Alternatively, I would like to get help finding a vector so that the sum is maximized, where . The vectors and are still in .
If you have points in , how can you find the "best fitting" hyperplane so that gets as small as possible?
is the distance from the hyperplane to the k:th point. If the equation of is , where is the normal of the hyperplane, then the distance is defined as
Ok, both problems are relevant when finding a transformation matrix for a random vector , so you can describe a random vector by the formula
has the dimension d, and so does . Each element in is normally distributed, but not independent of each other. on the other hand, has it's elements normal distributed around 0 with the standard derivation 1, and independently of each other. is a matrix with its columns pairwise orthogonal to each other, of which some could be . is the average of the random vectors and is simple to find for a set of sample random vectors, the difficulty is to find . The distribution of is called multivariate normal distribution.
If I, for example have about 100 points in a three-dimensional room, I may want to find the direction in which the points differ the most and how much they may differ in that direction. Do you have any idea of how to find that direction? Or maybe the transformation matrix from the problem above?
Any suggestions would be appreciated.