Regression on multi-variable data

Hello I have a set of data that looks like this:

I want a model of the form

M(x, y, x) = p*(x^0.5) + q(y^0.5) + s(z^0.5)

-> all the exponents are equal to 0.5

where p, q and s are the coefficients that I am looking for.

Each x, y, z, M is a experimental data point, therefore there exist some error in the data, and I want the model with the best fit from which I can calculate the overal error.

How can I obtain this ?

I greatly appreciate your suggestions

Answer 1

[Star Strider]

This is a linear problem, so a linear solution will work.

Try this:

M = [1.48 1 0.9 0.69 0];
x = [8.526 5.6 6.9654 7 0];
y = [0.105 2.8 1.7838 0 0];
z = [0.348 0.21 0 .02 0.25];

B = sqrt([x(:) y(:) z(:) ones(5,1)]) \ M(:);
p = B(1)
q = B(2)
s = B(3)
intcpt = B(4)

Mfit = sqrt([x(:) y(:) z(:) ones(5,1)]) * B;

Resid = M(:) - Mfit

producing:

p =
   0.443190863040845

q =
   0.075119847669093

s =
   0.931994338072853

intcpt =
  -0.488034944641904

Resid =
   0.099808696536123
  -0.113539064951291
   0.118034358405705
  -0.126341765596015
   0.022037775605478

There are too many dimensions to plot, however the residuals indicate the the fit is reasonably good.

You could also use regress, fitlm, and other functions.