Non-linear fitting for more than three variables

I have a heat transfer experimental data set whrere Nu=f(Re,Theta,Beta).The data file is attached to this question.I want to make a non-linear curve fitting for this data set, so I wrote the following script file.For clarification purpose,Firstly,I imported the data from excel sheet as a matrix named data and as column vectors in the second import,each column is named according to the name in the excel sheet.When I try to run the script ,many errors prevent the running.Is this script correct for running?

% Create an anonymous function that describes the expected relationship
% between X and Y
f=@(c,x) c(1)*(x(:,1).^c(2))*(x(:,2).^c(3))*(x(:,3).^c(4));
% data set 
% Specify x variables from data file,Re,Theta and Beta columns.
x=data;
% Specify y variable from data file ,(Nu)column.
y=Nu;
% Specify a vector of starting conditions for the solvers
c0=[1;1;1;1];
% Perform a nonlinear regression
Beta=nlinfit(x,y,f,c0);

[Alex Sha]

Refer to the results below:

Root of Mean Square Error (RMSE): 22.6077638837674
Sum of Squared Residual: 57244.4306363077
Correlation Coef. (R): 0.921067053007903
R-Square: 0.848364516136664
Adjusted R-Square: 0.842695899917474

Parameter	Best Estimate
----------	-------------
c1	0.273964728046078
c2	0.739698526865773
c3	0
c4	-0.416259686940384

[MOHAMED ABDULAZIM]

Did you use the code that I have attached to this post?

[Walter Roberson]

No, Alex used a commercial tool to find a good fit, very likely better than nlinfit could give.

[MOHAMED ABDULAZIM]

What is this tool?

[Walter Roberson]

1st Opt by 7th Dimension Software

Answer 1

[Alan Stevens]

Because of the form of the desired fit, one can also do a fit by first taking logs. See the coding and result below. (Note Theta is turned into radians and 2pi added in order to avoid having zero angles give problems when taking logs). I've assumed the data is in the workspace with the names Nu, Re, Theta and Beta:

% Logarithmic variables
nu = log(Nu);
re = log(Re);
theta = log(Theta);  %% Theta here is Theta*180/pi + 2pi to convert to radians and remove zeros.
beta = log(Beta);


% Construct vector and matrix
n = length(nu);

V = [sum(nu); sum(nu.*re); sum(nu.*theta); sum(nu.*beta)];

M = [n          sum(re)        sum(theta)       sum(beta);
     sum(re)    sum(re.^2)     sum(re.*theta)   sum(re.*beta);
     sum(theta) sum(theta.*re) sum(theta.^2)    sum(theta.*beta);
     sum(beta)  sum(beta.*re)  sum(beta.*theta) sum(beta.^2)];
 
 c = M\V;
 c1 = exp(c(1));
 

% Functions
nufit = @(re,theta,beta) c(1) + c(2)*re + c(3)*theta + c(4)*beta;
Nufit = @(Re,Theta,Beta) c1*(Re.^c(2)).*(Theta.^c(3)).*(Beta.^c(4));


% Plots

plot(nu), grid
hold on
nuvals = nufit(re,theta,beta);
plot(nuvals)
legend('log target','log fit')
hold off

figure(2)
plot(Nu), grid
hold on
Nuvals = Nufit(Re,Theta,Beta);
plot(Nuvals)
legend('target','fit')
hold off

Here is the final comparison graph:

Just out of interest I've calculated the Pearson correlation coefficient for this fit:

Answer 2

[Matt J]

When I try to run the script ,many errors prevent the running.Is this script correct for running?

No, you need to be using .* to do your vectorized multiplications. Also, you have many rows where x(:,2)=0 which cannot possibly agree with your model, and causes all kinds of NaNs to be generated during the iterative search.

Also, are the coefficients all supposed to be non-negative? If so, nlinfit will not let you apply positivity constraints. I would recommend lsqcurvefit instead,

load xyData

keep=all(x>0,2);
x=x(keep,:); y=y(keep);

X=log(x); Y=log(y);
e=ones(size(Y));

c0=[e,X]\Y; c0(1)=exp(c0(1));  %Use log-linearity of the model

f=@(c,x) c(1)*(x(:,1).^c(2)).*(x(:,2).^c(3)).*(x(:,3).^c(4));

[Beta,~,resid]=lsqcurvefit(f,c0,x,y); 

I assume Alex did something similar, since the result I get from this is very close to his,

>> c0.',Beta.'

ans =

    0.6438    0.7292   -0.2038   -0.4317


ans =

    0.2564    0.7363    0.0000   -0.4029