MATLAB: How to do non-linear regression for three varietals

fitnlm

Hi All,

I have matrix with three variables (x,y,z) I would like to get best non linear regression for these variables using like this equation: Eq=a*x+b*y+c*z+d

How can I get the constants and correlation coefficient?

Thanks in advance,

Riyadh

Best Answer

The equation you posted is linear. Assuming it is a stand-in for a nonlinear equation, the usual way of fitting a function of several variables is to create a matrix of the incependent variables and passing that as one argument to the objective and fitting functions.

Example:

% % % MAPPING: x = xyz(:,1),  y = xyz(:,2),  z = xyz(:,3),  a = b(1),  b= B(2),  c = b(3),  d = b(4)
xyz = [x(:) y(:) z(:)];
Eq = @(b,xyz) b(1).*xyz(:,1) + b(2).*xyz(:,2) + b(3)*zyz(:,3) + b(4);

Then just use them as arguments to whatever fitting function you want (such as nlinfit or lsqcurvefit).

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MATLAB: Fit model with 3 independent variables and many parameters to data

Yes, if you have the Statistics Toolbox you can use the nlinfit() function to do this. Here is a very simple example.

function [] = nlinfitExample()
% Here is an example of using nlinfit(). For simplicity, none of
% of the fitted parameters are actually nonlinear!
% Define the data to be fit
x=(0:1:10)'; % Explanatory variable
y = 5 + 3*x + 7*x.^2; % Response variable (if response were perfect)
y = y + 2*randn((size(x)));% Add some noise to response variable
% Define function that will be used to fit data
% (F is a vector of fitting parameters)
f = @(F,x) F(1) + F(2).*x + F(3).*x.^2;
F_fitted = nlinfit(x,y,f,[1 1 1]);
% Display fitted coefficients
disp(['F = ',num2str(F_fitted)])
% Plot the data and fit
figure
plot(x,y,'*',x,f(F_fitted,x),'g');
legend('data','fit')
end

In my case, I just have one explanatory variable vector, x, but that could have been a matrix with multiple variables in each column.

Hope that helps.

MATLAB: Determining coefficients of an equation using your experimental data

What you refer to as ‘coefficients’ are actually called ‘parameters’. Estimating them is relatively straightforward. I am using fminsearch here, but the function ‘Yfit’ will work with the Statistics Toolbox nlinfit and the Optimization Toolbox lsqcurvefit functions. See the relevant documentation for details.

% Original Equation: Y = X * (a + (b*(Z)^c) * (F^d))
% Variable Matrix: 
Y = Y(:);
XZF = [X(:) Z(:) F(:)];
% Parameters: a = b(1),  b = b(2),  c = b(3),  d = b(4)
Yfcn = @(b,XZF) XZF(:,1) .* (b(1) + (b(2).*(XZF(:,2).^b(3)) .* XZF(:,3).^b(4)));
SSECF = @(b) sum((Y - Yfcn(b,XZF)).^2);
B0 = rand(4,1)*100;                         % Initial Parameter Estimates
[B,SSE] = fminsearch(SSECF, B0);

The ‘XZF’ matrix is necessary because most functions take only two parameters, the parameter vector and the independent variable. The ‘SSECF’ function is the sum-squared-error cost function for a least-squares fit. The estimated parameters are in the ‘B’ vector, and ‘SSE’ is the sum-squared-error when fminsearch ends.

This code works with created data, but fitting the parameters may be very difficult. It is important for your function here that your initial parameter estimates are as close to their actual values as you can estimate them to be. I cannot guarantee that fitting your function will give you the most appropriate parameters, in terms of their physical validity. You may have to experiment with several — perhaps hundreds — of different initial parameter estimates before you get a set that converges on physically appropriate estimates.

Because your data are actually 4-dimensional (with ‘Y’, ‘X’, ‘Z’, and ‘F’), you will not be able to plot it, or plot the fitted curve to it.

Best Answer

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