3D Plotting with Vectors and Meshgrid

I want to create a 3D plot for something that is very similar to the example in meshgrid (below)

[X,Y] = meshgrid(-2:.2:2, -2:.2:2);                                
Z = X .* exp(-X.^2 - Y.^2);                                        
surf(X,Y,Z)

Basically, I have vectors in my definition of Z and I am unsure about how to handle this. My definition of Z is

    Z = mean((S - (X*A+Y)).^2);

S and A are vector quantities so this line doesn't make sense. The only solution I can think of here is to create a 3D meshgrid where the third dimension of the length of my vectors S and A (they are the same length). I think this could get tricky though. Is there an easier way.

Example: I have two vectors P and Q like these:

P = [zeros(1, 30) ones(1,40) zeros(1,30)] + .1*randn(1,100);
Q=3.4.*([zeros(1, 30) ones(1,40) zeros(1,30)] + .1.*randn(1,100))+5.3;

Because Q is scaled and offset finding the mean squared error between them directly is pointless.

    MSE = mean((P-Q).^2);

this MSE doesn't mean much because Q is offset and scaled. I re-define MSE:

myMSE = mean((P-(a*Q+b)).^2)

where a and b are scalars. What I would like to do is plot myMSE as a function of both a and b. IE a is on the x-axis and b is on the y axis and Z is the value for myMSE.

Answer 1

[Matt Tearle]

There are times when, much as I love to mess with matrices and/or bsxfun, a for-loop is just the easiest way to go:

Z = zeros(size(X));
n = length(A);
for k = 1:n
    Z = Z + (S(k) - (X*A(k)+Y)).^2;
end
Z = Z/n;

(I'm assuming that the length of A and S is not related to the size of X and Y, and that you want to average over that dimension -- ie average over the values of A and S.)

EDIT TO ADD: based on the comments below, here's the whole workflow in one easy script :)

EDIT AGAIN: using P & Q as added to the question

% Make example vectors P & Q
P = [zeros(1, 30) ones(1,40) zeros(1,30)] + .1*randn(1,100);
Q=3.4.*([zeros(1, 30) ones(1,40) zeros(1,30)] + .1.*randn(1,100))+5.3;

% Assuming Q = xP + y, look for best x and y
[X,Y] = meshgrid(-10:.2:10, -10:.2:10);
n = length(P);
Z = zeros(size(X));
for k = 1:n
    Z = Z + (Q(k) - (X*P(k)+Y)).^2;
end
Z = Z/n;

% Get initial guess by clicking on the graph
contour(X,Y,Z,20)
x0 = ginput(1)

f = @(x) mean((Q - (x(1)*P+x(2))).^2);
xmin = fminsearch(f,x0)
% xmin should be about [3.4,5.3]

% Easier way?  Do linear fit!
xmin2 = polyfit(P,Q,1)
% Look similar...?

[Devin]

This is actually an optimazation problem, I want to find the scalar values of X and Y that will give me the minimum mean squared error.

[Matt Tearle]

OK, so you mean: given a scalar X and Y value, you want (scalar) Z to be mean((S - (X*A+Y)).^2) -- so "mean" here refers to averaging over the vectors S and A, right? Then you want to do this for a bunch of X and Y values to plot the result and see where the smallest Z is. Correct?

[Devin]

I think I did a poor job explaining this question. I edited my iniital question to include an example.

[Matt Tearle]

OK, that doesn't really change the answer, except to add that if you're assuming Q is a linear scaling of P, why not just do a linear fit? Answer above edited accordingly...