Finding asymptotes given data

Question

Background: I asked this question on Stack Overflow about how to program in Java or VBA a method to calculate asymptotes given a range of data points. I believe the underlying question would be more appropriate here than on SO - if I understand the statistical way of solving the problem, I will be able to solve it programmatically.

Problem: We are given $n \in [5,15]$ numbers on the interval $]0,1]$ that are the measured approximations of some real-life phenomena, call them $s_1, s_2, \cdots, s_n$. They tend to be decreasing so that $s_i<s_{i+1}$ (although they are approximations so that it's not always so). Looking at them on a graph, we see that it appears they have a horizontal asymptote as $n \rightarrow \infty$. Example:

i   value
-   -
1   0.8232
2   0.6032
3   0.5012
4   0.4646
5   0.45001
6   0.44981

which gives the following chart

The horizontal asymptote would be $y=a$ with $a$ being some number less than $s_n$. In this case, it seems like $a$ is close to $0.44$. I have two questions:

1. How do we find this asymptote if we do not know the underlying distribution? (I guess we assume the formula $e^{ax}+b$, is this true?)

2. How do we find the asymptote for some confidence interval, say 95%? Do we then assume that the measurements are accurate or should we assume that each $s_i$ follows a normal distribution on its true value for some low standard deviation $\sigma_i$ (so that the chance that the true value corresponding to $s_i$ has a 67% change of being within $[s_i-\sigma_i,s_i+\sigma_i]$)?

Answer 1

Instead of fitting the function, fit the first (positive) differences $d_i=s_{i-1}-s_{i}$

Edits based on OP comments

While addressing OP's comments, I happened upon an easier approach that I liked better:

Since the sequence of $s_i$ are decreasing, let's model each $s_i$ as the asymptote $\theta$ plus a positive term $\epsilon_i$ such that $s_i=\epsilon_i+\theta$. This implies that $d_i=s_{i-1}-s_i = \epsilon_{i-1}-\epsilon_i$. Since your function that you are approximating appears to have a discrete domain, we should instead model the first positive differences as a geometric sequence: $d_i=ar^i$. This implies that $s_i=s_1-\sum\limits_{j=1}^{i-1} d_i = s_1-\sum\limits_{j=1}^{i-1} ar^i$.

Fitting the geometric model to the first differences and minimizing sum of squares, I get: $\hat d_i =0.519(0.428)^i$. Now, $\theta = \lim\limits_{i\rightarrow \infty} s_i = s_1-\sum\limits_{j=1}^{\infty} d_i = s_1-\sum\limits_{j=1}^{\infty} ar^i=s_1 - \frac{ar}{1-r}=0.8232-0.38753 = 0.435627$, which is quite close to your intuition. See here for geometric series sums

I did the same calculation starting at each of the other six $s_i$, where I subtract $\hat d_{i-1}$ from $0.38753$ since we are now starting at $i=2,3,4...$:

$\hat \theta_i = s_i-[\frac{ar}{1-r}- \sum\limits_{j=1}^{i-1} d_j]$ where $\theta_i$ is the estimate of $\theta$ starting at $s_i$.

I got the following values:

$\{0.435627133,\;0.437502142,\;0.430359695,\;0.434313858,\;0.437061857\}$

Therefore, our range of estimates is approx $[0.430,0.438]$. I don't think you have enough data points to really do much proper inference, but as you can see the range is pretty small (although these values are correlated, of course).

Anyway, I think this is a cleaner analysis than my first one and easier to implement on a non-statistical computer language (C++ or JAVA).