# fitrkernel

Fit Gaussian kernel regression model using random feature expansion

## Description

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

fitrkernel 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 regression models include regularized support vector machine (SVM) and least-squares regression models.

To train a nonlinear SVM regression model on in-memory data, see fitrsvm.

example

Mdl = fitrkernel(X,Y) returns a compact Gaussian kernel regression model trained using the predictor data in X and the corresponding responses in Y.

Mdl = fitrkernel(Tbl,ResponseVarName) returns a kernel regression model Mdl trained using the predictor variables contained in the table Tbl and the response values in Tbl.ResponseVarName.

Mdl = fitrkernel(Tbl,formula) returns a kernel regression model trained using the sample data in the table Tbl. The input argument formula is an explanatory model of the response and a subset of predictor variables in Tbl used to fit Mdl.

Mdl = fitrkernel(Tbl,Y) returns a kernel regression model using the predictor variables in the table Tbl and the response values in vector Y.

example

Mdl = fitrkernel(___,Name,Value) specifies options using one or more name-value pair arguments in addition to any of the input argument combinations in previous syntaxes. For example, you can implement least-squares regression, specify the number of dimension of the expanded space, or specify cross-validation options.

example

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

## Examples

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Train a kernel regression model for a tall array by using SVM.

When you perform calculations on tall arrays, MATLAB® uses either a parallel pool (default if you have Parallel Computing Toolbox™) or the local MATLAB session. To run the example using the local MATLAB session when you have Parallel Computing Toolbox, change the global execution environment by using the mapreducer function.

mapreducer(0)

Create a datastore that references the folder location with the data. The data can be contained in a single file, a collection of files, or an entire folder. Treat 'NA' values as missing data so that datastore replaces them with NaN values. Select a subset of the variables to use. Create a tall table on top of the datastore.

varnames = {'ArrTime','DepTime','ActualElapsedTime'};
ds = datastore('airlinesmall.csv','TreatAsMissing','NA',...
'SelectedVariableNames',varnames);
t = tall(ds);

Specify DepTime and ArrTime as the predictor variables (X) and ActualElapsedTime as the response variable (Y). Select the observations for which ArrTime is later than DepTime.

daytime = t.ArrTime>t.DepTime;
Y = t.ActualElapsedTime(daytime);     % Response data
X = t{daytime,{'DepTime' 'ArrTime'}}; % Predictor data

Standardize the predictor variables.

Z = zscore(X); % Standardize the data

Train a default Gaussian kernel regression model with the standardized predictors. Extract a fit summary to determine how well the optimization algorithm fits the model to the data.

[Mdl,FitInfo] = fitrkernel(Z,Y)
Found 6 chunks.
|=========================================================================
| Solver | Iteration  /  |   Objective   |   Gradient    | Beta relative |
|        | Data Pass     |               |   magnitude   |    change     |
|=========================================================================
|   INIT |     0 /     1 |  4.313465e+01 |  6.296907e-02 |           NaN |
|  LBFGS |     0 /     2 |  3.704335e+01 |  1.789316e-02 |  9.985854e-01 |
|  LBFGS |     1 /     3 |  3.703211e+01 |  2.880402e-02 |  1.044172e-03 |
|  LBFGS |     2 /     4 |  3.701616e+01 |  2.297788e-02 |  5.115891e-04 |
|  LBFGS |     2 /     5 |  3.700183e+01 |  1.750937e-02 |  1.023672e-03 |
|  LBFGS |     3 /     6 |  3.679055e+01 |  4.815047e-02 |  1.113182e-02 |
|  LBFGS |     4 /     7 |  3.637852e+01 |  1.058657e-01 |  2.994089e-02 |
|  LBFGS |     5 /     8 |  3.565372e+01 |  1.406536e-01 |  7.033477e-02 |
|  LBFGS |     6 /     9 |  3.478061e+01 |  1.479288e-01 |  1.185262e-01 |
|  LBFGS |     7 /    10 |  3.616955e+01 |  1.544917e-01 |  2.790848e-01 |
|  LBFGS |     7 /    11 |  3.459534e+01 |  1.212256e-01 |  1.229242e-01 |
|  LBFGS |     8 /    12 |  3.379859e+01 |  8.791025e-02 |  5.417481e-02 |
|  LBFGS |     9 /    13 |  3.339981e+01 |  3.077806e-02 |  4.638645e-02 |
|  LBFGS |    10 /    14 |  3.325224e+01 |  3.082755e-02 |  2.867793e-02 |
|  LBFGS |    11 /    15 |  3.320036e+01 |  4.168377e-02 |  9.376887e-03 |
|  LBFGS |    12 /    16 |  3.309321e+01 |  5.018195e-02 |  1.831484e-02 |
|  LBFGS |    13 /    17 |  3.288069e+01 |  4.506485e-02 |  3.732443e-02 |
|  LBFGS |    14 /    18 |  3.245691e+01 |  3.787163e-02 |  1.036929e-01 |
|  LBFGS |    15 /    19 |  3.210116e+01 |  2.418833e-02 |  1.190984e-01 |
|  LBFGS |    16 /    20 |  3.190585e+01 |  2.666398e-02 |  3.921991e-02 |
|=========================================================================
| Solver | Iteration  /  |   Objective   |   Gradient    | Beta relative |
|        | Data Pass     |               |   magnitude   |    change     |
|=========================================================================
|  LBFGS |    17 /    21 |  3.172622e+01 |  2.548259e-02 |  3.805655e-02 |
|  LBFGS |    18 /    22 |  3.154538e+01 |  1.280266e-02 |  4.363429e-02 |
|  LBFGS |    19 /    23 |  3.138533e+01 |  1.446779e-02 |  8.822868e-02 |
|  LBFGS |    20 /    24 |  3.283513e+01 |  2.218528e-01 |  1.318597e-01 |
|  LBFGS |    20 /    25 |  3.158782e+01 |  1.019184e-01 |  6.992082e-02 |
|  LBFGS |    20 /    26 |  3.136869e+01 |  4.678412e-02 |  3.603399e-02 |
|========================================================================|
Mdl =
RegressionKernel
PredictorNames: {'x1'  'x2'}
ResponseName: 'Y'
Learner: 'svm'
NumExpansionDimensions: 64
KernelScale: 1
Lambda: 8.5385e-06
BoxConstraint: 1
Epsilon: 5.9303

Properties, Methods

FitInfo = struct with fields:
Solver: 'LBFGS-tall'
LossFunction: 'epsiloninsensitive'
Lambda: 8.5385e-06
BetaTolerance: 1.0000e-03
ObjectiveValue: 31.3687
RelativeChangeInBeta: 0.0360
FitTime: 52.7929
History: [1x1 struct]

Mdl is a RegressionKernel model. To inspect the regression 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 responses 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 improved 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 of fitrkernel. 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.

Specify the predictor variables (X) and the response variable (Y).

X = [Acceleration,Cylinders,Displacement,Horsepower,Weight];
Y = MPG;

Delete rows of X and Y where either array has NaN values. Removing rows with NaN values before passing data to fitrkernel can speed up training and reduce memory usage.

R = rmmissing([X Y]); % Data with missing entries removed
X = R(:,1:5);
Y = R(:,end);

Standardize the predictor variables.

Z = zscore(X);

Cross-validate a kernel regression model using 5-fold cross-validation.

Mdl = fitrkernel(Z,Y,'Kfold',5)
Mdl =
RegressionPartitionedKernel
CrossValidatedModel: 'Kernel'
ResponseName: 'Y'
NumObservations: 392
KFold: 5
Partition: [1x1 cvpartition]
ResponseTransform: 'none'

Properties, Methods

numel(Mdl.Trained)
ans = 5

Mdl is a RegressionPartitionedKernel model. Because fitrkernel implements five-fold cross-validation, Mdl contains five RegressionKernel models that the software trains on training-fold (in-fold) observations.

Examine the cross-validation loss (mean squared error) for each fold.

kfoldLoss(Mdl,'mode','individual')
ans = 5×1

13.0610
14.0975
24.0104
21.1223
24.3979

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

Specify the predictor variables (X) and the response variable (Y).

X = [Acceleration,Cylinders,Displacement,Horsepower,Weight];
Y = MPG;

Delete rows of X and Y where either array has NaN values. Removing rows with NaN values before passing data to fitrkernel can speed up training and reduce memory usage.

R = rmmissing([X Y]); % Data with missing entries removed
X = R(:,1:5);
Y = R(:,end);

Standardize the predictor variables.

Z = zscore(X);

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

rng('default')
[Mdl,FitInfo,HyperparameterOptimizationResults] = fitrkernel(Z,Y,'OptimizeHyperparameters','auto',...
'HyperparameterOptimizationOptions',struct('AcquisitionFunctionName','expected-improvement-plus'))
|====================================================================================================================|
| Iter | Eval   | Objective:  | Objective   | BestSoFar   | BestSoFar   |  KernelScale |       Lambda |      Epsilon |
|      | result | log(1+loss) | runtime     | (observed)  | (estim.)    |              |              |              |
|====================================================================================================================|
|    1 | Best   |      4.8295 |     0.80892 |      4.8295 |      4.8295 |     0.011518 |   6.8068e-05 |      0.95918 |
|    2 | Best   |      4.1488 |      0.3025 |      4.1488 |      4.1855 |       477.57 |     0.066115 |     0.091828 |
|    3 | Accept |      4.1521 |      0.2449 |      4.1488 |      4.1747 |    0.0080478 |    0.0052867 |       520.84 |
|    4 | Accept |      4.1506 |      1.1322 |      4.1488 |      4.1488 |      0.10935 |      0.35931 |     0.013372 |
|    5 | Best   |      4.1446 |      1.1006 |      4.1446 |      4.1446 |       326.29 |       2.5457 |      0.22475 |
|    6 | Accept |      4.1521 |     0.85437 |      4.1446 |      4.1447 |       932.16 |      0.19667 |       873.68 |
|    7 | Accept |      4.1501 |     0.38435 |      4.1446 |      4.1461 |     0.052426 |       2.5402 |     0.051319 |
|    8 | Best   |      4.1408 |      1.0809 |      4.1408 |      4.1402 |       850.91 |      0.01462 |      0.37284 |
|    9 | Accept |      4.1521 |     0.91426 |      4.1408 |      4.1427 |     0.019352 |     0.012035 |       63.493 |
|   10 | Accept |      4.1521 |      0.7835 |      4.1408 |      4.1452 |       853.22 |       1.0698 |       55.679 |
|   11 | Accept |      4.1521 |       0.713 |      4.1408 |      4.1416 |       1.4548 |     0.022234 |       26.275 |
|   12 | Accept |      4.1509 |     0.37568 |      4.1408 |      4.1469 |       877.82 |    0.0071133 |     0.012021 |
|   13 | Accept |      4.1422 |      0.9009 |      4.1408 |      4.1455 |       944.08 |     0.011177 |      0.31055 |
|   14 | Accept |      4.2032 |     0.45839 |      4.1408 |      4.1405 |       979.21 |     0.010842 |       13.776 |
|   15 | Accept |      4.1438 |     0.44632 |      4.1408 |      4.1509 |     0.001234 |     0.018449 |     0.044225 |
|   16 | Best   |      4.1372 |     0.26882 |      4.1372 |      4.1511 |       1.7802 |       2.5477 |     0.014737 |
|   17 | Accept |      4.1521 |      0.3489 |      4.1372 |      4.1466 |    0.0015946 |       2.5474 |       590.35 |
|   18 | Accept |      4.1452 |     0.44156 |      4.1372 |      4.1464 |     0.058846 |       1.0766 |      0.20569 |
|   19 | Accept |      4.1521 |     0.39458 |      4.1372 |      4.1461 |        2.187 |   2.5531e-06 |       278.92 |
|   20 | Accept |      4.1451 |     0.48726 |      4.1372 |      4.1461 |    0.0050283 |     0.039894 |      0.14402 |
|====================================================================================================================|
| Iter | Eval   | Objective:  | Objective   | BestSoFar   | BestSoFar   |  KernelScale |       Lambda |      Epsilon |
|      | result | log(1+loss) | runtime     | (observed)  | (estim.)    |              |              |              |
|====================================================================================================================|
|   21 | Best   |      4.1362 |     0.38386 |      4.1362 |      4.1426 |    0.0029885 |     0.039099 |       6.3938 |
|   22 | Accept |      4.1521 |     0.59548 |      4.1362 |      4.1449 |     0.035949 |     0.038533 |       80.585 |
|   23 | Accept |      4.1399 |     0.37521 |      4.1362 |      4.1446 |       50.001 |     0.095432 |      0.19954 |
|   24 | Accept |      4.1487 |     0.55658 |      4.1362 |      4.1374 |     0.012199 |     0.089894 |     0.034773 |
|   25 | Accept |      4.1521 |     0.25824 |      4.1362 |      4.1447 |    0.0011871 |      0.30153 |       425.89 |
|   26 | Accept |      4.1466 |     0.30403 |      4.1362 |       4.145 |    0.0011773 |     0.052213 |     0.017592 |
|   27 | Accept |      4.1418 |     0.40176 |      4.1362 |       4.145 |        7.556 |        1.655 |     0.016225 |
|   28 | Accept |      4.1407 |     0.34744 |      4.1362 |       4.145 |      0.01201 |       1.6696 |      0.38806 |
|   29 | Accept |      5.4284 |      1.3672 |      4.1362 |      4.1365 |    0.0010531 |   1.1032e-05 |     0.034083 |
|   30 | Accept |      4.1521 |     0.26662 |      4.1362 |      4.1365 |       7.2684 |   8.4842e-06 |       887.63 |

__________________________________________________________
Optimization completed.
MaxObjectiveEvaluations of 30 reached.
Total function evaluations: 30
Total elapsed time: 39.4239 seconds
Total objective function evaluation time: 17.2983

Best observed feasible point:
KernelScale     Lambda     Epsilon
___________    ________    _______

0.0029885     0.039099    6.3938

Observed objective function value = 4.1362
Estimated objective function value = 4.1365
Function evaluation time = 0.38386

Best estimated feasible point (according to models):
KernelScale     Lambda     Epsilon
___________    ________    _______

0.0029885     0.039099    6.3938

Estimated objective function value = 4.1365
Estimated function evaluation time = 0.41779
Mdl =
RegressionKernel
ResponseName: 'Y'
Learner: 'svm'
NumExpansionDimensions: 256
KernelScale: 0.0030
Lambda: 0.0391
BoxConstraint: 0.0652
Epsilon: 6.3938

Properties, Methods

FitInfo = struct with fields:
Solver: 'LBFGS-fast'
LossFunction: 'epsiloninsensitive'
Lambda: 0.0391
BetaTolerance: 1.0000e-04
ObjectiveValue: 1.7716
RelativeChangeInBeta: 8.5572e-05
FitTime: 0.0150
History: []

HyperparameterOptimizationResults =
BayesianOptimization with properties:

ObjectiveFcn: @createObjFcn/inMemoryObjFcn
VariableDescriptions: [5x1 optimizableVariable]
Options: [1x1 struct]
MinObjective: 4.1362
XAtMinObjective: [1x3 table]
MinEstimatedObjective: 4.1365
XAtMinEstimatedObjective: [1x3 table]
NumObjectiveEvaluations: 30
TotalElapsedTime: 39.4239
NextPoint: [1x3 table]
XTrace: [30x3 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 to which the regression model is fit, specified as an n-by-p numeric matrix, where n is the number of observations and p is the number of predictor variables.

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

Data Types: single | double

Response data, specified as an n-dimensional numeric vector. The length of Y must be equal to the number of observations in X or Tbl.

Data Types: single | double

Sample data used to train the model, specified as a table. Each row of Tbl corresponds to one observation, and each column corresponds to one predictor variable. Optionally, Tbl can contain one additional column for the response variable. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

• If Tbl contains the response variable, and you want to use all remaining variables in Tbl as predictors, then specify the response variable by using ResponseVarName.

• If Tbl contains the response variable, and you want to use only a subset of the remaining variables in Tbl as predictors, then specify a formula by using formula.

• If Tbl does not contain the response variable, then specify a response variable by using Y. The length of the response variable and the number of rows in Tbl must be equal.

Data Types: table

Response variable name, specified as the name of a variable in Tbl. The response variable must be a numeric vector.

You must specify ResponseVarName as a character vector or string scalar. For example, if Tbl stores the response variable Y as Tbl.Y, then specify it as 'Y'. Otherwise, the software treats all columns of Tbl, including Y, as predictors when training the model.

Data Types: char | string

Explanatory model of the response variable and a subset of the predictor variables, specified as a character vector or string scalar in the form "Y~x1+x2+x3". In this form, Y represents the response variable, and x1, x2, and x3 represent the predictor variables.

To specify a subset of variables in Tbl as predictors for training the model, use a formula. If you specify a formula, then the software does not use any variables in Tbl that do not appear in formula.

The variable names in the formula must be both variable names in Tbl (Tbl.Properties.VariableNames) and valid MATLAB® identifiers. You can verify the variable names in Tbl by using the isvarname function. If the variable names are not valid, then you can convert them by using the matlab.lang.makeValidName function.

Data Types: char | string

Note

The software treats NaN, empty character vector (''), empty string (""), <missing>, and <undefined> elements as missing values, and removes observations with any of these characteristics:

• Missing value in the response variable

• At least one missing value in a predictor observation (row in X or Tbl)

• NaN value or 0 weight ('Weights')

### 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: Mdl = fitrkernel(X,Y,Learner="leastsquares",NumExpansionDimensions=2^15,KernelScale="auto") implements least-squares 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.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: Mdl = fitrkernel(X,Y,'Learner','leastsquares','NumExpansionDimensions',2^15,'KernelScale','auto')

Note

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

Kernel Regression Options

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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 (rows in X).

Example: 'BoxConstraint',100

Data Types: single | double

Half the width of the epsilon-insensitive band, specified as the comma-separated pair consisting of 'Epsilon' and 'auto' or a nonnegative scalar value.

For 'auto', the fitrkernel function determines the value of Epsilon as iqr(Y)/13.49, which is an estimate of a tenth of the standard deviation using the interquartile range of the response variable Y. If iqr(Y) is equal to zero, then fitrkernel sets the value of Epsilon to 0.1.

'Epsilon' is valid only when Learner is svm.

Example: 'Epsilon',0.3

Data Types: single | double

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 fitrkernel function selects the number of dimensions using 2.^ceil(min(log2(p)+5,15)), where p is the number of predictors.

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. MATLAB obtains the random basis for random feature expansion by using the kernel scale parameter. For details, see Random Feature Expansion.

If you specify 'auto', then MATLAB 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

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.

When Learner is 'svm', 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

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

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.

ValueAlgorithmResponse rangeLoss function
'leastsquares'Linear regression via ordinary least squaresy ∊ (-∞,∞)Mean squared error (MSE): $\ell \left[y,f\left(x\right)\right]=\frac{1}{2}{\left[y-f\left(x\right)\right]}^{2}$
'svm'Support vector machine regressionSame as 'leastsquares'Epsilon-insensitive: $\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,|y-f\left(x\right)|-\epsilon \right]$

Example: 'Learner','leastsquares'

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

ValueDescription
0fitrkernel does not display diagnostic information.
1fitrkernel 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

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

If fitrkernel requires more memory than the value of BlockSize to hold the transformed predictor data, then MATLAB 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 fitrkernel uses to transform the data in X 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')

Other Regression Options

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Categorical predictors list, specified as one of the values in this table.

ValueDescription
Vector of positive integers

Each entry in the vector is an index value indicating that the corresponding predictor is categorical. The index values are between 1 and p, where p is the number of predictors used to train the model.

If fitrkernel uses a subset of input variables as predictors, then the function indexes the predictors using only the subset. The CategoricalPredictors values do not count the response variable, observation weight variable, or any other variables that the function does not use.

Logical vector

A true entry means that the corresponding predictor is categorical. The length of the vector is p.

Character matrixEach row of the matrix is the name of a predictor variable. The names must match the entries in PredictorNames. Pad the names with extra blanks so each row of the character matrix has the same length.
String array or cell array of character vectorsEach element in the array is the name of a predictor variable. The names must match the entries in PredictorNames.
"all"All predictors are categorical.

By default, if the predictor data is in a table (Tbl), fitrkernel assumes that a variable is categorical if it is a logical vector, categorical vector, character array, string array, or cell array of character vectors. If the predictor data is a matrix (X), fitrkernel assumes that all predictors are continuous. To identify any other predictors as categorical predictors, specify them by using the 'CategoricalPredictors' name-value argument.

For the identified categorical predictors, fitrkernel creates dummy variables using two different schemes, depending on whether a categorical variable is unordered or ordered. For an unordered categorical variable, fitrkernel creates one dummy variable for each level of the categorical variable. For an ordered categorical variable, fitrkernel creates one less dummy variable than the number of categories. For details, see Automatic Creation of Dummy Variables.

Example: 'CategoricalPredictors','all'

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

Predictor variable names, specified as a string array of unique names or cell array of unique character vectors. The functionality of PredictorNames depends on the way you supply the training data.

• If you supply X and Y, then you can use PredictorNames to assign names to the predictor variables in X.

• The order of the names in PredictorNames must correspond to the column order of X. That is, PredictorNames{1} is the name of X(:,1), PredictorNames{2} is the name of X(:,2), and so on. Also, size(X,2) and numel(PredictorNames) must be equal.

• By default, PredictorNames is {'x1','x2',...}.

• If you supply Tbl, then you can use PredictorNames to choose which predictor variables to use in training. That is, fitrkernel uses only the predictor variables in PredictorNames and the response variable during training.

• PredictorNames must be a subset of Tbl.Properties.VariableNames and cannot include the name of the response variable.

• By default, PredictorNames contains the names of all predictor variables.

• A good practice is to specify the predictors for training using either PredictorNames or formula, but not both.

Example: "PredictorNames",["SepalLength","SepalWidth","PetalLength","PetalWidth"]

Data Types: string | cell

Response variable name, specified as a character vector or string scalar.

• If you supply Y, then you can use ResponseName to specify a name for the response variable.

• If you supply ResponseVarName or formula, then you cannot use ResponseName.

Example: "ResponseName","response"

Data Types: char | string

Response transformation, specified as either 'none' or a function handle. The default is 'none', which means @(y)y, or no transformation. For a MATLAB function or a function you define, use its function handle for the response transformation. The function handle must accept a vector (the original response values) and return a vector of the same size (the transformed response values).

Example: Suppose you create a function handle that applies an exponential transformation to an input vector by using myfunction = @(y)exp(y). Then, you can specify the response transformation as 'ResponseTransform',myfunction.

Data Types: char | string | function_handle

Observation weights, specified as the comma-separated pair consisting of 'Weights' and a vector of scalar values or the name of a variable in Tbl. The software weights each observation (or row) in X or Tbl with the corresponding value in Weights. The length of Weights must equal the number of rows in X or Tbl.

If you specify the input data as a table Tbl, then Weights can be the name of a variable in Tbl that contains a numeric vector. In this case, you must specify Weights as a character vector or string scalar. For example, if weights vector W is stored as Tbl.W, then specify it as 'W'. Otherwise, the software treats all columns of Tbl, including W, as predictors when training the model.

By default, Weights is ones(n,1), where n is the number of observations in X or Tbl.

fitrkernel normalizes the weights to sum to 1.

Data Types: single | double | char | string

Cross-Validation Options

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Cross-validation flag, specified as the comma-separated pair consisting of 'Crossval' and 'on' or 'off'.

If you specify 'on', then the software implements 10-fold cross-validation.

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 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 specify only one of these four name-value arguments: 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 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 specify only one of these four name-value arguments: 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 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 a k-by-1 cell vector in the Trained property of the cross-validated model.

To create a cross-validated model, you can specify only one of these four name-value arguments: 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

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

Example: 'HessianHistorySize',10

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 BlockSize. Otherwise, the default value is 100.

Example: 'IterationLimit',500

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','Epsilon'}.

• '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 fitrkernel by varying the parameters. To control the cross-validation type and other aspects of the optimization, use the HyperparameterOptimizationOptions name-value pair argument.

Note

The values of 'OptimizeHyperparameters' override any values you specify using other name-value arguments. For example, setting 'OptimizeHyperparameters' to 'auto' causes fitrkernel to optimize hyperparameters corresponding to the 'auto' option and to ignore any specified values for the hyperparameters.

The eligible parameters for fitrkernel are:

• Epsilonfitrkernel searches among positive values, by default log-scaled in the range [1e-3,1e2]*iqr(Y)/1.349.

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

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

• Learnerfitrkernel searches among 'svm' and 'leastsquares'.

• NumExpansionDimensionsfitrkernel 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('fitrkernel',[Horsepower,Weight],MPG);
params(2).Range = [1e-4,1e6];

Pass params as the value of 'OptimizeHyperparameters'.

By default, the 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). To control the iterative display, set the Verbose field of the 'HyperparameterOptimizationOptions' name-value argument. To control the plots, set the ShowPlots field of the 'HyperparameterOptimizationOptions' name-value argument.

For an example, see Optimize Kernel Regression.

Example: 'OptimizeHyperparameters','auto'

Options for optimization, specified as a structure. This argument modifies the effect of the OptimizeHyperparameters name-value 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' and 'randomsearch', and the entire grid for 'gridsearch'
MaxTime

Time limit, specified as a positive real scalar. The time limit is in seconds, as measured by tic and toc. The 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 observed objective function value against the iteration number. If you use Bayesian optimization (Optimizer is 'bayesopt'), then this field also plots the best estimated objective function value. The best observed objective function values and best estimated objective function values correspond to the values in the BestSoFar (observed) and BestSoFar (estim.) columns of the iterative display, respectively. You can find these values in the properties ObjectiveMinimumTrace and EstimatedObjectiveMinimumTrace of Mdl.HyperparameterOptimizationResults. If the problem includes one or two optimization parameters for Bayesian optimization, 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 at the command line:

• 0 — No iterative display

• 1 — Iterative display

• 2 — Iterative display with extra information

For details, see the bayesopt Verbose name-value argument and the example Optimize Classifier Fit Using Bayesian Optimization.

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 this field is false, the optimizer uses a single partition for the optimization.

The setting true usually gives the most robust results because it 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 options.
CVPartitionA cvpartition object, as created by cvpartition'Kfold',5 if you do not specify a 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 regression model, returned as a RegressionKernel model object or RegressionPartitionedKernel cross-validated model object.

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

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 regression models, and for economical memory usage, a RegressionKernel 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 mean squared error (MSE) or epsilon-insensitive, depending on the type of linear regression 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 regression 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 also includes the optimization information from training Mdl. 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.

Examine the information provided by 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)

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### Random Feature Expansion

Random feature expansion, such as Random Kitchen Sinks[1] or Fastfood[2], is a scheme to approximate Gaussian kernels of the kernel regression algorithm 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.

After mapping the predictor data into a high-dimensional space, the kernel regression algorithm searches for an optimal function that deviates from each response data point (yi) by values no greater than the epsilon margin (ε).

Some regression problems cannot be described adequately using a linear model. In such cases, obtain a nonlinear regression model by replacing the dot product x1x2 with a 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), the 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 Sinks[1] 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[2] scheme introduces another random basis V instead of Z using Hadamard matrices combined with Gaussian scaling matrices. This random basis reduces computation cost to O(mlogp) and reduces storage to O(m).

You can specify values for m and σ2, using the NumExpansionDimensions and KernelScale name-value pair arguments of fitrkernel, respectively.

The fitrkernel function uses the Fastfood scheme for random feature expansion and uses linear regression to train a Gaussian kernel regression model. Unlike solvers in the fitrsvm function, which require computation of the n-by-n Gram matrix, the solver in fitrkernel 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 observations that lie outside the epsilon margin (ε), and helps to prevent 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.

## Tips

• Standardizing predictors before training a model can be helpful. You can standardize training data and scale test data to have the same scale as the training data by using the normalize function.

## Algorithms

fitrkernel 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 fitrkernel 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 fitrkernel uses a block-wise strategy, fitrkernel implements LBFGS by distributing the calculation of the loss and gradient among different parts of the data at each iteration. Also, fitrkernel 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 fitrkernel displays diagnostic information for each data pass and stores the information in the History field of FitInfo.

When fitrkernel does not use a block-wise strategy, the initial estimates are zeros. If you specify 'Verbose',1, then fitrkernel 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.

## Version History

Introduced in R2018a