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

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abxz on 12 Nov 2019
Edited: abxz on 15 Nov 2019 and
a,k,c,b, and g are constants to be determined.
KALYAN ACHARJYA on 13 Nov 2019
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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: 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
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 ##### 2 CommentsShowHide 1 older comment
abxz on 13 Nov 2019
@ Alex Sha, Thanks a lot for your help.
will you please provide the code also.

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.
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.

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
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 abxz on 15 Nov 2019
@ Alex Sha , Thank you so much.