CompactRegressionGP

Namespace: classreg.learning.regr

Compact Gaussian process regression model class

Description

CompactRegressionGP is a compact Gaussian process regression (GPR) model. The compact model consumes less memory than a full model, because it does not include the data used for training the GPR model.

Because the compact model does not include the training data, you cannot perform some tasks, such as cross-validation, using the compact model. However, you can use the compact model for making predictions or calculate regression loss for new data (use predict and loss).

Construction

Create a CompactRegressionGP object from a full RegressionGP model object by using compact.

Properties

Fitting

FitMethod — Method used to estimate the parameters

'none' | 'exact' | 'sd' | 'sr' | 'fic'

Method used to estimate the basis function coefficients, β; noise standard deviation, σ; and kernel parameters, θ, of the GPR model, stored as a character vector. It can be one of the following.

Fit Method	Description
'none'	No estimation. fitrgp uses the initial parameter values as the parameter values.
'exact'	Exact Gaussian process regression.
'sd'	Subset of data points approximation.
'sr'	Subset of regressors approximation.
'fic'	Fully independent conditional approximation.

BasisFunction — Explicit basis function

'none' | 'constant' | 'linear' | 'pureQuadratic' | function handle

Explicit basis function used in the GPR model, stored as a character vector or a function handle. It can be one of the following. If n is the number of observations, the basis function adds the term H*β to the model, where H is the basis matrix and β is a p-by-1 vector of basis coefficients.

Explicit Basis	Basis Matrix
'none'	Empty matrix.
'constant'	$$H=1$$ H is an n-by-1 vector of 1s, where n is the number of observations.
'linear'	$$H=[1,X]$$ X is the expanded predictor data after the software creates dummy variables for the categorical variables. For details about creating dummy variables, see CategoricalPredictors.
'pureQuadratic'	$$H=\left[1,X,X_2\right],$$ where $$X_2=\left[\begin{array}{cccc}{x}_{11}^2& x_{12}^2& \cdots & x_{1d}^2\\ x_{21}^2& x_{22}^2& \cdots & x_{2d}^2\\ ⋮& ⋮& ⋮& ⋮\\ x_{n1}^2& x_{n2}^2& \cdots & x_{nd}^2\end{array}\right].$$ For this basis option, CompactRegressionGP does not support X with categorical predictors.
Function handle	Function handle, hfcn, that fitrgp calls as: $$H=hfcn(X),$$ where X is an n-by-d matrix of predictors, d is the number of predictors after the software creates dummy variables for the categorical variables, and H is an n-by-p matrix of basis functions.

Data Types: char | function_handle

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

Beta — Estimated coefficients

vector

Estimated coefficients for the explicit basis functions, stored as a vector. You can define the explicit basis function by using the BasisFunction name-value pair argument in fitrgp.

Data Types: double

Sigma — Estimated noise standard deviation

scalar value

Estimated noise standard deviation of the GPR model, stored as a scalar value.

Data Types: double

ModelParameters — Parameters used for training

GPParams object

Parameters used for training the GPR model, stored as a GPParams object.

Kernel Function

KernelFunction — Form of the covariance function

'squaredExponential' | 'matern32' | 'matern52' | 'ardsquaredexponential' | 'ardmatern32' | 'ardmatern52' | function handle

Form of the covariance function used in the GPR model, stored as a character vector containing the name of the built-in covariance function or a function handle. It can be one of the following.

Function	Description
'squaredexponential'	Squared exponential kernel.
'matern32'	Matern kernel with parameter 3/2.
'matern52'	Matern kernel with parameter 5/2.
'ardsquaredexponential'	Squared exponential kernel with a separate length scale per predictor.
'ardmatern32'	Matern kernel with parameter 3/2 and a separate length scale per predictor.
'ardmatern52'	Matern kernel with parameter 5/2 and a separate length scale per predictor.
Function handle	A function handle that fitrgp can call like this: Kmn = kfcn(Xm,Xn,theta) where Xm is an m-by-d matrix, Xn is an n-by-d matrix, and Kmn is an m-by-n matrix of kernel products such that Kmn(i,j) is the kernel product between Xm(i,:) and Xn(j,:). d is the number of predictor variables after the software creates dummy variables for the categorical variables. For details about creating dummy variables, see CategoricalPredictors. theta is the r-by-1 unconstrained parameter vector for kfcn.

Data Types: char | function_handle

KernelInformation — Information about the parameters of the kernel function

structure

Information about the parameters of the kernel function used in the GPR model, stored as a structure with the following fields.

Field Name	Description
Name	Name of the kernel function
KernelParameters	Vector of the estimated kernel parameters
KernelParameterNames	Names associated with the elements of KernelParameters.

Data Types: struct

Prediction

PredictMethod — Method used to make predictions

'exact' | 'bcd' | 'sd' | 'sr' | 'fic'

Method that predict uses to make predictions from the GPR model, stored as a character vector. It can be one of the following.

PredictMethod	Description
'exact'	Exact Gaussian process regression
'bcd'	Block Coordinate Descent
'sd'	Subset of Data points approximation
'sr'	Subset of Regressors approximation
'fic'	Fully Independent Conditional approximation

Alpha — Weights

numeric vector

Weights used to make predictions from the trained GPR model, stored as a numeric vector. predict computes the predictions for a new predictor matrix Xnew by using the product

$$K\left(X_{new},A\right)*\alpha .$$

$$K\left(X_{new},A\right)$$ is the matrix of kernel products between $$X_{new}$$ and active set vector A and α is a vector of weights.

Data Types: double

ResponseTransform — Transformation applied to predicted response

'none' (default)

Transformation applied to the predicted response, stored as a character vector describing how the response values predicted by the model are transformed. In RegressionGP, ResponseTransform is 'none' by default, and RegressionGP does not use ResponseTransform when making predictions.

Active Set Selection

ActiveSetVectors — Subset of training data

matrix

Subset of training data used to make predictions from the GPR model, stored as a matrix.

predict computes the predictions for a new predictor matrix Xnew by using the product

$$K\left(X_{new},A\right)*\alpha .$$

$$K\left(X_{new},A\right)$$ is the matrix of kernel products between $$X_{new}$$ and active set vector A and α is a vector of weights.

ActiveSetVectors is equal to the training data X for exact GPR fitting and a subset of the training data X for sparse GPR methods. When there are categorical predictors in the model, ActiveSetVectors contains dummy variables for the corresponding predictors.

Data Types: double

ActiveSetMethod — Method used to select the active set

'sgma' | 'entropy' | 'likelihood' | 'random'

Method used to select the active set for sparse methods ('sd','sr', or 'fic'), stored as a character vector. It can be one of the following.

ActiveSetMethod	Description
'sgma'	Sparse greedy matrix approximation
'entropy'	Differential entropy-based selection
'likelihood'	Subset of regressors log likelihood-based selection
'random'	Random selection

The selected active set is used in parameter estimation or prediction, depending on the choice of FitMethod and PredictMethod in fitrgp.

ActiveSetSize — Size of the active set

integer value

Size of the active set for sparse methods ('sd','sr', or 'fic'), stored as an integer value.

Data Types: double

Object Functions

lime	Local interpretable model-agnostic explanations (LIME)
loss	Regression error for Gaussian process regression model
partialDependence	Compute partial dependence
plotPartialDependence	Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
predict	Predict response of Gaussian process regression model
shapley	Shapley values

Examples

Compute Predictions and Regression Loss for Test Data

Generate example training data.

rng(1) % For reproducibility
n = 100000;
X = linspace(0,1,n)';
X = [X,X.^2];
y = 1 + X*[1;2] + sin(20*X*[1;-2]) + 0.2*randn(n,1);

Train a GPR model using the subset of regressors ('sr') approximation method and predict using the subset of data ('sd') method. Use 50 points in the active set and sparse greedy matrix approximation ('sgma') method for active set selection. Because the scales of the first and second predictors are different, it is good practice to standardize the data.

gprMdl = fitrgp(X,y,'KernelFunction','squaredExponential','FitMethod', ...
    'sr','PredictMethod','sd','Basis','none','ActiveSetSize',50, ...
    'ActiveSetMethod','sgma','Standardize',1,'KernelParameters',[1;1]);

fitrgp accepts any combination of fitting, prediction, and active set selection methods. In some cases it might not be possible to compute the standard deviations of the predicted responses, hence the prediction intervals. See Tips. And, in some cases, using the exact method might be expensive because of the size of the training data.

Create a compact GPR object.

cgprMdl = compact(gprMdl);

Generate the test data.

n = 4000;
Xnew = linspace(0,1,n)';
Xnew = [Xnew,Xnew.^2];
ynew = 1 + Xnew*[1;2] + sin(20*Xnew*[1;-2]) + 0.2*randn(n,1);

Use the compact object to predict the response in test data and the prediction intervals.

[ypred,~,yci] = predict(cgprMdl,Xnew);

Plot the true response, predicted response, and prediction intervals.

figure
plot(ynew,'r')
hold on
plot(ypred,'b')
plot(yci(:,1),'k--')
plot(yci(:,2),'k--')
legend('True responses','GPR predictions','95% prediction limits','Location','Best')
xlabel('x')
ylabel('y')
hold off

Compute the mean squared error loss on the test data using the trained GPR model.

L = loss(cgprMdl,Xnew,ynew)

L = 
0.0497

Copy Semantics

Value. To learn how value classes affect copy operations, see Copying Objects.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The predict function supports code generation.

For more information, see Introduction to Code Generation.

Version History

Introduced in R2015b