There is a branch of statistics called Response Surface Methodology, that deals with optimizing a potentially noisy surface. Usually the idea is to use some sort of approximation to smooth out the noise, then solve for the minimum of your approximation. In RSM, you might then generate further data in the area of where you have determined the optimum to be, then repeat the analysis.
If you have only a fixed amount of data, then the trick will be to intelligently smooth the data, with a tool that is easy to solve for the minimum value. So I might suggest a spline model like my SLM toolbox , which can allow you to control how well the curve is fit, perhaps by adding more knots where the fit seems inadequate, or by adjusting a smoothing parameter.
For example, consider the curve:
x = linspace(0,pi,250);
y = -sin(x) + randn(size(x))/20;
plot(x,y,'-')
Yes, you and I know the minimum happens to fall at pi/2. (Shh! It is a secret.)
slm = slmengine(x,y,'knots',10,'reg','cross','plot','on');
Now, use slmpar (included in the toolbox) to return the minimum value and the location of that min.
[funval,minloc] = slmpar(slm,'minfun')
funval =
-1.0002
minloc =
1.5348
There are various options in the tool to help you fit virtually any function.
