Stat 5810, Applied Spatial Statistics

Homework Assignment 4 (9/27/00)

25 Points - Due Fr 10/6/00 by 4pm in my office or as a Web page

You may work in groups of 2 people on this assignment and turn in a joint solution.

You have to use S-Plus and S+SpatialStats to answer all of the following questions. Your solutions should consist of the S-Plus commands you have typed, any S-Plus functions you have written, and the output (text and postscript graphics) produced in S-Plus, i.e., exactly the same things as for Homework Assignment 3.

1) Write a S-Plus function that creates a CSR pattern in the triangular region with vertices (x, y) = (0, 0), (1, 0), and (0.5, 1). Note that make.pattern only supports rectangular boundaries - so you have to develop a completely new function based on the simulation mechanism described in Section 2.4.1. It might be easiest to develop 2 functions instead of one, i.e., one function that checks whether a randomly created point falls into the triangular region and one function that creates points in the rectangle that encloses the triangle until the desired number of points has been obtained.

Simulate n = 1000 events inside this triangular region. Plot the raw data (and the triangular boundary) and provide at least two meaningful plots that verify that your simulated data indeed represents a homogeneous Poisson process in the triangular region. Comment on your results. (5 Points)
2) Write a S-Plus function for the Index of Dispersion Test discussed in Section 2.4.2. Work with randomly scattered quadrats in the region of interest (only use quadrats that fully fall inside this region). Input to this function should be the number m of quadrats, the identical area A of each quadrat, and an object of type spp. Make sure that the required conditions for this test are met (if not, prompt the user to increase m or use a larger area A). Test your function with the quakes.bay sample data set and different values for m and A. (5 Points)
3) Look at the other 3 sample point patterns that are available in S+SpatialStats, i.e., bramble, lansing, and quakes.wash. Graphically (using at least 2 exploratory plots) assess whether one of these point patterns might represent a homogeneous Poisson process. (5 Points)
4) Using the julian function that converts between Julian and Calendar dates (an additional handout will be provided in class on Monday), extract the difference in days between two consecutive earthquakes from the quakes.bay data set. The diff command might also be helpful.

Plot your obtained differences in days. Does your plot resemble any familiar distribution, e.g., exponential, gamma, log-normal, ...? If you think you identified the shape, use ML estimation to estimate the unknown parameter(s) of the assumed distribution. Use QQ plots (assuming a theoretical distribution based on your ML estimates) to further determine the distribution of the underlying data. (5 Points)
5) Argue why under H0, i.e., under CSR, the Byth & Ripley test statistic is approximately N(1/2, 1/(12m)) distributed. You can base your answer completely on simulations in S-Plus or formal arguments (without using S-Plus) or also combine simulation and formal arguments. (5 Points)