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I just started trying to undestand the notion of stationary in time series. Basically I have 2 questions:

  1. Can stationary time series contain regulary cycles and thus seasonality patterns? For exmaple in this tutorial it is stated that stationary time series can not have seasonal components (predicitable cycles) https://otexts.com/fpp2/stationarity.html whereas in this figures (https://i.imgur.com/3lKCxEn.png) the green time series that clearly has cycles (and thus seasonality) is labeled as 'stationary' (and I have seen these kind of figures quite often if you just google 'stationary time series)
  2. Can a stationary time series have periods with no fluctuations and periods with high fluctuations? As far as I understood the variance and the (aut)covarianz should not change over time making such a time series not stationary. But here in this picture (https://www.researchgate.net/profile/Hazrat_Ali3/publication/326619835/figure/fig10/AS:654171351044097@1532978012116/Non-stationary-and-stationary-time-series-As-CDR-activities-of-users-are-aggregated-on.png) the time series below is labeld as stationary altough it has periods with changing fluctuations.

I hope you can help me as I am confused about the concept of stationarity. I'd appreciate every comment.

The bounty is to expire quite soon. So I'd be happy if some could at least give me one answer to my questions. It will help me a lot.

Why is nobody answering the questions? Are they not clear enough? If so, please tell me. I think they are important and fundamental as the concept of stationarity is quite important.

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    $\begingroup$ If a series contains "seasonality patterns," then a fortiori it is not stationary. The underlying concept of stationarity is that statistical properties of the series do not change over time, while the underlying concept of seasonality is that those properties do change, but in a periodic way. The ResearchGate image you reference appears to focus solely on the location (perhaps the mean) of each marginal distribution, wholly ignoring other statistical properties (such as the dispersion, which obviously is not stationary). This is known as "first order" stationarity. $\endgroup$ – whuber Oct 13 '20 at 16:01
  • $\begingroup$ Thanks whuber for your answer. So would you say that the green time series in (i.imgur.com/3lKCxEn.png is not stationary because it is seasonal? And for me the ResearchGate picture is not stationary per definition. The variance it not constant over time (at all). I think it is wrong to say that this time series is stationary because if you only require the stationary attributes to be valid for a certain portion of the time series, the stationary requirements can be fullfield by almost every time series $\endgroup$ – PeterBe Oct 13 '20 at 16:38
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    $\begingroup$ As I indicated, there are various definitions of "stationary." They vary according to which statistical properties must remain unchanged. But none of the examples in your image would even satisfy one of the loosest definitions, "weak first order stationarity." Those images don't tell you much about how the variance changes over time, though. $\endgroup$ – whuber Oct 13 '20 at 17:40
  • $\begingroup$ Thanks whuber. So did I understand correctly that the two time series that are labeled stationary are not-stationary (meaning that the pictures are wrong)? I ask this because especially this figures i.imgur.com/3lKCxEn.png can be seen many times on the internet and they are always labeled as stationary (the green line). Is it not stationary because of the regular cycles because that is what I thought after reading the tutorial I gave a link to in the post? $\endgroup$ – PeterBe Oct 14 '20 at 7:15
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    $\begingroup$ A regular cycle is manifestly not stationary. However, it is difficult to interpret those pictures because they are extremely sketchy. If the colored curves are intended to represent either (a) data or (b) expected values, then their regular variation is strong, clear evidence of non-stationarity. $\endgroup$ – whuber Oct 14 '20 at 13:03
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  1. Stationary series cannot have a fixed seasonal component, which is saying, if you take a stationary series and you sum it to $sin(t)$, the result will not be stationary. Stationary series can be seasonally autocorrelated, which means that what happens one month is correlated with what will happen the next year in the same month. For stationarity to hold, however, in the long run (after some number of years) this autocorrelation must vanish. It's often hard to tell from sampled data if the time process behind it is stationary or not (it's a matter of statistical tests, not about precise measuring) but that green series in the imgur image does not seem stationary, not anymore than the lower-right red series anyway (the upper two red series show even worse behavior allright).

A premise for the second answer: there is more than one definition of stationarity, but generally both the unconditional mean and variance (and also the auto-covariance function) must be constant over time. This doesn't mean that fluctuations can't happen, but that if you don't know any value of the series around time $t$, knowing $t$ itself doesn't tell you anything about the moments of $Y_t$. This is weak stationarity, strong stationarity is similar but is not limited to first and second order moments (mean, variance, covariance), but the whole distribution. You can relate this to the first answer, as $sin(t)$ would tell you something about the expected value of $Y_t$, so that component makes the series non-stationary.

  1. In that image both series have strong fluctuations in mean (upper series) or variance (lower series), you may say that the lower series is stationary in mean (that's what Whuber has being saying in the comments), but, as we have seen, this is not sufficient for even the weaker commonly accepted definition of stationarity, that requires also second moment consistency. It's hard to say if either of the two plotted series is stationary, because fluctuations are possible in theory, but must be brief in relation to the length of the whole series, in order to have a good degree of confidence that the series is indeed stationary. In the case of both series plotted in the image that you linked, the "fluctiations" end before the end of the series, and that hints for stationarity, but those could also not be fluctuations at all, but random wandering, they are to long to be considered just fluctuations.
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  • $\begingroup$ Thanks carlo for your answer (I upvoted it). I have follow up questions: to 1) So whenever we have regular cycles throughout the whole time series we can say that it is not stationary? And when we have regular cycles for a short time span, it can still be stationary (when there is no trend)? Is that correct? To 2) In the lower time series you can clearly see that the fluctuations (and thus the variance) between 0-40 is way lower than the ones between e.g. 60-100. Thus imho, per definition it can't be stationary as the statistical properties are definitely not constant over time. $\endgroup$ – PeterBe Oct 22 '20 at 9:15
  • $\begingroup$ Further: do you know a good website where examples of stationary and non-stationary time series are displayed? Maybe this would help me understanding the notion of stationarity. Of course I also did an Internet search but all of the websites I found (see the links in the question) seem to use different (seemingly wrong) concepts of stationarity which is a main reason for my confusion. $\endgroup$ – PeterBe Oct 22 '20 at 9:18
  • $\begingroup$ Sorry, I don't know such a site. Anyway, regular cycles are not such a great problem for analysis generally, the period must be regular though (the covariance structure is constant). The intensity of the "wave signal" is considered to be random and autocorrelated, but the fact that it never switches sign is left ignored. This typical example is about a non-stationary process very often trated as stationary: depending on what are your goals, this flaw in your methodology may or may not be a problem. $\endgroup$ – carlo Oct 22 '20 at 12:48
  • $\begingroup$ Thanks carlo for your answer and effort. You said that "regular cycles are not such a great problem". But if we have regular cycles the time series is not stationary, right? $\endgroup$ – PeterBe Oct 22 '20 at 13:04
  • $\begingroup$ And what about the 2nd additional quesiton "In the lower time series you can clearly see that the fluctuations (and thus the variance) between 0-40 is way lower than the ones between e.g. 60-100. Thus imho, per definition it can't be stationary as the statistical properties are definitely not constant over time. " From what I read about stationarity the time series can't be regarded as stationary $\endgroup$ – PeterBe Oct 22 '20 at 13:05

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