How to test for the symmetry of a finite sequence? I have a finite sequence of real numbers ${\{a_n\}}_{n=0}^{N-1}$, for the sake of simplicity I assume $N\gt1$ is even.
The sequence is symmetric (I would say even like an even function) iff
$a_k=a_{N-1-k}, \forall k=0..\frac{N}{2}-1$
So for example $\{1,2,3,3,2,1\}$ is a symmetric sequence while $\{1,2,3,3,2,2\}$ is not a symmetric sequence.
Now, my real data are measurement and they are affected by errors.
For example, in the following R script s1 sequence is not symmetric while s2 sequence is symmetric:
s1<-rnorm(n = 100,mean = 0,sd = 1)
s2_a<-rnorm(n=50,mean=0,sd=1)
s2_b<-s2_a+rnorm(n=50,mean=0,sd=1/7)
s2<-c(s2_a,rev(s2_b))
plot(s1,type='l')
plot(s2,type='l')



I can invent by myself a "score" of symmetry like the following one
$$\sum_{k=0}^{\frac{N}{2}-1} |a_k-a_{N-1-k}|$$
and the lower is the score and the higher is the symmetry of the sequence... but I would like to know whether there is some classic method to deal with this kind of problem.
 A: I call your problem "inexact palindrome detection." Look at the example from genetics, here the paper "A method to find palindromes in nucleic acid sequences." Palindromes can be exact and inexact, in latter case the match is not precise but close.
Your problem is different from the cited paper because it seems that your data is real, while DNA sequences are obviously categorical. The real palindromes, i.e. $a_i\in {\rm I\!R}$, will be more difficult to deal with than conventional palindromes where $a_i$ are categorical or integer. However, you may find the approaches similar and the references in the paper to relevant methods of detection. Since you allow for errors, your palindromes are inexact.
Algorithmically, if these were categorical sequences such as texts and genes, I'd start with Hamming distances on first and inverted second half of the sequences: $a_1,a_2,\dots,a_{N/2}$ and $a_{N/2+1},\dots,a_{N-1},a_N$. 
Your score looks like Manhattan distance where each position in sequence is a dimension, and can be more appropriate for real data.
One thing about these distance measures is that if you compare scores of palindromes of different length then you may need to scale the distance measure, say by $1/N$, because long inexact palindromes will tend to have bigger distances than shorter ones.
