How do I fit two power law varying independent variables ( X1 and X2) in the form Y= Constant * X1^n1 * X2^n2. Appreciate the solution with an example for simple data

I am trying to fit tow Power law based variables x1 and x2 in the form of Y=constant * (x1^n1) * (x2^^n2). Kindly suggest the way forward. Kumar RK

[dpb]

Which toolboxes do you have available? If have Curve Fitting Toolbox, there are examples of using the interactive tool to fit custom nonlinear models; follow along there and modify to match your own specific form should be able to get you going...

Answer 1

[John D'Errico]

As dpb says, the curvefitting TB allows you to fit that model. If you lack that TB, then you can use other tools, like the optimization or stats TB.

Note that a very simple transformation allows you to get pretty good estimates of the parameters in that model. Just take the log. Then your model becomes linear.

log(y) = log(constant) + n1*log(x1) + n2*log(x2)

for that model, giving you good estimates for starting values. Or possibly, this may be the right way to do the fit, if y spans several orders of magnitude. In that case, additive Gaussian noise is probably not a good assumption, but proportional noise will make sense. (Nonlinear regression tools implicitly assume additive Gaussian noise.) Then taking the log transforms a proportional noise structure into additive noise.

[kumar rk]

Dear John Thanks for your suggestion. The values of Y data varies from 10e-4 to 10e-9. The fits of additive nature is possible in minitab and other tools. As we require to export the fit equation to a Engineering software ANSYS CFX whcih accepts only the productive rather than the additive equations. i.e Y= Constant * (x**n1)* (x2**n2). The output of the fit constant, n1 and n2 are to be coded further in Intel Fortran so as to use it in CFX.

Appreciate the methodology to be followed..

[dpb]

Well, you didn't give us the input to know what you've got at your disposal...what's wrong with the CF'ing Toolbox methodology???

[John D'Errico]

Gosh, I did not know it was impossible to fit a model to the log problem, and then convert back. I guess I'll need to remember how to solve for C, if I only know the value of log(C). Be serious.

The point is that IF your data spans 5 powers of 10 in Y, then you will NEVER be happy with the result from the curve fitting toolbox, with that model in exponential for as you have it.

What will happen is the model will fit some of your data EXTREMELY well, and ignore the rest of your data.

While you CAN do the fit, no matter where you do it, the result will be pure crapola. Sorry, but it will. Your next question will be an anguished "Why does my curve fit so poorly down at the bottom end?" I really don't care if you use MATLAB, Minitab or Excel for the fit. As a power-law model as you want to fit it, you will not be happy. But hey, I've only been doing this for 40 years.

Answer 2

[Sachin Bharadwaj M]

clc; clear all;

filename = 'D:\set12.xlsx';

sheet = 1;

xRange = 'A2:E71';

yRange = 'F2:F71';

X = xlsread(filename,sheet,xRange);

Y = xlsread(filename,sheet,yRange);

modelfun = @(b,x)b(1) .* X(:,1).^b(2) .* X(:,2).^b(3)

beta0 = [0 0 0];

mdl = fitnlm(X,Y,modelfun,beta0)

Y_model = mdl.Fitted;

R_sqr = mdl.Rsquared.Ordinary;

per_dev = (Y-Y_model).*100./Y;

for i = 1:mdl.NumCoefficients

b(i,1) = mdl.Coefficients.Estimate(i,1);

end

% Y_model = b(1) .* (X(:,1).^b(2)) .* (X(:,2).^b(3)) .* ( X(:,3).^b(4))

%% PLOTTING OF RESULTS:

figure

plot(Y,Y,'k-', Y,Y+((max(Y)-0)*5/100),'g-.', Y,Y-((max(Y)-0)*5/100),'g-.', Y,Y+((max(Y)-0)*10/100),'r:', Y,Y-((max(Y)-0)*10/100),'r:');

hold on

scatter(Y, Y_model, 'b.') % ,'MarkerEdgeColor','k' , 'filled', 'MarkerFaceColor', 'b'

hold off

legend('No Error','+5%','-5%','+10%','-10%','Data points', 'Location','southeast')

xlabel('Actual', 'FontWeight', 'bold', 'FontSize', 14)

ylabel('Prediction', 'FontWeight', 'bold', 'FontSize', 14)

zlabel('T ^{o}C', 'FontWeight', 'bold', 'FontSize', 14)

title('Scatter Plot of Actual versus Fitted')