how to solve a system of linear equation in matlab for gridded data

I have a linear equation log(P) -log(diff(S)/dt) =log(z) + log(a)+ b*log(S) and gridded values for log(P),log(diff(S)/dt) and log(S) of size 364x9. in which 364 stands for no of days and 9 stands for the no of grids . so i have to solve the above equation for 9 grids to get a, b and z for the available grds in matlab

[Dyuman Joshi]

If I understand it correctly, you want to solve 1 equation with 3 unknown variables? or 364 equations with 3 unknown variables?

What is expected size of z, a and b?

[FATHIMA JAMSHEENA P]

i want to solve the given equation with 3 unknown variables and i have 3 known data set of size 364*9 in which 364 stands for temporal resolution so i have to solve the equation for 9 grids (which will give 9 equation ) to get a final value for a b and z. that is here no. of equations are more than that of no. of unknowns

[Torsten]

z and a cannot be solved for separately. So you have 2 unknowns, namely log(az) and b.

And you have 364 x 9 measurement data for 2 unknowns ? Did I understand this correctly ?

log(P(i,j)) - log(diff(S)/dt(i,j)) = log(az)+ b*log(S(i,j))

for 1 <= i <= 364 and 1 <= j <= 9 ?

[FATHIMA JAMSHEENA P]

z, a and b are unknown parameters and for finding them we have to use the above given relation in which P and S data of size 364x9 are available. so if we construct a forloop to find these parameters we have to get a, b and z data of size 9*1 as solution.

Answer 1

[Torsten]

As said, you can't estimate z and a separately. Say you have

y = z + a

and data for y. Say you get z = 4 and a = 2 as solution. Then all combinations of z and a with z + a = 6 would be solutions, too. One says that the problem is overfitted.

If you want to get solutions for the revised problem, use

for i = 1:9
    A = [ones(364,1) log(S(:,i))];
    b = log(P(:,i))-log(gradient(S(:,i)));
    sol = A\b;
    za(i) = sol(1);
    b(i) = sol(2);
end

I assume that the unit per days for dS/dt is correct. Otherwise, you will have to scale gradient(S(:,i)) by the correct unit.