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How to curve fit following summation equation in MATLAB with given experimental data.

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abxz
abxz on 12 Nov 2019
Edited: abxz on 15 Nov 2019
and
a,k,c,b, and g are constants to be determined.

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KALYAN ACHARJYA
KALYAN ACHARJYA on 13 Nov 2019
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Accepted Answer

Alex Sha
Alex Sha on 13 Nov 2019
d(K/ (1+exp(c-b*t)))/dt=k*exp(c-b*t)*b/sqr(1+exp(c-b*t)),so your fitting function become:
eq.png
is the above correct? if yes, you may see the function is over-fit, that means the parameters will not be unique, one solution likes below:
Root of Mean Square Error (RMSE): 11257.9862380487
Sum of Squared Residual: 1647649303.76923
Correlation Coef. (R): 0.891727403408187
R-Square: 0.795177761989108
Adjusted R-Square: 0.726903682652144
Determination Coef. (DC): 0.795103266693536
F-Statistic: 7.64708134900884
Parameter Best Estimate
-------------------- -------------
a 67082.9685482823
k 2.55264230357405E16
c 39.1820152654189
b 0.0511178327965598
g 1.74671169710137E-249
p1.jpg

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KALYAN ACHARJYA
KALYAN ACHARJYA on 13 Nov 2019
@Alex
Why you are providing the two answers?
You can add the modified answer as a comments in the first answer.

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More Answers (2)

Alex Sha
Alex Sha on 12 Nov 2019
Hi, Yadav, in your function "a*d/dt(k/1+exp(c-b*t))", what is "d/dt"? does "k/1" equal to "k"? Please describe clearly.

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abxz
abxz on 13 Nov 2019
@ Alex Sha , Please send code . I need code so if i have to do similar kind of fitting , i can take the help from the code.
Thank you.

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Alex Sha
Alex Sha on 14 Nov 2019
Hi, Yadav, I actually use a software package other than Matlab, named 1stOpt, it is much easy for using without guessing initial start-values, since it adopts global optimization algorithm. The code looks like below:
Parameter a,k,c,b,g;
ConstStr f=a*(k*exp(c-b*t(i))*b)/(1+exp(c-b*t(i)))^2;
Variable t,z[OutPut],y;
StartProgram [Basic];
Sub MainModel
Dim as integer i, j, n
Dim as double temd1, temd2
for i = 0 to DataLength - 1
n = t(i)
temd1 = 0
for j = 0 to n
temd1 = temd1 + f
next
temd2 = 0
for j = 0 to n
temd2 = temd2 + g*y(j)
next
z(i) = temd1 + temd2
Next
End Sub
EndProgram;
Data;
t=[0,2,4,6,8,10,12,14,16,18,20,22,24];
z=[0,0,666.6,5333.33,10666.6,21333,42666.6,4666.6,42666.6,42666.6,42666.6,42666.6,85333.3];
y=[1.25*10^8,1.*10^8,1.25*10^8,2.2*10^10,1.3*10^11,1.4*10^11,1.25*10^11,4.7*10^10,7.9*10^10,9.5*10^10,9.4*10^10,8.8*10^10,9.4*10^10];
a much better result:
Root of Mean Square Error (RMSE): 9901.69220512954
Sum of Squared Residual: 1274565610.8266
Correlation Coef. (R): 0.918681151445272
R-Square: 0.84397505802081
Adjusted R-Square: 0.791966744027747
Determination Coef. (DC): 0.841498837497943
F-Statistic: 9.7840155772957
Parameter Best Estimate
-------------------- -------------
a -277156.527610417
k -10698197.9029827
c 28.5533367811484
b 0.35649699061682
g 3.06293808255498E-8
c231.jpg

  3 Comments

KALYAN ACHARJYA
KALYAN ACHARJYA on 14 Nov 2019
@Alex Is this package supports by Matlab? You are giving multiple answers for same question. you can add extentions answer in the comment section of respective answer.
Alex Sha
Alex Sha on 14 Nov 2019
Sorry for giving multi-answers.
1stOpt is an independent software package I think, not the add-in or toolbox of Matlab.

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