Melb2018

Lab 7: Envelopes and Monte Carlo tests

This session is concerned with evelopes of summary statistics and Monte Carlo tests. The lecturer’s R script is available here (right click and save).

library(spatstat)
## Loading required package: spatstat.data

## Loading required package: methods

## Loading required package: nlme

## Loading required package: rpart

## 
## spatstat 1.56-1.007       (nickname: 'Damn You Autocorrect') 
## For an introduction to spatstat, type 'beginner'

Exercise 1

For the swedishpines data:

  1. Plot the (K) function along with pointwise envelopes from 39 simulations of CSR:

    plot(envelope(swedishpines, Kest, nsim=39))
    

    OK,

    plot(envelope(swedishpines, Kest, nsim=39), main = "")
    
    ## Generating 39 simulations of CSR  ...
    ## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
    ##  39.
    ## 
    ## Done.
    

  2. Plot the (L) function along with pointwise envelopes from 39 simulations of CSR.

    Like above now with Lest:

    plot(envelope(swedishpines, Lest, nsim=39), main = "")
    
    ## Generating 39 simulations of CSR  ...
    ## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
    ##  39.
    ## 
    ## Done.
    

  3. Plot the (L) function along with simultaneous envelopes from 19 simulations of CSR, using ginterval=c(0,5).

    plot(envelope(swedishpines, Lest, nsim = 19, global = TRUE, ginterval=c(0,5)), main = "")
    
    ## Generating 19 simulations of CSR  ...
    ## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,  19.
    ## 
    ## Done.
    

  4. Plot the (L) function for along with simultaneous envelopes from 99 simulations of CSR using ginterval=c(0,5). What is the significance level of the associated test?

    plot(envelope(swedishpines, Lest, nsim = 99, global = TRUE, ginterval=c(0,5)), main = "")
    
    ## Generating 99 simulations of CSR  ...
    ## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
    ## 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76,
    ## 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,  99.
    ## 
    ## Done.
    

    Which yields an 1% significance test.

Exercise 2

To understand the difficulties with the (K)-function when the point pattern is not spatially homogeneous, try the following experiment (like in the previous lab session).

  1. Generate a simulated realisation of an inhomogeneous Poisson process, e.g.

    X <- rpoispp(function(x,y){ 200 * exp(-3 * x) })
    

    OK,

    X <- rpoispp(function(x,y){ 200 * exp(-3 * x) })
    
  2. Compute simulation envelopes (of your favorite type) of the (K)- or (L)-function under CSR. They may well indicate significant departure from CSR.

    There indeed often seems to be a departure from CSR:

    par(mfrow=c(1,2))
    plot(envelope(X, Kest, global = TRUE))
    
    ## Generating 99 simulations of CSR  ...
    ## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
    ## 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76,
    ## 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,  99.
    ## 
    ## Done.
    
    plot(envelope(X, Lest, global = TRUE))
    
    ## Generating 99 simulations of CSR  ...
    ## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
    ## 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76,
    ## 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,  99.
    ## 
    ## Done.
    

  3. Fit a Poisson point process model to the japanesepines data with log-quadratic trend (formula ~polynom(x,y,2)). Plot the (L)-function of the data along with simultaneous envelopes from 99 simulations of the fitted model.

    This can be done by the code:

    fit <- ppm(japanesepines ~ polynom(x, y, 2))
    plot(envelope(fit, Linhom, global = TRUE), main = "")
    
    ## Generating 198 simulated realisations of fitted Poisson model (99 to 
    ## estimate the mean and 99 to calculate envelopes) ...
    ## 1, 2, 3, 4.6.8.10.12.14.16.18.20.22.24.26.28.30.32.34.36.38.
    ## 40.42.44.46.48.50.52.54.56.58.60.62.64.66.68.70.72.74.76.78
    ## .80.82.84.86.88.90.92.94.96.98.100.102.104.106.108.110.112.114.116.
    ## 118.120.122.124.126.128.130.132.134.136.138.140.142.144.146.148.150.152.154.156
    ## .158.160.162.164.166.168.170.172.174.176.178.180.182.184.186.188.190.192.194.
    ## 196. 198.
    ## 
    ## Done.