MATLAB: Calculating the matrix K at test inputs after training a Gaussian Process with fitrgp

fitrgpgaussian processkrigingmachine learningregressiongpstatisticsStatistics and Machine Learning Toolbox

If I trained a GP using training data D = {X, y} with fitrgp and I obtained my gprMdl:

gprMdl = fitrgp(data.X, data.Y, 'KernelFunction', 'squaredexponential', ...
                'BasisFunction', 'none', 'verbose', 1, 'FitMethod', 'exact')
[ystar, ysd, yint] = predict(gprMdl, Xstar)

How can I obtain the matrix K(Xstar, Xstar)? I can not find the subfunction of the RegressionGP that calculates the matrices K.

Thanks

Best Answer

Hi Umberto,

There is an undocumented way of calculating what you want. Here is an example:

rng(0,'twister');
N = 100;
x = linspace(-10,10,N)';
y = 1 + x*5e-2 + sin(x)./x + 0.2*randn(N,1);
gpr = fitrgp(x,y,'FitMethod','Exact','PredictMethod','Exact');
kfcn = gpr.Impl.Kernel.makeKernelAsFunctionOfXNXM(gpr.Impl.ThetaHat)
K = kfcn(x(1:5,:),x(1:7,:))

K(i,j) kernel function evaluated for x(i,:) and x(j,:). For example,

K(3,6)
kfcn(x(3,:),x(6,:))

I would be interested in knowing why you want to compute K.

Hope that helps,

Gautam

Related Solutions

MATLAB: How to choose the best kernel for a Gaussian process regression, possibly using “bayesopt” function? How to predict unseen data with the resulting model

There are two ways you could do this. One of them involves "bayesopt" function.

(1) The easiest way is to use the "fitrgp" function. There is an example of how to optimize a fitrgp hyperparameter (Sigma) on the "fitrgp" function Documentation page:

https://www.mathworks.com/help/stats/fitrgp.html?searchHighlight=fitrgp

To optimize ‘KernelFunction’, using default values for all other hyperparameters, you could do the following (using data from the documentation example):

load(fullfile(matlabroot,'examples','stats','gprdata2.mat'))
model1 = fitrgp(x,y,'OptimizeHyperparameters',{'KernelFunction'});

Because fitrgp tends to be sensitive to the Sigma parameter, you will probably get a better result if you simultaneously optimize KernelFunction and Sigma. Also, searching categorical variables inherently requires more function evaluations, so you should probably run it longer:

model2 = fitrgp(x,y,'OptimizeHyperparameters',{'KernelFunction','Sigma'},...
    'HyperparameterOptimizationOptions', struct('MaxObjectiveEvaluations',100));

The "fitrgp" function documentation page lists the hyperparameters that are eligible for optimization, and what values are searched:

https://www.mathworks.com/help/stats/fitrgp.html#input_argument_namevalue_OptimizeHyperparameters.

You can use this model to make predictions on unseen data directly with the "predict" function:

model2 = fitrgp(x,y,'OptimizeHyperparameters',{'KernelFunction','Sigma'},...
    'HyperparameterOptimizationOptions', struct('MaxObjectiveEvaluations',100));
% Predict on unseen data

xNew = 0.2
ypred = predict(model2, xNew)

(2) A second approach would be to use "bayesopt" function directly, defining your own objective function. Here, the objective function is the 5-fold crossvalidation loss of the model.

load(fullfile(matlabroot,'examples','stats','gprdata2.mat')) 
% Define kernel variable
kernel = optimizableVariable('KernelFunction',{'exponential','squaredexponential','matern32','matern52',...
'rationalquadratic','ardexponential','ardsquaredexponential','ardmatern32','ardmatern52','ardrationalquadratic'},...
'Type','categorical') 
% Call bayesopt, capturing x and y in the objective function

bo = bayesopt(@(T)objFcn(T,x,y), kernel)
% Define an objective function
function Loss = objFcn(Vars, x, y)
m = fitrgp(x, y, 'KernelFunction', char(Vars.KernelFunction), 'KFold', 5);Loss = kfoldLoss(m);
end

(2.1) To optimize both KernelFunction and Sigma using "bayesopt" function, you can do the following:

% Define 2 variables
kernel = optimizableVariable('KernelFunction',{'exponential','squaredexponential','matern32','matern52',...
'rationalquadratic','ardexponential','ardsquaredexponential','ardmatern32','ardmatern52','ardrationalquadratic'},...
'Type','categorical')
sigma = optimizableVariable('Sigma',[1e-4,10],'Transform','log')
 
% Call bayesopt, capturing x and y in the objective function
bo = bayesopt(@(T)objFcn(T,x,y), [sigma, kernel], 'MaxObjectiveEvaluations', 100)
 
function Loss = objFcn(Vars, x, y)
m = fitrgp(x, y, 'KernelFunction', char(Vars.KernelFunction), ...
                 'Sigma', Vars.Sigma, 'ConstantSigma', true,...
                 'KFold', 5);
Loss = kfoldLoss(m);
end

To get a model that can make predictions, you need to fit a model without passing the ‘kfold’ argument, but keeping the optimal kernel function obtained from "bayesopt" function.

kernel= bo.XAtMinObjective.KernelFunction
model = fitrgp(x, y, 'KernelFunction', char(kernel));
% Predict on unseen data
xNew = 0.2
ypred = predict(model2, xNew)

MATLAB: Sampling from Posterior Distribution of GPR Model from fitrgp()

Hi Sterling,

Here's an example illustrating how to sample from the posterior distribution of a GPR model. The code uses an undocumented function predictExactWithCov. If you have categorical predictors, you would need to convert them to dummy variables before using that function.

Hope this helps,

Gautam

function testPosteriorCovarianceGP()
% 1. Some dummy data.
rng(0,'twister');
X = [-10,-8,-5,-1,1,3,7,10]';
N = numel(X);
y = 1 - X*5e-2 + sin(X)./X + 1e-4*randn(N,1);
M = 1000;
Xnew = linspace(-10,10,M)';
% 2. Fit a GPR model.
gpr = fitrgp(X,y,'Sigma',0.01,'SigmaLowerBound',1e-6);
% 3. Access the posterior covariance matrix by calling the undocumented
% predictExactWithCov method on the gpr.Impl object. Vectors ynew can be
% simulated from a Normal distribution with mean pred and covariance covmat
% using cholcov or svd and randn.
alpha = 0.05;
[pred,covmat,ci] = predictExactWithCov(gpr.Impl,Xnew,alpha);
figure(1);
plot(Xnew,pred,'b');
hold on;
plot(Xnew,ci(:,1),'m');
plot(Xnew,ci(:,2),'g');
plot(X,y,'ko','MarkerFaceColor','k');
xlabel('x');
ylabel('y');
figure(2);
T = cholcov(covmat);
numSamples = 50;
ynew = pred + T'*randn(M,numSamples);
plot(Xnew,ynew);
hold on;
plot(X,y,'ko','MarkerFaceColor','k');
xlabel('x');
ylabel('y');
end

Best Answer

Related Solutions

MATLAB: How to choose the best kernel for a Gaussian process regression, possibly using “bayesopt” function? How to predict unseen data with the resulting model

MATLAB: Sampling from Posterior Distribution of GPR Model from fitrgp()

Related Question