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Questions tagged [point-process]

A point process is a stochastic process in which the data are sets of points ordered in a mathematical space. A common example is the Poisson process, in which points are ordered in time with the interarrival times exponentially distributed.

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Can any thinning of a Poisson point process be thinned further into a Poisson point process?

Is the following true? Let $\Phi$ be a homogenous Poisson point process of positive intensity. Let $X$ be a thinning of $\Phi$ where $X$ is stationary and has positive intensity. Then $X$ can be ...
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Can any point process be thinned into a Poisson point process?

Is the following true? For any point process $X$ there exists a procedure $\mathcal{P}$ and a (homogenous) Poisson point process $Y$ with intensity $\lambda > 0$ such that the process of drawing a ...
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Why do we use only event history for temporal point processes?

I'm trying to understand temporal points processes. In particular, neural TPPs. In all the works I've read, the only features fed into the model are a sequence of event timestamps and marks if they ...
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Limit of Ripley's K function as r goes to infinity?

Considering the Ripley's K function as used in spatial point-process analysis, or the closely related L function, I am wondering what the limit of the function is as r approaches infinity. I am aware ...
Sriram Ramamurthy's user avatar
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Questions around modelling arrival process of randomly sized groups

I have the following situation: I'm trying to model groups arriving to some location by some process. I assume the distribution on some interval $T$ is a Gamma-Poisson mixture where $\Lambda \sim ...
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Is there a correction for samples from a (linear) Prophet model when trained on an inhomogenous Poisson point process?

Facebook's Prophet is a popular modelling choice for time series forecasting in production due to many steps being automated (and thus convenient). This can sometimes lead to over-reliance on it when ...
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Confirming the validity of a point process model

I have a question regarding a point process model I generated for a biological point pattern exhibiting repulsion and attraction at different scales. My goal for this model was to generate simulated ...
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Statistical test whether data conforms to a spatial point process--gaza bombing locations

I came across this image on twitter, and it made me think about testing a point process hypothesis. Now this is a politically sensitive image, and I don't want to run afoul of any SE posting ...
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Are these two equivalent forms for the likelihood of a Poisson point process?

I have a Poisson point process in a bounded region $W$. I'm trying to calculate the likelihood of observing a particular set of points within $W$. I'm told that there are two equivalent forms of ...
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Normalizing constant calculation of Strauss Process

Suppose that I have the following Strauss Process up to a proportionality constant $$p(\mu_{1}, \mu_{2},..., \mu_{K},K)\propto \xi^{K}\prod_{i=1}^{K} I(\mu_{i}\in R) *a^{\sum_{i,j}|\mu_{i}-\mu_{j}|<...
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(Using conditional expectation to calculate) expected value of the product of two dependent random variables

Let $\mathbf{X}$ be Binomial point process in $W = [0, 6] \times [0, 4]$ with $n$ points. Let $A_1 = [0, 2] \times [0, 4]$, $A_2 = [0, 6] \times [0, 2]$, and $A_3 = [2, 6] × [2, 4]$. I want to find $E[...
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Tail of the maximum of a time-varying Poisson-GP marked process

Consider a time-varying version of the Poisson-GP marked process on the real line as commonly used in Peak Over Threshold (POT) modelling of a variable $Y$. More precisely we have a given time-varying ...
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A hard core spatial point process that a vehicle encounters as Poisson arrivals?

Does there exist a point process $X$ in the plane with the following two properties? $X$ is hard core. Discs of radius $h$ can be centered on the points in $X$ without overlapping. $X$ is ...
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Likelihood for a log Gaussian Cox process (LGCP)

Suppose I have a log Gaussian Cox process (LGCP) $X$ with log intensity function $\lambda(x)=S(x)$ where $S$ follows a Gaussian process. Since LGCP still falls under the umbrella of inhomogeneous ...
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Simulation envelopes for Foxall J not containing the line y=1 (fully reproducible example)

During some exploratory analysis of my data I noticed that when Jfox was calculated for a point pattern relative to some polygons, I frequently obtained that the pooled envelopes area did not include ...
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Convergence of intensity function of self-exciting point processes imply convergence of processes?

For every $n \in \mathbb{N}$, let $N_n$ be the point process on $[0,1]$ defined by the conditional intensity function $\lambda_n: [0,1] \rightarrow \mathbb{R_{\geq 0}}$, where $$ \lambda_n(t) = \mu_n(...
José Kling's user avatar
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pointwise envelopes not including Theoretical line Foxall J

I am computing pointwise envelopes for the Foxall's J function to investigate the whether some point patterns of interest are clustered, avoid or are independent from other point patterns or polygons. ...
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Algorithm for generating a Poisson process on a complicated 2d geometry

I am looking at some count data by geographic counties in California. As a starting point, a Poisson process came to mind--though there are other good choices like negative binomial, etc. Given a $\...
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Definition of point processes

We say a finite point process $X$ on $[N]=\{1,…,N\}$ can be understood as a random subset. It is defined either via its inclusion probabilities, that is $\mathbb{P}(S \subseteq X)$ for $S \subseteq [N]...
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Is the following definition of the variance of the number of points correct?

I have been asked to calculate "the variance of the number of points" in the following pictures, inside the circles: I was looking for the definition of "the variance of the number of ...
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Doing a temporal cluster analysis for terrorism

I am trying to conduct a temporal cluster analysis on a terrorism dataset, but I've hit a bit of a brick wall regarding the method. The dataset spans between 01-01-2015 and 31-12-2019, and for my 3 ...
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Knowing when to stop with spatial point thinning -- is an approach with Clark-Evans test valid?

I need a random sample of presence points from the species' area to use in distribution modeling. I have an excessive amount of spatially clustered presence records, so I use minimum nearest neighbor ...
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Is Nearest Neighbor Distance Ratio (NNDR) test good for assessing point pattern?

Regularity of point pattern is routinely checked in ArcGIS with the Nearest Neighbor Distance Ratio test providing significance estimations: https://pro.arcgis.com/en/pro-app/latest/tool-reference/...
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Best models to relate a frequent point process and a rare point process

I want to relate two processes and understand their cause-and-effect relationship. The first process is a point process with frequent occurrences (more than 90%) and has support [0,1] and truncated ...
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Predict background counts given past observations and assumption of linear variation of the rate parameter in time

Consider the following problem. We have a time series of counts (poisson-distributed) data. In this time series we can select an off-pulse window in which only background is present and a subsequent ...
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Simple point processes and the expected value

When considering simple point processes, how is $$P(t_{n+1} \in [t;t+dt]) = E[N([t;t+dt])]$$, where $N(A)$ is the number of points in interval $A$ and $dt$ is the length of an infinitesimal interval. ...
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1 vote
1 answer
398 views

KS Test for Hawkes Process

I am analyzing some data by applying 1d-Hawkes processes. To evaluate the fit of the model itself (and particularly the chosen kernel, i.e. exponential), I am performing a KS test. However, it is not ...
Giant Steps's user avatar
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spatstat::plot.envelope observed line longer than significance band [duplicate]

Using the R package spatstat I'm plotting pooled pointwise envelopes of the function Jfox calculated for different point pattern in an hyperframe. The workflow is ...
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spatstat::plot.envelope observed line longer than shading

Using the R package spatstat I'm plotting pooled pointwise envelopes of the function Jfox calculated for different point pattern in an hyperframe. The workflow is ...
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3 votes
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What is the "sensible default" for the Kest function of the spatstat package?

I'm reading the documentation on the Kest function (page 731) from the spatstat package. For the argument $r$, that is the "Vector of values for the argument $...
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2 votes
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Uniform distribution versus a Poisson homogeneous process with Ripley's K

I'm testing astropy's RipleysKEstimator. I create a 2-dimensional uniform distribution in [0, 1), and obtain its K value for a radius $r=0.5$. It was my ...
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How to simulate spatial point patterns that have spatial structure similar to that of given spatial point pattern?

I have some spatial point pattern X distributed in polygon wind and I wonder how can I simulate different point patterns that by their spatial properties (for example, number of points, spatial ...
Liudas Daumantas's user avatar
1 vote
1 answer
134 views

Point Pattern Analysis: L-r function

I'm new to pattern analysis. And I got a result of L-r function as below. So, how should I interpret the plot? Does it mean the points are cluster when L-r value is less than 0 and dispersed when the ...
andrew's user avatar
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2 votes
1 answer
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trimming window in Foxall G-function and J-function in R spatstat

I am using the Jfox function in spatstat to explore the distance of points in a point pattern to the nearest polygon between a ...
Filippo's user avatar
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2 votes
1 answer
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Does it make sense to consider ratio of pair correlation functions of two point patterns?

I have two point patterns, one is real data, the other one is simulated data from a model I constructed for the real data. Now I want to see if the second order property of the model matches that of ...
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108 views

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-...
ABIM's user avatar
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3 votes
1 answer
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Point process model diagnostic: Nearest-Neighbor Distance Distribution or Pair Correlation Function?

I have a point pattern which is clearly inhomogeneous. Furthermore, the inhomogeneity has two components: a large scale effect and a local scale effect. I have constructed a Markov point process model ...
NamelessGods's user avatar
2 votes
1 answer
140 views

How should one simulate a continuously-marked point process by Birth and Death MCMC?

As the question states, I am not sure how I should go about doing that. The main issue is how do I draw a sample from the marked space? The way I am doing things now is as follow: If we are in a ...
NamelessGods's user avatar
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1 answer
92 views

Lcross Confidence Envelopes - R software

I have performed an Lcross examination in R with the following code: ...
Alaina M De Biasi's user avatar
8 votes
2 answers
856 views

Spatial Point Process: Does an inhomogeneous first order intensity function affect the second order dependence?

As the title suggests, I am having a bit of confusion on the effect of first order intensity function. If I have a first order intensity function that says in a certain region the points are much more ...
NamelessGods's user avatar
1 vote
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93 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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Distance between point process realizations

Is there a valid distance metric for measuring how similar are the realisations of two point processes? E.g. let's say we simulate two histories $ h_1, h_2 $ in the time interval $ [0, T] $ for two ...
Dion's user avatar
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4 votes
1 answer
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Ripley K function value in a specific radius and dataset using R's Kest function

I'm having general trouble with calculating Ripley's K function values. The following is a simple spatial point pattern, where both X and Y range from 0 to 200: Here's its corresponding Ripley K ...
Delano762's user avatar
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311 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 ...
Patrick Taggart's user avatar
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Area interaction model sufficient statistics calculation algorithm

I'm currently trying to write a simulation algorithm in Rcpp for a point process model which requires calculating the sufficient statistics for an Area interaction point process model. It entails a ...
NamelessGods's user avatar
1 vote
1 answer
84 views

Maximum Pseudo-Likelihood estimator for soft-core point process

I am trying to fit a soft-core point process model on a set of point pattern using maximum pseudo-likelihood. I followed the instructions given in this paper by Baddeley and Turner And here is the R-...
NamelessGods's user avatar
1 vote
0 answers
364 views

Unexpected behavior in inhomogeneous Cross K Function (Kcross.inhom) [closed]

I am currently analyzing a point pattern in R using the "spatstat" package. I am comparing two different areas, therefore I made two plots for each area (first two plot-left-area1; second ...
Beloiu Mirela's user avatar
5 votes
1 answer
498 views

Point Pattern Analysis: Assumptions for Hopkins-Skellam Index

I am currently analyzing a point pattern in R using the "spatstat" package. My analyses are primarily exploratory, as I do not have any strong reason to suspect either clustering or regularity, though ...
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4 votes
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Point process - intensity function vs probability density function

Suppose we have a point process in $\mathbb{R}$ with intensity $\lambda(x)$. Then, for a given compact set ${ S}$ we have $$\Lambda({ S})=\int_{\rm S} \lambda(x) \, dx,$$ where $\Lambda({ S})$ is ...
dynamic89's user avatar
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2 answers
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Fit Hawkes process to 1d data using python package TICK

How can one fit the 1-dimensional Hawkes process with exponential kernel to the experimental 1d dataset (t1,t2,t3...tn) and check the goodness-of-fit via tick python3 package? I found on official ...
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