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.

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Confusion about Poisson rectangular pulse model

I am reading a paper by Rodriguez-Iturbe et al. from 1986 and am confused by the below derivation. The model they are working with is a Poisson process with rate $\lambda$ in which each occurrence in ...
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How to calculate the average inter-arrival time from a series of inter-arrival time?

I am currently handling count data and just look into the poisson process. It maybe a trivial question but I am really confused on it. I have collected count data and the inter-arrival time between ...
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what are the parameters of a poisson term in a jump diffussion SDE

I am trying to simulate a Jump diffusion process in python and not sure what i should specify as the parameters of the poisson term. Here is what i have ...
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24 views

Boundary estimation using statistical techniques

Do you know a good methodology to estimate the boundary between two sets? Here are the specifics of the problem: I am studying a recursion defined as $$ 2(n+q) x_{n+2}= (r(n+q) +s)x_{n+1}+((2-r)(n+q)...
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How to model the likelihood of an inhomogeneous Poisson process with “uncertain” event values

Using the example of an inhomogeneous Poisson process in 1 dimension for simplicity, with a varying rate parameter $\lambda (t)$. Let's say I am trying to find the form of $\lambda (t)$, using data ...
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How to Simulate Pure-Jump Process associated Marked Point Process in R

Fix $M>0$ and let $(\tau_i,\zeta_i)_{i =1}^{\infty}$ be a marked point process associated to the poisson-random measure $\mu$ on $[0,1]\times [-M,M]^2\times [0,M]\times [-M,M]$ with uniform finite-...
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Exercise on Poisson model: how to get fitted values?

Setting: Let $C_{it}$ denote the number of claims of individual $i$ in a certain period $t$. Assume that $C_{it}$ is distributed as a Poisson with claim rate (mean) $\gamma_{it}$. Let $\log(\...
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63 views

Poisson process on an $n$-sphere

I have an algorithm that embeds data points into Euclidean space. If I norm these points then they will lie on the unit $n$-sphere, where $n+1$ is the dimensionality of the embedding space (generally ...
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28 views

Conditional Poisson Process

I cannot reach a correct answer and I don't know why. I am trying to calculate this by conditioning on $N(t)=n$ and I ended up with $e^{-At(z^s)}$. However, the correct answer is $\dfrac{e^{-At(z^s)}-...
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Is there a closed-form expression for the Poisson Lognormal pmf?

Suppose the $\lambda$ parameter of a Poisson distribution is generated from a $LogNormal(\mu, \sigma)$ distribution. Can the final pmf be expressed with only elementary functions? $$f(x;\mu,\sigma)=\...
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Can we consider arrival of pest outbreaks reports as Poisson Process?

I have a scenario where given a geographical region, farmers within that region can generate pest outbreak reports (like twitter messages) using a mobile phone. Also, a particular pest can spread to ...
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71 views

Finding the probability of survival of an insurance company

I was given as a homework exercise the following problem: however, I came into a disagreement with one of my classmates. Given that the solution is not shown, I was wondering whether mine was correct....
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Poisson Process conditional probability

Let $N(t)$ be a Poisson Process with rate $\lambda$ Find $\displaystyle P(N(4) \le 2N(2) \mid N(2) = 1) = \frac{\sum_{i = 0 }^2P(N(4) = i, N(2) = 1)}{P(N(2) = 1)} = ?$ Can I split this up using the ...
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How can I predict the total number of cell with Poisson process

I detected cells in circulation with our own system, the cell is very rare compared to other blood cells, I acquired the temporal information of each cell and calculated intervals between 2 cells. The ...
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17 views

Multivariate Bayesian Car Model Result

I have developed a multivariate Bayesian Car model for three crash severity level analysis. I found that the covariance for both heterogenous effects and the spatial effect is not significant for any ...
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70 views

Confidence interval over multiple Poisson distributions

There are five different periods (one following each other) of unequal lengths where one specific event is happening such that it follows Poisson distributions (e.g. meteorite falls). It is always the ...
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380 views

Probability of a 500 year flood occuring in the next 100 years - comparison of approaches

I'm looking at this problem A $500$-year flood is one that occurs once in every $500$ years. a) What is the probability of having at least $3$ such floods in $500$ years? b) What is the ...
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Poisson process as a spatial process

Let's consider a Poisson process on the line with rate parameter $\lambda$. There are two ways to think about this: In any interval $[a,b)$ the expected number of events is distributed as a Poisson ...
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37 views

Simultaneous Poisson processes

The arrival of taxis at a taxi stand is Poisson at rate $\lambda$ per hour. The arrival of people to the stand is also Poisson but at rate $\mu$ per hour. Taxis do not stop (they leave empty) if ...
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75 views

Pareto/NBD with time-varying covariates

I am trying to incorporate time-varying covariates into the Pareto/NBD model (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.597.3165&rep=rep1&type=pdf) Model assumptions start at ...
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Does it make sense to infer a rate (as a probability distribution or upper limits) for a Poisson process if there are “no events”

I have an inhomogeneous Poisson process with a rate $\lambda (\mathbf{t})$ defined on some parameters $\mathbf{t}$. I am trying to infer $\lambda (\mathbf{t})$ from some data, which are events (really ...
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34 views

Smallest gap between $n$ Poisson events

Given a Poisson process with parameter $\lambda,$ run time forward until the first $n>1$ events occur. What is the distribution of the smallest time interval between two events? This is surely ...
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32 views

MLE estimation of poisson intensity for a normal compounded poisson process

I was working on a project where a time series is modeled as compound poisson distribution with normally distributed jump levels and I want to get MLE estimator for the poisson intensity, jump mean, ...
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40 views

Poisson regression for count data that is not Poisson distributed?

Is it true that Poisson regression is used to model count data? But not all count data follow a Poisson distribution? Then you can still use Poisson regression in that case?
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39 views

Comparison of Poisson cumulative probabilities between models with different rates

I'm building an anomaly detection model using the Poisson distribution. I'm calculating the Cumulative Probability: P(X >= x) for thousands of count data, all of them with different rates. I need to ...
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Poisson Fun exercise question

Based on your understanding of the Poisson process, determine the numerical values of $a$ and $b$ in the following expression. $$ \int_{t}^∞ {λ^6τ^5e^{−λτ} \over 5!}dτ= \sum_{k = a}^b{(λt)^ke^{-...
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19 views

Which distribution or process should be used for wearout reliability modeling?

When modeling the reliability of a system, it is usual to use exponential distribution to model errors that occur randomly throughout the system's useful lifetime (the middle part of the well-known ...
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1answer
26 views

Extension of Poisson Parameter for Different Temporal Interval

Suppose $Y$ follows a Poisson distribution with parameter $ \lambda $ that explains a temporal Poisson process over an interval of 30 seconds. Now, it stands to reason that for an interval of 60 ...
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Test and CI for unstable Poisson process

I have empirical data on a process that I assume is Poisson with a given mean, say $\mu$ (unknown). The data is of the form $(x_i, 1\leq i\leq n)$ for $n$ consecutive time periods. I am concerned ...
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64 views

Upper confidence bound for Poisson process rate parameter

I am interested in computing an upper confidence bound for the rate parameter, $\lambda$, in a Poisson process. Specifically, I have a set of observations $$ X_\text{obs} = \{(n_1,t_1), \ldots, (n_N,...
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346 views

Alternating between two states {A, B} each with exp distributed durations. What's the probability of state=A at time t?

Say I have a light bulb that can be on (A) or off (B). It alternates between being state A or B. It will be in state A for a duration a ~ exp(α), and in state B for duration b ~ exp(β), (...
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Conditional distribution of arrival times in Poisson process

Suppose I know over a window $[0, T)$ that I have observed $n$ samples from a poisson process $N_t \sim p(n|\lambda t) = \frac{1}{n!}(\lambda t)^{n}\exp(-\lambda t)$. What is the conditional ...
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Poisson Process With Two Different Rates

I am looking to investigate a Poisson Process that has two separate rates; essentially, there is a season of games that each have their own points rate (per minute). Comprehensively, over the season, ...
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Overdispersion under different longitudinal constraints

I have panel data (police districts observed across many months) and I am modeling a count outcome. Fitting a Poisson model (simple pre/post comparison with 6 months of data) results in significantly ...
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23 views

How to compute the expected number of events in the following conditional renewal process?

I have a stochastic point process with event times $\{x_1, x_2, ...\} $ and I want to compute the expected number of events $n(T)$ over the interval $[0,T]$. The point process is generated as follows: ...
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How to show the variance of the inter-arrival time of a Cox process driven by a Poisson process of constant intensity $\lambda$ is $3\lambda$

Ideas on how to show that the variance of a doubly-stochastic Poisson process(aka a Cox process) driven by a homogeneous(stationary) Poisson process of intensity $\lambda$ is $3\lambda$ ? I've come ...
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To detect which of the three processes more-aptly fits Poisson process

There happens to be a type of generic poll, which can be described as:- A participant can vote only once and for only 1 of 3 running candidates - Red, Blue and Green. A random voter starts the ...
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Solving an equation with logarithms of negative numbers

I'm modelling a bunch of non-homogeneous poisson processes. I'm impossing on them a linear-log functional form. you end up with two parameters, theta1 and theta0 for each estimated process. The ...
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Understanding the solution to a problem about a homogeneous Poisson process

This is probably easy, but right now I can't figure it out, so bear with me. The question is: Let $\{N(t),t\ge 0\}$ be a homogeneous Poisson process on $(0,\infty)$ with rate $\lambda$. Let $\{S_i, i=...
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304 views

Manually simulating Poisson Process in R

The following problem tells us to generate a Poisson process step by step from $\rho$ (inter-arrival time), and $\tau$ (arrival time). One of the theoretical results presented in the lectures ...
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Is this problem worked out correctly? [closed]

Calls are received at a company call center according to a Poisson process at the rate of five calls per minute. (a) Find the probability that no call occurs over a 30-second period. (b) Find the ...
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29 views

Expectation of arrival times

Let $(N_t)_t$ be a Poisson process with parameter λ = 2. By $τ_k$ denote the time of the k-th arrival (k = 1, 2, . . .), and by $ρ_k = τ_k −τ_{k−1}$ - the interarrival time between the (k−1)th and kth ...
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71 views

How to check if a data is poisson sampled?

I was reading one article which develops a theory for the Poisson sampled data. That is the data is collected over time-points $\{T_k, k>1\}$, which are jump-moments of a homogeneous Poisson ...
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104 views

What is the difference between time, arrival-time, and inter-arrival-time is poisson process?

Let $(N_t)_t$ be a Poisson process with parameter λ = 2. By $τ_k$ denote the time of the k-th arrival (k = 1, 2, . . .), and by $ρ_k = τ_k −τ_{k−1}$ - the interarrival time between the (k−1)th and kth ...
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How to properly truncate exponential distribution to represent random memory-less arrivals?

Typically, Poisson and exponential distributions are used to represent random memory-less arrival processes. It has come to my attention, however, that a more realistic distribution is a truncated (or ...
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29 views

Distribution of the y-coordinate of a 2D Poisson Process

Consider a Poisson process with rate 1 over $R_+ \times R$. I found this term in a scientific article, I assume it means : -The number of points in a subset $A$ of $R_+ \times R$ has the Poisson ...
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1answer
33 views

Generating Arrival Counts Given Time

I'm looking to generate arrival counts given a length of time for a Poisson process. This is similar to another question, however rather than generate the particular arrivals, I'd like to just get the ...
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53 views

Poisson process - measuring empirical changes

I was recently discussing Poisson processes in industrial settings with a colleague and he came up with a great question. Let's say that defective products on a manufacturing line tend to occur ...
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79 views

Simulating a (discretized) Cox process via binomial sampling

Let X be a Cox process (doubly-stochastic Poisson process) driven by a Poisson process with fixed intensity(rate) $\lambda=50$ , and choose some small time interval $dt=0.01$ . Is the proper way to ...
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40 views

Inhomogeneous K-function to indicate need for spatial dependence/interaction term in Poisson point process model

I am mapping and modelling a disease of sheep. I have approx 4200 point locations in my dataset, each of which represents the centroid of a given sheep farm. I have created a K-function difference ...

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