Talk:Poisson process

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I've taken quite a few liberties on my last set of edits so a justification would probably be in order. The major change I made was to treat spatial processes as an extension of Poisson processes instead of treating temporal Poisson processes as a special case of multi-dimensional Poisson processes. I believe this change is justified, as a Poisson process is after all a stochastic process which is tradiationally a random function of time. I believe multidimensional Poisson processes are worth mentioning, but perhaps would warrant their own article as there are characteristic differences about them that aren't done justice in this article. For example, the number of events in a given time interval is given as N(b)-N(a), but the formulation for the number of events in a (not necessarily rectangular) region of space based on the function N(x,y) is not nearly as straightforward. Also, the memorylessness property of multidimensional P.P. is not well motivated. (I omitted further discussion on spatial P.P. to keep from pushing this important concept of memorylessness even further down the article.) Until and unless this separation proposition reaches some consensus, I've left spatial P.P. as a subsection. (See also comments on the bomb analogy below.) Willem 22:23, 7 October 2006 (UTC)

I've also substantially shortened the intro, moving most of the material into the body of the article. I removed the discussion on memorylessness up through and including the bus analogy as it duplicates material covered in said article. Last, I noticed that this article treated certain processes (e.g. the stars in the sky example) as multi-dimensional Poisson processes that could be more simply characterized by an equivalent probability density function. Using a 1-D case as an example, it can be shown that sampling N events from an i.i.d. uniform distribution on an interval is equivalent to sampling a homogeneous Poisson process on that interval given that there are N points. A similar result holds for more general distributions by modifying the rate function. However these things may be better represented using a spatial probability density function instead of some cooked up rate function for a multi-dimensional Poisson process. Willem 22:23, 7 October 2006 (UTC)

This article seems to have got somewhat confused between Poisson Process and Poisson Distribution. The bombs in areas of London is certainly not a Poisson process (for a start, they were likely to fall at night more than in the day and if a bomb had just arrived there was a good chance of an air-raid being on so a good chance of another soon).

You misunderstand. It's a 2-dimensional spacial Poisson process; there is no time involved! Think of just ONE bombing raid: the number of bombs hitting a particular square mile has a Poisson distribution; the number hitting another, non-overlapping square mile also has a Poisson distribution, and they are independent of each other. The number of bombs falling during any specified time period has nothing at all to do with this. Michael Hardy 20:17, 14 Feb 2005 (UTC)

And now I've read it again, and it seems perfectly clear: it says "two-dimensional" and it talks about non-overlapping plots of ground, so I don't understand how you got so confused. The number of bombs falling during any particular time period was not one of the random variables in this process at all. Michael Hardy 20:20, 14 Feb 2005 (UTC)

I removed the bomb analogy. For the most part, it is a fine analogy, but I thought the rainfall example worked better. After all, the number of bombs is often set a priori, as is sometimes the intended bomb distribution. I also removed the stars analogy for the reasons discussed above relating the uniform distribution to the Poisson process. Willem 22:23, 7 October 2006 (UTC)

It is a classic (and well known) example of a Poisson distribution. I know that classical telephone models are Poisson Processes but the stars example I am not certain of --- surely this is a Poisson distribution again? Have I misunderstood or is there a couple of genuine errors here? If these are errors would anyone mind me rectifying this page?

I think it would also be worth adding the derivation of a Poisson Process as a pure Birth process. Any thoughts?

--Richard Clegg 19:36, 14 Feb 2005 (UTC)

If I may chime in here... It may help to think in terms not of the arrival times, but the interarrival times.

Imagine a spatial Poisson process in one dimension. The distances between the 'bombs' should be exponential, just like the interarrival times of the Poisson process. Instead of making a cumulative count of the number of 'bombs' up to distance x, we just mark down the points at which they dropped, but otherwise the characteristics of this spatial 'process' are just the same. What seems to be the trick here is separating the concept of a stochastic process from the procession in time of a random variable. Think of a stochastic process, instead, as simply a random variable indexed by a (non-random) variable, the latter potentially indicating time, but equally validly, space or some other dimension.

Now think about the pattern of bombings in two dimensions. If the distances between the bombs are described by two-dimensional (independent) exponential distributions (if I recall correctly), we have a two-dimensional Poisson 'process'—it doesn't involve time, but otherwise is Poisson. The number of bombs in a particular area is of course described by a Poisson distribution.

I don't agree that the text as it stands is "perfectly" clear, except perhaps to those with a decent amount of training in the area; no doubt even some of those would disagree. It seems to me that a little more detail would benefit the lay-reader, perhaps by referring back to the introductory comments about Poisson processes in general. Ben Cairns 22:48, 14 Feb 2005 (UTC).

What you are saying (I think) is that the number of bombs N(x,y) in some notional area defined by x,y and an aribtrary origin is a two dimensional Poisson process? I agree. You must admit that this intent is hard to guess from the text. Indeed the text positively argues against that interpretation by saying the number of bombs "on a specified area" which heavily implies to me a constant area. I wouldn't say I was untrained in this area, I just finished teaching a masters class about Poisson processes. I have come across the bombs example and the stars example before as examples of Poisson distributions. Obviously these can be summed to Poisson processes but it is in no way clear from the text that this was what was intended hence my thinking it was the classic confusion of Poisson distribution and process. The text given does nothing to clear up the distinction between a Poisson process and a Poisson distribution and, I would say obfuscates it. However, I don't want to be too pushy about this.

--Richard Clegg 00:45, 15 Feb 2005 (UTC)

I was referring to Michael's comment about it being "perfectly clear", which would only be so if it already made perfect sense to someone before they read it. I did not intend to imply anything about any particular person's training in the area, except that of a 'typical' reader of the Wikipedia. I think that the introductory text does give a definition of a Poisson process that is sufficient to support the bombs example, but it seems too formal for the likely audience. At least, a more intuitive definition could be added to the text for those who just want an idea of what's going on. However, Michael's point is, I think, that the reference to non-overlapping areas and Poisson distributions relates back to the introduction where this stuff is explained. Some of the lack of clarity may be due to this reference, which does no good if a reader is looking at the examples to try and understand the introduction (which, let's say, they largely skipped over). Ben Cairns 02:57, 15 Feb 2005 (UTC).

[edit] Proposed edits

OK. For what it's worth I think the introduction does little to clarify the situation. "The number of arrivals in each interval of time or region in space..." from the word arrival one strongly gets the impression of a fixed space and a change over time which is misleading. The introduction is technically correct but I think it manages to give the conditions for a Poisson Process without actually saying what a Poisson Process is. However, I'm very reluctant to make a change now since the opinion here seems to be against. How about this text very early in the introduction:

A one dimensional Poisson process is a stochastic process defined on some interval. The Poisson process can be thought of as defining the number of occurrences of some event within a subset of this interval. The process A(t) is a Poisson process if the probability of arrivals in some subset of the interval is given by

where k is the number of events occurring within [t, t + τ] and λ is a parameter known as the rate parameter. It can be seen that the probability defined is a Poisson distribution with parameter λ τ.

I would also like to slightly edit the examples to at the very least try define "This" in the sentence "This is a two-dimensional Poisson Process"... perhaps to "The number of bombs falling in some area A is Poisson process defined over the area A as it varies. (Though I'm not sure about the last three words, I want to clarify the idea that the area is varying.)

Looking further, this equation is already given on the page for Poisson distribution. Indeed I think the page for Poisson distribution is somewhat clearer to a novice about what a Poisson Process is than this page. However, I am new to wikipedia and reluctant to be too heavy handed. --Richard Clegg 10:21, 15 Feb 2005 (UTC)

I like your text above, although the second sentence is a bit vague and doesn't seem to clarify the distinction between Poisson processes and distributions much more effectively than what is already there. The rest is quite clear. I am coming around to thinking that the whole article could use some rearranging, such as the division of examples into 1-D and multi-D sections, the latter coming after most of the article. Find a place to fit your text in, delete and/or rearrange superceded content, and we'll take it from there. For your suggestion about changes to the bombs example, it seems to me that 'this' needs to be clarified in the rest of the article first, so that the example is clearer, rather than the other way around. Ben Cairns 12:16, 15 Feb 2005 (UTC).

I have made the edit but I am not wholly sure it is clear though I think it does improve clarity, particularly for people not too familiar with the subject. Further editing would be gratefully welcomed. --Richard Clegg 17:10, 15 Feb 2005 (UTC)

[edit] Clarity, examples

There are a few things that are not stated clear enough in the article.

This is what I understand, though I may be wrong in some aspects:

Poisson processes are stateless, that is, the occurrence of past events at particular moments in time has no effect on the chances of new events occurring.

Events are distributed in an uniform manner (assuming constant λ, this should also be explained better in the beginning), but without enforcing uniformity. The uniformity is just a result of randomness. There must be no rule linking different occurrences.

I do not believe stars are distributed by a Poisson law, as they tend to form clusters. This is because they interact with each other by gravitational attraction, which has an effect on the way they are distributed.

For the falling bombs to be distributed according to a Poisson process, they would have to be fired independently. If they were dropped in an air raid, they would fall more or less along a straight line. Now, if the trajectory of the plane and the point of drop are known facts you can still consider this a Poisson process in which the random variable is the deviation caused by the aerodynamics of the bombs and wind, but when the plane trajectory is a random variable, this is no longer a Poisson process.

The rest of the examples I think are OK, but a common characteristic should be pointed out. In general you obtain the Poisson distribution/Poisson process when the events are produced by many independent sources.

I would really like feedback on this, at least to find out if I'm right.

Radu124 11:19, 26 September 2006 (UTC)

==There is an innate lack of knowledge of astronomy in the above, as well as a lack of knowledge of Poisson processes. As for the visible starts in the night sky, with very few exceptions, the appearance of stars clost to one another has NOTHING to do with physical clustering of stars in space. When two stars appear to be close to each other in the sky, almost always they are at drasticly-different radial distances from the Earth and from each other. Hence, their "closeness" is merely an illusion. Let me repeat myself: it had nothing to do with gravitational attractions, physical clustering, etc.

The only known exceptions to this are the Pleiades, which really are a star cluster in our Galaxy, and the two Magellanic Clouds, which are close to our Galaxy, and are also only visible in the Southern Hemisphere, anyway.

Also, when it comes to one-dimensional Poisson processes, a feature of their very nature is that "clusters" of events occur. This follows from the definitions of the Poisson process, and namely from the independence of non-overlapping time intervals. Your mind may rebel against this notion, but this is the way that it is.

Likewise, in two-dimensional (spatial) Poisson processes, the number of events occurring in any non-overlapping areas is independent, and from the very nature, clusters occur. These clusters aren't made by any cause: they just happen, and looking for reasons is a useless exercise.

First of all, although I do not agree with you, thank you for the feedback.

I do not know why you made the assumption that my observation was based on the visible starts in the night sky. I believe that we both should be past that level of education by now. My comment was based mostly on common sense rather than knowledge in astronomy, and my reasoning was that, since the stars interact through gravity, the independence assumption is not met. Of course you don't have to believe me, but you could look for a paper called "Cosmological Distances and Fractal Statistics of Galaxy Distribution". Although not specifically about stars it could prove my point.

Also I didn't do the calculations to prove it, I believe you are also wrong about Poisson Processes. Although clustering may occur on the small scale, averaging over large intervals should decrease the variance. Think about noise reduction in digital photography for example.

By the way, innate means "existing in one from birth; inborn". You didn't expect me to be born with a knowledge in astronomy, did you?

Radu124 13:05, 14 December 2006 (UTC)

sorry, by mistake I have removed something:

[edit] Mistake in one of the examples

Original text: A common assumption in the study of simple queueing systems is that the times-of-arrival of "customers" is a Poisson process, and that the "service time" for each "customer" is exponentially-distributed. This sees application in a wide range of queueing systems, including computer communication networks.

As far as I know "service times" are not exponentially distributed as the exponential distribution is generally inappropriate for modelling process delay times (you could use Erlang or Weibull for this). It's the inter-arrival time that is exponentially distributed (inter-event time of random arrivals).

Is this correct?

Posiebers 04:07, 08 November 2006 (UTC)

I think it often makes sense to model interarrival times as exponentially distributed. It's far less plausible for service times. In some cases it may be true of service times. That wouldn't necessarily be considered a reason not to use exponentially distributed service times in a mere math exercise to assign to students. Michael Hardy 17:13, 14 December 2006 (UTC)