fitrkernel
Fit Gaussian kernel regression model using random feature expansion
Syntax
Description
fitrkernel
trains or crossvalidates 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 lowdimensional space into a
highdimensional space, then fits a linear model in the highdimensional space by
minimizing the regularized objective function. Obtaining the linear model in the
highdimensional space is equivalent to applying the Gaussian kernel to the model in the
lowdimensional space. Available linear regression models include regularized support
vector machine (SVM) and leastsquares regression models.
To train a nonlinear SVM regression model on inmemory data, see fitrsvm
.
returns a kernel regression model Mdl
= fitrkernel(Tbl
,ResponseVarName
)Mdl
trained using the
predictor variables contained in the table Tbl
and the
response values in Tbl.ResponseVarName
.
specifies options using one or more namevalue pair arguments in addition to any
of the input argument combinations in previous syntaxes. For example, you can
implement leastsquares regression, specify the number of dimension of the
expanded space, or specify crossvalidation options.Mdl
= fitrkernel(___,Name,Value
)
[
also returns the hyperparameter optimization results when you optimize
hyperparameters by using the Mdl
,FitInfo
,HyperparameterOptimizationResults
] = fitrkernel(___)'OptimizeHyperparameters'
namevalue pair argument.
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  4.313465e+01  6.296907e02  NaN   LBFGS  0 / 2  3.704335e+01  1.789316e02  9.985854e01   LBFGS  1 / 3  3.703211e+01  2.880402e02  1.044172e03   LBFGS  2 / 4  3.701616e+01  2.297788e02  5.115891e04   LBFGS  2 / 5  3.700183e+01  1.750937e02  1.023672e03   LBFGS  3 / 6  3.679055e+01  4.815047e02  1.113182e02   LBFGS  4 / 7  3.637852e+01  1.058657e01  2.994089e02   LBFGS  5 / 8  3.565372e+01  1.406536e01  7.033477e02   LBFGS  6 / 9  3.478061e+01  1.479288e01  1.185262e01   LBFGS  7 / 10  3.616955e+01  1.544917e01  2.790848e01   LBFGS  7 / 11  3.459534e+01  1.212256e01  1.229242e01   LBFGS  8 / 12  3.379859e+01  8.791025e02  5.417481e02   LBFGS  9 / 13  3.339981e+01  3.077806e02  4.638645e02   LBFGS  10 / 14  3.325224e+01  3.082755e02  2.867793e02   LBFGS  11 / 15  3.320036e+01  4.168377e02  9.376887e03   LBFGS  12 / 16  3.309321e+01  5.018195e02  1.831484e02   LBFGS  13 / 17  3.288069e+01  4.506485e02  3.732443e02   LBFGS  14 / 18  3.245691e+01  3.787163e02  1.036929e01   LBFGS  15 / 19  3.210116e+01  2.418833e02  1.190984e01   LBFGS  16 / 20  3.190585e+01  2.666398e02  3.921991e02  =========================================================================  Solver  Iteration /  Objective  Gradient  Beta relative    Data Pass   magnitude  change  =========================================================================  LBFGS  17 / 21  3.172622e+01  2.548259e02  3.805655e02   LBFGS  18 / 22  3.154538e+01  1.280266e02  4.363429e02   LBFGS  19 / 23  3.138533e+01  1.446779e02  8.822868e02   LBFGS  20 / 24  3.283513e+01  2.218528e01  1.318597e01   LBFGS  20 / 25  3.158782e+01  1.019184e01  6.992082e02   LBFGS  20 / 26  3.136869e+01  4.678412e02  3.603399e02  ========================================================================
Mdl = RegressionKernel PredictorNames: {'x1' 'x2'} ResponseName: 'Y' Learner: 'svm' NumExpansionDimensions: 64 KernelScale: 1 Lambda: 8.5385e06 BoxConstraint: 1 Epsilon: 5.9303 Properties, Methods
FitInfo = struct with fields:
Solver: 'LBFGStall'
LossFunction: 'epsiloninsensitive'
Lambda: 8.5385e06
BetaTolerance: 1.0000e03
GradientTolerance: 1.0000e05
ObjectiveValue: 31.3687
GradientMagnitude: 0.0468
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 namevalue 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'
namevalue pair argument.
CrossValidate Kernel Regression Model
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);
Standardize the predictor variables.
Z = zscore(X);
Crossvalidate a kernel regression model using 5fold crossvalidation.
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 fivefold crossvalidation, Mdl
contains five RegressionKernel
models that the software trains on trainingfold (infold) observations.
Examine the crossvalidation 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 Kernel Regression
Optimize hyperparameters automatically using the 'OptimizeHyperparameters'
namevalue pair argument.
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);
Standardize the predictor variables.
Z = zscore(X);
Find hyperparameters that minimize fivefold crossvalidation loss by using automatic hyperparameter optimization. Specify 'OptimizeHyperparameters'
as 'auto'
so that fitrkernel
finds the optimal values of the 'KernelScale'
, 'Lambda'
, and 'Epsilon'
namevalue pair arguments. For reproducibility, set the random seed and use the 'expectedimprovementplus'
acquisition function.
rng('default') [Mdl,FitInfo,HyperparameterOptimizationResults] = fitrkernel(Z,Y,'OptimizeHyperparameters','auto',... 'HyperparameterOptimizationOptions',struct('AcquisitionFunctionName','expectedimprovementplus'))
====================================================================================================================  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  6.8068e05  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  2.5531e06  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  1.1032e05  0.034083   30  Accept  4.1521  0.26662  4.1362  4.1365  7.2684  8.4842e06  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: 'LBFGSfast'
LossFunction: 'epsiloninsensitive'
Lambda: 0.0391
BetaTolerance: 1.0000e04
GradientTolerance: 1.0000e06
ObjectiveValue: 1.7716
GradientMagnitude: 0.0051
RelativeChangeInBeta: 8.5572e05
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
X
— Predictor data
numeric matrix
Predictor data to which the regression model is fit, specified as an nbyp 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
Tbl
— Sample data
table
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 inTbl
as predictors, then specify the response variable by usingResponseVarName
.If
Tbl
contains the response variable, and you want to use only a subset of the remaining variables inTbl
as predictors, then specify a formula by usingformula
.If
Tbl
does not contain the response variable, then specify a response variable by usingY
. The length of the response variable and the number of rows inTbl
must be equal.
Data Types: table
ResponseVarName
— Response variable name
name of variable in Tbl
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
formula
— Explanatory model of response variable and subset of predictor variables
character vector  string scalar
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
orTbl
)NaN
value or0
weight ('Weights'
)
NameValue Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Namevalue 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 leastsquares 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 crossvalidation namevalue argument together with the
'OptimizeHyperparameters'
namevalue argument. You can modify the
crossvalidation for 'OptimizeHyperparameters'
only by using the
'HyperparameterOptimizationOptions'
namevalue argument.
BoxConstraint
— Box constraint
1
(default)  positive scalar
Box
constraint, specified as the commaseparated 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
Epsilon
— Half width of epsiloninsensitive band
'auto'
(default)  nonnegative scalar value
Half the width of the epsiloninsensitive band, specified as the
commaseparated 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
NumExpansionDimensions
— Number of dimensions of expanded space
'auto'
(default)  positive integer
Number of dimensions of the expanded space, specified as the
commaseparated 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
KernelScale
— Kernel scale parameter
1
(default)  'auto'
 positive scalar
Kernel scale parameter, specified as the commaseparated 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
Lambda
— Regularization term strength
'auto'
(default)  nonnegative scalar
Regularization term strength, specified as the commaseparated 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
Learner
— Linear regression model type
'svm'
(default)  'leastsquares'
Linear regression model type, specified as the commaseparated pair
consisting of 'Learner'
and 'svm'
or 'leastsquares'
.
In the following table, $$f\left(x\right)=T(x)\beta +b.$$
x is an observation (row vector) from p predictor variables.
$$T(\xb7)$$ is a transformation of an observation (row vector) for feature expansion. T(x) maps x in $${\mathbb{R}}^{p}$$ to a highdimensional space ($${\mathbb{R}}^{m}$$).
β is a vector of coefficients.
b is the scalar bias.
Value  Algorithm  Response range  Loss function 

'leastsquares'  Linear regression via ordinary least squares  y ∊ (∞,∞)  Mean squared error (MSE): $$\ell \left[y,f\left(x\right)\right]=\frac{1}{2}{\left[yf\left(x\right)\right]}^{2}$$ 
'svm'  Support vector machine regression  Same as 'leastsquares'  Epsiloninsensitive: $$\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,\leftyf\left(x\right)\right\epsilon \right]$$ 
Example: 'Learner','leastsquares'
Verbose
— Verbosity level
0
(default)  1
Verbosity level, specified as the commaseparated pair consisting of
'Verbose'
and either 0
or
1
. Verbose
controls the amount
of diagnostic information fitrkernel
displays at
the command line.
Value  Description 

0  fitrkernel does not display
diagnostic information. 
1  fitrkernel 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
BlockSize
— Maximum amount of allocated memory
4e^3
(4GB) (default)  positive scalar
Maximum amount of allocated memory (in megabytes), specified as the
commaseparated 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 blockwise strategy. For details about the
blockwise strategy, see Algorithms.
Example: 'BlockSize',1e4
Data Types: single
 double
RandomStream
— Random number stream
global stream (default)  random stream object
Random number stream for reproducibility of data transformation,
specified as the commaseparated 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 highdimensional space. For details, see
Managing the Global Stream Using RandStream and Creating and Controlling a Random Number Stream.
Example: 'RandomStream',RandStream('mlfg6331_64')
CategoricalPredictors
— Categorical predictors list
vector of positive integers  logical vector  character matrix  string array  cell array of character vectors  'all'
Categorical predictors list, specified as one of the values in this table.
Value  Description 

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 If 
Logical vector 
A 
Character matrix  Each 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 vectors  Each 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'
namevalue 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
PredictorNames
— Predictor variable names
string array of unique names  cell array of unique character vectors
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
andY
, then you can usePredictorNames
to assign names to the predictor variables inX
.The order of the names in
PredictorNames
must correspond to the column order ofX
. That is,PredictorNames{1}
is the name ofX(:,1)
,PredictorNames{2}
is the name ofX(:,2)
, and so on. Also,size(X,2)
andnumel(PredictorNames)
must be equal.By default,
PredictorNames
is{'x1','x2',...}
.
If you supply
Tbl
, then you can usePredictorNames
to choose which predictor variables to use in training. That is,fitrkernel
uses only the predictor variables inPredictorNames
and the response variable during training.PredictorNames
must be a subset ofTbl.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
orformula
, but not both.
Example: "PredictorNames",["SepalLength","SepalWidth","PetalLength","PetalWidth"]
Data Types: string
 cell
ResponseName
— Response variable name
"Y"
(default)  character vector  string scalar
Response variable name, specified as a character vector or string scalar.
If you supply
Y
, then you can useResponseName
to specify a name for the response variable.If you supply
ResponseVarName
orformula
, then you cannot useResponseName
.
Example: "ResponseName","response"
Data Types: char
 string
ResponseTransform
— Response transformation
'none'
(default)  function handle
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
Weights
— Observation weights
vector of scalar values  name of variable in Tbl
Observation weights, specified as the commaseparated 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
CrossVal
— Crossvalidation flag
'off'
(default)  'on'
Crossvalidation flag, specified as the commaseparated pair
consisting of 'Crossval'
and 'on'
or 'off'
.
If you specify 'on'
, then the software implements
10fold crossvalidation.
You can override this crossvalidation setting using the
CVPartition
, Holdout
,
KFold
, or Leaveout
namevalue pair argument. You can use only one crossvalidation
namevalue pair argument at a time to create a crossvalidated
model.
Example: 'Crossval','on'
CVPartition
— Crossvalidation partition
[]
(default)  cvpartition
partition object
Crossvalidation partition, specified as a cvpartition
partition object
created by cvpartition
. The partition object
specifies the type of crossvalidation and the indexing for the training and validation
sets.
To create a crossvalidated model, you can specify only one of these four namevalue
arguments: CVPartition
, Holdout
,
KFold
, or Leaveout
.
Example: Suppose you create a random partition for 5fold crossvalidation on 500
observations by using cvp = cvpartition(500,'KFold',5)
. Then, you can
specify the crossvalidated model by using
'CVPartition',cvp
.
Holdout
— Fraction of data for holdout validation
scalar value in the range (0,1)
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:
Randomly select and reserve
p*100
% of the data as validation data, and train the model using the rest of the data.Store the compact, trained model in the
Trained
property of the crossvalidated model.
To create a crossvalidated model, you can specify only one of these four namevalue
arguments: CVPartition
, Holdout
,
KFold
, or Leaveout
.
Example: 'Holdout',0.1
Data Types: double
 single
KFold
— Number of folds
10
(default)  positive integer value greater than 1
Number of folds to use in a crossvalidated model, specified as a positive integer value
greater than 1. If you specify 'KFold',k
, then the software completes
these steps:
Randomly partition the data into
k
sets.For each set, reserve the set as validation data, and train the model using the other
k
– 1 sets.Store the
k
compact, trained models in ak
by1 cell vector in theTrained
property of the crossvalidated model.
To create a crossvalidated model, you can specify only one of these four namevalue
arguments: CVPartition
, Holdout
,
KFold
, or Leaveout
.
Example: 'KFold',5
Data Types: single
 double
Leaveout
— Leaveoneout crossvalidation flag
'off'
(default)  'on'
Leaveoneout crossvalidation flag, specified as the commaseparated 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:
Reserve the observation as validation data, and train the model using the other n – 1 observations.
Store the n compact, trained models in the cells of an nby1 cell vector in the
Trained
property of the crossvalidated model.
To create a crossvalidated model, you can use one of these
four namevalue pair arguments only: CVPartition
, Holdout
, KFold
,
or Leaveout
.
Example: 'Leaveout','on'
BetaTolerance
— Relative tolerance on linear coefficients and bias term
1e4
(default)  nonnegative scalar
Relative tolerance on the linear coefficients and the bias term (intercept), specified as the commaseparated 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 $${\Vert \frac{{B}_{t}{B}_{t1}}{{B}_{t}}\Vert}_{2}<\text{BetaTolerance}$$, then optimization terminates.
If you also specify GradientTolerance
, then optimization terminates when the software satisfies either stopping criterion.
Example: 'BetaTolerance',1e6
Data Types: single
 double
GradientTolerance
— Absolute gradient tolerance
1e6
(default)  nonnegative scalar
Absolute gradient tolerance, specified as the commaseparated pair consisting of 'GradientTolerance'
and a nonnegative scalar.
Let $$\nabla {\mathcal{L}}_{t}$$ be the gradient vector of the objective function with respect to the coefficients and bias term at optimization iteration t. If $${\Vert \nabla {\mathcal{L}}_{t}\Vert}_{\infty}=\mathrm{max}\left\nabla {\mathcal{L}}_{t}\right<\text{GradientTolerance}$$, then optimization terminates.
If you also specify BetaTolerance
, then optimization terminates when the
software satisfies either stopping criterion.
Example: 'GradientTolerance',1e5
Data Types: single
 double
HessianHistorySize
— Size of history buffer for Hessian approximation
15
(default)  positive integer
Size of the history buffer for Hessian approximation, specified as the
commaseparated 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
IterationLimit
— Maximum number of optimization iterations
positive integer
Maximum number of optimization iterations, specified as the
commaseparated 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
OptimizeHyperparameters
— Parameters to optimize
'none'
(default)  'auto'
 'all'
 string array or cell array of eligible parameter names  vector of optimizableVariable
objects
Parameters to optimize, specified as the commaseparated 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 ofhyperparameters
.
The optimization attempts to minimize the crossvalidation loss
(error) for fitrkernel
by varying the parameters.
To control the crossvalidation type and other aspects of the
optimization, use the
HyperparameterOptimizationOptions
namevalue
pair argument.
Note
The values of 'OptimizeHyperparameters'
override any values you specify
using other namevalue 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:
Epsilon
—fitrkernel
searches among positive values, by default logscaled in the range[1e3,1e2]*iqr(Y)/1.349
.KernelScale
—fitrkernel
searches among positive values, by default logscaled in the range[1e3,1e3]
.Lambda
—fitrkernel
searches among positive values, by default logscaled in the range[1e3,1e3]/n
, wheren
is the number of observations.Learner
—fitrkernel
searches among'svm'
and'leastsquares'
.NumExpansionDimensions
—fitrkernel
searches among positive integers, by default logscaled in the range[100,10000]
.
Set nondefault parameters by passing a vector of
optimizableVariable
objects that have nondefault
values. For example:
load carsmall params = hyperparameters('fitrkernel',[Horsepower,Weight],MPG); params(2).Range = [1e4,1e6];
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 + crossvalidation loss). To control the iterative display, set the Verbose
field of
the 'HyperparameterOptimizationOptions'
namevalue argument. To control the
plots, set the ShowPlots
field of the
'HyperparameterOptimizationOptions'
namevalue argument.
For an example, see Optimize Kernel Regression.
Example: 'OptimizeHyperparameters','auto'
HyperparameterOptimizationOptions
— Options for optimization
structure
Options for optimization, specified as a structure. This argument modifies the effect of the
OptimizeHyperparameters
namevalue argument. All fields in the
structure are optional.
Field Name  Values  Default 

Optimizer 
 'bayesopt' 
AcquisitionFunctionName 
Acquisition functions whose names include
 'expectedimprovementpersecondplus' 
MaxObjectiveEvaluations  Maximum 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  Inf 
NumGridDivisions  For '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 
ShowPlots  Logical 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 
SaveIntermediateResults  Logical 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:
For details, see the  1 
UseParallel  Logical 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 crossvalidation at every
iteration. If this field is The setting
 false 
Use no more than one of the following three options.  
CVPartition  A cvpartition object, as created by cvpartition  'Kfold',5 if you do not specify a crossvalidation
field 
Holdout  A scalar in the range (0,1) representing the holdout fraction  
Kfold  An integer greater than 1 
Example: 'HyperparameterOptimizationOptions',struct('MaxObjectiveEvaluations',60)
Data Types: struct
Output Arguments
Mdl
— Trained kernel regression model
RegressionKernel
model object  RegressionPartitionedKernel
crossvalidated model
object
Trained kernel regression model, returned as a RegressionKernel
model object or RegressionPartitionedKernel
crossvalidated model
object.
If you set any of the namevalue pair arguments
CrossVal
, CVPartition
,
Holdout
, KFold
, or
Leaveout
, then Mdl
is a
RegressionPartitionedKernel
crossvalidated 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).
FitInfo
— Optimization details
structure array
Optimization details, returned as a structure array including fields described in this table. The fields contain final values or namevalue pair argument specifications.
Field  Description 

Solver  Objective function minimization technique:

LossFunction  Loss function. Either mean squared error (MSE) or
epsiloninsensitive, depending on the type of linear
regression model. See Learner . 
Lambda  Regularization term strength. See
Lambda . 
BetaTolerance  Relative tolerance on the linear coefficients and the
bias term. See BetaTolerance . 
GradientTolerance  Absolute gradient tolerance. See
GradientTolerance . 
ObjectiveValue  Value of the objective function when optimization terminates. The regression loss plus the regularization term compose the objective function. 
GradientMagnitude  Infinite norm of the gradient vector of the objective
function when optimization terminates. See
GradientTolerance . 
RelativeChangeInBeta  Relative changes in the linear coefficients and the bias
term when optimization terminates. See
BetaTolerance . 
FitTime  Elapsed, wallclock time (in seconds) required to fit the model to the data. 
History  History 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.
HyperparameterOptimizationResults
— Crossvalidation optimization of hyperparameters
BayesianOptimization
object  table of hyperparameters and associated values
Crossvalidation 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'
namevalue pair
argument:
Value of Optimizer Field  Value of HyperparameterOptimizationResults 

'bayesopt' (default)  Object of class BayesianOptimization 
'gridsearch' or 'randomsearch'  Table of hyperparameters used, observed objective function values (crossvalidation loss), and rank of observations from lowest (best) to highest (worst) 
More About
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 highdimensional space, the kernel regression algorithm searches for an optimal function that deviates from each response data point (y_{i}) 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 x_{1}x_{2}′ with a nonlinear kernel function $$G({x}_{1},{x}_{2})=\langle \phi ({x}_{1}),\phi ({x}_{2})\rangle $$, where x_{i} is the ith observation (row vector) and φ(x_{i}) is a transformation that maps x_{i} to a highdimensional space (called the “kernel trick”). However, evaluating G(x_{1},x_{2}), 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,
$$G({x}_{1},{x}_{2})=\langle \phi ({x}_{1}),\phi ({x}_{2})\rangle \approx T({x}_{1})T({x}_{2})\text{'},$$
where T(x) maps x in $${\mathbb{R}}^{p}$$ to a highdimensional space ($${\mathbb{R}}^{m}$$). The Random Kitchen Sinks[1] scheme uses the random transformation
$$T(x)={m}^{1/2}\mathrm{exp}\left(iZx\text{'}\right)\text{'},$$
where $$Z\in {\mathbb{R}}^{m\times p}$$ is a sample drawn from $$N\left(0,{\sigma}^{2}\right)$$ and σ^{2} 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(mlog
p) and reduces storage to O(m).
You can specify values for m and σ^{2}, using the NumExpansionDimensions
and
KernelScale
namevalue 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
nbyn Gram matrix, the solver in
fitrkernel
only needs to form a matrix of size
nbym, 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 Limitedmemory BroydenFletcherGoldfarbShanno (LBFGS) solver with ridge (L_{2}) regularization. To find the type of LBFGS solver used for training, type FitInfo.Solver
in the Command Window.
'LBFGSfast'
— LBFGS solver.'LBFGSblockwise'
— LBFGS solver with a blockwise strategy. Iffitrkernel
requires more memory than the value ofBlockSize
to hold the transformed predictor data, then it uses a blockwise strategy.'LBFGStall'
— LBFGS solver with a blockwise strategy for tall arrays.
When fitrkernel
uses a blockwise 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 blockwise 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 LargeScale 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.
Extended Capabilities
Tall Arrays
Calculate with arrays that have more rows than fit in memory.
Usage notes and limitations:
fitrkernel
does not support talltable
data.Some namevalue pair arguments have different defaults compared to the default values for the inmemory
fitrkernel
function. Supported namevalue pair arguments, and any differences, are:'BoxConstraint'
'Epsilon'
'NumExpansionDimensions'
'KernelScale'
'Lambda'
'Learner'
'Verbose'
— Default value is1
.'BlockSize'
'RandomStream'
'ResponseTransform'
'Weights'
— Value must be a tall array.'BetaTolerance'
— Default value is relaxed to1e–3
.'GradientTolerance'
— Default value is relaxed to1e–5
.'HessianHistorySize'
'IterationLimit'
— Default value is relaxed to20
.'OptimizeHyperparameters'
'HyperparameterOptimizationOptions'
— For crossvalidation, tall optimization supports only'Holdout'
validation. By default, the software selects and reserves 20% of the data as holdout validation data, and trains the model using the rest of the data. You can specify a different value for the holdout fraction by using this argument. For example, specify'HyperparameterOptimizationOptions',struct('Holdout',0.3)
to reserve 30% of the data as validation data.
If
'KernelScale'
is'auto'
, thenfitrkernel
uses the random stream controlled bytallrng
for subsampling. For reproducibility, you must set a random number seed for both the global stream and the random stream controlled bytallrng
.If
'Lambda'
is'auto'
, thenfitrkernel
might take an extra pass through the data to calculate the number of observations inX
.fitrkernel
uses a blockwise strategy. For details, see Algorithms.
For more information, see Tall Arrays.
Automatic Parallel Support
Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.
To perform parallel hyperparameter optimization, use the
'HyperparameterOptimizationOptions', struct('UseParallel',true)
namevalue argument in the call to the fitrkernel
function.
For more information on parallel hyperparameter optimization, see Parallel Bayesian Optimization.
For general information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).
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