Questions tagged [poisson-process]

For questions about the theory or applications of the Poisson process, one of the most widely applied point processes in statistics and elsewhere.

45
votes
8answers
14k views

Is there any gold standard for modeling irregularly spaced time series?

In field of economics (I think) we have ARIMA and GARCH for regularly spaced time series and Poisson, Hawkes for modeling point processes, so how about attempts for modeling irregularly (unevenly) ...
73
votes
5answers
20k views

Please explain the waiting paradox

A few years ago I designed a radiation detector that works by measuring the interval between events rather than counting them. My assumption was, that when measuring non-contiguous samples, on ...
25
votes
2answers
50k views

How to know if a data follows a Poisson Distribution in R?

I am an undergrad student and I have a project for my probability class. Basically, I have a dataset about the hurricanes that impacted my country for a series of years. In my probability Book, (...
12
votes
1answer
11k views

What are the differences between survival analysis and Poisson regression?

I'm working on a classical churn prediction problem using the number of visits of a given user to a site and I thought that Poisson Regression was the right tool for modelling the future engagement of ...
4
votes
1answer
401 views

Poisson processes

I have two realizations of a poisson stochastic process, they are over the same space with rate $\lambda_{1}$ and $\lambda_{2}$. What is the probability that N elements in both sequences are the same, ...
4
votes
1answer
2k views

Testing for Poisson process

I have some discrete times of events and I would like to do a test to see if they are likely to have come from a homogeneous Poisson process. From this pdf, I see: REMARK 6.3 ( TESTING POISSON )...
2
votes
1answer
2k views

Nonhomogeneous poisson process simulation

I've been looking at ways to generate a Nonhomogeneous Poisson Process (NHPP) including the nonlinear time transformation (using a rate-1 process and inverting the cumulative rate function). I've also ...
3
votes
2answers
87 views

Any expression for the probability of a hard sphere in Boolean model

I am working hard on a problem of Boolean model. In a example of Boolean model, points are scattered in the plane according to a homogeneous Poisson process of intensity λ. On each of these points a ...
14
votes
1answer
8k views

How to estimate Poisson process using R? (Or: how to use NHPoisson package?)

I have a database of events (i.e. a variable of dates) and associated covariates. The events are generated by the non-stationary Poisson process with parameter being an unknown (but possibly linear) ...
2
votes
3answers
2k views

Bayesian parameter estimation of a Poisson process with change/no-change observations at irregular intervals

Consider a Poisson process with unknown parameter $\lambda$. We perform a sequence of $n$ observations at intervals $\overline{t}=t_1,\,t_2,\,\dots,\,t_n$. Each observation is a binary variable $x_i$ ...
3
votes
2answers
3k views

Simulation of a Poisson Process

I am trying to simulate the compound Poisson process using the next algorithm that I found in a textbook on stochastic processes. Let $S_0 = 0$. Generate i.i.d. exponential random ...
4
votes
1answer
416 views

Mean service time of a $M/E_2/1$ queueing system?

Consider a $M/E_2/1$ queueing system, where the customer arrival rate is $\lambda$ and the service time distribution has a gamma distribution with parameters $2$ and $\mu$, i.e. with p.d.f. $\mu^2te^{-...
2
votes
1answer
2k views

Ripley's K Function and L Function for Point Patterns

The following is a spatial point pattern: and these are the corresponding Ripley's K function and L function for this data: How are these functions interpreted?
2
votes
0answers
145 views

Conditions for Poisson approximation of the superposition of non-Poisson processes

It is well known that the superposition of $N$ Poisson processes is itself a Poisson process with an intensity given by $\sum_{n=1}^{N} \lambda _{n}$. Conversely a superposition including any non-...
4
votes
1answer
88 views

Mean length of time spent queueing in $M/E_2/1$ system?

Context: Consider a $M/E_2/1$ queueing system, where the customer arrival rate is $\lambda$ and the service time distribution has a gamma distribution with parameters $2$ and $\mu$, i.e. with p.d.f. $\...
2
votes
2answers
433 views

Poisson process: getting a poisson from an exponential assumption

In a poisson process, getting an exponential distribution from a poisson assumption (for eg. in a given period of time $t$, if number of books that arrive is given by a Poisson, then the time until ...
1
vote
1answer
239 views

Time up to $n$th event in Poisson process distributed as $\frac{1}{2\lambda} \chi^2_{2n}$

Let's assume that a number X of some events over time $t$ is modeled by Poisson distribution with rate $\lambda$ (here, it's rate, not mean): $$ X \sim Poisson(\lambda \cdot t) ~~~~ (\lambda t ~\text{...
3
votes
1answer
380 views

Is it valid to simulate a shot noise Poisson process with discretization (R)

I have a two-factor shot noise process as follows (sorry for the picture of text): I want to simulate from this process. Is it valid to do so in a discretized way? The solution to these SDEs is a ...
3
votes
1answer
522 views

Poisson Process

I would appreciate a hint on this problem: A pedestrian wishes to cross a single lane of fast-moving traffic. Suppose the number of vehicles that have passed by time $t$ is a Poisson process of rate $...
2
votes
3answers
3k views

Can I estimate the parameter of a Poisson arrival process from a low-incidence observation period?

If I know only that the arrival process is Poisson, and I observe it for a pre-chosen (say, unit) period of time, observing $k$ arrivals, is it meaningful to describe an estimate of its time parameter ...
2
votes
1answer
411 views

Non homogenous Poisson process with simple rates

I am trying to stimulate number of claims in the next 12 months using a non-homogeneous poisson process. The rates are: ...
1
vote
0answers
42 views

The probability of disc that fully overlaps discs in a Boolean model

In a example of Boolean model, points are scattered in the plane according to a homogeneous Poisson process of intensity $λ$. On each of these points a disc of fixed radius $r$ is placed. Similar to ...
0
votes
0answers
48 views

Binomial/Poisson Problem - Pizza Orders

Suppose that the number of orders per hour at a pizza shop follows a Poisson process with rate 5 per 30 minutes. Suppose that the pizza orders are large with probability 2/3, small with probability 1/...
0
votes
0answers
9 views

Coverage of a Boolean model

The following problem I am working on involves the coverage of Boolean Model (BM). The grains in the BM are discs with fixed radius $r$. $A$ is the set of this BM in space $ \mathbb R^2 $ and $K$ ...