Talk:Stochastic process

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Stochasticity is from archery[edit]

The article should mention that no matter how fancy the statistics get, it's all present in the way arrows group themselves around their intended target, which was a pointed stick stuck in a Greek hillside --- a stochos. 69.105.38.7 (talk) 06:14, 22 January 2013 (UTC)

This etymology is already mentioned in the Stochastic article, which is where I think it belongs. Statisfactions (talk) 16:18, 23 January 2013 (UTC)

Random fields vs. Stochastic processes[edit]

This article says that a "basic type of a stochastic process is a random field". But the random field article states that it is a generalization of a stochastic process. Which is which? Statisfactions (talk) 16:33, 23 January 2013 (UTC)

"C*-algebras of random variables"[edit]

There is an opaque mentioning of "C*-algebras of random variables", and the GNS construction in the construction section. Maybe it should say von Neumann algebras to be more specific? Could someone knowledgeable elaborate on that in the article? Presumably this refers to the C*-approach to conditional expectations (although I don't know what GNS has to do with it). Mct mht (talk) 11:04, 22 March 2013 (UTC)

Moreover, all this should go to some article about noncommutative probability, or at least to some article about different approaches to probability theory as whole. Why is it mentioned in the article on stochastic processes? Boris Tsirelson (talk) 13:31, 22 March 2013 (UTC)

Historical information?[edit]

It would be nice to have a note on when the concept first appeared in the mathematical literature, and by whom. RMPK (talk) 22:35, 10 April 2013 (UTC)

random variables are not natural numbers ?[edit]

The article says:"If both t and X_t belong to N, the set of natural numbers, ..." Is this correct ? How can the random variable X_t be a natural number ? — Preceding unsigned comment added by 213.54.36.209 (talk) 10:22, 14 June 2013 (UTC)

It is non-rigorous but quite usual, to say such things as "almost surely, X is a number between a and b", "almost surely, X is an integer" etc., when X is a random variable. Even more non-rigorous and still usual (especially among non-mathematicians) is, to omit the "almost surely". Boris Tsirelson (talk) 10:51, 14 June 2013 (UTC)

Reading Level[edit]

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)

Heterogeneous process explanation[edit]

The Heterogeneous process page routes to this one, but then is not included in the explanation. This would be good to include for those rewriting this. Thanks. Alrich44 (talk) 15:10, 30 July 2014 (UTC)

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The Introduction is flawed in a number of ways.[edit]

1. Wiki recommend starting with a simple explanation, and gradually adding more complexity; the present Introduction is too complex. Casual readers will be content to read the Introduction, and skim through the rest of the article, but interested students, and we are all students, will move on and read and re-read the article, in conjunction with others.

2. There are factual errors in the Introduction, which need to be rectified. This is important, because the concept under discussion is fundamental, not just to study of probability and statistics, but to engineering as well.

I propose to rewrite the Introduction, but I do not want to do so, without getting agreement from those who feel that they have a stake in maintaining this page; can anyone with such a stake make themselves known? - we can then work jointly on revising the Introduction, to achieve consensus. Systems Analyst 2 (talk) 03:07, 21 September 2016 (UTC)

As for me, it does start with a simple explanation. Do you think otherwise? And what are the factual errors? Boris Tsirelson (talk) 05:15, 21 September 2016 (UTC)
Firstly, I propose that we address four levels of reader: 1. casual reader 2. new student 3. intermediate student 4. advanced student. In my view, the Introduction is confusing to a casual reader. I further propose that we address this by using the following model: 1. Provide an example. 2. Formalise the approach. 3. Give examples of use. This can be done for each level of reader. Secondly, there are factual errors: 1. As the term Stochastic Process is used in a generic sense, it is not true to say that it is a time sequence; think of casting dice, for example. The four classifications need to be covered in the Introduction in an orderly way. 2. The Introduction does not clarify the distinction between a Stochastic system and a Deterministic system; they have a degree of overlap; for example, a specific sample from a stochastic process could be indistinguishable from the output of Finite State Machine, or the equivalent Turing Machine, both being purely deterministic; in fact, there is a non-denumerable set of such samples, and their equivalent FSMs. This is of practical significance in cryptography, where keys or one-time-pads need to be 'strong'. I can provide more detail, if this approach meets with agreement.Systems Analyst 2 (talk) 06:03, 22 September 2016 (UTC)
On the first: your "1. Provide an example. 2. Formalise the approach. 3. Give examples of use." is good, but should all that be done in the lead? Isn't it too long and detailed for the lead (especially, "3")? Well, try to do so, and we'll see.
On the second: I am puzzling, why "casting dice" is not a time sequence? Isn't it a sequence of i.i.d random variables? Also, the problem of pseudo-randomness versus true randomness is good, but again, why already in the lead? Doesn't it overload the lead? Boris Tsirelson (talk) 10:06, 22 September 2016 (UTC)
Now I note that you speak about "introduction", not "lead". Many articles contain a lead, then introduction section, then other sections. Do you mean this architecture? Boris Tsirelson (talk) 10:09, 22 September 2016 (UTC)
I will try a first draft Introduction, over the next week, and see how long it is; as you say, there could be a Lead, which could be a short summary of the Introduction.
As for the dice; you could cast six fair dice simultaneously, and that would be one sample from a Stochastic Process; there does not have to be an independent variable (time). This is an example of a discrete/discrete case.
Downloading a share-price history is also a sample from an S.P. for that share; it is not a Random Variable, though it is a time history of course; both are simple examples that are intelligible to a general reader, I would think. I tend to find that the formalism is off-putting at first; often it seems too abstract, and even obtuse, and an example breaks the ground.
I checked for other Maths articles of level-B quality and Top Importance, and found the following:
Probability Theory refers to Stochastic Process, hence we need to fit in with that. (I noticed a 'howler' in there.)
Probability Space has an intersection with Probability Theory, but does not refer to S.P.
I am not sure why those two articles are not combined, or why there is not a common 'Head' Article for the two.
Below S.P. in the development of ideas lie: Bayes Theorem, Probability Density Function, Probability Distribution and Random Variable, all level-B & Top Importance.
Markov Chain sits to one side of S.P. Again, I am not sure why it is not combined with S.P.
We need to fit in with this context, I think, as per Wiki's recommendation. — Preceding unsigned comment added by Systems Analyst 2 (talkcontribs) 03:37, 24 September 2016 (UTC)

The concept of the Stochastic Process is based on axiomatic Probability Theory [Maybeck, chapter 3]. There are four classifications of Stochastic Process, summarised here [Jazwynski]:

1. Discrete time and discrete state-space, known as Markov Chains.

Examples include repeatedly selecting a card from a full, well shuffled pack, tossing a coin, or repeatedly tossing a die [Maybeck, chapter 3]; the result is a sequence of heads or tails for a fair coin, or a sequence of integers, each from one to six, for a fair die, for example. Any such experiment, giving a specific sequence of fixed length, generates a sample ω from a sample space Ω, of probability P(ω), where P is the Probability Function; it is the ensemble of such samples that constitutes the Stochastic Process [Maybeck, chapter 3,4]. It is important to realise that the intuitive concept of what is 'random' can be misleading here; for example, the coin-tossing experiment could yield a sample which is all 'heads'; this does not seem random, in itself. Any other recurring sequence of head or tails could occur with equal probability, or the sequence could be non-recurring; this is analogous to the occurence of rational numbers in the real domain. It is just that there are many more non-recurring sequences in a long but finite sequence, and hence the non-recurring sequences outnumber the recurring sequences, but all individual sequences are equally likely, given a fair coin or a fair die. There is no requirement for the points of the sample space to be numbers, or sets of numbers; each could be made up of letters or arbitrary symbols, or sets of symbols, depending on how the card, die or coin is marked. Applications include a Roulette Wheel, a Pack of Cards or a Slot Machine; other areas of application are Simulation and Cryptography. It is worth noting here that there is an overlap between the behaviour of Deterministic Systems, and Stochastic Systems; for example, Pseudo-random Number Generators [Knuth, 1981] are deterministic Finite State Machines [Knuth], or Turing machines [Turing], that can produce finite runs of numbers, of more or less 'random' appearance, but in parts of the sequence the numbers do not look 'random'; such sequences are used in running numerical simulations of systems [Marse and Roberts], where noise needs to be simulated. The 'randomness of such numbers is considered in [Hull and Dobell], in an earlier survey of the literature. In Cryptography [Welsh], the intention is to convert a meaningful message into an apparently random sequence, a sample from a stochastic process, using an algorithm; there is also an algorithm to reverse the encryption. One method is to use a one-time pad, but this relies on the 'randomness' of the pad, which has to be generated in some way. In other words, deterministic systems in the form of random-number generators, are used to mimic samples from a Stochastic Process; thus stochastic systems and deterministic systems are not entirely separate from each other, in terms of the output samples generated. 2. Continuous time and continuous state-space. Here, for example, time is specified by a real number, and the values of the state variable are also real numbers. An alternative example could involve distance, instead of time. In this case, there is no discrete sequence of samples being taken from the sample space; instead, there is a continuum of values, taken continuously in time, say. Here, time is increasing monotonically, and the state variable is varying, in general. Denoting the ensemble of samples taken from the sample space by X(\[FilledSmallCircle],\[FilledSmallCircle]), where the first argument denotes time, and the second denotes the sample taken, then X(t,\[FilledSmallCircle]) is the ensemble of samples taken at the specific time t, and X(\[FilledSmallCircle],\[Omega]) is the specific sample \[Omega], taken over time. The function X(t,\[FilledSmallCircle]) is referred to as a Random Variable, at the time t, with a mean value and variance at time t. X(\[FilledSmallCircle],\[Omega]) will look like a specific time plot, the properties of which can be specified; for example, it has a mean-value function, which can itself vary with time, and it has a variance, also varying with time, in general. The stochastic process is the ensemble of X(\[FilledSmallCircle],\[Omega]). The mean-value function is the mean of the ensemble, and the variance is the variance of the ensemble. It should be noted that a specific sample X(\[FilledSmallCircle],\[Omega]) could be a 'random constant', a sinusoid of any frequency, a square-wave or any regular cyclic function, but it is far more likely to be what conforms to the intuitive notion of a random quantity, varying in time. However, there is no requirement for X(\[FilledSmallCircle],\[Omega]) to be continuous, as time varies; it is only continuous if X is correlated in time. Brownian motion is the classic example of the stochastic process X(\[FilledSmallCircle],\[FilledSmallCircle]). The scalar X can be generalised to the vector X. The scalar mean is then replaced by a mean-vector, and the scalar variance is replaced by the covariance matrix. If the vector stochastic process is correlated in time, then the scalar correlation function is replaced by the auto-cross correlation matrix, and the covariance matrix generalises to the auto-cross covariance matrix. Applications could involve modelling the noise from an analogue electronic sensor, or circuit. If the noise from a sensor is recorded and plotted, this would be a sample from the stochastic process involved. Re-running the experiment repeatedly would give an ensemble of samples from the stochastic process, and its statistics could be investigated. Another application could involve modelling the tilts of the local gravity vector, relative to the earth's reference ellipsoid; in this example, the tilts will be correlated in distance, as a vehicle moves over the ellipsoid, which can affect the accuracy of an inertial navigation system. Itô stochastic calculus is based on this notion of a stochastic process, and enables the modelling of stochastic systems of diverse forms, using stochastic differential equations.The differential equations can combine systematic 'signal', due to the deterministic nature of the differential equations, with added 'noise'. The histories of state variables of such a system are samples from the stochastic processes; thus a stochastic process can include a deterministic contribution, along with random noise. In addition, if the system model includes 'random' constants, these can integrate up to a ramp in a particular sample time history X(\[FilledSmallCircle],\[Omega]), but the gradient of the ramp will vary from sample to sample. Feed-back loops can also generate sinusoids, and these too can vary from sample to sample. These examples indicate that a stochastic process can model a wide range of dynamic variables, hence the power of the concept. 3. Discrete time and continuous state-space. A sensor may measure a quantity at a fixed sampling rate; this can involve measurement noise, due to the sensor itself, but also process noise, due to the system being monitored. The sampling process yields discrete-time data. Many engineering systems are implemented in this way. They can be modelled as stochastic difference quations. This has led to the development of the Kalman Filter, which is used in diverse disciplines. If a satellite's position is measured in orbit, then the measurements involve noise, but in place of time, the noise arises at different positions of the satellite measured. An equity or share price is available in discrete time; a history of a share price is a sample from a discrete-time stochastic process; it is not the stochastic process irself. Investors may use price-data to estimate the mean of a share-price, and look for deviations to trade on; there is an assumption that there will be no rapid change in the mean, but this does occur occasionally, and is usually unpredictable. To reduce risk, portfolios of shares are employed, usually in different sectors of the market; the linear combination of share holdings at a particular time averages noise across the portfolio. There is often dynamics involved in a share price; for example it may ramp up, oscillate, or even exponentiate for a time. The price is still a sample from a stochastic process; a Kalman filter can be used to model linear dynamics in the presence of noise. Systems Analyst 2 (talk) 21:35, 1 October 2016 (UTC)

Wow... Interesting. And problematic.
Interesting, since it could be a helpful introductory/explanatory section in a textbook.
Problematic, since Wikipedia is not a textbook.
Likewise, you could rewrite the article "function" like that: "One classification (or do you mean, class?) of functions, x=f(t), are functions from time to space; these describe a motion of a point (in particular, the barycenter of a material body). Another class..."
However, given a function, say you cannot say, whether its argument is time, or money, or something else. Thus, your-style classification is not a classification of functions (as mathematical objects), but rather a classification of their real-world applications.
Yes, I am a mathematician and you are not, and accordingly, I treat mathematical objects and you treat their real-world applications. This itself is OK. However, your thinking (cogitation) may appear in Wikipedia only if it can be sourced. Have you a "reliable source" (preferably textbook) that conforms to your text? If you have, please provide. Otherwise, try to publish your cogitation elsewhere. Maybe, on Wikiversity? Boris Tsirelson (talk) 05:37, 2 October 2016 (UTC)
A good reference is "Stochastic Models, Estimation and Control, Volume 1" by Peter S. Maybeck, Academic Press 1979. What I wrote is a bit long for an Intro, but it could be used in the section "Classification" of the main article; a precis might serve as an Introduction. I agree that the mathematical viewpoint and the systems analysis viewpoint are different, as in s.a. we use less rigour, but stochastic processes are very important in engineering and financial engineering, hence we need to embrace both viewpoints in the article, I feel. I could use the above to revise the Classification section, and provide a draft here in Talk. Systems Analyst 2 (talk) 20:26, 10 October 2016 (UTC)
OK, I'll look at this book. Boris Tsirelson (talk) 21:18, 10 October 2016 (UTC)
Looking at Sect. 4 of the book by Maybeck, I do not see any essential deviation of his terminology from the terminology used by mathematicians. In particular, "stochastic process" is still a mathematical object (rather than a real-world application of the mathematical object). I do not see coin tossing there, but still, a finite family of independent random variables, each with two equiprobable values (0 and 1), is a mathematical object (often called Bernoulli process); its real-world applications include both repeated toss of a single coin, and a one-time toss of a finite collection of coins.
Many other thoughts contained in your text above also are not contained in that book, as far as I see for now. Boris Tsirelson (talk) 06:00, 11 October 2016 (UTC)

Applications Section[edit]

I think the entirety of the article is well-cited and gives a good picture of what stochastic processes are. My one concern after reading the article is that I don't know how they are used. I'd like to propose an applications section, with a couple of examples from different fields. For example, we could have a few sentences about financial applications, a few about weather prediction models, and a few more sentences about some scientific (biology or chemistry?) application. Anyone who has worked with an application should feel free to contribute to that. — Preceding unsigned comment added by Chaley17 (talkcontribs) 17:41, 11 October 2016 (UTC)