6
votes
Are these two equivalent forms for the likelihood of a Poisson point process?
I would consider only the first one as being the likelihood of a Poisson point process. It is unclear what the second expression is aiming at.
Point processes are complex and so is the formulation of ...
- 1,514
5
votes
Stationarity assumption for a point process
Stationarity implies that the statistical parameters of an underlying process do not vary over space (or time), and is often formally described as "invariance under translation" (not to be confused ...
5
votes
Accepted
Definition of point processes
There'a a lot going on here, but it is primarily a matter of mathematical notation and language. I will discuss a bunch of related in issues in the context of a simple running example. I won't ...
- 334k
4
votes
Accepted
Spatial Point Process: Does an inhomogeneous first order intensity function affect the second order dependence?
First order intensity and second order intensity measure different aspects of a process that can be almost independently varied. In particular, not every point process can be regarded as an ...
- 334k
4
votes
Accepted
Point Pattern Analysis: Assumptions for Hopkins-Skellam Index
I just want to chime in with @whuber and thank you for a well written question.
I can add that the null hypothesis of the Hopkins-Skellam test
is CSR, so your result is consistent with your other ...
- 1,228
3
votes
Fit Hawkes process to 1d data using python package TICK
the tick module only allows you to estimate your Hawkes process for a fixed value of the decay $\beta$ for an exponential kernel $\phi(t) = \alpha \beta \exp (-\...
- 131
3
votes
Accepted
Hawkes-Process for non binary events
One approach to answer your question is to use a multivariate Hawkes process to handle events associated with different volumes (or counts in your case). Instead of considering a one-dimensional ...
- 146
3
votes
Measuring correlation of point processes
There are algorithms to compute cross-correlation of two point process (times of events) directly without any binning of the input. A classical one is the multi-tau algorithm that is used in ...
- 463
3
votes
Accepted
How to test change in rate of event occurrences?
The kind of data you are asking about represent a point process. There are various kinds, but the most common kind to work with are (homogeneous) Poisson processes, which seems to be what you have in ...
3
votes
Is there any gold standard for modeling irregularly spaced time series?
In spatial data analysis data is most of the time sampled irregularly in space. So one idea would be to see what is done there, and implement variogram estimation, kriging, and so on for one-...
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3
votes
Accepted
What is the "sensible default" for the Kest function of the spatstat package?
The answer is in spatstat's FAQ:
How are the r values determined in Kest(X) ?
The default r values for Kest are computed as follows:
The maximum r value is ...
- 4,362
3
votes
Is the following definition of the variance of the number of points correct?
The answer is in a paper by Torquato on hyperuniformity, linked from the Wikipedia page that you cite. In 2 dimensions as in your diagram, it's the variance of the number of particles among multiple ...
- 102k
3
votes
Is the following definition of the variance of the number of points correct?
I am hopeful that the following code and resulting graph will be helpful. I believe that what you're reading is spreading the points over a wide area and varying where the circle is centered. I ...
- 67.2k
3
votes
Algorithm for generating a Poisson process on a complicated 2d geometry
A very simple solution (in the same spirit as the answer by @jbowman) is to generate a Poisson point process in the bounding box and then restrict to the irregular region. This is valid since for a ...
- 1,228
2
votes
Accepted
Averaging Ripley's K or L function for several samples
May I suggest that you look at chapter 16 of Baddeley, Rubak & Turner (2015) Spatial Point Patterns: Methodology and Applications with R. Specifically Section 16.8 Analysing summary functions from ...
- 1,228
2
votes
Accepted
Is the Matérn covariance function associated with the Matérn cluster process?
Is the Matérn covariance function associated with the Matérn cluster
process?
No. Except from the fact that they are named after the same person.
The Matérn cluster process is treated in many ...
- 1,228
2
votes
Does it make sense to consider ratio of pair correlation functions of two point patterns?
The standard procedure is to generate several (e.g. 99) simulations based on the fitted model and then use these to make envelopes of the estimated pair correlation function. This allows you to see ...
- 1,228
2
votes
Fit Hawkes process to 1d data using python package TICK
i found it very frustrating that all the samples i found in the tick doc used simulated data which i couldn't use to effectively understand how to feed real data in. i had to run one of their ...
- 121
2
votes
Spatial Point Process: Does an inhomogeneous first order intensity function affect the second order dependence?
In broad terms your understanding sounds right. In particular, you are right that it is basically impossible to distinguish "first order inhomogeneity" and "second order clustering due to interactions ...
- 1,228
2
votes
Accepted
trimming window in Foxall G-function and J-function in R spatstat
Thanks for a well described problem. The code for Jfox (together with most of spatstat) was written by Adrian Baddeley, and I just passed your question to him. The ...
- 1,228
2
votes
How to simulate spatial point patterns that have spatial structure similar to that of given spatial point pattern?
This is very hard to answer in general. How closely do you want the new pattern to resemble the original pattern? If you want the exact same number of points most standard point process models (such ...
- 1,228
2
votes
Accepted
Uniform distribution versus a Poisson homogeneous process with Ripley's K
Following whuber's advice I checked the methods for edge effects correction:
none:
this method does not take into account any edge effects whatsoever.
...
- 4,362
2
votes
spatstat::plot.envelope observed line longer than shading
This behaviour is correct. The shaded region is the envelope of 39 curves which were each computed by generating a completely random pattern and computing the estimate of Foxall's J function ...
- 176
2
votes
Predict background counts given past observations and assumption of linear variation of the rate parameter in time
You have a Poisson point process with intensity (rate) function $\lambda(t)$, say. Assume the observation window is contained in the interval $[0, T]$ and the observed points $t_1, t_2, \dotsc, t_n$. ...
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2
votes
Algorithm for generating a Poisson process on a complicated 2d geometry
You can find the areas of the counties in California from https://en.wikipedia.org/wiki/List_of_counties_in_California. This will enable you to generate a count $n_i$ from the appropriate Poisson ...
- 41.1k
2
votes
pointwise envelopes not including Theoretical line Foxall J
Please provide a minimal working example.
Pooling of objects (such as simulation envelopes) is only justified when the individual objects (envelopes) were created under identical conditions. In your ...
- 176
2
votes
Accepted
Simulation envelopes for Foxall J not containing the line y=1 (fully reproducible example)
No.
No.
No.
A simulation envelope is constructed by finding the largest and smallest values of the simulated summary functions. The simulated function values are random, so there is no guarantee that ...
- 176
2
votes
Accepted
(Using conditional expectation to calculate) expected value of the product of two dependent random variables
If you know how to compute the expectation for two disjoint sets, then you can define $A_{1, 1} = [0, 2] \times [2, 4]$ and $A_{1, 2} = [0, 2] \times [0, 2]$, and $A_{2, 1} = [2, 6] \times [0, 2]$. ...
- 741
2
votes
Accepted
Normalizing constant calculation of Strauss Process
There are many X validated entries on the approximation of a normalizing constant given a sample from the associated distribution:
Can MCMC algorithm estimate partition function (normalizing constant)...
- 108k
2
votes
Statistical test whether data conforms to a spatial point process--gaza bombing locations
Some thoughts, quite naive perhaps. Use or modify as you see fit.
If you have information on which regions are built and which are not, you can make use of simulation. Compute a suitable statistic of ...
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