2nd Order Polynomial Coefficient Solving

I have been having very good success with guidance from the community with using Curve Fitter. I now would like to produce coefficients with an added variable.

I have the Data for R,S,P,T.

T = a + b*P + c*S + d*S^2 + e*(S*P) + f*(S^2*P)

Trying to find coefficients a,b,c,d,e,f.

The coefficients would a median seperated by 'R' breakpoints [550 725 950] , so realistically I would have a 6x3 matrix of coefficients.

At the moment linear regression will work, but the affect of P on T is linear, and the affect of R and S on T is non-linear in reality.

I have a 63000 x 4 matrix for data.

 R             S                    P                   T
716	    28.5000000000000	291.727272727300	184.407961051320
721	    28.5000000000000	291.625000000000	187.140145995908
721	    28.5000000000000	291.625000000000	187.220504376631
722.5	28.5000000000000	291.625000000000	187.140145995908
722.5	28.5000000000000	291.625000000000	187.140145995908

I tried this along with a couple other methods to no avail.

Ignore the column indices, that was another iteration from the above data.

TRQ1 = @(c,VAR) c(1).*VAR(:3) + c(2) + c(3).*VAR(:,2) + c(4).*VAR(:,4) + c(5).*VAR(:,5) + c(6).*VAR(:,6);
F_TRQ= linsolve(VAR(:,[1 3] ) , TRQ, TRQ1)

Answer 1

[Torsten]

I don't see R in your regression equation

T = a + b*P + c*S + d*S^2 + e*(S*P) + f*(s^2*P)

And I assume that "s" means "S".

M = [ones(63000,1),P,S,S.^2,S.*P,S.^2.*P];
y = T;
x = M\y;
a = x(1)
b = x(2)
c = x(3)
d = x(4)
e = x(5)
f = x(6)

[Eric]

Yes, I edited the s to S. The constant "a" would change if between each set of coefficients based of R

[Torsten]

Thus problem solved ?

[Eric]

Thank You!. I am going to dig deeper now on how to add constraints to the coefficients and how to incorporate the constant related to R

[Torsten]

If you have to put constraints on the coefficients, use "lsqlin" to solve the overdetermined linear system M*x = y.

E.g.

M = [ones(63000,1),P,S,S.^2,S.*P,S.^2.*P];
y = T;
lb = [...]; % lower bounds on the coefficients
ub = [...]; % upper bounds on the coefficients
x = lsqlin(M,y,[],[],[],[],lb,ub);
a = x(1)
b = x(2)
c = x(3)
d = x(4)
e = x(5)
f = x(6)

[Eric]

So far I havent needed constraints yet, which I was surprised about. Still hung up on using R for an index/value. I tried replacing 'ones' with R and it seemed to be in the right direction but R I turned my constant into an coefficient which I know is incorrect.

[Eric]

After more research, I cant seem to make the data do what I want.

I am trying to have only the constant x(1) linked to data from R. So it would be the "offset" based off R.

Thank you again!

[Torsten]

From what I understand, you want something similar to this ?

T1 = T(R<=550);
P1 = P(R<=550);
S1 = S(R<=550);
n1 = numel(T1);
T2 = T(R>550 & R<=725);
P2 = P(R>550 & R<=725);
S2 = S(R>550 & R<=725);
n2 = numel(T2);
T3 = T(R>725);
P3 = P(R>725);
S3 = S(R>725);
n3 = numel(T3);
M = [[ones(n1,1);zeros(n2,1);zeros(n3,1)],[zeros(n1,1);ones(n2,1);zeros(n3,1)],...
     [zeros(n1,1);zeros(n2,1);ones(n3,1)],[P1;P2;P3],[S1;S2,S3],[S1.^2;S2.^2;S3.^2],...
     [S1.*P1;S2.*P2;S3.*P3],[S1.^2.*P1;S2.^2*P2;S3.^2*P3]];
y = [T1;T2;T3];
x = M\y;
a1 = x(1)
a2 = x(2)
a3 = x(3)
b = x(4)
c = x(5)
d = x(6)
e = x(7)
f = x(8)

[Eric]

I think so, but need to keep the total amount of coefficients including the constant to 6.

[Walter Roberson]

M = [[ones(n1,1);zeros(n2,1);zeros(n3,1)],[zeros(n1,1);ones(n2,1);zeros(n3,1)],...
     [zeros(n1,1);zeros(n2,1);ones(n3,1)],[P1;P2;P3],[S1;S2,S3],[S1.^2;S2.^2;S3.^2],...
     [S1.*P1;S2.*P2;S3.*P3],[S1.^2.*P1;S2.^2*P2;S3.^2*P3]];

I think that sould be [S1;S2;S3] instead of [S1;S2,S3]

[Torsten]

@Walter Roberson

Thank you.

@Eric

You get 6 coefficients. The first coefficient depends on R. The regression equation solved above is

T = x(1) + x(4)*P + x(5)*S + x(6)*S^2 + x(7)*(S*P) + x(8)*(S^2*P) if R <= 550

T = x(2) + x(4)*P + x(5)*S + x(6)*S^2 + x(7)*(S*P) + x(8)*(S^2*P) if R > 550 & R <=725

T = x(3) + x(4)*P + x(5)*S + x(6)*S^2 + x(7)*(S*P) + x(8)*(S^2*P) if R > 725

[Eric]

Thank you so much. I will now keep my head down and keep working on learning.