Quote:
Originally Posted by saikiranra
I believe you can use the SVD to find the psuedo inverse matrix using the Moore-Penrose method. I used a Moore-Penrose psuedo matrix method in the Python Numpy library, although I believe that the general solve method in Numpy does something similar. I'm also new at this entire statistics world, so I'm sure one of the guru's here can fill us in.  |
Yeah, that. For a real positive semi-definite symmetric matrix (like A'A for any A), the SVD is something like U D U' where U is orthogonal and D is diagonal. In our case, A'A is not full rank, so the last diagonal value of D is 0. Using the method in the link above, the pseudo-inverse is computed as U E U' where E is diagonal with elements Ei = 1/Di except where Di=0, in which case Ei=0 too. This makes the product of a matrix and its pseudo-inverse equal to U' F U where F is diagonal with Fi = 1 if Di is non-zero and Fi=0 if Di = 0.
So if the original matrix is full rank, the pseudo-inverse is just the regular inverse and the product of the matrix and pseudo-inverse is U U' = I. If the original matrix is not full rank, then multiplying by the pseudo inverse basically zeros out the component of the vector in the zero-rank projection and multiplies by the inverse of the remaining subspace. Or, the product of the matrix and its pseudo-inverse is U U' except that you replace the vector of U corresponding to Di=0 with a zero vector.
In this case, the zero-rank projection (or the row vector of U' that corresponds to the Di that is zero) is something like 1/T[ 1 1 1 1 ...1] which computes the mean, because the direction in the C vector corresponding to its mean is the direction that cannot be determined.
One other formulation for CCWMOA would just be:
if we have T teams, have T-1 unknown values C1, C2, ..., C(T-1) and set CT = -Sum(C1, C2,... C(T-1)) in all of the equations (thus enforcing that all T values of Ci are zero mean). Then we only have T-1 equations with T-1 unknowns and everything is full rank. This is just another way of saying we want to find the values of C1, C2, ... CT that minimize the prediction error subject to the constraint that the resulting set of Ci values have zero mean.