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I have a discrete series $x[n]$. It is extracted from real life and I do not have probability distribution of each value $x[n]$. Is there any computational method to prove whether the series is stationary or not?

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    $\begingroup$ It depends on the process, the source of the data. Depending on the number of samples and if any assumption may be made you can get to a probability value (and significance) that will tell you if it could be stationary. $\endgroup$ – Moti Jul 10 '16 at 5:52
  • $\begingroup$ I took the liberty to comment on the terms you use in the answer – not because I didn't want to improve your original question, but because I think the real question here is a bit of confusion what things can be said about processes (and can't be said) based on an observation. $\endgroup$ – Marcus Müller Jul 10 '16 at 11:41
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1 Proof

You're misusing the word proof. Remember that a proof, even one led with stochastic methods, always leads to an absolute "If A, then B, no doubt" statement.

Since your $x$ is a realization of a stochastic process of which you don't know many properties, pretty much everything is possible – for example, a perfectly white process might have a realization that by sheer luck is extremely periodic. Now, if you have a finite observation with discrete values, which is exactly what computational implies, you can only make a statement that seeing such a realization is extremely unlikely, yet not impossible – which shows the difference between a proof and a strong suspicion based on good indication¹.

You can often calculate likelihoods that a specific observation stems from a given emitting quality (here: realization of a stochastic process being caused by a stationary process), and say that with a confidence of $x \%$ you can say that the process behind this finite observation is stationary.

Having a confidence that you need to incorporate into your statements validity is the crux of every descriptive science – and, having worked with medical scientists before, it happens very easily to claim you've proven e.g. effectiveness of a specific cure, but that is a different, non-mathematical, non-DSP meaning of proving things. Just because something worked 100 out of 100 times doesn't mean it's right. It might just imply some high probability it is, and understanding that has very interesting consequences for understanding scientific publications – for example, even for high-quality medical papers whose statements are true with a confidence of $0.98$, the probability that of 35 papers all are correct is less than $\frac12$. Social sciences often have to work with lesser confidences, due to many things only being observable on smaller sample sets.

2 Inference from a single Observation

Without making a very strong statement on the stochastic process², you can't say anything about different realizations of the same process by looking at a single realization.

In other words, having a single $x_k[n]$ will not allow you to make a statement on $x_l[n],\,l\ne k$, unless you claim that you can deduct how different realizations will behave just by looking at one limited one. That's a statement that I personally find just as strong (or actually, stronger) than stationarity.

3 Stationarity

This is a property of a stochastic process. If you just say "I want to show whether this is stationary", people will assume you mean strictly³ stationary.

This implies that the joint probability distribution⁴ is invariant to shifts in time. Well, your discrete observation started at a specific time, so no statement can be given (see 2).

Now, since you're looking at a digital signal (with finite observation duration/data points), this isn't as much of a setback as one would instantly presume: due to the observation duration being limited, we obviously can't make any statements about how things act when looking at things that happen at scales much larger than the observation.

Hence, here, the term of wide-sense⁵ stationarity (WSS) saves the day by reducing the statement from

"this process' joint probability distribution is time-invariant"

to

"when comparing a realization of the process with a time-shifted version of itself, the stochastic properties of that only depend on the time shift, not the absolute times."

Now, this second statement is based on autocorrelation – and that is easily estimated/computable.

In essence: if different realizations have the same autocorrelation function, the underlying process is wide-sense stationary.

Being limited observations, the estimated autocorrelation function will not be identical – but it can be similar.

By defining a metric for similarity and a threshold for that metric based on error probability, statements on the probability of that process being WSS can be made.

In the most intuitive case, similarity could be understood as the root difference of the squares of the autocorrelation estimates.

By definition, the power spectral density (PSD) of a process is the Fourier transform of the autocorrelation – and happily, thanks to the Wiener-Chintschin theorem, this is, for WSS processes, equivalent to the expectation value of the squared Fourier transform.

Hence, comparing the PSD estimate done by squaring the discrete Fourier transform (DFT) of observations of different realizations of the same process can be a way to describe how likely a process is stationary – especially if your underlying phenomenon is harmonic/periodic, doing the DFT is something that might be helpful, anyway, and with the FFT, calculating $\left(\text{DFT}(x_i)\right)^2$ is more computationally efficient than estimating autocorrelation based on convolution.


¹ This is really a side note: Watch 12 Angry Men, the 1957 Movie. Claiming something is proven is a big statement.

² ergodicity

³ strictly, or strongly stationary – I must admit, not being a native speaker, I'm not 100% sure there's no difference here; if someone could comment on that, I'd be grateful.

⁴ your signal is 1D, so this is equivalent to the probability density function being invariant

weak or wide-sense – same as ³

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