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?


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    $\begingroup$ Regarding the problem of checking a sequence for randomness: dilbert.com/strip/2001-10-25 $\endgroup$ – MBaz Sep 1 '16 at 0:07
  • $\begingroup$ The spike occurs when the shift is zero. The autocorrelation of a sequence with itself is large. I wouldn't conclude that the samples are independent from either plot; certainly not from the green one. $\endgroup$ – MBaz Sep 1 '16 at 0:10

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.

  • $\begingroup$ I think for this purpose mutual information is a stronger measure compared to correlation. You can find a couple of resources online by looking up: mutual information test randomness $\endgroup$ – msm Sep 1 '16 at 15:32
  • $\begingroup$ Thank you for the updated answer. It addresses most of my Questions but I request you to kindly comment on this sentence, (1) " you should note that the autocorrelation of a white process (i.e. a process with uncorrelated random variables) is a pure delta function." Does this mean that in my graph, the autocorrelation is a pure delta function? (2) To check for independence, in general for samples obtained from a linear system do we check auto correlation between samples or cross-correlation between two different sequences obtained from the same system? $\endgroup$ – Srishti M Sep 3 '16 at 3:09
  • $\begingroup$ (1) The smaller the ACF at nonzero lags, better it looks like a pure delta. (2) I think both. $\endgroup$ – msm Sep 3 '16 at 7:45

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