Sampling from Posterior Distribution of GPR Model from fitrgp()

I can get a gprMdl via:

rng(0,'twister');
N = 100;
x = linspace(-10,10,N)';
y = 1 + x*5e-2 + sin(x)./x + 0.2*randn(N,1);
gprMdl = fitrgp(x,y,'FitMethod','Exact','PredictMethod','Exact');

and the posterior covariance matrix via:

kfcn = gprMdl.Impl.Kernel.makeKernelAsFunctionOfXNXM(gprMdl.Impl.ThetaHat)
K = kfcn(x(1:5,:),x(1:7,:))

(credit: Gautam Pendse in MATLAB Answers)

How can I go about sampling a model from the posterior distribution of gprMdl?

To clarify, I mean like the curves from the 2nd tile of:

Source: https://commons.wikimedia.org/wiki/File:Gaussian_Process_Regression.png

Being able to do this is a matter of being able to finish up a graduate program, so any help is much appreciated!

[Sterling Baird]

Perhaps all I need is to use the mean from predict and the covariance matrix from the undocumented function along with mvnrnd, similar to what's described in: https://www.mathworks.com/matlabcentral/answers/442492-sampling-based-on-a-correlation-matrix-for-multiple-parameters

Though I'm not sure if this is sampling from the posterior..

Thoughts?

Answer 1

[Gautam Pendse]

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

[Sterling Baird]

What is alpha? (Edit: I just realized it controls the confidence intervals)

[Jitin Malhotra]

@Gautam Pendse, I din't understand what exactly you did and how i should use it on my data. If possible can you please provide more details about this undocumented function? predictExactWithCov(gpr.Impl,Xnew,alpha);

My main motive is to calculate CI_width and CI_var, which is basically the 95% confidence interval width and variance acc. to given formulae

well I got my model as output from regression learner app, How should I move forward on this?