Fitting exponential model to data I have a series of measurements of some value, about 10000 numbers. The numbers come timestamped in real-life clock seconds, at regular time intervals. All real-life measurements have errors, both systematic and random sensor noise, this one is no exception.
An oversimplified mathematical model of the physical process says that that at least the middle portion of the data is probably described by the following differential equation:
$$\frac{df(t)}{dt} = \alpha·S^{\beta}·f(t)^{\beta}$$
where $\alpha$ and $\beta$ are some unknown constants i.e. they don’t change over time (we also know $0 < \beta < 1$), and S is a known constant $10^{-5}$. The formula basically says the time derivative of the function is proportional to some power of the functions’s current value.
How can I find values of $\alpha$ and $\beta$ which would best fit (e.g. minimum squared error or similar) the measured experimental data of $f(t)$ function?
Update: according to a symbolic math software, the above differential equation has the following solution:
$$f(t) = {(t·\alpha·S^\beta·( 1 - \beta ) + C1)^{\frac{1}{1-\beta}} }$$
But the question remains, how to best fit that function to the measured data to get these constant numbers?
Here's the source data
 A: So, you have $n$ data points $(t_i,y_i)$ and you want to fit them as well as possible using the model
$$y = \big[\alpha( 1 - \beta ) \,t + c\big]^{\frac{-1}{\beta-1}} $$ which is much better conditioned letting $a=\alpha( 1 - \beta )$ and $b=\frac{-1}{\beta-1}$  and then rewrite
$$y=(a t+c)^b$$ It is highly nonlinear and, as usual, you need at least consistent guesses to start the nonlinear regression.
The model cannot be linearized as it is. But suppose that you give an arbirary value to $b$. Then, for this value, you have
$$z=y^{\frac 1b}=a t+c$$ and you compute the sum of squares
$$\text{SSQ}(b)=\sum_{i=1}^n \big[a t_i+c-z_i\big]^2$$ which is a basic linear regression. So, for this value of $b$, you have $a(b)$, $c(b)$ and $\text{SSQ}(b)$.
Run several values of $b$ and locate where, more or less, $\text{SSQ}(b)$ is minimum. AT this point, you have your guesses.
But, now, you must run the nonlinear regression since what is measured is $y$ and not any of its possible transforms.
All of that should not require more that very few minutes with Excel or any similar tool.
A: For this specific dataset, the standard nonlinear optimisation module in Mathematica reaches a solution without much trouble. After I scaled the data by a factor of $10^7$, the best fit was reported as
$$g(t)=(0.149176 t+0.621692)^{3.84546}.$$
In general, it would be decisive to have good and consistent initial approximations for the parameters.
