$$ f(t_1) = m + k\cdot e^{\dfrac{-t_1}{\tau}} \hspace{1cm} (1)\\ f(t_2) = m + k\cdot e^{\dfrac{-t_2}{\tau}} \hspace{1cm} (2)\\ f(t_3) = m + k\cdot e^{\dfrac{-t_3}{\tau}} \hspace{1cm} (3) $$ The $t_1$, $f(t_1)$, $t_2$, $f(t_2)$, $t_3$, $f(t_3)$ are known (from three different measurements) and I would like to find the unknown parameters $m$, $k$ and $\tau$ from the above system of equations (1-3). What is the easiest solution of the above problem ?
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In the right hand side of each equation, don't you mean $e^{\dfrac{-x_i}{\tau}}$, for $i=1,2,3$? – Marra Nov 16 '17 at 11:24
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1Hi Marra, You have right. I have already corrected the equations. Thank you ! – Theo D. Nov 16 '17 at 11:29
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Eliminate $m$ $$f(t_2)-f(t_1)=k\left( e^{-\frac{t_2}{\tau}}- e^{-\frac{t_1}{\tau}} \right)$$ $$f(t_3)-f(t_1)=k\left( e^{-\frac{t_3}{\tau}}- e^{-\frac{t_1}{\tau}} \right)$$ Make the ratio to eliminate $k$ $$\frac{f(t_3)-f(t_1) } {f(t_2)-f(t_1) }=\frac{ e^{-\frac{t_3}{\tau}}- e^{-\frac{t_1}{\tau}} }{ e^{-\frac{t_2}{\tau}}- e^{-\frac{t_1}{\tau}}}$$ and you are left with one nonlinear equation in $\frac 1 \tau$.
When solved, go back to get $k$ and then $m$.
Edit
If you are able to select the $t_i$'s, it would be very nice in arithmetic progression ! Let $t_2=t_1+a$, $t_3=t_2+a=t_1+2a$ and the last equation becomes $$\frac{f(t_3)-f(t_1) } {f(t_2)-f(t_1) }=1+e^{-\frac{a}{\tau }}$$ then $\tau$.
Claude Leibovici
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and the main Problem is how we get that $\tau$ from this equation? – Dr. Sonnhard Graubner Nov 16 '17 at 11:38
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Except for very specific values, you would require a root finder (Newton method would be the simplest). Are the $t_i$'s in arithmetic progression ? – Claude Leibovici Nov 16 '17 at 11:40
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Thank you very much for these first answers. t1, t2, and t3 are three different time points, where i make my measurements. There are randomly selected. Would it be helpful if I selected these measurements time points after an arithmetic progresseion rule ? – Theo D. Nov 16 '17 at 12:15
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@TheoD. This would be wonderful ! See my edit. However, what I suppose is that you wake more than three measurements. So, I suspect that some nonlinear regression would be required at a time. If the errors are small, the above procedure would give more than "reasonable" estimates to start. – Claude Leibovici Nov 16 '17 at 12:25
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@Claude Leibovici. Thank you very much. This is a very nice idea. I will start with this approach and I see how it is going. – Theo D. Nov 16 '17 at 12:34
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@TheoD. You are very welcome ! Please, let me know. As an experimentalist, you face here a very interesting situation, be sure. Cheers. – Claude Leibovici Nov 16 '17 at 12:37