Best equation for Curve Fitting

Question

Vishal Dolas 2021-6-18

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/859390-best-equation-for-curve-fitting

编辑： Scott MacKenzie 2021-6-19

Can someone give me the best equation for fitting this curve

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% Shear Stress Data (tau)
x = [0	0.004451043	0.038688186	0.735062819	0.782019317	1.594521919	1.642507629	2.59312869	2.643649113	3.753413617	3.808892339	5.101477848	5.161995222	6.596302962	6.657818541	8.139665372	8.202526359	9.683129426	9.748762511	11.19273216	11.25754175	12.53313276	12.59651093	13.69044727	13.75079095	14.68565326];
x = x*10^-3;  % converting to meters
y = [-1.66E+07	-1.65E+07	-1.62E+07	-1.02E+07	-1.01E+07	-6868466.274	-6778223.204	-5199368.229	-5166179.72	-4430907.666	-4420065.933	-4168450.441	-4168500.008	-4167024.178	-4172591.221	-4306588.629	-4318151.713	-4594452.961	-4624282.626	-5296317.693	-5366938.191	-6848554.377	-7016084.319	-1.02E+07	-1.08E+07	-2.07E+07];
fcn = @(b,x) b(1)+ b(2).*x - b(3).*exp(-b(4)*x);  % suggest a better equation please
B0 = rand(4,1);
B = lsqcurvefit(fcn, B0, x, y)
figure
plot(x, y, 'p')
hold on
plot(x, fcn(B,x), '-r')
hold off                                            
grid
xlabel('X')
ylabel('Y')
%legend('Data', sprintf('y = %.3f\\cdotx^{%.3f}', B), 'Location','E')

I want to capture the data points as precisely as possible, it doesn't matter if equation is complex.

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Alex Sha 2021-6-18

在 MATLAB Online 中打开

The equation below seems to be good enough:

y = 1/(p1*sin(p2*x+p3))+p4*x+p5

Root of Mean Square Error (RMSE): 165925.431217479
Sum of Squared Residual: 715812466842.366
Correlation Coef. (R): 0.999366806323172
R-Square: 0.998734013580577
Parameter	Best Estimate
----------	-------------
p1	3.28885892497527E-7
p2	190.268111842121
p3	-2.95337291125282
p4	-71910402.0440169
p5	-449707.42534779

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Answer 1

Scott MacKenzie 2021-6-18

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链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/859390-best-equation-for-curve-fitting#answer_727785

编辑：Scott MacKenzie 2021-6-19

在 MATLAB Online 中打开

How about r = .9999?

Linear model Poly9:
     f(x) = p1*x^9 + p2*x^8 + p3*x^7 + p4*x^6 + 
                    p5*x^5 + p6*x^4 + p7*x^3 + p8*x^2 + p9*x + p10
Coefficients (with 95% confidence bounds):
       p1 =  -1.997e+26  (-3.775e+26, -2.186e+25)
       p2 =   9.985e+24  (-1.688e+24, 2.166e+25)
       p3 =  -1.829e+23  (-5.031e+23, 1.372e+23)
       p4 =   1.157e+21  (-3.601e+21, 5.914e+21)
       p5 =   7.562e+18  (-3.392e+19, 4.904e+19)
       p6 =  -1.788e+17  (-3.934e+17, 3.592e+16)
       p7 =   1.324e+15  (6.891e+14, 1.958e+15)
       p8 =  -5.242e+12  (-6.207e+12, -4.277e+12)
       p9 =   1.177e+10  (1.118e+10, 1.236e+10)
       p10 =   -1.66e+07  (-1.668e+07, -1.651e+07)
Goodness of fit:
  SSE: 7.084e+10
  R-square: 0.9999
  Adjusted R-square: 0.9998
  RMSE: 6.654e+04

This is a bit of an odd question. It's a bit like asking, Can someone tell me what the best song is? Really, by what criteria? If the only criterion for the question herein is achieving the highest r (or R-squared), then the question is unanswerable, since for any proposed equation and very high r, you can always just add more terms to get an even higher r.