Fitting exponential curve with three parameters to some sample points.

Question

I've been trying to solve this problem for days without involving a stat program or anything similar (including least squares fitting) but I've been unsuccessful. I attempted to substitute the given points, form three equations but I couldn't solve them. How do I get around this problem? In advance thank you for your help.

This question answers the question without the constant $C$, and gives a reference for the general case. — rogerl, Nov 07 '19 at 19:32
Do you want the curve equation or do you want how the values are obtained? — NoChance, Nov 07 '19 at 20:08
@NoChance, both would be fine but I believe J.G. has answered my question below. — Meghan C, Nov 08 '19 at 12:55

J.G. · Accepted Answer · 2019-11-07T22:32:50.153

2

Define $B:=Ae^{-14k},\,t:=e^{-5k}$ so $B+C=9176,\,Bt^{11}+C=7681,\,Bt^{13}+C=7542$ and $B(1-t^{11})=1495,\,Bt^{11}(1-t^2)=139$. Hence $1495t^{13}-1634t^{11}+139=0$. The only root in $(0,\,1)$ is $t=0.9082$. Now use$$k=-\frac15\ln t,\,\left(\begin{array}{c} B\\ C \end{array}\right)=\left(\begin{array}{cc} 1 & 1\\ t^{11} & 1 \end{array}\right)^{-1}\left(\begin{array}{c} 9176\\ 7681 \end{array}\right),\,A=Bt^{-2.8}.$$

edited Nov 07 '19 at 22:32

answered Nov 07 '19 at 19:43

J.G.

115,835

It is not clear to me how did you find $x=e^{-5k}$. Thanks. – NoChance Nov 07 '19 at 20:11
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@NoChance That was a poor choice of name on my part for the variable; I've changed it to $t$. All I really did was change to a more convenient variable for comparing the differences between data. – J.G. Nov 07 '19 at 20:19
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Thanks for the clarification, good answer. – NoChance Nov 07 '19 at 20:20
Many thanks for your solution, however, I believe the first equation on the second line in incorrect. – Meghan C Nov 07 '19 at 22:24
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@MeghanC Thanks; fixed. – J.G. Nov 07 '19 at 22:32
One more question, why choose a value of t between 0 and 1? – Meghan C Nov 08 '19 at 16:49
@MeghanC Increasing $x$ reduces $y$, so you'll need $t$ in that range unless $A<0$. – J.G. Nov 08 '19 at 16:54
Thanks, it makes sense. – Meghan C Nov 08 '19 at 20:45

Fitting exponential curve with three parameters to some sample points.

1 Answers1