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in the answers given to this question Can stationary time series contain regulary cycles and periods with different fluctuations it is said that a stationary time series is not allowed to have regular cycles. According to my search a stationary time series has 3 properties:

  • Expactation value is constant for every t
  • Variance is constant for every t
  • Autocovariance is constant for every t

Which of those is violated by a time series with regular cycles? I'd appreciate every comment.

PeterBe
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  • I cannot see any meaningful way in which this hasn't been answered in your previous question. What distinction are you trying to make? – whuber Nov 17 '20 at 16:37
  • Thanks whuber for your answer. Basically with the answers given in the previous post I can't know which of those properties is violated. If you have a look at the green line here https://i.imgur.com/3lKCxEn.png which is wrongly labeled as stationary while in the previous post the others were saying that it is not stationary as it contains cycles. But I think it does not violate one of the three properties mentioned above: it has a constant expactation value, variance and autocovariance. – PeterBe Nov 17 '20 at 17:11
  • "Contains cycles" is a special form of **not** having a constant expectation. If you disagree, then please explain what you mean by "contains cycles." – whuber Nov 17 '20 at 17:12
  • Thanks whuber for your answer. Does the green line not have a constant expectation value? For me it has a constant expectation value (just take the mean). With cycles I am just taking about cycles in the data if you plot it. For example the green line has cycles (but also the red lines). So it is more or less something like a sin(t) function – PeterBe Nov 17 '20 at 17:23
  • When people are referring to an "expectation" in a time series, they are referring to the expected value of a *random variable* that describes the distribution of possible values *at one fixed time.* The expectation is not an average over time. – whuber Nov 17 '20 at 17:39
  • Thanks whuber for your answer and effort. Basically I do not understand what your are meaning. The green line for me definitely has a constant expectation value for every fixed time t if you see the individual values of each time t as a realization of a random variable. So if the green line is not regarded to have a constant expectation value, would you mind telling me how a time series with a constand expectation values looks like? – PeterBe Nov 18 '20 at 10:16
  • "Constant expectation" means the expected value at one time equals the expected value at another. Assuming the "green line" plots the expectation versus time, a time series with non-constant expectation will have a non-horizontal "green line." – whuber Nov 18 '20 at 13:43
  • Thanks whuber for your comment. I still do not understand what you are saying. We have a time series that has by defintion a different value for every time t. Behind the time series there is some process that generates this data. In the green line the expactation value is clear for me. The process creating the green line has an expactation value of the mean – PeterBe Nov 18 '20 at 15:25
  • You wrote ""Constant expectation" means the expected value at one time equals the expected value at another." -->How can you calculate an expactation value of a single time slot. There is only one realization of a value for that time slot. If I for instance have a time series that plots the sales of a company then the value for January 2020 is a one time realized value. How could (and why would) I calculate the expectation value of such a single time slot. For me this does not make sense. – PeterBe Nov 18 '20 at 15:29
  • You need a definition: see https://stats.stackexchange.com/questions/126791. Please do not confuse a *time series* with a *time series process.* – whuber Nov 18 '20 at 15:34
  • Thanks for your comment and effort whuber. But still I do not see how you could and would calculate the expectation value of a single event as you wrote with "expected value at one time ". For me this does not make sense at all. – PeterBe Nov 18 '20 at 16:15
  • I don't think we can meaningfully carry on this conversation until you demonstrate you have access to rigorous definitions of the concepts--especially of a time series--and begin referring to those definitions. Otherwise, it looks unlikely that we will be able to communicate. – whuber Nov 18 '20 at 16:20
  • Thanks whuber for your comment. Bascially I a lot about time series and I have knowledge about statistics. What confuses me is that you say that the green line with regular cycles does not have a constant expectation value. The values generated by the statistical process clearly have a constant expectation value (the mean). – PeterBe Nov 18 '20 at 16:24
  • But I have to admit that I have problems with the defintion "Expactation value is constant for every t". I do not know how one can calculate the expectation values for a single timeslot that only has one realization. In 'conventional' statistics this is meaningless and I still think that defining a expectation value for a single time slot (e.g. sales in January) either does not make sense or only makes sense if you calculate the expectation value of the process behind it. – PeterBe Nov 18 '20 at 16:27
  • Would you mind giving me a link to an example where the expectation value of a time series for different t are calculated with real values such that I can have a look at the formula and/or the approach? I'd highly appreciate it as I am quite confused about this one at the moment :-( – PeterBe Nov 18 '20 at 16:29
  • See https://stats.stackexchange.com/a/160733/919 for definitions and https://stats.stackexchange.com/a/411541/919 for an explicitly constructed process with explicit calculations (demonstrating a distinction between weak stationarity and stationarity). – whuber Nov 18 '20 at 17:13
  • Thanks whuber for your comments and help and for posting the links. When looking at the plot of the second link how did you calculate the expectation value of 3/2. As far as I understood you used the forumula of the probability distribution for calculating it or did you calculate the expectation value by only using the sampled time series? If so, how did you do this. Because this is exactly the problem that I am facing. I do not have a predefined probability distribtution for my time series. I only see the values for a certain time slot and do not have any additional information. – PeterBe Nov 19 '20 at 11:08
  • I used the formula for expectation. Once again, there's a crucial difference between a *time series* (the data) and a *time series process.* Given a time series, *there is nothing whatsoever you can do* to estimate the properties of the underlying process *until you adopt a model for it and make assumptions.* The underlying idea is to use that model and those assumptions to combine data at different times to deduce properties of the hypothesized process. The most basic assumption is [ergodicity](https://en.wikipedia.org/wiki/Ergodic_process). – whuber Nov 19 '20 at 15:28
  • Thanks whuber for your answer and effort. I really appreicate it. As you said, you need a model to calculate the expectation values (as you did with the formula). But when looking at a time series with regular cycles and not having any further information, how can you say that the expectation value for every t is not constant. I think you cannot say that without having more information. This brings me back to my basic quesiton. Why does a time series with regular cycles violate stationarity properties? – PeterBe Nov 19 '20 at 15:53
  • Because it is *extremely* unlikely (although not completely impossible) that any single realization of a stationary series would exhibit cycles. – whuber Nov 19 '20 at 15:57
  • Thanks whuber for your answer. Yes I got to know that a stationary time series does not contain cycles. But the question is: Why? And which of the properties of stationarity mentioned above is violated. We can't say anything about the expectation value of a time series when not having any further information. – PeterBe Nov 19 '20 at 16:26
  • Hi whuber. Can you tell me which property of stationarity is violated by a time series with regular cycles when not having more information about the time series itself except the values of the time series which makes it not possible to calculate the expectation value (as you also stated in one of the comments). I'd appreciate any further comments from you. – PeterBe Nov 20 '20 at 10:53
  • Unfortunately this question was closed because it is stated that on an older question of mine there is an answer to this question which is simply not true. There is no answer to the question why a time series with a regular cycle (and no other information or assumptions) violates the property of stationariy "Expactation value is constant for every t" and especially it is not shown how you can deduce that – PeterBe Nov 20 '20 at 11:29
  • The present question doesn't ask that. When you edit your question to make it clear and original, the community will be able to vote to reopen it. – whuber Nov 20 '20 at 13:58
  • Thanks whuber for your comment and effort. I really appreciate it. What about one of my last comments "Yes I got to know that a stationary time series does not contain cycles. But the question is: Why? And which of the properties of stationarity mentioned above is violated. We can't say anything about the expectation value of a time series when not having any further information. " --> Furhter would you advice me to ask a new question or to edit the existing one if I want to have answers from the community? But if maybe you could answer my question then there is no need for that – PeterBe Nov 20 '20 at 17:27

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