Curve fitting with multiple variables?

I have three different curves for S=2, S=1 and S=0.5 as shown in Figure below. The Book1.xlsx is attached.

I want to use curve fitting to find out one common equation such as, HF = f(S,T).

I used the following comman line to get the equation which I got from previous post (link). But I am getting following error:

Error using nlinfit (line 213)

Error evaluating model function '@(b,x)exp(b(1)).*x(:,1).^b(2)+b(3).*x(:,2)

What should I do? Is there any other better method as well?

T1 = readtable('Book1.xlsx', 'VariableNamingRule','preserve')

T1 = 10x4 table
    T         S2            S1           S0.5   
    __    __________    __________    __________

    10    2.2639e+05      1.63e+05         44925
    12    3.2139e+05    2.3122e+05    1.3253e+05
    14    4.0539e+05    2.9211e+05    2.0969e+05
    16    4.7838e+05    3.4568e+05     2.764e+05
    18    5.4038e+05    3.9192e+05    3.3265e+05
    20    5.9137e+05    4.3084e+05    3.7846e+05
    22    6.3136e+05    4.6243e+05    4.1383e+05
    24    6.6034e+05     4.867e+05    4.3874e+05
    26    6.7833e+05    5.0363e+05     4.532e+05
    28    6.8531e+05    5.1325e+05    4.5722e+05

VN = T1.Properties.VariableNames

VN = 1x4 cell array
    {'T'}    {'S2'}    {'S1'}    {'S0.5'}

Viv = regexp([VN{:}], '(\d*\.\d*)|\d*', 'match')

Viv = 1x3 cell array
    {'2'}    {'1'}    {'0.5'}

Viv = str2double(Viv)

Viv = 1x3

2.0000 1.0000 0.5000

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Viym = T1{:,2:4};
Viyv = Viym(:);                                                 % 'Vi' Vector

Vivm = repmat(Viv, size(T1,1), 1);
Vivv = Vivm(:);                                                 % 'Viv' Vector

y = Viyv;
SoCv = repmat(T1.T, size(Viym,2), 1)                           % 'SoC' Vector

Error using . (line 229)
Unrecognized table variable name 'deltaT'.

x = [SoCv Vivv];                                                % Concatenate Vectors

objfcn = @(b,x) exp(b(1)).*x(:,1).^b(2) + b(3).*x(:,2);

mdl = fitnlm(x, y, objfcn, rand(3,1))

[Matt J]

Your code results in a different error from the one you mention, see the Run output above.

[Torsten]

@Liril Silvi

x must be the matrix

x = [10 0.5;12 0.5;...;28 0.5;10 1;12 1;...;28 1;10 2;12 2;...;28 2] 

and y must be the matrix

y = [S0.5;S1;S2]

in your code.

And why do you use exp(b(1)) in your model function ? Why not simply b(1) ?

[Liril Silvi]

@Matt J

deltaT should be T (I edited this one in the original question). Now, you will get the error as I said.

Answer 1

[Sam Chak]

Hi @Liril Silvi,

See if the polynomial surface function is acceptable

tb  = readtable('Book2.xlsx', 'VariableNamingRule','preserve');
S   = tb{1:30,2};
T   = tb{1:30,1};
HF  = tb{1:30,3};
[sf, gof] = fit([S, T], HF, 'poly24')

sf = 
     Linear model Poly24:
     sf(x,y) = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2 + p21*x^2*y 
                    + p12*x*y^2 + p03*y^3 + p22*x^2*y^2 + p13*x*y^3 + p04*y^4
     Coefficients (with 95% confidence bounds):
       p00 =  -1.028e+06  (-1.028e+06, -1.028e+06)
       p10 =   1.173e+06  (1.173e+06, 1.173e+06)
       p01 =   1.108e+05  (1.108e+05, 1.108e+05)
       p20 =  -4.329e+05  (-4.329e+05, -4.329e+05)
       p11 =  -9.663e+04  (-9.663e+04, -9.662e+04)
       p02 =       -2110  (-2110, -2110)
       p21 =   4.004e+04  (4.004e+04, 4.004e+04)
       p12 =        2021  (2021, 2021)
       p03 =  -5.616e-12  (-5.367e-11, 4.243e-11)
       p22 =      -826.9  (-826.9, -826.9)
       p13 =  -1.061e-12  (-5.705e-12, 3.584e-12)
       p04 =   8.182e-14  (-5.452e-13, 7.089e-13)

gof = struct with fields:
           sse: 2.0286e-17
       rsquare: 1
           dfe: 18
    adjrsquare: 1
          rmse: 1.0616e-09

plot(sf, [S, T], HF), xlabel('S'), ylabel('T'), zlabel('HF')

[Liril Silvi]

This polynomial equation has excellent fit. But, I was more into a simple and concise equation rather than a long one with lots of coefficients. Something like as you said about paraboloid function.

[Sam Chak]

Hi @Liril Silvi

The Hyperbolic Paraboloid has this polynomial form when expanded. Thus, you can select 'poly22'.

syms a [1 4]
syms x y

%% Hyperbolic Paraboloid
hp  = a1*(x - a3)^2 + a2*(y - a4)^2;
expand(p)

ans = 

tb  = readtable('Book2.xlsx', 'VariableNamingRule','preserve');
S   = tb{1:30,2};
T   = tb{1:30,1};
HF  = tb{1:30,3};
[sf, gof] = fit([S, T], HF, 'poly22')

sf = 
     Linear model Poly22:
     sf(x,y) = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2
     Coefficients (with 95% confidence bounds):
       p00 =  -5.246e+05  (-6.033e+05, -4.459e+05)
       p10 =   8.896e+04  (2.158e+04, 1.563e+05)
       p01 =   6.545e+04  (5.808e+04, 7.282e+04)
       p20 =        2105  (-2.134e+04, 2.555e+04)
       p11 =        2343  (827.7, 3858)
       p02 =       -1199  (-1386, -1012)

gof = struct with fields:
           sse: 4.9789e+09
       rsquare: 0.9936
           dfe: 24
    adjrsquare: 0.9922
          rmse: 1.4403e+04

plot(sf, [S, T], HF), xlabel('S'), ylabel('T'), zlabel('HF')