- Sample the function at various points.
- Compute the zeroth-order term, which is the mean value of the function over the domain.
- Compute the first-order terms, which are the effects of each variable individually.
- Compute the second-order term, which is the interaction effect between the variables.
- Use Lagrange interpolation to approximate the function values.
How do I use Cut-HDMR in MATLAB?
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Hi, I wish to use Cut-HDMR (HDMR = High Dimensional Model Representaion) to solve multivariate piecewise continuous functions. I found a code at https://in.mathworks.com/matlabcentral/fileexchange/92890-bivariate-cut-hdmr but could not understand their application.
Could someone please help me out with a sample code for a bivariate piecewise continuous function so that I can understand the process application in MATLAB? The expansion may be taken only upto the second order term with Lagrange interpolation.
Thanks
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Answers (1)
Naren
on 17 Jan 2024
Edited: Naren
on 17 Jan 2024
Hello Rohit,
Inorder to apply bivariate CUT-HDMR (High Dimensional Model Representation) to a piecewise continuous function, follow the steps given below:
Given that you've mentioned using Lagrange interpolation and limiting the expansion to second-order terms, let's consider a simple bivariate piecewise continuous function as an example:
function z = piecewise_function(x, y)
if x < 0.5
if y < 0.5
z = sin(x) + cos(y);
else
z = cos(x) + sin(y);
end
else
if y < 0.5
z = x^2 + y^2;
else
z = x*y;
end
end
end
To apply CUT-HDMR to this function, we need to create a script that will perform the following steps:
Here's a simple MATLAB script that demonstrates these steps:
% Define the domain for the variables x and y
x_domain = linspace(0, 1, 10); % 10 points between 0 and 1
y_domain = linspace(0, 1, 10);
% Sample the function at the points in the domain
[X, Y] = meshgrid(x_domain, y_domain);
Z = arrayfun(@piecewise_function, X, Y);
% Compute the zeroth-order term (mean of the function)
h0 = mean(Z, 'all');
% Compute the first-order terms
hx = mean(Z, 2) - h0; % Mean over y, for each x
hy = mean(Z, 1) - h0; % Mean over x, for each y
% Compute the second-order term
hxy = Z - (h0 + hx + hy');
% Perform Lagrange interpolation for the first-order terms
% Note: For simplicity, we are doing interpolation at the sampled points
% In practice, you would use interpolation to estimate values at new points
hx_interp = interp1(x_domain, hx, x_domain, 'linear', 'extrap');
hy_interp = interp1(y_domain, hy, y_domain, 'linear', 'extrap');
% The CUT-HDMR approximation of the function is then:
Z_approx = h0 + hx_interp + hy_interp' + hxy;
figure;
subplot(1, 2, 1);
surf(X, Y, Z);
title('Original Function');
xlabel('x');
ylabel('y');
zlabel('z');
subplot(1, 2, 2);
surf(X, Y, Z_approx);
title('CUT-HDMR Approximation');
xlabel('x');
ylabel('y');
zlabel('z_approx');
Note that this is a simplified example. In practice, you would need to consider the continuity of the function and the appropriate interpolation method to apply. Also, the CUT-HDMR method can be more complex, especially for higher-dimensional functions and higher-order expansions.
Regards,
Naren
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