How to compare/analyze the behavior of two functions (mathematically)?

Question

Python

There are 2 functions that take the following values:
a=[68570,63700,51834,77696,62736,66166,57270,49269,53630,56640,59073,58786] and b=[47407,53878,46567,60152,62329, 67344,61022,50006,54662,62554,65570,64833]
If you build graphs, it is easy to understand that their behavior is basically similar, especially at the end, periods of ups and downs almost coincide, etc. But how do you figure it out mathematically?

I want a program in python that will give the similarity of functions, periods of increase and decrease, etc.

Answer 1

In mathematics, this is called hypothesis testing. Specifically, you have the fact that 2 implementations belong to the same random process (moreover, if you are interested in comparing recessions and ascents, then you need to look for a theory on non-stationary random processes).
In short, since you do not know the true state of the process, then the arithmetic mean between 2 implementations at each point is taken as it. Then the distribution function of deviations of each implementation from this empirical estimate is calculated. Next, you must specify the characteristics of the noise (most often normal, and its RMSD is the main evaluation parameter), which you consider acceptable and check whether all the observed deviations fit the hypothesis with a certain level of significance, for example, 0.05.
If you do not know the parameters of the noise, then on the contrary, its MSWO can become for you one of the simplest indicators of similarity / difference between two implementations, at a given significance level (for example, limit it to 0.05 and look for the maximum MSWD at which the homogeneity hypothesis is not rejected ).