Talk:Poisson point process

Confusion about definition
I find the first sentence very confusing, as a non-statistician (but accomplished computer scientist and writer).

In probability, statistics and related fields, a Poisson point process or Poisson process (also called a Poisson random measure, Poisson random point field or Poisson point field) is a type of random mathematical object that consists of points randomly located on a mathematical space.

Shortened grammatically, this says

a...Poisson process is

Definition coming...

a type of random mathematical object

OK, a process can be an object; how is this one conceived?

that consists of points randomly located on a mathematical space.

Or, even shorter:

a....process...consists of points.


 * Thanks for the suggestions/comments (also, perhaps sign your comments). I just made a separate sentence for the terms. I hope that makes it easier grammatically Improbable keeler (talk) 10:07, 11 September 2018 (UTC)

A "process" necessarily denotes "time", or at least transitions from state to state. (A set of?) "randomly located points" denotes some state of the mathematical space - the state with those points in it. A state in this case has no relation to time - it's just a static thing (unless for example one were to differentiate the points by when they were placed). A set of points by itself isn't a "process".

The confusion deepens in the "also called" clause, because all of the examples are things, not processes that create those things ("Poisson random measure, Poisson random point field or Poisson point field"). Again, how is a "process" "also called" a "field"? Is it a common mathematical shorthand to conflate processes with the objects those processes create? If so, let's not do it in a definitional article. (And, what's the relation to "measure" i.e. how is a collection of points a "measure"? I understand the term "Poisson random measure" may be widely used, but here the word measure is highly conceptually diverting without explanation or crossref.)

This confusion continues further into the article:

"In the plane, the point process, also known as a spatial Poisson process,[13] can represent the locations of scattered objects such as  transmitters in a wireless network,[10][14][15][16] particles  colliding into a detector, or trees in a forest.[17]"

How does a process "represent" things (e.g., "locations" above)? A Poisson process might be a process that created that those things (chose random locations to place trees). But to me, again, this article contains confusions between the process, and the characteristics and uses of the object that results. (E.g. a map of where trees are, that happens to have certain properties ("Poisson properties?") that somehow result from it's being randomly distributed. If random distribution of points in a mathematical space is the only characteristic then what differentiates "Poisson" from "random"?)

Maybe something like one of these is what is meant (again I am NOT an expert on Poisson distributions or processes):

"A Poisson point process or Poisson process is a mathematical object representing a process that creates points randomly located on a  mathematical space. Such a collection of points is also called a  Poisson random measure (because...), a Poisson random point field or  a Poisson point field."

"A Poisson point process or Poisson process is a mathematical object representing the activity of a process that creates points randomly  located on a mathematical space." Such a collection of points is also called a Poisson random measure (because...), a Poisson random point field or a Poisson point field."

"....This randomly placed set of points has the following characteristics...." "A process that places points randomly in a mathematical space is called a Poisson Process. Such a collection of points is also called a  Poisson random measure (because...), a Poisson random point field or  a Poisson point field. These randomly-placed points form a trace of  the random process. In respect to the particular mathematical space,  they lie in a distribution called a Poisson distribution. The  characteristics of a Poisson distribution are X,Y,Z."


 * Yes. This is an old (and much debated) issue. The original use of the term "point process" (by Palm) referred to something in time (ie phone call arrivals) ie how they were located in time. The term was then kept for other mathematical spaces. A random point field is a better term. I agree it does not make sense, but point process is definitely the most used term. I tried to explain this in the terminology subsection Improbable keeler (talk) 10:07, 11 September 2018 (UTC)

My point is that I think 4 concepts should be differentiated in sequence: (1) a process, (2) points that result from the process, (3) the distribution of those points defining what is means to be "Poisson", and (4) the characteristics of the Poisson distribution.

The rhetorical aim is to show how a Poisson process (if my inferences are correct) leads to a distribution with Poisson characteristics.

I hope this helps describe my confusion and a possible fix for it.


 * Points 1) and 2) touches on something that is rarely ever discussed or written about: in the 1-D the term "Poisson process" is used for two different but highy related objects. The first (and most common) is the Poisson (stochastic) process, which describes the (discrete) random number of objects at some place in time. The second is the the Poisson (point) process, which describes the points in time when the (discrete) random number of objects increases (usually by one). Of course, knowing one gives the other (that is to say, this is two different ways at looking at/describing the same random object). But this makes only sense in 1-D, I think, where stochastic processes live. Now I know no paper/book that talks about this terminology confusion for the Poisson process, but two books discuss how point processes are associated with/related to/interpreted as discrete-valued stochastic processes. See this section of the article on stochastic processes. Improbable keeler (talk) 10:07, 11 September 2018 (UTC)


 * I should also add that this confusion in terminology/interpretation has already been mentioned by me in separate sections on this talk page (Sections: Splitting the article and Different interpretations of point processes) -- I don't know how to hyperlink those sections. My suggestion was to separate the single article (back) into two articles. Improbable keeler (talk) 10:14, 11 September 2018 (UTC)

Shortening the Article
Thank you very much for the article. I was wondering whether the article could be shortened in some areas, specifically in the introduction and the definition, so that people who just want to quickly look up what a poisson process is can have a glance and understand the gist of it. I thought the german version is very good in that respect: https://de.wikipedia.org/wiki/Poisson-Prozess.

Splitting the article
I was chiefly responsible for writing the very long article on the Poisson point process and merging it with the Poisson process article. I did this because I didn't want people to think that the Poisson process as just something that lives on the real line, as it seems in many probability books for engineers. The German version looks like that, and it has more maths than words, so seems like a quick guide where to look for certain key results or properties, but not an encyclopedia article. The German version is quite technical too, introducing cadlag functions etc.

I suppose the issue is that people consider a "Poison process" as a stochastic process on the real line and a "Poisson point process" as a point process on the plane (or higher dimensions). But I never seen a reference saying precisely that.

All that said, perhaps it would be good to create two articles again. All the 1-D stuff from this article can go into the Poisson process article. But then we write one is a natural generalization of the other. And often people use the term Poisson process to refer to a Poisson point process defined on abstract spaces. We also say "this article refers to the Poisson point process defined on the real line". Improbable keeler (talk) 15:42, 6 January 2017 (UTC)

Different interpretations of point processes
There are different mathematical interpretations of a point process, such a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear. Others consider a point process as a stochastic process, where the process is indexed by sets of the underlying space on which it is defined, such as the real line or $$n$$-dimensional Euclidean space. Other stochastic processes such renewal and counting processes are studied in the theory of point processes. Sometimes the term "point process" is not preferred, as historically the word "process" denoted an evolution of some system in time, so point process is also called a random point field. Improbable keeler (talk) 15:42, 6 January 2017 (UTC)

Article re-written
Great work and thank you. I promise to try to get back here and help as much as a physicist can. Juan Riley (talk) 21:55, 15 November 2015 (UTC)

Ah, somebody created this article. I've actually been writing an article on the Poisson point process for the last couple of weeks (but been sidetracked with the history section and some related articles). I plan to upload it later this month or next. It will extend on this article considerably, and I plan to merge it with the article on "Poisson process" which I think needs a lot of work and is misleading in that it focuses too much on the Poisson point process on the real line whereas the process on the plane has gathered considerable research interest over the years. Improbable keeler (talk) 07:42, 10 January 2014 (UTC)

A merger with Spatial Poisson process might be required first. Juan Riley (talk) 00:05, 22 May 2014 (UTC)

Excuse me for being a physicist and not a mathematician....but there are point processes of any number of dimensions...there is the special case of these which are, I believe, known as Poisson point processes.. and a Spatial Poisson process is nothing more than a Poisson point process in a specific number of dimensions. Thus I would prefer to start over with a general article Poisson point process..and go from there. If one is going to generalize from the typical 1 dimensional thingy called a Poisson process..why do it in stages. Juan Riley (talk) 00:39, 24 June 2014 (UTC)

"Poisson process" and "Poisson point process" mean the same thing. I agree with JuanRiley that merging this article into spatial Poisson process might be the best thing to do, because "Poisson process" article focuses on the 1D case while "Spatial point process" focuses on the multidimensional case, and already covers it with plenty of detail. If you want to propose a renaming of "spatial Poisson process" article then that might be a thing to do. --mcld (talk) 13:26, 9 October 2014 (UTC)

Almost always, "Poisson process" means "Poisson point process". However, there are "Poisson processes" which consist of other objects, such as lines, as covered in the monograph "Poisson processes" by Kingman. I sill have the article I have been working on, but I got distracted. I'll try to upload it soon -- I am just working on the history section and also sort out the merger suggestion. Improbable keeler (talk) 23:26, 14 December 2014 (UTC)

Keeler, I too got distracted. Looking forward to your revision. Juan Riley (talk) 23:53, 14 December 2014 (UTC)

There is no point in calling the points "isolated points". They are just points. — Preceding unsigned comment added by 78.97.140.148 (talk) 17:43, 21 December 2014 (UTC)

So I have *finally*  revised -- well, completely re-wrote -- the article. I think I should somehow now merge the articles. I may have got carried away and it could be too long and need some reducing. Still a lot of work required...Improbable keeler (talk) 14:32, 13 November 2015 (UTC)

Kudos User:Improbable keeler
Great work and thank you. I promise to try to get back here and help as much as a physicist can. Juan Riley (talk) 21:55, 15 November 2015 (UTC)

Many thanks. I should have done uploaded it sooner, but I had planned to improve it. The foundation is there, it just needs polishing such as inserting page numbers to citations to books. I'll make a couple diagrams too. I originally studied physics, but I never encountered the Poisson process there -- perhaps because there is no interaction between the points. I then saw a (inhomogeneous) Poisson process used in a energy model developed by David Ruelle -- it's covered in a recent book on the Sherrington-Kirkpatrick model. Some googling suggests that perhaps the term Poisson random field is used more in physics. Physicists defined and studied determinantal point processes (another article that needs improving), which have repulsion between points, in order to model fermions -- technically Poisson is a type of DPP (but on the very edge ie the repulsion is zero). Improbable keeler (talk) 08:13, 16 November 2015 (UTC)

Dubious applications
We need some applications of the Poisson point process, particularly in one-dimension. There are many, but people tend to confuse the Poisson distribution with the Poisson process. One can't infer occurrences of certain phenomena can be modelled with a Poisson process just because the number of occurrences that happened over a large time frame (eg wars over a number of decades) -- more important is (relatively strong) independence between occurrences/points. I've copied pasted the applications section from the old Poisson process article, but I think the actual examples/citations are not appropriate.

From previous article on Poisson process
The following examples are also well-modeled by the Poisson process:


 * Number of road crashes (or injuries/fatalities) at a site or in an area
 * Goals scored in a association football match.
 * Requests for individual documents on a web server.
 * Particle emissions due to radioactive decay by an unstable substance. In this case the Poisson process is non-homogeneous in a predictable manner—the emission rate declines as particles are emitted.
 * Action potentials emitted by a neuron.
 * L. F. Richardson showed that the outbreak of war followed a Poisson process from 1820 to 1950.
 * Photons landing on a photodiode, in particular in low light environments. This phenomenon is related to shot noise.
 * Opportunities for firms to adjust nominal prices.
 * Arrival of innovations from research and development.
 * Requests for telephone calls at a switchboard.
 * In queueing theory, the times of customer/job arrivals at queues are often assumed to be a Poisson process.
 * The evolution (changes on pages) of Internet, in general (although not in the particular case of Wikipedia)

Improbable keeler (talk) 13:33, 16 November 2015 (UTC)

Space-time necessary?
I am not insure if a section on space-time is required (it's just a space case of the n-dimensional Poisson point process), but I've copied it from the old Poisson process article:

Space-time
A further variation on the Poisson process, the space-time Poisson process, allows for separately distinguished space and time variables. Even though this can theoretically be treated as a pure spatial process by treating "time" as just another component of a vector space, it is convenient in most applications to treat space and time separately, both for modeling purposes in practical applications and because of the types of properties of such processes that it is interesting to study.

In comparison to a time-based inhomogeneous Poisson process, the extension to a space-time Poisson process can introduce a spatial dependence into the rate function, such that it is defined as $$\lambda(x,t)$$, where $$x \in V$$ for some vector space V (e.g. R2 or R3). However a space-time Poisson process may have a rate function that is constant with respect to either or both of x and t. For any set $$S \subset V$$ (e.g. a spatial region) with finite measure $$\mu(S)$$, the number of events occurring inside this region can be modeled as a Poisson process with associated rate function λS(t) such that


 * $$\lambda_S(t) = \int_S \lambda(x,t)\,d\mu(x).$$

Separable space-time processes
In the special case that this generalized rate function is a separable function of time and space, we have:


 * $$\lambda(x,t) = f(x) \lambda(t) \,$$

for some function $$f(x)$$. Without loss of generality, let


 * $$\int_V f(x) \, d\mu(x)=1.$$

(If this is not the case, λ(t) can be scaled appropriately.) Now, $$f(x)$$ represents the spatial probability density function of these random events in the following sense. The act of sampling this spatial Poisson process is equivalent to sampling a Poisson process with rate function λ(t), and associating with each event a random vector $$X$$ sampled from the probability density function $$f(x)$$. A similar result can be shown for the general (non-separable) case.

Improbable keeler (talk) 13:33, 16 November 2015 (UTC)

Invalidity of the translated equation under inhomogeneous Poisson process
I do not think that the claim made in the discussion on Inhomogeneous Poisson processes that translates the expected number of events that occur in the interval [t,t+h) to [0,h) is accurate. The rate function, by definition depends on the exact value of t, and is changing relative to the value of t, meaning that stationarity of the type required for such a translation claim to be valid will not hold. I.e., if I expect most of my customers will arrive between 10 and 11 AM, and no customers will arrive at 1 – 2 AM (because the shop is closed), I cannot say that merely because the time interval's length is 1 hour (h), it is the case that E[N(1AM,2AM)]=E[N(10AM,11AM)] even if they are driven by the same inhomogeneous process.

Mdpacer (talk) 03:29, 31 March 2016 (UTC)


 * That claim is not made for the inhomogeneous Poisson process -- only the homogeneous one -- or am I missing something? The inhomgeneous process depends on the interval [t,t+h), as you say, and that is said. Perhaps it could be made clearer? Improbable keeler (talk) 14:45, 26 January 2017 (UTC)

Inter-arrival times and simulation
I found this article useless because I could not find what I was looking for - a good way of simulating the Poisson process using inter-arrival times.

The Poisson process has a crucial property - interarrival times are exponentially distributed. This fact is very briefly mentioned in the article, while it deserves a separate section.

Moreover, this fact can be used to simulate a non-stationary Poisson process that has a transition of the intensity from one value to another. This can be done using the interarrival times, by changing the interarrival time distribution. The part of the article that describes simulations of the Poisson process is useful only for the stationary case. — Preceding unsigned comment added by Skryba2000 (talk • contribs) 14:04, 24 July 2020 (UTC)

Source wrong
Daley and Vere-Jones Vol. 2 is listed for [47] in "Interpreted as a point process on the real line," but it should be Vol. 1. — Preceding unsigned comment added by 134.164.64.28 (talk) 16:32, 30 June 2021 (UTC)

Move to Poisson process?
Surely that is the WP:COMMONNAME for this concept, isn't it? Danstronger (talk) 23:04, 17 December 2021 (UTC)

Bad writing
In the section History of terms, this sentence appears:

"Feller worked from 1936 to 1939 alongside Harald Cramér at Stockholm University, where Lundberg was a PhD student under Cramér who did not use the term Poisson process in a book by him, finished in 1936, but did in subsequent editions, which his has led to the speculation that the term Poisson process was coined sometime between 1936 and 1939 at the Stockholm University"

From the extremely poor grammar, the text makes it appear as if the author of the 1936 book is Lundberg, although I am not aware of any book by him in 1936. I am guessing the book referred to is "Mathematical Methods of Statistics" by Cramér. 2601:200:C000:1A0:B4D9:2B4B:27D8:5BAE (talk) 21:28, 30 July 2022 (UTC)