How can I perform nonlinear regression with two input variables and one dependent variable

I have a data set with three columns, say, Y, X1 and X2 of which Y is the dependent variable (on X1 and X2). I need to know how to use MATLAB to perform nonlinear regression with two input variables for a particular model

y=f(X1,X2)=(1+aX1+bX2)/(1+cX1+dX2).

Answer 1

[John D'Errico]

Simpler than using nonlinear regression is to use LINEAR regression.

If you have

y=f(X1,X2)=(1+aX1+bX2)/(1+cX1+dX2)

then multiply by (1+cX1+dX2).

y*(1+cX1+dX2) = 1+aX1+aX2

or

y - 1 = a*X1 + a*X2 -c*y*x1 -d*y*x2

You can do it using no extra functions, than just some basic MATLAB.

So assume that X1, X2, and Y are all COLUMN vectors. We implement it in MATLAB as:

coefs = [X1, X2, -Y.*X1, -Y.*X2]\(Y-1);
a = coefs(1);
b = coefs(2);
c = coefs(3);
d = coefs(4);

Basically one line of code. No iterative routine needed. No starting values. Or, you can use these estimates as very good estimators of the coefficients for a nonlinear estimation.

[Ubong Abia]

Where do I start from? I have never used Matlab, so when I open the software, how do I get to using these codes?

[Amos]

very elegant solution

[John D'Errico]

(TY)

As others have said, you need to learn to use MATLAB then. I cannot teach you a complete course in basic MATLAB skills, especially when that is done so much more ably in other places. Read the getting started tutorials.

Answer 2

[Star Strider]

This will work:

X1 = randi(9, 1, 10);                               % Create Data
X2 = randi(9, 1, 10);                               % Create Data
Y = randi(99, 10, 1);                               % Create Data

XY = [X1(:)  X2(:)];                                % Create Independent Variable Matrix
Y  = Y(:);                                          % Dependent Variable

% % % MAPPING: p(1) = a,  p(2) = b,  p(3) = c,  p(4) = d, 
f = @(p,x) (1+p(1).*x(:,1)+p(2).*x(:,2))./(1+p(3).*x(:,1)+p(4)*x(:,2));

SSECF = @(p) sum((Y-f(p,XY)).^2);                   % Sum-Squared-Error Cost Function
P = fminsearch(SSECF, [1; 1; 1; 1]);                % Estimate Parameters

[Amos]

nice ;-)

[Ubong Abia]

Many thanks Star Strider. The only challenge is I have never used Matlab and I don't know where to type that in or how to get results. This could be stretching a bit but could you help with a step by step approach? I mean from when open the software and how to import my excel data etc. Please help me.

[Star Strider]

@Amos — Thank you!

-----------------

@Ubong Abia — My pleasure.

First read Getting Started to understand MATLAB, how it works, and how to make it work for you. Please take your time reading it, run the examples, and get comfortable with it. Be sure you understand how to create, save, and run script files (using the .m extension). Don’t worry about the strange looking code I wrote (it uses anonymous functions), since there will be time for you to learn about Function Basics later when you’re more comfortable with MATLAB. (I linked to it here to make it easier for you to find and read.) MATLAB is a very large environment, so make extensive use of the documentation. I refer to it frequently. To bring that up, type doc in the Command Window.

Second, to read an Excel file, use the xlsread function. See the documentation on xlsread to understand how it works.

Third, with respect to the full MATLAB code, I don’t have your Excel file, so I can’t write specific code. Assuming the first two columns are ‘X1’ and ‘X2’ and the third is ‘Y’, the actual code would go something like this:

[d,s,r] = xlsread('Your_Data_File.xlsx');

XY = d(:,1:2);
Y = d(:,3);

% % % MAPPING: p(1) = a,  p(2) = b,  p(3) = c,  p(4) = d, 
f = @(p,x) (1+p(1).*x(:,1)+p(2).*x(:,2))./(1+p(3).*x(:,1)+p(4)*x(:,2));

SSECF = @(p) sum((Y-f(p,XY)).^2);                   % Sum-Squared-Error Cost Function
P = fminsearch(SSECF, [1; 1; 1; 1]);                % Estimate Parameters

I had xlsread return all three outputs here because you may need that other information, and having it in your workspace makes it easier for you to access it.

I’m not certain what you want to do, so I didn’t go further than importing the data and estimating the parameters of your function.

This should get you started.

[Ubong Abia]

You have been most helpful Star Strider. Quick one, how different are the codes from the curve fitting tool as I believe there is a platform for one to type a custom equation and obtain parameters. Is there a difference between the regression codes and the curve fitting tool in terms of getting accurate coefficients?

[Star Strider]

I do not have the Curve Fitting Toolbox. (I have the Optimization and Statistics Toolboxes, so don’t need it for what I do.) I prefer to use fminsearch in my Answers because everyone has it.

My code will give you accurate coefficients for your model. It may be more robust that code using the gradient-descent algorithms.

Answer 3

[Amos]

You could use lsqcurvefit for that, where prbably X1 and X2 need to be put together into a single array, and a,b,c,d also need to be put together into a parameter array p.