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Point process models

Two of the popular models for analyzing point process data are Cox and Hawkes processes. My question is how do we compare the statistical properties of these two processes as they both can be ...
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1answer
12 views

Bivariate K function for inhomogeneous spatial point processes

I have some inhomogeneous spatial point patterns of individuals in a cactus population. I also have marks, such as "diseased"x "healthy" individuals, and "adult" x "juvenile". I've already computed ...
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Polygon pattern analysis -> point pattern analysis?

Is it correct to convert polygons to points in order to do a point pattern analysis ? I have a set of polygons that tend to be clustered around roads in a study area. I just want to show this fact. ...
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1answer
62 views

Use of Poisson distribution to analyse distribution of individuals in space

Dytham 2010 suggests using the Poisson distribution to establish whether individuals are evenly distributed in space. Say we end up with a map of individuals in a study site that looks like the ...
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1answer
38 views

Homogeneous vs. Inhomogeneous Poisson point process

What are the main theorical differences between the homogeneous and inhomogeneous Poisson point process? What are the aspects and condition of my data that I can determine which point process best ...
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23 views

Implementing equation for Conditional Intensity Function

I have point processes for which I would like to compute the conditional intensity functions. I know R has a package for doing point process analysis, but I have already done a lot of work on this ...
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1answer
17 views

How to combine MLEs for a point process

I am estimating parameters of a point process by doing an MLE calculation using event time data. However I in fact have several distinct sets of data from different locations. I can estimate the ...
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46 views

Understanding Poisson Point Process [closed]

Could someone explain the Poisson Point Process? Is it simply an Integration over a Poisson Process for higher dimensions? If so, how do you derive the function, say in two-dimensional space, and ...
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37 views

Simulation of a process consist of Brownian motion and Poisson process

I am trying to simulate the following process: h(t)=B(t)+e[P1(t)-P2(t)] in which B(t) is a Brownian motion and P1, P2 are Poisson process with ...
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29 views

Point processes: Calculating the probability that a system survives a specified interval

A system is subjected to shocks of type 1, 2 and 3, which are generated by independent homogeneous Poisson processes with respective intensities $\lambda_1$=0.2, $\lambda_2$=0.3 and $\lambda_3$=0.4. A ...
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40 views

Is there a minimum sample size for Besag's transformation of Ripley's K-function?

I am using Besag's L function to test complete spatial randomness for plant populations. Specifically, I am implementing the univariate L-function Lest in R on a ...
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25 views

A union bound on a continuous variable

Assume a continuous point process (say Poisson) in [0,t]. Assume that it is given that only three jumps occurred in [0,t], however, their exact time coordinate is unknown. In addition, it is known ...
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1answer
107 views

Residuals plot from ripley K function on spatstat

I am a rookie on R and on spatstat package. I would like some help with the Kres function on spatstat. As is always wise to plot the residuals from any kind of ...
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21 views

Likelihood Lévy process

Consider af complete observation sample path of a Levy process $X_t$ (wtr a poisson randommeasure $\mu$) on [0,t] concentrated on the integers. The parametrization is a family of Levy measures ...
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2answers
203 views

Under what circumstances is the log likelihood function of a point process concave?

I am trying to understand under what circumstances the log likelihood function of a point process concave. Assume that the process can be defined by a conditional intensity function and that the log ...
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1answer
107 views

Test hypothesis point process is Poisson [duplicate]

I have some data and I would like to test the hypothesis that they come from a homogeneous Poisson process. I can of course look at the inter event times and test if these are exponentially ...
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1answer
61 views

A modelling question about point processes with heavy tails

I am trying to model a number of point processes for which I have data. If I choose to model each one using a (different) homogeneous Poisson process and estimate the rate using MLE then for some of ...
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1answer
282 views

What is the difference between (universal) kriging and spatial autoregressive models?

As part of a course on missing observations in social/survey statistics I am trying to explore existing methods of predicting either point pattern or polygon data. I got quite confused by all the ...
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97 views

How to simulate a etas spatio-temporal model?

My question is how simulate the spatio-temporal ETAS point process (a point process used for earthquake prediction) defined by a conditional intensity function. More precisely: let $\mathbf{V}$ be a ...
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2answers
99 views

Similarity measures for point processes

I have multiple measurements of a point process: vectors of 0's and 1's. I'm trying to gauge the similarity of the measurements, but have no idea how to proceed. Any suggestions? Thanks!
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94 views

Maximam r distance for Ripley's K-function

I am using R's package spatstat to study the locational pattern of conflict events in Africa (around 8.000 points) using point pattern analysis techniques. I was able to obtain the plot of g(r), the ...
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1answer
101 views

Predicting a continuous outcome using point process descriptors

I have measured a series of times for discrete events along with a continuous variable. So essentially I measure a point process $P: t_1, t_2, \dots, t_n$ and values $A_1(t=x_1), A_2(t=x_2), \dots, ...
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1answer
295 views

What's so Poisson about a Poisson Point Process? (or, can I generate one using random ordered pairs?)

I know there is an R spatstat function to generate a ppp (Poisson Point Process), but I'm working in python, and I am not clear what spatstat.ppp is doing behind the scenes. If I generate a an ...
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2answers
137 views

Measuring correlation of point processes

There is a huge literature on time series analysis. My data does not seem to fit into the standard model in that it consists of event times, that is the times at which an event occurs. What is a ...
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94 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 ...
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4answers
951 views

Superposition of two dependent Poisson processes

$N_1(t)$ and $N_2(t)$ are two independent Poisson processes with intensities $λ_1$ and $λ_2$ respectively. $v_1$ and $v_2$ are two dependent positive random variables, and $p=P(v_1<c)=P(v_2<c)$, ...
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1answer
506 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?
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1answer
154 views

Are there models for “censored” spatial point processes?

This is a problem I'm encountering in the context of analyzing a data set comprised of all crime locations in a city over a fixed time interval, although it could potentially arise in other types of ...
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3answers
2k views

Measure the uniformity of distribution of points in a 2D square

I have a 2D square, and I have a set of points inside it, say, 1000 points. I need a way to see if the distribution of points inside the square are spread out (or more or less uniformly distributed) ...
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1answer
186 views

Interpretation of spatial Gcross plots

On page 192 of Analysing spatial point patterns in 'R' (Baddeley 2011), there are plots of the Gcross function for the amacrine dataset. I am looking for an interpretation of the plot. off/off and ...
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342 views

Edge correction of Ripley's K-function for two 1D point processes?

I am just beginng an investigation involving characterizing the dependence between two 1D stochastic point processes $x$, $y$. The natural approach seems to involve Ripley's K-function: $$ K(t) = ...
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6answers
3k 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) ...
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55 views

Learning models for dependent point processes

This is largely a literature request. I have some data consisting of, for each use of a credit card, the card owner, the store, and the time of the use. This could be viewed as a collection of ...
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82 views

Markov point processes and mobility

I want to model the case where a particular message is forwarded from a source to multiple nodes with transmission radius r (in multiple hops), until the message reaches a particular destination. The ...
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178 views

How to estimate the intensity of a multidimensional point process?

Note that for a homogeneous point process the density is just a number, while for an in-homogeneous process it is a function. In addition, how can I distribute that function on a larger study region ...
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2answers
165 views

Edge effects in K-function

I am trying to write some codes for distance-based evaluation of point process such as $G$, $F$ and $K$ functions. In the implementation of $K$ there is a case to consider edge effect to have an ...
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2answers
653 views

What are similarity measures between a line and a set of points?

A colleague discussed about a concept of similarity between two different entities line and a set of points. My first guess for solution was considering distances squared ($d^2$) as in LS. So for the ...
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1answer
210 views

What is conditioning in spatial statistics?

Could someone explain to me: the concept of "conditioning" in spatial statistics in a fairly advanced context? Here is an example to clarify the question: Step 1) generate a 2D point process, here 6 ...
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337 views

Dependent thinning Poisson process

If $N_1$ and $N_2$ are independent Poisson processes then the superposition is a Poisson process. Is it possible to construct two dependent Poisson processes such that the superposition is a Poisson ...
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1answer
123 views

Attractive pairwise interaction point process

I'm doing some spatial pattern analysis. After looking over some work on Markov Point Processes, I'm finding that all of the pairwise interaction processes are 'repulsive'. In "Statistical Inference ...
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156 views

How to condition a marked point processes?

Conditioning a point process is the concept being investigated in this post. The marked point processes are of interest. Figure A demonstrates a phenomenon being investigated for modelling. We ...
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1answer
404 views

Mixing and dividing point processes

At the following figure at left side two realizations of point processes with different density (intensity) $\lambda_1$ and $\lambda_2$ is being mixed matching the center of the belonging areas to ...
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1answer
2k views

What are density and intensity of point pattern?

A simplified explanation with the focus on the following questions is being inquired. Appreciation goes in advance to whom provides scientific--simple--practical explanation. Having an area A ...
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167 views

How to condition a point process to the actual sampling data?

Figure A demonstrates a point process (object=rectangle) with marks as the length and width of the rectangle. Figure B shows a realization of the point process regarding the information extracted from ...
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2answers
4k views

How to generate nD point process?

Context: Poisson point processes (PPP) are widely discussed in the literature. In the following figure a framework to generate two-dimensional PPP is demonstrated. First the area being studied (part ...
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2answers
745 views

How to define marked point processes?

Background: In mining engineering, sampling is used to get the rate of concentration of minerals in rocks. Sampling procedure is carried out in the field on the area being explored at the predefined ...
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297 views

Moments of function of Poisson process

Mirror thread on Mathoverflow. (I'm new to Poisson processes, so please edit if my terminology is incorrect.) This is a special case of a problem I'm working on; hoping for intuition that will ...
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141 views

How to denoise a “Poissonous” time series

I have $N$ time series each of which can be modeled as $$y_{kt}=Ax_{kt}+b+\varepsilon_{kt}\quad(1\le k\le N,1\le t\le T),$$ where $x_{kt}\sim\text{Pois}(\lambda\Delta t)$ and $\varepsilon_{kt}\sim ...
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1answer
263 views

When doing quadrat counting, how do you construct the quadrats?

I want to perform quadrat count analysis on several point processes (or one marked point process), to then apply some dimensionality reduction techniques. The marks are not identically distributed, ...
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632 views

How to find relationships between different types of events (defined by their 2D location)?

I have a dataset of events that happened during the same period of time. Each event has a type (there are few different types, less then ten) and a location, represented as a 2D point. I would like ...