# 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

expand all

### Kernel Regression Properties

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

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

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

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

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

• β is a vector of coefficients.

• b is the scalar bias.

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

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

Data Types: single | double

Kernel scale parameter, specified as a positive scalar.

Data Types: single | double

Box constraint, specified as a positive scalar.

Data Types: double | single

Regularization term strength, specified as a nonnegative scalar.

Data Types: single | double

Since R2023b

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

Since R2023b

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

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

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

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.

ValueDescription
'lasso (L1)'Lasso (L1) penalty: $\lambda \sum _{j=1}^{p}|{\beta }_{j}|$
'ridge (L2)'Ridge (L2) penalty: $\frac{\lambda }{2}\sum _{j=1}^{p}{\beta }_{j}^{2}$

λ specifies the regularization term strength (see Lambda).

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

### Other Regression Properties

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

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

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

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

Response variable name, specified as a character vector.

Data Types: char

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 $f\left(x\right)=T\left(x\right)\beta +b.$

• $T\left(·\right)$ 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

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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 |  4.307833e+01 |  4.345788e-02 |           NaN |
|  LBFGS |     0 /     2 |  3.705713e+01 |  1.577301e-02 |  9.988252e-01 |
|  LBFGS |     1 /     3 |  3.704022e+01 |  3.082836e-02 |  1.338410e-03 |
|  LBFGS |     2 /     4 |  3.701398e+01 |  3.006488e-02 |  1.116070e-03 |
|  LBFGS |     2 /     5 |  3.698797e+01 |  2.870642e-02 |  2.234599e-03 |
|  LBFGS |     2 /     6 |  3.693687e+01 |  2.625581e-02 |  4.479069e-03 |
|  LBFGS |     2 /     7 |  3.683757e+01 |  2.239620e-02 |  8.997877e-03 |
|  LBFGS |     2 /     8 |  3.665038e+01 |  1.782358e-02 |  1.815682e-02 |
|  LBFGS |     3 /     9 |  3.473411e+01 |  4.074480e-02 |  1.778166e-01 |
|  LBFGS |     4 /    10 |  3.684246e+01 |  1.608942e-01 |  3.294968e-01 |
|  LBFGS |     4 /    11 |  3.441595e+01 |  8.587703e-02 |  1.420892e-01 |
|  LBFGS |     5 /    12 |  3.377755e+01 |  3.760006e-02 |  4.640134e-02 |
|  LBFGS |     6 /    13 |  3.357732e+01 |  1.912644e-02 |  3.842057e-02 |
|  LBFGS |     7 /    14 |  3.334081e+01 |  3.046709e-02 |  6.211243e-02 |
|  LBFGS |     8 /    15 |  3.309239e+01 |  3.858085e-02 |  6.411356e-02 |
|  LBFGS |     9 /    16 |  3.276577e+01 |  3.612292e-02 |  6.938579e-02 |
|  LBFGS |    10 /    17 |  3.234029e+01 |  2.734959e-02 |  1.144307e-01 |
|  LBFGS |    11 /    18 |  3.205763e+01 |  2.545990e-02 |  7.323180e-02 |
|  LBFGS |    12 /    19 |  3.183341e+01 |  2.472411e-02 |  3.689625e-02 |
|  LBFGS |    13 /    20 |  3.169307e+01 |  2.064613e-02 |  2.998555e-02 |
|=========================================================================
| Solver | Iteration  /  |   Objective   |   Gradient    | Beta relative |
|        | Data Pass     |               |   magnitude   |    change     |
|=========================================================================
|  LBFGS |    14 /    21 |  3.146896e+01 |  1.788395e-02 |  5.967293e-02 |
|  LBFGS |    15 /    22 |  3.118171e+01 |  1.660696e-02 |  1.124062e-01 |
|  LBFGS |    16 /    23 |  3.106224e+01 |  1.506147e-02 |  7.947037e-02 |
|  LBFGS |    17 /    24 |  3.098395e+01 |  1.564561e-02 |  2.678370e-02 |
|  LBFGS |    18 /    25 |  3.096029e+01 |  4.464104e-02 |  4.547148e-02 |
|  LBFGS |    19 /    26 |  3.085475e+01 |  1.442800e-02 |  1.677268e-02 |
|  LBFGS |    20 /    27 |  3.078140e+01 |  1.906548e-02 |  2.275185e-02 |
|========================================================================|
Mdl =
RegressionKernel
PredictorNames: {'x1'  'x2'}
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
ObjectiveValue: 30.7814
RelativeChangeInBeta: 0.0228
FitTime: 33.7022
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.

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

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

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

Xtrain = X(idxTrn,:);
Ytrain = Y(idxTrn);
[Ztrain,tr_mu,tr_sigma] = zscore(Xtrain); % Standardize the training data
tr_sigma(tr_sigma==0) = 1;
Mdl = fitrkernel(Ztrain,Ytrain,'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.

Standardize the test data using the same mean and standard deviation of the training data columns. Estimate the epsilon-insensitive error for the test set.

Xtest = X(idxTest,:);
Ztest = (Xtest-tr_mu)./tr_sigma; % Standardize the test data
Ytest = Y(idxTest);

L = loss(Mdl,Ztest,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,Ztrain,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,Ztest,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.

## Version History

Introduced in R2018a

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