Help in interpreting the auto correlation graph

Question

I want to check if a time series is (a) random (b) independent. For these I am using the autocorrelation (AC). Autocorrelation refers to the correlation of a time series with its own past and future values.

If random variables are independent then there should be no correlation between them. The reverse is not true.

I have two different vectors as observations obtained from sensor measurement that are a sequence of samples and I treat this as a time series -denoted by variables phi and phit respectively.

• Question 1: I cannot interpret the graph, what the spike indicates. Please help

• Question 2: For time series generated from linear system in order to check for randomness, what is the test to perform and what is the value to expect??

• Question 3: For time series obtained from linear system in order to check for independence between samples, what test to perform and what is the value to expect?

• Question 4: The given time series is obtained from nonlinear dynamical system that is chaotic. If correlation test is not valid, then what tests should I perform to check for (a) randomness (b) independence?

Answer 1

In my opinion this question is too broad. The first point is that you want to see if a given time series is random or not. I have doubt on how we can define this term here. Also you don't mention any distribution. Normally, we are interested to see if a time-series belongs to a specific distribution or not, for which there are standard techniques.

Regarding dependency, you should note that correlation only captures linear dependencies. You can have random variables that are highly dependent but have very low correlation. So it is important that what specific type of dependency you are looking for.

Edit: That said, referring to the ACF a peak at lag zero is visible. The reason is that at zero lag the time-series shows maximum similarity which is not surprising since it is obviously similar to itself! Note that the amplitude of this peak for a general complex signal $x[n]$ is $$R_{XX}[0]=\sum{x[n]x^*[n]}=\sum{|x[n]|^2}$$ which is the signal's power. You can easily show (theoretically) that $$R_{XX}[k]\le R_{XX}[0], \,\forall k$$

In addition, you should note that the autocorrelation of a white process (i.e. a process with uncorrelated random variables) is a pure delta function. So closer your autocorrelation to a delta means it is more uncorrelated, which is a good thing for a PRNG.

Regarding, your question about mutual information and entropy, yes both of them are used. Entropy is a well known measure for randomness. I cannot give you more details in here but can suggest you looking at this or similar resources.