How to determine constants using a linear solve of logarithmic transformations which minimize the sum of the residual squares for the below equations?

I have a multivariate power equation in the form y = a1 * (x1^a2) * (x2^a3) * (x3^a4) * (x4^a5) * (x5^a6) , where I have an array of values for each x-term and for the y-term that are each 19775x1. I need to determine the values for the constants a1, a2, a3, a4, and a5 and was told to use a linear solve of logarithmic transformations which minimize the sum of the residual squares. Is there a way to do this in MATLAB?

Attached is my code that I have thus far:

%% Non-dimensional empirical model

% fit used to create correlation:

% pi_theta = a1 .* (pi_y).^a2 .* (pi_z).^a3 .* (pi_twp).^a4 .* ...
%     (pi_tcb).^a5 .* (pi_alpha_cb).^a6;

% log10(pi_theta) = log10(a1) + a2 * log10(pi_y) + a3 * log10(pi_z)...
%     + a4 * log10(pi_twp) + a5 * log10(pi_tcb) + a6 * log10(pi_alpha_cb);

% solve constants using linear solve of logarithmic transformations of
% above equation.

% a1 = 
% a2 = 
% a3 = 
% a4 = 
% a5 = 
% a6 = 

X = [ones(size(pi_y)), log10(pi_y), log10(pi_z), log10(pi_twp), ...
    log10(pi_tcb), log10(pi_alpha_cb)];

Y = log10(pi_theta);

[b, bint,r,rint,stats] = mvregress(Y,X);

I keep getting this error, though:

Error using mvregress (line 456)
Covariance is not positive-definite.

Error in Dimensionless_parameters_analysis (line 266)
[b, bint,r,rint,stats] = mvregress(Y,X);
 

Answer 1

[Star Strider]

REPOSTED —

I decided to let ga have a go at your data and function.

This code :

D = load('X-Y.mat '); 
X = D.X ; 
Y = D.Y; 

objfcn = @(a,x) a(1) .* (x(:,1).^a(2)) .* (x(:,2).^a(3)) .* (x(:,3).^a(4)) .* (x(:,4).^a(5)) .* (x(:,5).^a(6 )); 
ftns = @(a) norm(Y - objfcn(a,X)); 

PopSz = 500 ; 
Parms = 6 ; 
opts = optimoptions('ga', 'PopulationSize',PopSz, 'InitialPopulationMatrix',randi(1E+4,PopSz,Parms)*1E-1, 'MaxGenerations',2E3, 'PlotFcn',@gaplotbestf, 'PlotInterval',1 ); 
[theta,fval,exitflag,output] = ga(ftns, Parms, [],[],[],[],zeros(Parms,1),Inf(Parms,1),[],[],opts ) 

converged fairly quicklly (00:06:21) in 407 generations to produce these parameter estimates :

Parameters = 

1.0e+02 * 

0.002865234375000 
8.700138633728027 
0.027473449707031 
0.009913467407227 
0.000003921508789 
0.002913574218750 

with a fitness value :

fval = 
66.598426312801081 

that is also the best of several runs, each with different 'InitialPopulationMatrix' amplitudes. The predicted ‘Y’ values are all complex.

[Geoff Doback]

Thank you, sir! I actually found a flaw in my original data and fixed it, and this worked perfectly!

[Star Strider]

As always, my pleasure!

I very much appreciate your contacting me and asking me to re-post my Answer.

Answer 2

[Matt J]

The fit is ill-posed (has infinite solutions) because your data pi_tcb is basically 3.2 times pi_tmp:

pi_tcb=10.^(X(:,5));
pi_twp=10.^(X(:,4));

>> norm(pi_tcb-3.2*pi_twp)

ans =

     1.906978550450982e-14