some question about optimization

I have a function: A=W*S, A is a matrix of 4*1000, and S is a matrix of 2*1000, then I want to get a optimized matrix W of 4*2 to make the equation as good as possible, so I want to know how to solve this problem, can I use the optimization toolbox

Answer 1

[Stephan]

Hi,

try this command:

W = A/S

Here is an example for small data:

>> A = [1 2; 3 5; 3 8; 2 7; 8 11]

A =

         1     2
         3     5
         3     8
         2     7
         8    11

>> S = [1 5; 3 6; -4 9]

S =

         1     5
         3     6
        -4     9

>> W = A/S

W =

             0    0.3333    0.0000
             0    0.9216   -0.0588
             0    1.1569    0.1176
             0    0.9020    0.1765
             0    2.2745   -0.2941

>> x = int8(W*S)

x =

      5×2 int8 matrix

        1    2
        3    5
        3    8
        2    7
        8   11

>> Test = A == x

Test =

      5×2 logical array

       1   1
       1   1
       1   1
       1   1
       1   1

That's what you wanted to do?

The result of this procedure will depend strongly on how well the matrices A and S correlate and how large the measurement errors / noise in the data is i think. Therefore, I recommend checking the result with the test A = B with B = W * S as shown. The fraction of zeros in the logical matrix resulting from the review is an indicator of the goodness of this approach, I think. You should be aware that there are rounding errors and include them in the evaluation.

For bigger examples with added errors which disturb the linear dependencies it still works good i think:

S = (randi(25, 2, 1000));
A = ones(4, 1000);
A(1,:) = sin(S(1,:));
A(2,:) = 2 * S(2,:);
A(3,:) = 0.5 * S(2,:) + log(S(2,:));
A(4,:) = -5 * S(1,:);
W = vpa(A/S);
A = int8(A);
x = sum(sum(int8(W*S) == A)) / 4000
same = int8(W*S) == A;

Lines 1 to 6 just construct a problem of your dimensions with built in non linearity to simulate bad data / noise for testing.

x is the ratio of correct calculated elements by the operation A/S and reached about 0.748...0.759 for this example.

I hope this works for your purpose.

best regards

Stephan