How can I create a best fit polynomial for my data?

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Syed Ali Zaryab 2018-9-4

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评论： Titus Edelhofer 2018-9-9

I have a set of data points that I received from a supplier which gives relative efficiency based on values of x and y. I want to create a polynomial similar to the one given below using the data provided Efficiency = a * (x)^2 + b * (x*y) + c * (y)^2 + d * (x)+ e * (y) + f

So basically what I want is a function which optimizes the value of a,b,c,d,e and f for me and gives me results that are almost similar to the one in the data table.

Attached herewith is the picture of the graph that the supplier provided along with the data points. Can you please tell me what functions can I use in Matlab to help me achieve this task.

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Answer 1

Titus Edelhofer 2018-9-4

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Hi Syed,

if I assume you have three (column) vectors x, y and Efficiency, then it's simply a linear regression problem:

A = [x.^2 x.*y y.^2 x y ones(size(x))];
sol = A \ Efficiency;

Then sol contains the (best approximation) values a,...,f.

Titus

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Syed Ali Zaryab 2018-9-7

编辑：Syed Ali Zaryab 2018-9-7

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Dear Titus,

I tried to implement your suggestion and wrote the following bit of code

x = [0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1;
        0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1;
        0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1;
        0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1];
eff = [.9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9;
        .95 .95 .95 .95 .95 .95 .95 .95 .95 .95 .95 .95 .95;
        .9667 .9667 .9667 .9667 .9667 .9667 .9667 .9667 .9667 .9667 .9667 .9667 .9667;
        .9833 .9833 .9833 .9833 .9833 .9833 .9833 .9833 .9833 .9833 .9833 .9833 .9833];
y= [0.93 0.91 0.89 0.87 0.85 0.84 0.835 0.83 0.83 0.835 0.84 0.85 0.86;       
       0 0 0 1.27 1.1 1.04 1 0.99 0.98 0.975 0.97 0.98 0.99 ;
       0 0 0 0 0 0 0 0 1.07 1.03 1.025 1.025 1.03;
       0 0 0 0 0 0 0 0 0 0 0 1.1 1.07 ];
A = [x.^2 x.*y y.^2 x y ones(size(x))];
sol = A \ eff;
sol;