RegressionKernel

Gaussian kernel regression model using random feature expansion

Description

RegressionKernel is a trained model object for Gaussian kernel regression using random feature expansion. RegressionKernel 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.

Unlike other regression models, and for economical memory usage, RegressionKernel model objects do not store the training data. However, they do store information such as the dimension of the expanded space, the kernel scale parameter, and the regularization strength.

You can use trained RegressionKernel models to continue training using the training data, predict responses for new data, and compute the mean squared error or epsilon-insensitive loss. For details, see resume, predict, and loss.

Creation

Create a RegressionKernel object using the fitrkernel function. This function 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 machines (SVM) and least-squares regression models.

Properties

Kernel Regression Properties

Epsilon — Half width of epsilon-insensitive band

nonnegative scalar

Half the width of the epsilon-insensitive band, specified as a nonnegative scalar.

If Learner is not 'svm', then Epsilon is an empty array ([]).

Data Types: single | double

Learner — Linear regression model type

'svm' (default) | 'leastsquares'

Linear regression model type, specified as 'leastsquares' or 'svm'.

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

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

• $$T(·)$$ 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.

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

NumExpansionDimensions — Number of dimensions of expanded space

positive integer

Number of dimensions of the expanded space, specified as a positive integer.

Data Types: single | double

KernelScale — Kernel scale parameter

positive scalar

Kernel scale parameter, specified as a positive scalar.

Data Types: single | double

BoxConstraint — Box constraint

positive scalar

Box constraint, specified as a positive scalar.

Data Types: double | single

Lambda — Regularization term strength

nonnegative scalar

Regularization term strength, specified as a nonnegative scalar.

Data Types: single | double

Mu — Predictor means

numeric vector | []

Since R2023b

This property is read-only.

Predictor means, specified as a numeric vector. If you specify Standardize as 1 or true when you train the kernel model, then the length of the Mu vector is equal to the number of expanded predictors (see ExpandedPredictorNames). The vector contains 0 values for dummy variables corresponding to expanded categorical predictors.

If you set Standardize to 0 or false when you train the kernel model, then the Mu value is an empty vector ([]).

Data Types: double

Sigma — Predictor standard deviations

numeric vector | []

Since R2023b

This property is read-only.

Predictor standard deviations, specified as a numeric vector. If you specify Standardize as 1 or true when you train the kernel model, then the length of the Sigma vector is equal to the number of expanded predictors (see ExpandedPredictorNames). The vector contains 1 values for dummy variables corresponding to expanded categorical predictors.

If you set Standardize to 0 or false when you train the kernel model, then the Sigma value is an empty vector ([]).

Data Types: double

FittedLoss — Loss function used to fit linear model

'epsiloninsensitive' | 'mse'

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

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

Regularization — Complexity penalty type

'lasso (L1)' | 'ridge (L2)'

Complexity penalty type, specified as 'lasso (L1)' or 'ridge (L2)'.

The software composes the objective function for minimization from the sum of the average loss function (see FittedLoss) and a regularization value from this table.

Value	Description
'lasso (L1)'	Lasso (L1) penalty: $$\lambda {\displaystyle \sum _{j=1}^p\left|{\beta }_j\right|}$$
'ridge (L2)'	Ridge (L2) penalty: $$\frac{\lambda }{2}{\displaystyle \sum _{j=1}^p{\beta }_j^2}$$

λ specifies the regularization term strength (see Lambda).

The software excludes the bias term (β₀) from the regularization penalty.

Other Regression Properties

CategoricalPredictors — Indices of categorical predictors

vector of positive integers | []

Categorical predictor indices, specified as a vector of positive integers. CategoricalPredictors contains index values indicating that the corresponding predictors are categorical. The index values are between 1 and p, where p is the number of predictors used to train the model. If none of the predictors are categorical, then this property is empty ([]).

Data Types: single | double

ModelParameters — Parameters used for training model

structure

Parameters used for training the RegressionKernel model, specified as a structure.

Access fields of ModelParameters using dot notation. For example, access the relative tolerance on the linear coefficients and the bias term by using Mdl.ModelParameters.BetaTolerance.

Data Types: struct

PredictorNames — Predictor names

cell array of character vectors

Predictor names in order of their appearance in the predictor data, specified as a cell array of character vectors. The length of PredictorNames is equal to the number of columns used as predictor variables in the training data X or Tbl.

Data Types: cell

ExpandedPredictorNames — Expanded predictor names

cell array of character vectors

Expanded predictor names, specified as a cell array of character vectors.

If the model uses encoding for categorical variables, then ExpandedPredictorNames includes the names that describe the expanded variables. Otherwise, ExpandedPredictorNames is the same as PredictorNames.

Data Types: cell

ResponseName — Response variable name

character vector

Response variable name, specified as a character vector.

Data Types: char

ResponseTransform — Response transformation function to apply to predicted responses

'none' | function handle

Response transformation function to apply to predicted responses, specified as 'none' or a function handle.

For kernel regression models and before the response transformation, the predicted response for the observation x (row vector) is $$f\left(x\right)=T(x)\beta +b.$$

• $$T(·)$$ is a transformation of an observation for feature expansion.

• β corresponds to Mdl.Beta.

• b corresponds to Mdl.Bias.

For a MATLAB^® function or a function that you define, enter its function handle. For example, you can enter Mdl.ResponseTransform = @function, where function accepts a numeric vector of the original responses and returns a numeric vector of the same size containing the transformed responses.

Data Types: char | function_handle

Object Functions

incrementalLearner	Convert kernel regression model to incremental learner
lime	Local interpretable model-agnostic explanations (LIME)
loss	Regression loss for Gaussian kernel regression model
partialDependence	Compute partial dependence
plotPartialDependence	Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
predict	Predict responses for Gaussian kernel regression model
resume	Resume training of Gaussian kernel regression model
shapley	Shapley values

Examples

Train Gaussian Kernel Regression Model

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 |  4.307833e+01 |  9.925486e-02 |           NaN |
|  LBFGS |     0 /     2 |  2.782790e+01 |  7.202403e-03 |  9.891473e-01 |
|  LBFGS |     1 /     3 |  2.781351e+01 |  1.806211e-02 |  3.220672e-03 |
|  LBFGS |     2 /     4 |  2.777773e+01 |  2.727737e-02 |  9.309939e-03 |
|  LBFGS |     3 /     5 |  2.768591e+01 |  2.951422e-02 |  2.833343e-02 |
|  LBFGS |     4 /     6 |  2.755857e+01 |  5.124144e-02 |  7.935278e-02 |
|  LBFGS |     5 /     7 |  2.738896e+01 |  3.089571e-02 |  4.644920e-02 |
|  LBFGS |     6 /     8 |  2.716704e+01 |  2.552696e-02 |  8.596406e-02 |
|  LBFGS |     7 /     9 |  2.696409e+01 |  3.088621e-02 |  1.263589e-01 |
|  LBFGS |     8 /    10 |  2.676203e+01 |  2.021303e-02 |  1.533927e-01 |
|  LBFGS |     9 /    11 |  2.660322e+01 |  1.221361e-02 |  1.351968e-01 |
|  LBFGS |    10 /    12 |  2.645504e+01 |  1.486501e-02 |  1.175476e-01 |
|  LBFGS |    11 /    13 |  2.631323e+01 |  1.772835e-02 |  1.161909e-01 |
|  LBFGS |    12 /    14 |  2.625264e+01 |  5.837906e-02 |  1.422851e-01 |
|  LBFGS |    13 /    15 |  2.619281e+01 |  1.294441e-02 |  2.966283e-02 |
|  LBFGS |    14 /    16 |  2.618220e+01 |  3.791806e-03 |  9.051274e-03 |
|  LBFGS |    15 /    17 |  2.617989e+01 |  3.689255e-03 |  6.364132e-03 |
|  LBFGS |    16 /    18 |  2.617426e+01 |  4.200232e-03 |  1.213026e-02 |
|  LBFGS |    17 /    19 |  2.615914e+01 |  7.339928e-03 |  2.803348e-02 |
|  LBFGS |    18 /    20 |  2.620704e+01 |  2.298098e-02 |  1.749830e-01 |
|=========================================================================
| Solver | Iteration  /  |   Objective   |   Gradient    | Beta relative |
|        | Data Pass     |               |   magnitude   |    change     |
|=========================================================================
|  LBFGS |    18 /    21 |  2.615554e+01 |  1.164689e-02 |  8.580878e-02 |
|  LBFGS |    19 /    22 |  2.614367e+01 |  3.395507e-03 |  3.938314e-02 |
|  LBFGS |    20 /    23 |  2.614090e+01 |  2.349246e-03 |  1.495049e-02 |
|========================================================================|

Mdl = 
  RegressionKernel
              ResponseName: 'Y'
                   Learner: 'svm'
    NumExpansionDimensions: 64
               KernelScale: 1
                    Lambda: 8.5385e-06
             BoxConstraint: 1
                   Epsilon: 5.9303

FitInfo = struct with fields:
                  Solver: 'LBFGS-tall'
            LossFunction: 'epsiloninsensitive'
                  Lambda: 8.5385e-06
           BetaTolerance: 1.0000e-03
       GradientTolerance: 1.0000e-05
          ObjectiveValue: 26.1409
       GradientMagnitude: 0.0023
    RelativeChangeInBeta: 0.0150
                 FitTime: 25.1639
                 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.

Estimate Sample Loss and Resume Training

Resume training a Gaussian kernel regression model for more iterations to improve the regression loss.

Load the carbig data set.

load carbig

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); 

Reserve 10% of the observations as a holdout sample. Extract the training and test indices from the partition definition.

rng(10)  % For reproducibility
N = length(Y);
cvp = cvpartition(N,'Holdout',0.1);
idxTrn = training(cvp); % Training set indices
idxTest = test(cvp);    % Test set indices

Train a kernel regression model. Standardize the training data, set the iteration limit to 5, and specify 'Verbose',1 to display diagnostic information.

Xtrain = X(idxTrn,:);
Ytrain = Y(idxTrn);

Mdl = fitrkernel(Xtrain,Ytrain,'Standardize',true, ...
    'IterationLimit',5,'Verbose',1)

|=================================================================================================================|
| Solver |  Pass  |   Iteration  |   Objective   |     Step      |    Gradient   |    Relative    |  sum(beta~=0) |
|        |        |              |               |               |   magnitude   | change in Beta |               |
|=================================================================================================================|
|  LBFGS |      1 |            0 |  5.691016e+00 |  0.000000e+00 |  5.852758e-02 |                |             0 |
|  LBFGS |      1 |            1 |  5.086537e+00 |  8.000000e+00 |  5.220869e-02 |   9.846711e-02 |           256 |
|  LBFGS |      1 |            2 |  3.862301e+00 |  5.000000e-01 |  3.796034e-01 |   5.998808e-01 |           256 |
|  LBFGS |      1 |            3 |  3.460613e+00 |  1.000000e+00 |  3.257790e-01 |   1.615091e-01 |           256 |
|  LBFGS |      1 |            4 |  3.136228e+00 |  1.000000e+00 |  2.832861e-02 |   8.006254e-02 |           256 |
|  LBFGS |      1 |            5 |  3.063978e+00 |  1.000000e+00 |  1.475038e-02 |   3.314455e-02 |           256 |
|=================================================================================================================|

Mdl = 
  RegressionKernel
              ResponseName: 'Y'
                   Learner: 'svm'
    NumExpansionDimensions: 256
               KernelScale: 1
                    Lambda: 0.0028
             BoxConstraint: 1
                   Epsilon: 0.8617

Mdl is a RegressionKernel model.

Estimate the epsilon-insensitive error for the test set.

Xtest = X(idxTest,:);
Ytest = Y(idxTest);

L = loss(Mdl,Xtest,Ytest,'LossFun','epsiloninsensitive')

L = 
2.0674

Continue training the model by using resume. This function continues training with the same options used for training Mdl.

UpdatedMdl = resume(Mdl,Xtrain,Ytrain);

|=================================================================================================================|
| Solver |  Pass  |   Iteration  |   Objective   |     Step      |    Gradient   |    Relative    |  sum(beta~=0) |
|        |        |              |               |               |   magnitude   | change in Beta |               |
|=================================================================================================================|
|  LBFGS |      1 |            0 |  3.063978e+00 |  0.000000e+00 |  1.475038e-02 |                |           256 |
|  LBFGS |      1 |            1 |  3.007822e+00 |  8.000000e+00 |  1.391637e-02 |   2.603966e-02 |           256 |
|  LBFGS |      1 |            2 |  2.817171e+00 |  5.000000e-01 |  5.949008e-02 |   1.918084e-01 |           256 |
|  LBFGS |      1 |            3 |  2.807294e+00 |  2.500000e-01 |  6.798867e-02 |   2.973097e-02 |           256 |
|  LBFGS |      1 |            4 |  2.791060e+00 |  1.000000e+00 |  2.549575e-02 |   1.639328e-02 |           256 |
|  LBFGS |      1 |            5 |  2.767821e+00 |  1.000000e+00 |  6.154419e-03 |   2.468903e-02 |           256 |
|  LBFGS |      1 |            6 |  2.738163e+00 |  1.000000e+00 |  5.949008e-02 |   9.476263e-02 |           256 |
|  LBFGS |      1 |            7 |  2.719146e+00 |  1.000000e+00 |  1.699717e-02 |   1.849972e-02 |           256 |
|  LBFGS |      1 |            8 |  2.705941e+00 |  1.000000e+00 |  3.116147e-02 |   4.152590e-02 |           256 |
|  LBFGS |      1 |            9 |  2.701162e+00 |  1.000000e+00 |  5.665722e-03 |   9.401466e-03 |           256 |
|  LBFGS |      1 |           10 |  2.695341e+00 |  5.000000e-01 |  3.116147e-02 |   4.968046e-02 |           256 |
|  LBFGS |      1 |           11 |  2.691277e+00 |  1.000000e+00 |  8.498584e-03 |   1.017446e-02 |           256 |
|  LBFGS |      1 |           12 |  2.689972e+00 |  1.000000e+00 |  1.983003e-02 |   9.938921e-03 |           256 |
|  LBFGS |      1 |           13 |  2.688979e+00 |  1.000000e+00 |  1.416431e-02 |   6.606316e-03 |           256 |
|  LBFGS |      1 |           14 |  2.687787e+00 |  1.000000e+00 |  1.621956e-03 |   7.089542e-03 |           256 |
|  LBFGS |      1 |           15 |  2.686539e+00 |  1.000000e+00 |  1.699717e-02 |   1.169701e-02 |           256 |
|  LBFGS |      1 |           16 |  2.685356e+00 |  1.000000e+00 |  1.133144e-02 |   1.069310e-02 |           256 |
|  LBFGS |      1 |           17 |  2.685021e+00 |  5.000000e-01 |  1.133144e-02 |   2.104248e-02 |           256 |
|  LBFGS |      1 |           18 |  2.684002e+00 |  1.000000e+00 |  2.832861e-03 |   6.175231e-03 |           256 |
|  LBFGS |      1 |           19 |  2.683507e+00 |  1.000000e+00 |  5.665722e-03 |   3.724026e-03 |           256 |
|  LBFGS |      1 |           20 |  2.683343e+00 |  5.000000e-01 |  5.665722e-03 |   9.549119e-03 |           256 |
|=================================================================================================================|
| Solver |  Pass  |   Iteration  |   Objective   |     Step      |    Gradient   |    Relative    |  sum(beta~=0) |
|        |        |              |               |               |   magnitude   | change in Beta |               |
|=================================================================================================================|
|  LBFGS |      1 |           21 |  2.682897e+00 |  1.000000e+00 |  5.665722e-03 |   7.172867e-03 |           256 |
|  LBFGS |      1 |           22 |  2.682682e+00 |  1.000000e+00 |  2.832861e-03 |   2.587726e-03 |           256 |
|  LBFGS |      1 |           23 |  2.682485e+00 |  1.000000e+00 |  2.832861e-03 |   2.953648e-03 |           256 |
|  LBFGS |      1 |           24 |  2.682326e+00 |  1.000000e+00 |  2.832861e-03 |   7.777294e-03 |           256 |
|  LBFGS |      1 |           25 |  2.681914e+00 |  1.000000e+00 |  2.832861e-03 |   2.778555e-03 |           256 |
|  LBFGS |      1 |           26 |  2.681867e+00 |  5.000000e-01 |  1.031085e-03 |   3.638352e-03 |           256 |
|  LBFGS |      1 |           27 |  2.681725e+00 |  1.000000e+00 |  5.665722e-03 |   1.515199e-03 |           256 |
|  LBFGS |      1 |           28 |  2.681692e+00 |  5.000000e-01 |  1.314940e-03 |   1.850055e-03 |           256 |
|  LBFGS |      1 |           29 |  2.681625e+00 |  1.000000e+00 |  2.832861e-03 |   1.456903e-03 |           256 |
|  LBFGS |      1 |           30 |  2.681594e+00 |  5.000000e-01 |  2.832861e-03 |   8.704875e-04 |           256 |
|  LBFGS |      1 |           31 |  2.681581e+00 |  5.000000e-01 |  8.498584e-03 |   3.934768e-04 |           256 |
|  LBFGS |      1 |           32 |  2.681579e+00 |  1.000000e+00 |  8.498584e-03 |   1.847866e-03 |           256 |
|  LBFGS |      1 |           33 |  2.681553e+00 |  1.000000e+00 |  9.857038e-04 |   6.509825e-04 |           256 |
|  LBFGS |      1 |           34 |  2.681541e+00 |  5.000000e-01 |  8.498584e-03 |   6.635528e-04 |           256 |
|  LBFGS |      1 |           35 |  2.681499e+00 |  1.000000e+00 |  5.665722e-03 |   6.194735e-04 |           256 |
|  LBFGS |      1 |           36 |  2.681493e+00 |  5.000000e-01 |  1.133144e-02 |   1.617763e-03 |           256 |
|  LBFGS |      1 |           37 |  2.681473e+00 |  1.000000e+00 |  9.869233e-04 |   8.418484e-04 |           256 |
|  LBFGS |      1 |           38 |  2.681469e+00 |  1.000000e+00 |  5.665722e-03 |   1.069722e-03 |           256 |
|  LBFGS |      1 |           39 |  2.681432e+00 |  1.000000e+00 |  2.832861e-03 |   8.501930e-04 |           256 |
|  LBFGS |      1 |           40 |  2.681423e+00 |  2.500000e-01 |  1.133144e-02 |   9.543716e-04 |           256 |
|=================================================================================================================|
| Solver |  Pass  |   Iteration  |   Objective   |     Step      |    Gradient   |    Relative    |  sum(beta~=0) |
|        |        |              |               |               |   magnitude   | change in Beta |               |
|=================================================================================================================|
|  LBFGS |      1 |           41 |  2.681416e+00 |  1.000000e+00 |  2.832861e-03 |   8.763251e-04 |           256 |
|  LBFGS |      1 |           42 |  2.681413e+00 |  5.000000e-01 |  2.832861e-03 |   4.101888e-04 |           256 |
|  LBFGS |      1 |           43 |  2.681403e+00 |  1.000000e+00 |  5.665722e-03 |   2.713209e-04 |           256 |
|  LBFGS |      1 |           44 |  2.681392e+00 |  1.000000e+00 |  2.832861e-03 |   2.115241e-04 |           256 |
|  LBFGS |      1 |           45 |  2.681383e+00 |  1.000000e+00 |  2.832861e-03 |   2.872858e-04 |           256 |
|  LBFGS |      1 |           46 |  2.681374e+00 |  1.000000e+00 |  8.498584e-03 |   5.771001e-04 |           256 |
|  LBFGS |      1 |           47 |  2.681353e+00 |  1.000000e+00 |  2.832861e-03 |   3.160871e-04 |           256 |
|  LBFGS |      1 |           48 |  2.681334e+00 |  5.000000e-01 |  8.498584e-03 |   1.045502e-03 |           256 |
|  LBFGS |      1 |           49 |  2.681314e+00 |  1.000000e+00 |  7.878714e-04 |   1.505118e-03 |           256 |
|  LBFGS |      1 |           50 |  2.681306e+00 |  1.000000e+00 |  2.832861e-03 |   4.756894e-04 |           256 |
|  LBFGS |      1 |           51 |  2.681301e+00 |  1.000000e+00 |  1.133144e-02 |   3.664873e-04 |           256 |
|  LBFGS |      1 |           52 |  2.681288e+00 |  1.000000e+00 |  2.832861e-03 |   1.449821e-04 |           256 |
|  LBFGS |      1 |           53 |  2.681287e+00 |  2.500000e-01 |  1.699717e-02 |   2.357176e-04 |           256 |
|  LBFGS |      1 |           54 |  2.681282e+00 |  1.000000e+00 |  5.665722e-03 |   2.046663e-04 |           256 |
|  LBFGS |      1 |           55 |  2.681278e+00 |  1.000000e+00 |  2.832861e-03 |   2.546349e-04 |           256 |
|  LBFGS |      1 |           56 |  2.681276e+00 |  2.500000e-01 |  1.307940e-03 |   1.966786e-04 |           256 |
|  LBFGS |      1 |           57 |  2.681274e+00 |  5.000000e-01 |  1.416431e-02 |   1.005310e-04 |           256 |
|  LBFGS |      1 |           58 |  2.681271e+00 |  5.000000e-01 |  1.118892e-03 |   1.147324e-04 |           256 |
|  LBFGS |      1 |           59 |  2.681269e+00 |  1.000000e+00 |  2.832861e-03 |   1.332914e-04 |           256 |
|  LBFGS |      1 |           60 |  2.681268e+00 |  2.500000e-01 |  1.132045e-03 |   5.441369e-05 |           256 |
|=================================================================================================================|

Estimate the epsilon-insensitive error for the test set using the updated model.

UpdatedL = loss(UpdatedMdl,Xtest,Ytest,'LossFun','epsiloninsensitive')

UpdatedL = 
1.8933

The regression error decreases by a factor of about 0.08 after resume updates the regression model with more iterations.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™. (since R2023a)

Usage notes and limitations:

• The predict function supports code generation.

• When you train a kernel regression model by using fitrkernel, the value of the ResponseTransform name-value argument cannot be an anonymous function.

For more information, see Introduction to Code Generation.

Version History

Introduced in R2018a

R2023b: Kernel models support standardization of predictors

The fitckernel and fitrkernel functions support the standardization of numeric predictors. That is, in the call to either function, you can specify the Standardize value as true to center and scale each numeric predictor variable. Kernel models include Mu and Sigma properties that contain the means and standard deviations, respectively, used to standardize the predictors before training. The properties are empty when the fitting function does not perform any standardization.

R2023a: Generate C/C++ code for prediction

You can generate C/C++ code for the predict function.