Best equation for Curve Fitting

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Vishal Dolas
Vishal Dolas on 18 Jun 2021
Edited: Scott MacKenzie on 19 Jun 2021
Can someone give me the best equation for fitting this curve
clc
close all
clear all
% 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.
  1 Comment
Alex Sha
Alex Sha on 18 Jun 2021
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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Answers (1)

Scott MacKenzie
Scott MacKenzie on 18 Jun 2021
Edited: Scott MacKenzie on 19 Jun 2021
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.

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