Talk:Stochastic process

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Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 10:14, 17 January 2022 (UTC)

Reading Level
Simple as it may be, this article, or at least the introduction, are filled with jargon and are not written at an acceptable reading level. I recommended that those who are knowledgeable on the subject matter revise the introduction and make it more accessible to the public. As is, this is unreadable unless the reader has an extensive learning history with statistics. — Preceding unsigned comment added by 173.162.169.233 (talk) 23:13, 6 August 2013 (UTC)
 * Hopefully, the writing for the specialists can be maintained, and writing for beginners added right after. — Charles Edwin Shipp (talk) 14:41, 30 September 2013 (UTC)
 * No, hopefully the writing for beginners can be enhanced. Specialist already know the material-- you don't write an encyclopedia article for specialists, you write it for beginners. Put technical details at the end.  Geoffrey.landis (talk) 13:45, 8 July 2016 (UTC)
 * Agreed. AND Wikipedia has a specific and agreed mandate to write at the level of beginners, or, at least, a high school junior or senior.  THAT is concensus on this project, and has been for almost two decades.  Mathematics and statistics/probability are not the exception to that rule, nor should they be.  I am not unschooled in mathematics nor in statistics, and I find this jargony lede circular and unintelligible.  We can do better.  rags (talk) 16:43, 7 November 2018 (UTC)

Article completely re-written
I have completely re-written the article from scratch. I tried to capture all the key parts from the previous version of the article. Apologies if I have left out key contributions. Apologies also if I have made any mistakes or broken any rules or guidlines (I'm relatively new at wikipedia). I will now detail reasons for the re-write and possible issues of the new version. Improbable keeler (talk) 13:25, 4 January 2017 (UTC)

Mathematical object versus mathematical model
Although 95+% of people who are interested in stochastic processes are probably interested in models of something, a stochastic process does not, strictly speaking, need to be a model of anything. So I changed the opening sentence used in the previous article version. Improbable keeler (talk) 13:25, 4 January 2017 (UTC)

Traditional mathematical definition
I have used the traditional mathematical definition of a stochastic process (ie an index collection of random variables). (I say "traditional" because two separate books wrote that and many other books use this mathematical definition). I have then given the traditional interpretation of this (ie a random system evolving over time). I later write a stochastic process can also be interpreted as a random function. Improbable keeler (talk) 13:25, 4 January 2017 (UTC)

Words versus mathematics
To describe individual stochastic processes, I think using more words than mathematics is a better approach in such a general article. Admittedly, we then don't know the precise mathematical definition of each process, but then the person can click on the main article of each process covered.Improbable keeler (talk) 13:25, 4 January 2017 (UTC)

Abundance of citations
I have tried to be very generous in citing. (Perhaps too much in some cases). I did this partly to strengthen claims like "commonly used", but ultimately I am trying to make it easier for contributors to use the citations to improve other related articles. It also makes it easier for everybody to find a book or resource if there are more options.Improbable keeler (talk) 13:25, 2 January 2017 (UTC)

For books, I have tried to always cite the page number, unless referring to a subject covered by the whole book. I have also done the same, mostly, for longer articles, with the exception of a couple of articles, such as the historical pieces/memoires by Harald Cramér (1976) and Paul-André Meyer (translated in 2009) as well as some of the articles used in the history section of statistical physics.Improbable keeler (talk) 13:25, 2 January 2017 (UTC)

Diagrams
The article needs more diagrams. I think there are some good diagrams on other stochastic process articles (eg Wiener process), which can be re-used.Improbable keeler (talk) 13:25, 4 January 2017 (UTC)

Links
The article needs more links to other Wikipedia articles.Improbable keeler (talk) 13:25, 4 January 2017 (UTC)

Mathematical defintions need checking
I think the defintions are all correct, but a second (or third, fourth etc) opinion would be nice, particularly on finite-dimensional probability distributions and separability. Improbable keeler (talk) 13:25, 4 January 2017 (UTC)

Applications
I didn't include many applications. Each major stochastic process (at least all the ones covered) have an abundance of applications, so I thought it would not be difficult for contributors to find applications. Also, perhaps, detailed examples of applications belong on the articles of the individual processes.Improbable keeler (talk) 13:25, 4 January 2017 (UTC)


 * Hi, I think this article as it stands is quite outstanding for wiki standard. While I did not find the lack of applications bothersome, I do wonder if mentioning some of the typical problems/questions that are investigated in the study of stochastic process would be worthwhile. Things like first passage time (e.g. in gambler's ruin problems, diffusion process), etc. Manoguru (talk) 22:44, 10 April 2017 (UTC)


 * I sort of tried to do that with the classic processes, Brownian/Wiener and Poisson, giving their original applications in the history section. But I suppose one can cite other typical examples earlier on, where the articles are first described in detail.Improbable keeler (talk) 09:03, 16 March 2018 (UTC)

Abundance of history
The article has a large history section. If that's an issue, I suggest moving sections to the articles on individual articles. Improbable keeler (talk) 13:25, 4 January 2017 (UTC)

Construction towards the end
The article previous version had a signicant section (perhaps a third) dedicated to the construction (and its issues) of stochastic processes. Although important for some mathematicians, I think most readers will not be interested in this, so I have put it at the end. Improbable keeler (talk) 13:25, 4 January 2017 (UTC)

Definitions
I tried to put the important definitions towards the start of the article and the less important ones at the end. I have tried to make the order of the defintions flow logically (eg we need to separability before we can talk about how to resolve construction issues).

Stochastic processes versus random fields
The previous article version had the index set as being one-dimensional (or some totally-ordered set T), so the interpretation was that stochastic processes evolve over time, which meant if we replace the index set with the Cartesian plane (or some manifold), then the resulting collection of random variables is known as a random field, which is then a generalization of a stochastic/random process. This is a popular convention. But many books also keep the index set for a stochastic processes very general, which means that a random field is just a specific case of a stochastic/random process. I have tried to include both viewpoints or conventions. I think the article on random fields should be kept and then extended and improved, using some of the references cited in the stochastic process article (eg the books by Alder). Improbable keeler (talk) 13:25, 4 January 2017 (UTC)

Stochastic processes versus point processes
Point processes have a couple of interpretations, with literature (for many years now) favouring the random measure one. Some, perhaps most, authors consider points processes and stochastic/random processes as different objects, while acknowledging that some stochastic processes (eg ones with discrete state spaces) can be re-considered by looking at their corresponding point processes. Some consider point process as types of stochastic processes.

For example, to some people, a Poisson (point) process is something that evolves over time (ie it is defined on the real line), while for others it's something that can be defined on the real line or much more general spaces. In the Lévy process literature, often people say "Poisson process" to refer to the random object on the real line, while reserving "Poisson random measure" for the Poisson point process on the higher dimensions (usually the Cartesian product between the half-line and the space that the state space of the Lévy process).

I have tried to inclde these different viewpoints or conventions. Improbable keeler (talk) 13:25, 4 January 2017 (UTC)

Wiener process versus Brownian motion (process)
I am not sure which term is better. Part of me thinks Wiener process is more appropriate for the mathematical object. But then many people, perhaps most, use simply "Brownian motion" when referring to the stochastic process.Improbable keeler (talk) 13:25, 4 January 2017 (UTC)

End of my discussion
Thanks for reading. Improbable keeler (talk) 15:25, 4 January 2017 (UTC)


 * Thank you for the excellent work!
 * A small remark on the phrase "Historically, the random variables were indexed by some set of increasing numbers" (in the lead): I feel what you mean, but still, "increasing number" sounds bad. Every set of real numbers is ordered naturally, which is not a reason to call it "increasing". Boris Tsirelson (talk) 19:21, 4 January 2017 (UTC)

Many thanks. I wrote the article a few months ago, thinking a re-write was the best approach, but I just didn't have the time to do the citations until now. As "increasing numbers", sure. It's a fair point. It actually took some effort to write that introduction. I wanted it to be understandable to normal people. But I guess most people will assume the numbers are increasing. Improbable keeler (talk) 20:03, 4 January 2017 (UTC)

Martingale in continuous time - incorrect
"A martingale is a discrete-time or continuous-time stochastic process with the property that, given all the previous values of the process, the expectation of the next value of a martingale is equal to the current value." – For discrete time it is OK, but for continuous time it is not. What could be meant by "next value" and "current value" when the time is continuous? If these are $$X_s$$ and $$X_t$$ where $$s<t,$$ then t is not "next" to s. And if s=t, then again, t is not "next" to s. We should either formulate it for discrete time only (and send the reader to the main article for more), or give a correct formulation for continuous time. If we prefer a simple but incorrect formulation for accessibility (which I doubt we should) then AT LEAST we should warn the reader by such words as "roughly" etc. Boris Tsirelson (talk) 10:17, 14 September 2018 (UTC)

I propose: "A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values." Boris Tsirelson (talk) 19:51, 15 September 2018 (UTC)

No objections, no remarks? Well, now implemented. Boris Tsirelson (talk) 15:41, 17 September 2018 (UTC)


 * Sounds good. No objections. Thanks. Improbable keeler (talk) 06:38, 20 September 2018 (UTC)

Poorly written
The article is poorly written and not very accessible to the layman. I did what I could to improve the opening paragraph, but it could use more work — Preceding unsigned comment added by Unified field (talk • contribs) 23:46, 16 October 2020 (UTC) https://stats.stackexchange.com/questions/126791/is-a-time-series-the-same-as-a-stochastic-process — Preceding unsigned comment added by Unified field (talk • contribs) 23:40, 16 October 2020 (UTC)