Statistics for Point Patterns in Space and Beyond

This is the website for participants in the summer school Statistics for Point Patterns in Space and Beyond given by Jesper Møller, Rasmus Waagepetersen, Jakob Gulddahl Rasmussen, Christophe Biscio and Ege Rubak as part of the 1st International Summer School for PhD Students and Other Young Researchers in Aalborg, August 22-26 2022.

Location: Aalborg University, Fredrik Bajers Vej 7A room 4-108, DK-9200 Aalborg Ø

Getting credit for the course

To receive a course diploma (4 ECTS) the participants will be evaluated by successful participation during the lectures, including the solution of exercises. For the final assessment, a satisfactory solution of selected exercises (henceforth called hand-in exercises) are required with deadline September 11, 2022. The hand-in exercises and to whom they should be send are listed as part of the description of each day of the workshop below. Please write your name clearly on the handin exactly as it appears on PhD Moodle, so we can easily cross reference with the participant list.

Required preparation for the course

You will need a working install of R (and RStudio) with the package spatstat to participate, so please go through these steps:

1. If you don’t have R (mandatory) and Rstudio (optional, but highly recommended) installed please do so as described here (note as described you need to choose a non-standard install option for R during install on Windows to be sure to avoid problems)

2. Watch this video about using RStudio (feel free to only skim it if you are already RStudio user or consciously choose another interface to R): https://youtu.be/mAtAHwiP204

3. When you have R installed on your own computer, please install spatstat (http://cran.r-project.org/package=spatstat) and all its dependencies by running this R command in the R console: install.packages("spatstat", dependencies = TRUE)

Monday and Tuesday (lecturers Rasmus Waagepetersen and Ege Rubak)

These days are divided into a morning and afternoon session for a total of four sessions.

Each session consists of:

1. A lecture based on the same slide set (updated Aug. 22) for all sessions
2. Exercises from the slides using “pen and paper”
3. Software demonstration based on lab notes (see below)
4. Lab exercises on the computer

Topic	Lab notes	Lab exercises	Lab solutions	Hand-in exercise
Intro and moments	Notes 1	Lab 1	Lab solutions 1	Exerc. 1 in Intro sec. of slides
Poisson and envelopes	Notes 2	Lab 2	Lab solutions 2	Exerc. 2 in Poisson sec. of slides
Cox/cluster and estimating eqns.	Notes 3	Lab 3	Lab solutions 3	Exerc. 1 in Cox/cluster sec. of slides or comp. exerc. html,pdf,Rmd
Determinantal and Markov models	Notes 4	Lab 4	Lab solutions 4	Computer exercise html,pdf,Rmd

Additional slides about determinantal point processes for session 4.

Hand-in exercises for Monday and Tuesday

Possible hand-in exercises are listed in the table above. You need to hand in a total of three exercises for this part of the workshop. There has to be at least one theoretical exercise and one computer exercise in your hand in. Solutions for the theoretical exercises should be send to Rasmus ([email protected]) while the solution to the computer exercises should be send to Ege ([email protected]).

Wednesday (lecturer Jakob Gulddahl Rasmussen)

• Slides (spacetime-slides.pdf) about temporal and spatio-temporal point processes
• R-code for demonstrating temporal and spatio-temporal point processes

Hand-in exercises for Wednesday

• One of the exercises at slide 11, 17, 33 or 35 in spacetime-slides.pdf should be handed in to Jakob ([email protected]).

Optional reading for Wednesday

This is not required but may be helpful when solving exercises:

• Lecture notes on marked temporal point processes

Thursday (lecturers Jakob Gulddahl Rasmussen and Jesper Møller)

Thursday consists of four lectures

• First lecture: We continue with the slides and R-code from day 3 (Jakob).
• Second lecture: Slides (lecture_networks_no_pause.pdf) about point processes on linear networks (Jesper).
• Third lecture: Slides (lecture_spheres_no_pause.pdf) about point processes on the sphere (Jesper).
• Fourth lecture: Slides (dln-slides.pdf) about point processes on directed linear networks (Jakob).

Hand-in exercises for Thursday

• One of the exercises at slide 11, 18, 19, 30 or 44 in lecture_spheres_no_pause.pdf should be handed in to Jesper ([email protected]).
• One of the exercises at slide 23 or 44 in lecture_networks_no_pause.pdf should be handed in to Jesper ([email protected]).
• One of the exercises at slide 5, 10 or 15 in dln-slides.pdf should be handed in to Jakob ([email protected]).

Optional reading for Thursday

This is not required but may be helpful when solving exercises:

• Paper on log Gaussian Cox processes on the sphere (second lecture)
• Review paper on point processes on linear networks (third lecture)
• Paper on point processes on directed linear network (also at arXiv: 1812.09071) (fourth lecture)

Friday (lecturer Christophe Biscio)

Topological data analysis (TDA) is a new field that aims to use tools from topology to draw information on the “shape” of data. It has in the past years been successfully used in various fields, including in spatial statistics.

However, the mathematical background and theory in TDA are often unfamiliar for most statisticians.

The aim of this session is to give you an informal introduction to TDA, with a clear understanding of its main concepts and mathematical objects.

This will be illustrated with applications in spatial statistics to assess the goodness-of-fit of a point process.

The session will be accompanied by tutorials and exercices with the R package TDA.

Material for the session

Please find below the slides and codes reviewed during the course. Please do not hesitate to send me an email if you have any questions.

• TDA_Exercices_session.pdf
• TDA_Lecture1_part1.pdf
• TDA_Lecture1_part2.pdf
• Lecture1_Tutorial_TDA.html

Hand-in exercises for Friday

• Your solution to this exercise should be handed in to Christophe ([email protected]).

Optional reading for Friday

This is not required but may be helpful when solving exercises:

• Articles:
  • Biscio, C.A.N., Chenavier N., Hirsch C., Svane A.M. Testing goodness of fit for point processes via topological data analysis. Electronic Journal of Statistics, 14, 1024–1074, 2020.
  • Biscio, C.A.N., Møller, J. The accumulated persistence function, a new useful functional summary statistic for topological data analysis, with a view to brain artery trees and spatial point process applications. Journal of Computational and Graphical Statistics, 28, 671–681, 2019.
• Books:
  • Tamal K. Dey and Yusu Wang Computational Topology for Data Analysis, Cambridge University Press. Freely available online: https://www.cs.purdue.edu/homes/tamaldey/book/CTDAbook/CTDAbook.html
  • Herbert Edelsbrunner and John Harer Computational Topology - An Introduction, American Mathematical Society, Providence, RI. Freely available online: https://www.maths.ed.ac.uk/~v1ranick/papers/edelcomp.pdf

Information for developers

The Rmarkdown source code for some parts is available at https://github.com/spatstat/Aalborg2022

This material is Copyright (C) Jesper Møller, Rasmus Waagepetersen, Jakob Gulddahl Rasmussen, Christophe Biscio, Ege Rubak and Adrian Baddeley 2022