Background
I am doing a PhD in interdisciplinary mathematics and I have been working on an optimization problem related to the following function:
$$ f(x) = c_{1}(1-e^{-c_{2}(x-x_{0})})^{2}$$
where $c_{1}$, $c_{2}$ and $x_{0}$ are constants. Assume that we have an interval $[x_{a}, x_{b}]$ and then we sample a number $n$ of points $x_{k}$ from this interval and calculate the corresponding value of $f(x_{k})$ and then we sum over these values, i.e. $\Sigma = \sum_{k=1}^{n}{f(x_{k})}$. We then repeat this sampling process $M$ times. This leaves us with the following system of equations (i.e. one equation for each of these sums):
\begin{cases} \Sigma_{1} & = \sum_{k=1}^{n}{f(x_{1,k})} & = f(x_{1,1}) + f(x_{1,2}) + \dots f(x_{1,n}) \\ \Sigma_{2} & = \sum_{k=1}^{n}{f(x_{2,k})} & = f(x_{2,1}) + f(x_{2,2}) + \dots f(x_{2,n}) \\ & & \vdots \\ \Sigma_{M} & = \sum_{k=1}^{n}{f(x_{M,k})} & = f(x_{M,1}) + f(x_{M,2}) + \dots f(x_{M, n})\\ \end{cases}
Now, assume that we do not know what $f(x)$ actually looks like and that we want to recreate the function $f(x)$. To do so, we can solve this system of equations using linear algebra. We can not do this using a continous $f(x)$ so we will have to discretize our interval into $N$ bins with a binwidth of $h$, such that:
\begin{cases} f_{1} & \text{if } x \in [x_a, & x_a+h) \\ f_{2} & \text{if } x \in [x_a+h, & x_a+2h) \\ & \vdots \\ f_{N} & \text{if } x \in [x_a+(N-1)h, & x_b] \\ \end{cases}
If we treat this as a vector $\vec{f} = [f_{1}, f_{2}, \dots, f_{N}]^{T}$ of size ($N$-by-1) and bin all $x_{i,k}$ to their column corresponding to $f_j$ in a count matrix $C$ of size ($M$-by-$N$) where $M >> N$. We can then rewrite our system of equation as:
$$C\vec{f}=\vec{\Sigma}$$
where $\vec{\Sigma}=[\Sigma_{1}, \Sigma_{2}, \dots, \Sigma_{M}]^{T}$ of size ($M$-by-$1$). We can then use e.g. SVD to approximate the left inverse $C^{-1}$ to solve for $\vec{f}=C^{-1}\vec{\Sigma}$ to get an idea of what the true $f(x)$ might look like.
Problem
I want to know how to bin the interval $[x_{a}, x_{b}]$ and I want to do so by minimizing the error of approximating $f(x)$ using $\vec{f}$. In my real application RAM is a limiting factor (I have several different $f(x)$ I want to estimate at once, making $C^{-1}$ expensive to compute) so I want to have a fixed number of bins. So, given an $N$ number of bins, how do I select the width of each individual bin so that I minimize the error between $f(x)$ and $\vec{f}$? The values of $x$ is Poisson-distributed and $x$ and $f(x)$ can be very different in orders of magnitude.
What I've tried:
- Equidistant binning w.r.t. $x$ (best result so far)
- Equidistant binning w.r.t. $f(x)$
- Equidistant binning w.r.t. $ln(1+f(x))$
- Equidistant binning w.r.t. length of curve
- Binning where the count in each bin is the same for all bins
It does not make sense to me that equidistant binning w.r.t. $x$ (i.e. $h_{1}=h_{2}=\dots=h_{N}$) provides the best results, especially when the $\Delta f(x)$ is so different between the left and the right half of the bin. But when the resolution at the position where the $|f'(x)|$ gets higher then the error around the minimum of $f(x)$ gets significantly higher instead.
So, how do I select my $H = \{h_{1}, h_{2}, \dots, h_{N}\}$ for all bins and a fixed number of bins $N$?
Example
Here is an example of what $f(x)$ and equidistant binning w.r.t. $x$ look like (and arbitrary values for $c_{1}$, $c_{2}$ and $x_{0}$). Please note that there is no approximation here, just the exact values of $\vec{f}$.
