How can I know the volume behind a surface?

Hi! I imported some data x,y,z. z plotted against x and y creates a surface similar to a Gaussian one. I approximated my data with cftool. I obtained a surface in the space. Now I would like to know the volume behind the interpolant surface because it can approximate the spacial surface I am looking for.

Have you got any ideas? 'Cause I can only find in the guide cfit/integrate help but anything concerning sfit/integrate.

Many thanks...

Answer 1

[Star Strider]

If you have a matrix of data, I would use trapz to do the integration. You will have to use it two times — once in each dimension of the matrix — to get a scalar value for the volume.

[Francesca]

Thanks a lot. Actually I used: I=trapz(z,y)+trapz(x,z) but then I noticed that the result obtained in this way is irrespective of the interpolating surface used! I mean, it can be calulated just after importing x,y,z and there is not correlation with cftool... So it is not the volume I am looking for...

[Star Strider]

My pleasure!

If you use trapz to do double integration, you have to use it as:

fxy = ...              % Data Matrix
IntX = trapz(fxy);     % Integrate along one dimension to get a vector
IntY = trapz(IntX);    % Integrate to get the scalar value for the volume

It should produce the result you want.

If you have functions (with parameters estimated by one of the curve-fitting functions) rather than matrix data, use the integral2 function. It takes function arguments. That will work unless it is array-valued. If it is array-valued (the function is defined as and returns a vector or matrix rather than scalar) you will have to use two calls to integral instead, setting 'ArrayValued' to true with both calls.

[Francesca]

Unfortunately, using cftool for surfaces doesn't provide an equation for the interpolating surface. So I cannot use integral2.

However x,y,z are vectors of the same dimension.

[Star Strider]

I don’t have the Curve Fitting Toolbox (I have the Optimization and Statistics Toolboxes), so I cannot address the problems with it directly.

If you have a function that you fit that is a function of x, y, and parameters, you can substitute in it the parameters that you fit and use that (as an anonymous function or a function file) for your function argument to integral2.

If you have an anonymous function and a set of fitted parameters (here in vector b), for instance, you can use integral2 with it:

b = [b(1) ... b(n)];          % Parameter vector from fitting routine
f = @(b,x,y) b(1).*exp(-(b(2).*x).^2) + b(3).*exp(-(b(4).*y).^2);

(the function is hypothetical but is probably something close to yours), then use integral2:

vol = integral2(@(x,y) f(b,x,y), x1, x2, y1,y2);

where x1 ... y2 are the relevant integration limits. Your f function will pick up the parameter vector from your workspace. It seems like strange syntax, but it creates a function only in (x,y) from f(b,x,y) so that integral2 can use it. (I tested this code with my own set of b, x1 ... y2, and it works.)