Polynomial fitting with multiple independent variables

Can someone provide example how to perform Polynomial fitting (let's say of 2 or 3-rd order) with multiple independent variables? I have 3 variables: pressure, temperature and concentration (p,t,c) and expectation values of rate of reaction (r) depending on this 3 variables. My question is how to find functional form f(p,t,c)=r and how to perform fitting. (all three variables separetely f(p)=r etc. agree well with linear regresion model).

Thanks a lot

Answer 1

[Matt J]

There are several multi-dimensional polynomial fitting routines on the File Exchange. To name a few,

http://www.mathworks.com/matlabcentral/fileexchange/34765-polyfitn

http://www.mathworks.com/matlabcentral/fileexchange/13719-2d-weighted-polynomial-fitting-and-evaluation

[Oleksandr]

in my case it is 4D problem (3 independent and 1 dependent value)

[Matt J]

Are you saying that invalidates my answer somehow? As far as I can tell, polyfitn is applicable to any dimension.

[Oleksandr]

there are only 2D examples (3D max)

[Matt J]

Even if the examples are 2D, the tool is applicable to higher.

[Oleksandr]

could you please be more specific? Let's say I have P(1,2,3,4,5), T(5,4,3,2,1), C(-2,-1,0,1,2), then P(1)T(1)C(1)=R(1)=2; R(2)=4 ; R(3)=6 R(4)=8 ; R(5)=10.

[Matt J]

I don't know what your notation means. What is the form of the polynomial you are trying to fit? Do you want all possible polynomial terms in 3 independent variables. If so

 p=polyfitn([p(:),t(:),c(:)],r(:),  3);

[Oleksandr]

Undefined function 'polyfitn' for input arguments of type 'double'.

[Matt J]

Probably because you didn't download it from the link I gave you.

[Oleksandr]

great! many thanks! I have only few questions: what is the "default model" (all 3 polynomial terms)? In the output I've got 20 coefficients... Is it possible to do curve fitting or somehow visualize the results with these 3independent+1dependent variables?

[Matt J]

The default model includes all possible terms in a 3rd order 3-variable polynomial. To plot the results, you might use this FEX file

http://www.mathworks.com/matlabcentral/fileexchange/23623-ezimplot3-implicit-3d-functions-plotter

If you fix r to a known constant in your equation f(p,t,c)=r, you obtain an implicit equation for a surface in 3D, which the above will help you plot.

[Oleksandr]

thank you! and if I may to ask you what is the command to get the actual (fitted) values of the functions , not the coefficients?

[Matt J]

I see that there is a polyvaln.m that comes with the POLYFITN toolbox.

Answer 2

[dpb]

Z=zeros(size(p));  % intercept term
X=[Z p t c p.*p t.*t c.*c p.*t p.*c t.*c];  % 2nd order design matrix
c=r\X;                                      % LS solution

You will need a good-sized dataset to have sufficient DOF left after estimating all the terms and while it's a good sign that the "one at a time" plots seem to fit reasonably well that doesn't guarantee a good fit overall.

One would wish that Matlab would have all this built into one of the Toolboxes with a resulting ANOVA table and all but afaict while there are some additional tools in Curve Fitting and Stat toolboxes they really didn't build a general regression model toolset a la SAS, say, unfortunately. You're still on your own for that portion AFAIK.

[Oleksandr]

so what is the expression for function?

[dpb]

Z p t c p.*p t.*t c.*c p.*t p.*c t.*c

In order, as written above the design matrix is

intercept
3 variables
3 quadratic terms
three cross terms

The coefficients will be in that order in the return vector. You can reorder in whatever order suits you.