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

and

a,k,c,b, and g are constants to be determined.

[KALYAN ACHARJYA]

Define sysm and solve, any issue?

[abxz]

Not able to write code.. will you please help with the code

[abxz]

@ Kalyan Acharjya, will you please help out with code

[KALYAN ACHARJYA]

For assignments or homework, we help only after looking at your efforts to solve the question.

Answer 1

[Alex Sha]

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
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

[KALYAN ACHARJYA]

@Alex

Why you are providing the two answers?

You can add the modified answer as a comments in the first answer.

[abxz]

@ Alex Sha, Thanks a lot for your help.

will you please provide the code also.

Answer 2

[Alex Sha]

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]

d/dt is derivative wrt t. K/ (1+exp(c-b*t))

[abxz]

@Alex Sha , will you please help out with code.

[abxz]

@Alex Sha , will you please help out with code.

[abxz]

@ 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.

Answer 3

[Alex Sha]

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

[KALYAN ACHARJYA]

@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]

Sorry for giving multi-answers.

1stOpt is an independent software package I think, not the add-in or toolbox of Matlab.

[abxz]

@ Alex Sha , Thank you so much.