ {–1,1}; 1 for the positive class and –1 otherwiseHinge: $\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,1-yf\left(x\right)\right]$
'logistic'Logistic regressionSame as 'svm'Deviance (logistic): $\ell \left[y,f\left(x\right)\right]=\mathrm{log}\left\{1+\mathrm{exp}\left[-yf\left(x\right)\right]\right\}$

Example: 'Learner','logistic'

Number of dimensions of the expanded space, specified as the comma-separated pair consisting of 'NumExpansionDimensions' and 'auto' or a positive integer. For 'auto', the fitckernel function 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.

Example: 'NumExpansionDimensions',2^15

Data Types: char | string | single | double

Kernel scale parameter, specified as the comma-separated pair consisting of 'KernelScale' and 'auto' or a positive scalar. The software obtains a random basis for random feature expansion by using the kernel scale parameter. For details, see Random Feature Expansion.

If you specify 'auto', then the software selects an appropriate kernel scale parameter using a heuristic procedure. This heuristic 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.

Example: 'KernelScale','auto'

Data Types: char | string | single | double

Box constraint, specified as the comma-separated pair consisting of 'BoxConstraint' and a positive scalar.

This argument is valid only when 'Learner' is 'svm'(default) and you do not specify a value for the regularization term strength 'Lambda'. You can specify either 'BoxConstraint' or 'Lambda' because the box constraint (C) and the regularization term strength (λ) are related by C = 1/(λn), where n is the number of observations.

Example: 'BoxConstraint',100

Data Types: single | double

Regularization term strength, specified as the comma-separated pair consisting of 'Lambda' and 'auto' or a nonnegative scalar.

For 'auto', the value of 'Lambda' is 1/n, where n is the number of observations.

You can specify either 'BoxConstraint' or 'Lambda' because the box constraint (C) and the regularization term strength (λ) are related by C = 1/(λn).

Example: 'Lambda',0.01

Data Types: char | string | single | double

#### Cross-Validation Options

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Flag to train a cross-validated classifier, specified as the comma-separated pair consisting of 'Crossval' and 'on' or 'off'.

If you specify 'on', then the software trains a cross-validated classifier with 10 folds.

You can override this cross-validation setting using the CVPartition, Holdout, KFold, or Leaveout name-value pair argument. You can use only one cross-validation name-value pair argument at a time to create a cross-validated model.

Example: 'Crossval','on'

Cross-validation partition, specified as the comma-separated pair consisting of 'CVPartition' and a cvpartition partition object created by cvpartition. The partition object specifies the type of cross-validation and the indexing for the training and validation sets.

To create a cross-validated model, you can use one of these four name-value pair arguments only: CVPartition, Holdout, KFold, or Leaveout.

Example: Suppose you create a random partition for 5-fold cross-validation on 500 observations by using cvp = cvpartition(500,'KFold',5). Then, you can specify the cross-validated model by using 'CVPartition',cvp.

Fraction of the data used for holdout validation, specified as the comma-separated pair consisting of 'Holdout' and a scalar value in the range (0,1). If you specify 'Holdout',p, then the software completes these steps:

1. Randomly select and reserve p*100% of the data as validation data, and train the model using the rest of the data.

2. Store the compact, trained model in the Trained property of the cross-validated model.

To create a cross-validated model, you can use one of these four name-value pair arguments only: CVPartition, Holdout, KFold, or Leaveout.

Example: 'Holdout',0.1

Data Types: double | single

Number of folds to use in a cross-validated model, specified as the comma-separated pair consisting of 'KFold' and a positive integer value greater than 1. If you specify 'KFold',k, then the software completes these steps:

1. Randomly partition the data into k sets.

2. For each set, reserve the set as validation data, and train the model using the other k – 1 sets.

3. Store the k compact, trained models in the cells of a k-by-1 cell vector in the Trained property of the cross-validated model.

To create a cross-validated model, you can use one of these four name-value pair arguments only: CVPartition, Holdout, KFold, or Leaveout.

Example: 'KFold',5

Data Types: single | double

Leave-one-out cross-validation flag, specified as the comma-separated pair consisting of 'Leaveout' and 'on' or 'off'. If you specify 'Leaveout','on', then, for each of the n observations (where n is the number of observations excluding missing observations), the software completes these steps:

1. Reserve the observation as validation data, and train the model using the other n – 1 observations.

2. Store the n compact, trained models in the cells of an n-by-1 cell vector in the Trained property of the cross-validated model.

To create a cross-validated model, you can use one of these four name-value pair arguments only: CVPartition, Holdout, KFold, or Leaveout.

Example: 'Leaveout','on'

#### Convergence Controls

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Relative tolerance on the linear coefficients and the bias term (intercept), specified as the comma-separated pair consisting of 'BetaTolerance' and a nonnegative scalar.

Let ${B}_{t}=\left[{\beta }_{t}{}^{\prime }\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{t}\right]$, that is, the vector of the coefficients and the bias term at optimization iteration t. If ${‖\frac{{B}_{t}-{B}_{t-1}}{{B}_{t}}‖}_{2}<\text{BetaTolerance}$, then optimization terminates.

If you also specify GradientTolerance, then optimization terminates when the software satisfies either stopping criterion.

Example: 'BetaTolerance',1e–6

Data Types: single | double

Absolute gradient tolerance, specified as the comma-separated pair consisting of 'GradientTolerance' and a nonnegative scalar.

Let $\nabla {ℒ}_{t}$ be the gradient vector of the objective function with respect to the coefficients and bias term at optimization iteration t. If ${‖\nabla {ℒ}_{t}‖}_{\infty }=\mathrm{max}|\nabla {ℒ}_{t}|<\text{GradientTolerance}$, then optimization terminates.

If you also specify BetaTolerance, then optimization terminates when the software satisfies either stopping criterion.

Data Types: single | double

Maximum number of optimization iterations, specified as the comma-separated pair consisting of 'IterationLimit' and a positive integer.

The default value is 1000 if the transformed data fits in memory, as specified by the BlockSize name-value pair argument. Otherwise, the default value is 100.

Example: 'IterationLimit',500

Data Types: single | double

#### Other Kernel Classification Options

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Maximum amount of allocated memory (in megabytes), specified as the comma-separated pair consisting of 'BlockSize' and a positive scalar.

If fitckernel requires more memory than the value of 'BlockSize' to hold the transformed predictor data, then the software uses a block-wise strategy. For details about the block-wise strategy, see Algorithms.

Example: 'BlockSize',1e4

Data Types: single | double

Random number stream for reproducibility of data transformation, specified as the comma-separated pair consisting of 'RandomStream' and a random stream object. For details, see Random Feature Expansion.

Use 'RandomStream' to reproduce the random basis functions that fitckernel uses to transform the predictor data to a high-dimensional space. For details, see Managing the Global Stream (MATLAB) and Creating and Controlling a Random Number Stream (MATLAB).

Example: 'RandomStream',RandStream('mlfg6331_64')

Size of the history buffer for Hessian approximation, specified as the comma-separated pair consisting of 'HessianHistorySize' and a positive integer. At each iteration, fitckernel composes the Hessian approximation by using statistics from the latest HessianHistorySize iterations.

Example: 'HessianHistorySize',10

Data Types: single | double

Verbosity level, specified as the comma-separated pair consisting of 'Verbose' and either 0 or 1. Verbose controls the display of diagnostic information at the command line.

ValueDescription
0fitckernel does not display diagnostic information.
1fitckernel displays and stores the value of the objective function, gradient magnitude, and other diagnostic information. FitInfo.History contains the diagnostic information.

Example: 'Verbose',1

Data Types: single | double

#### Other Classification Options

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Names of classes to use for training, specified as the comma-separated pair consisting of 'ClassNames' and a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors. ClassNames must have the same data type as Y.

If ClassNames is a character array, then each element must correspond to one row of the array.

Use ClassNames to:

• Order the classes during training.

• Specify the order of any input or output argument dimension that corresponds to the class order. For example, use ClassNames to specify the order of the dimensions of Cost or the column order of classification scores returned by predict.

• Select a subset of classes for training. For example, suppose that the set of all distinct class names in Y is {'a','b','c'}. To train the model using observations from classes 'a' and 'c' only, specify 'ClassNames',{'a','c'}.

The default value for ClassNames is the set of all distinct class names in Y.

Example: 'ClassNames',{'b','g'}

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

Misclassification cost, specified as the comma-separated pair consisting of 'Cost' and a square matrix or structure.

• If you specify the square matrix cost ('Cost',cost), then cost(i,j) is the cost of classifying a point into class j if its true class is i. That is, the rows correspond to the true class, and the columns correspond to the predicted class. To specify the class order for the corresponding rows and columns of cost, use the ClassNames name-value pair argument.

• If you specify the structure S ('Cost',S), then it must have two fields:

• S.ClassNames, which contains the class names as a variable of the same data type as Y

• S.ClassificationCosts, which contains the cost matrix with rows and columns ordered as in S.ClassNames

The default value for Cost is ones(K) – eye(K), where K is the number of distinct classes.

fitckernel uses Cost to adjust the prior class probabilities specified in Prior. Then, fitckernel uses the adjusted prior probabilities for training and resets the cost matrix to its default.

Example: 'Cost',[0 2; 1 0]

Data Types: single | double | struct

Prior probabilities for each class, specified as the comma-separated pair consisting of 'Prior' and 'empirical', 'uniform', a numeric vector, or a structure array.

This table summarizes the available options for setting prior probabilities.

ValueDescription
'empirical'The class prior probabilities are the class relative frequencies in Y.
'uniform'All class prior probabilities are equal to 1/K, where K is the number of classes.
numeric vectorEach element is a class prior probability. Order the elements according to their order in Y. If you specify the order using the 'ClassNames' name-value pair argument, then order the elements accordingly.
structure array

A structure S with two fields:

• S.ClassNames contains the class names as a variable of the same type as Y.

• S.ClassProbs contains a vector of corresponding prior probabilities.

fitckernel normalizes the prior probabilities in Prior to sum to 1.

Example: 'Prior',struct('ClassNames',{{'setosa','versicolor'}},'ClassProbs',1:2)

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

Score transformation, specified as the comma-separated pair consisting of 'ScoreTransform' and a character vector, string scalar, or function handle.

This table summarizes the available character vectors and string scalars.

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 you define, use its function handle for score transform. The function handle must accept a matrix (the original scores) and return a matrix of the same size (the transformed scores).

Example: 'ScoreTransform','logit'

Data Types: char | string | function_handle

Observation weights, specified as the comma-separated pair consisting of 'Weights' and a positive numeric vector of length n, where n is the number of observations in X. The fitckernel function weighs the observations in X with the corresponding values in Weights.

The default value is ones(n,1).

fitckernel normalizes Weights to sum up to the value of the prior probability in the respective class.

Example: 'Weights',w

Data Types: single | double

#### Hyperparameter Optimization Options

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Parameters to optimize, specified as the comma-separated pair consisting of 'OptimizeHyperparameters' and one of these values:

• 'none' — Do not optimize.

• 'auto' — Use {'KernelScale','Lambda'}.

• 'all' — Optimize all eligible parameters.

• Cell array of eligible parameter names.

• Vector of optimizableVariable objects, typically the output of hyperparameters.

The optimization attempts to minimize the cross-validation loss (error) for fitckernel by varying the parameters. To control the cross-validation type and other aspects of the optimization, use the HyperparameterOptimizationOptions name-value pair argument.

### Note

'OptimizeHyperparameters' values override any values you set using other name-value pair arguments. For example, setting 'OptimizeHyperparameters' to 'auto' causes the 'auto' values to apply.

The eligible parameters for fitckernel are:

• KernelScalefitckernel searches among positive values, by default log-scaled in the range [1e-3,1e3].

• Lambdafitckernel searches among positive values, by default log-scaled in the range [1e-3,1e3]/n, where n is the number of observations.

• Learnerfitckernel searches among 'svm' and 'logistic'.

• NumExpansionDimensionsfitckernel searches among positive integers, by default log-scaled in the range [100,10000].

Set nondefault parameters by passing a vector of optimizableVariable objects that have nondefault values. For example:

params = hyperparameters('fitckernel',meas,species);
params(2).Range = [1e-4,1e6];

Pass params as the value of 'OptimizeHyperparameters'.

By default, iterative display appears at the command line, and plots appear according to the number of hyperparameters in the optimization. For the optimization and plots, the objective function is log(1 + cross-validation loss) for regression and the misclassification rate for classification. To control the iterative display, set the Verbose field of the 'HyperparameterOptimizationOptions' name-value pair argument. To control the plots, set the ShowPlots field of the 'HyperparameterOptimizationOptions' name-value pair argument.

For an example, see Optimize Kernel Classifier.

Example: 'OptimizeHyperparameters','auto'

Options for optimization, specified as the comma-separated pair consisting of 'HyperparameterOptimizationOptions' and a structure. This argument modifies the effect of the OptimizeHyperparameters name-value pair argument. All fields in the structure are optional.

Field NameValuesDefault
Optimizer
• 'bayesopt' — Use Bayesian optimization. Internally, this setting calls bayesopt.

• 'gridsearch' — Use grid search with NumGridDivisions values per dimension.

• 'randomsearch' — Search at random among MaxObjectiveEvaluations points.

'gridsearch' searches in a random order, using uniform sampling without replacement from the grid. After optimization, you can get a table in grid order by using the command sortrows(Mdl.HyperparameterOptimizationResults).

'bayesopt'
AcquisitionFunctionName

• 'expected-improvement-per-second-plus'

• 'expected-improvement'

• 'expected-improvement-plus'

• 'expected-improvement-per-second'

• 'lower-confidence-bound'

• 'probability-of-improvement'

Acquisition functions whose names include per-second do not yield reproducible results because the optimization depends on the runtime of the objective function. Acquisition functions whose names include plus modify their behavior when they are overexploiting an area. For more details, see Acquisition Function Types.

'expected-improvement-per-second-plus'
MaxObjectiveEvaluationsMaximum number of objective function evaluations.30 for 'bayesopt' or 'randomsearch', and the entire grid for 'gridsearch'
MaxTime

Time limit, specified as a positive real. The time limit is in seconds, as measured by tic and toc. Run time can exceed MaxTime because MaxTime does not interrupt function evaluations.

Inf
NumGridDivisionsFor 'gridsearch', the number of values in each dimension. The value can be a vector of positive integers giving the number of values for each dimension, or a scalar that applies to all dimensions. This field is ignored for categorical variables.10
ShowPlotsLogical value indicating whether to show plots. If true, this field plots the best objective function value against the iteration number. If there are one or two optimization parameters, and if Optimizer is 'bayesopt', then ShowPlots also plots a model of the objective function against the parameters.true
SaveIntermediateResultsLogical value indicating whether to save results when Optimizer is 'bayesopt'. If true, this field overwrites a workspace variable named 'BayesoptResults' at each iteration. The variable is a BayesianOptimization object.false
Verbose

Display to the command line.

• 0 — No iterative display

• 1 — Iterative display

• 2 — Iterative display with extra information

For details, see the bayesopt Verbose name-value pair argument.

1
UseParallelLogical value indicating whether to run Bayesian optimization in parallel, which requires Parallel Computing Toolbox™. Due to the nonreproducibility of parallel timing, parallel Bayesian optimization does not necessarily yield reproducible results. For details, see Parallel Bayesian Optimization.false
Repartition

Logical value indicating whether to repartition the cross-validation at every iteration. If false, the optimizer uses a single partition for the optimization.

true usually gives the most robust results because this setting takes partitioning noise into account. However, for good results, true requires at least twice as many function evaluations.

false
Use no more than one of the following three field names.
CVPartitionA cvpartition object, as created by cvpartition.'Kfold',5 if you do not specify any cross-validation field
HoldoutA scalar in the range (0,1) representing the holdout fraction.
KfoldAn integer greater than 1.

Example: 'HyperparameterOptimizationOptions',struct('MaxObjectiveEvaluations',60)

Data Types: struct

## Output Arguments

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Trained kernel classification model, returned as a ClassificationKernel model object or ClassificationPartitionedKernel cross-validated model object.

If you set any of the name-value pair arguments CrossVal, CVPartition, Holdout, KFold, or Leaveout, then Mdl is a ClassificationPartitionedKernel cross-validated classifier. Otherwise, Mdl is a ClassificationKernel classifier.

To reference properties of Mdl, use dot notation. For example, enter Mdl.NumExpansionDimensions in the Command Window to display the number of dimensions of the expanded space.

### Note

Unlike other classification models, and for economical memory usage, a ClassificationKernel model object does not store the training data or training process details (for example, convergence history).

Optimization details, returned as a structure array including fields described in this table. The fields contain final values or name-value pair argument specifications.

FieldDescription
Solver

Objective function minimization technique: 'LBFGS-fast', 'LBFGS-blockwise', or 'LBFGS-tall'. For details, see Algorithms.

LossFunctionLoss function. Either 'hinge' or 'logit' depending on the type of linear classification model. See Learner.
LambdaRegularization term strength. See Lambda.
BetaToleranceRelative tolerance on the linear coefficients and the bias term. See BetaTolerance.
ObjectiveValueValue of the objective function when optimization terminates. The classification loss plus the regularization term compose the objective function.
RelativeChangeInBetaRelative changes in the linear coefficients and the bias term when optimization terminates. See BetaTolerance.
FitTimeElapsed, wall-clock time (in seconds) required to fit the model to the data.
HistoryHistory of optimization information. This field is empty ([]) if you specify 'Verbose',0. For details, see Verbose and Algorithms.

To access fields, use dot notation. For example, to access the vector of objective function values for each iteration, enter FitInfo.ObjectiveValue in the Command Window.

A good practice is to examine FitInfo to assess whether convergence is satisfactory.

Cross-validation optimization of hyperparameters, returned as a BayesianOptimization object or a table of hyperparameters and associated values. The output is nonempty when the value of 'OptimizeHyperparameters' is not 'none'. The output value depends on the Optimizer field value of the 'HyperparameterOptimizationOptions' name-value pair argument:

Value of Optimizer FieldValue of HyperparameterOptimizationResults
'bayesopt' (default)Object of class BayesianOptimization
'gridsearch' or 'randomsearch'Table of hyperparameters used, observed objective function values (cross-validation loss), and rank of observations from lowest (best) to highest (worst)

## Limitations

• fitckernel does not accept initial conditions for the vector of coefficients beta (β) and bias term (b) used to determine the decision function, $f\left(x\right)=T\left(x\right)\beta +b.$

• fitckernel does not support standardization and cross-validation options.

• fitckernel does not accept table inputs.

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### Random Feature Expansion

Random feature expansion, such as Random Kitchen Sinks[1] and Fastfood[2], is a scheme to approximate Gaussian kernels of the kernel classification algorithm to use for big data in a computationally efficient way. Random feature expansion is more practical for big data applications that have large training sets, but can also be applied to smaller data sets that fit in memory.

The kernel classification algorithm searches for an optimal hyperplane that separates the data into two classes after mapping features into a high-dimensional space. Nonlinear features that are not linearly separable in a low-dimensional space can be separable in the expanded high-dimensional space. All the calculations for hyperplane classification use only dot products. You can obtain a nonlinear classification model by replacing the dot product x1x2' with the nonlinear kernel function $G\left({x}_{1},{x}_{2}\right)=〈\phi \left({x}_{1}\right),\phi \left({x}_{2}\right)〉$, where xi is the ith observation (row vector) and φ(xi) is a transformation that maps xi to a high-dimensional space (called the “kernel trick”). However, evaluating G(x1,x2) (Gram matrix) for each pair of observations is computationally expensive for a large data set (large n).

The random feature expansion scheme finds a random transformation so that its dot product approximates the Gaussian kernel. That is,

$G\left({x}_{1},{x}_{2}\right)=〈\phi \left({x}_{1}\right),\phi \left({x}_{2}\right)〉\approx T\left({x}_{1}\right)T\left({x}_{2}\right)\text{'},$

where T(x) maps x in ${ℝ}^{p}$ to a high-dimensional space (${ℝ}^{m}$). The Random Kitchen Sink scheme uses the random transformation

$T\left(x\right)={m}^{-1/2}\mathrm{exp}\left(iZx\text{'}\right)\text{'},$

where $Z\in {ℝ}^{m×p}$ is a sample drawn from $N\left(0,{\sigma }^{-2}\right)$ and σ2 is a kernel scale. This scheme requires O(mp) computation and storage. The Fastfood scheme introduces another random basis V instead of Z using Hadamard matrices combined with Gaussian scaling matrices. This random basis reduces the computation cost to O(mlogp) and reduces storage to O(m).

The fitckernel function uses the Fastfood scheme for random feature expansion and uses linear classification to train a Gaussian kernel classification model. Unlike solvers in the fitcsvm function, which require computation of the n-by-n Gram matrix, the solver in fitckernel only needs to form a matrix of size n-by-m, with m typically much less than n for big data.

### Box Constraint

A box constraint is a parameter that controls the maximum penalty imposed on margin-violating observations, and aids in preventing overfitting (regularization). Increasing the box constraint can lead to longer training times.

The box constraint (C) and the regularization term strength (λ) are related by C = 1/(λn), where n is the number of observations.

## Algorithms

fitckernel minimizes the regularized objective function using a Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) solver with ridge (L2) regularization. To find the type of LBFGS solver used for training, type FitInfo.Solver in the Command Window.

• 'LBFGS-fast' — LBFGS solver.

• 'LBFGS-blockwise' — LBFGS solver with a block-wise strategy. If fitckernel requires more memory than the value of BlockSize to hold the transformed predictor data, then it uses a block-wise strategy.

• 'LBFGS-tall' — LBFGS solver with a block-wise strategy for tall arrays.

When fitckernel uses a block-wise strategy, fitckernel implements LBFGS by distributing the calculation of the loss and gradient among different parts of the data at each iteration. Also, fitckernel refines the initial estimates of the linear coefficients and the bias term by fitting the model locally to parts of the data and combining the coefficients by averaging. If you specify 'Verbose',1, then fitckernel displays diagnostic information for each data pass and stores the information in the History field of FitInfo.

When fitckernel does not use a block-wise strategy, the initial estimates are zeros. If you specify 'Verbose',1, then fitckernel displays diagnostic information for each iteration and stores the information in the History field of FitInfo.

## References

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

[2] 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.

[3] 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.