Example: Data

Lecture 12: Point processes Spatial and Image Analysis

David Bolin University of Gothenburg

Gothenburg May 15, 2019 The locations of the tree species Beilschmiedia Pendula in the tropical rainforest plot on Barro Colorado Island.

Point processes David Bolin

Types of spatial data Example: Covariates

Three main types of data in spatial statistics Continuously indexed data. • s ,...,s • We have observations at some fixed locations 1 N of a random field X(s) where s is in some region D R2. X is a ✓ random field defined on D. Discretely indexed data. • s ,...,s • We have observations at some fixed locations 1 N of a random field X(s) where s is in some discrete set of locations D˜, such as a regular lattice. X is a random field defined on D˜. data. • s ,...,s • We have observations 1 N indicating where something occurred. We want to draw conclusions about the process based on these locations, which are now considered to be random. Possible covariates that can be used for drawing conclusions on the association of habitat preferences.

Point processes David Bolin Point processes David Bolin Different types of point patterns Example: Swedish pine saplings

One of the main questions for point pattern data is usually to • determine if we have clustering or repulsion. The completely random case corresponds to the Poisson • process. One typically use the K function to decide if we have • clustering or repulsion.

Point processes David Bolin The K function David Bolin

Example: Swedish pine saplings Neyman-Scott process

Hierarchical construction Simulate a “mother process” Z as a Poisson process. • For each point z Z, simulate n independent “daughter • i 2 i points” x from a distribution ⇡(x z ) and remove Z. j | i For example: Let n =5and take x z be uniform in a disc of • i | i radius 0.05 centered in zi. Extension: take n from some distribution ⇡(n). • i The K function David Bolin More advanced models David Bolin Matérn Type 1 inhibition process Inhomogeneous Poisson process

Simulate a homogeneous Poisson process Z. • Delete any point in Z that lies closer than a distance r from Apoissonprocesswithaspatiallyvaryingintensityfunction. • • the nearest other point. Example (x, y) = 100 exp( 3x). • Thus, pairs of close neighbours annihilate each other. • r is called the hard core distance. •

More advanced models David Bolin More advanced models David Bolin

Matérn Type 2 inhibition process Log-Gaussian

Simulate a homogeneous Poisson process Z. • Mark each point in Z by “ages”, which are independent and • Hierarchical model, where X is a Gaussian random field and uniformly distributed numbers in [0, 1]. • Z X is an inhomogeneous Poisson process where Delete any point in Z that lies closer than a distance r from | • (x, y)=exp(X(x, y)). another point that has has a higher age. Example: X is a Gaussian random field with mean 3 and an This process was originally used to model forests. • • exponential function. More advanced models David Bolin More advanced models David Bolin Example: Barro Colorado Island Models for grayscale images based on point processes

Inhom. Poisson Proc. 4

Point pattern 3

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1 LGCP

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-1 We can extend the boolean model to a model for grayscale • -2 images by replacing the discs with kernel functions. Example: Let the value at location s in the image be • For the inhomogeneous Poisson process, the intensity function • I(s)= ⇡ (s; z ,r I) is a regression on the soil covariates. G i i i For the LGCP the intensity function is the regression plus a X • mean-zero Gaussian random field. where ri are some positive random variables.

More advanced models David Bolin More advanced models David Bolin

Marked point processes and boolean models Project report

Parts 1 and 2 should be documented as lab reports, describing • what you did and the results you obtained but it does not need adetailedintroductionanddiscussion. The ideal form of the report for Part 3 is in principle a journal • paper, containing: 1 Project title, author names, course name, date of report. 2 Abstract/Summary: about 10-15 lines For each point z in the point pattern, assign a random “mark” • i 3 Introduction: Statement of problem, earlier work with Mi. references. Can be used to include more information in spatial point 4 Data description and source. • pattern data (such as tree size or age). 5 Methods: Mathematical, statistical, computational, image Models like this can also be used to build more complicated analysis. • spatial models, such as Boolean models: 6 Results with tables and figures. 7 Discussion: include if possible here also comparison with For each zi, define a disc Di centered at the point, with a • results from literature. random radius ri. 8 Conclusions, suggested continued studies. • Define a binary image by letting the pixels have value one in the region D ,andzeroelsewhere. 9 References. [i i More advanced models David Bolin Project David Bolin Deadlines Preliminary schedule Monday 10:00-12:00 Monday 13:15-15:00 For Parts 1 and 2: G14 1persons,9minutes G15 1person,9minutes G5 2persons,16minutes G23 1person,9minutes A PDF containing these two parts should be submitted via • PINGPONG at the latest May 17, 23:55. Include the code as a G12 2persons,16minutes G13 3persons,23minutes zip-file. G20 3persons,23minutes G6 2person,16minutes G2 1person,9minutes G4 2persons,16minutes G25 1person,9minutes G7 1persons,9minutes For Part 3: G11 1persons,9minutes G17 1persons,9minutes A PDF containing a preliminary version of the report should be • submitted together with Parts 1 and 2 at the latest May 17, Wednesday 10:00-12:00 Wednesday 13:15-15:00 23:55. It does not need to be complete. G24 3persons,23minutes G8 1persons,9minutes The final version of the report should be submitted at the G9 1persons,9minutes G22 2persons,16minutes • latest May 31, 23:55, together with the code as a zip-file, as a G10 3persons,23minutes G18 1persons,9minutes revision of the submitted project in PINGPONG. G16 1persons,9minutes G1 3persons,23minutes G3 3persons,23minutes G21 2persons,16minutes G19 2persons,16minutes Project David Bolin Project David Bolin

Project seminars Exam

The lectures and computer exercises next week will be devoted • Takes place on June 5: 8:30 - 12:30. to project seminars. • In the seminar there will be 7 minutes allotted to each group The exam can give 20 points, and the project reports another • • participant, where you should describe what you have done so 20 points. The final grade is based on the sum of the points from the far and what you plan to do. • After this there will be 2 minutes left for discussion for each exam and the project. • You need to have at least 8 points on the exam, as well as 8 group. • The object of the seminar is that you should get feedback from points on the project to pass the course. • The only aid you may bring to the exam is a “Chalmers the audience, both to point out parts that are less clear and • suggestions on what to do, the goal being to help you to write allowed” calculator (will likely not be needed). as good a report as possible.

Project David Bolin Exam David Bolin