4  Simulation

ImportantDisclaimer

These packages (Note 1) are a one-person project undergoing rapid evolution. Backward compatibility (per Hadley Wickham) is provided as a courtesy rather than a guarantee.

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Note

The examples in Chapter 4 require

library(groupedHyperframe.random)

4.1 Simulated Point-Pattern

The function .rppp() (Chapter 36) simulates superimposed point-patterns (ppp.object, Chapter 24) via vectorized parameterization, which is supported in the parameter-specification of both the random point-pattern and the random marks.

Listing 4.1 simulates an \(x\)- and \(y\)-coordinates only, two superimposed Matérn (1986)’s cluster processes \((\kappa, \mu, s)\) with parameters \((\kappa_1, \mu_1, s_1) = (10,8,.15)\) and \((\kappa_2, \mu_2, s_2) = (5,4,.06)\) using a vectorized parameterization of Table 4.1.

Table 4.1: Vectorized parameterization of Listing 4.1 and parameter restriction

	kappa \(\kappa\)	mu \(\mu\)	scale \(s\)
Vecterization	\((\kappa_1, \kappa_2)\)	\((\mu_1, \mu_2)\)	\((s_1, s_2)\)
Restriction	\(\kappa_i>1\)	\(\mu_i>1\)	\(s_i>0\)

Listing 4.1: Superimposed Matérn (1986)’s cluster processes

set.seed(12); r = .rppp(rMatClust(kappa = c(10, 5), mu = c(8, 4), scale = c(.15, .06)))

Point-pattern simulated by `spatstat.random::rMatClust()`

Listing 4.2 simulates two superimposed marked point-patterns,

1. one marked point-pattern with
   • \(x\)- and \(y\)-coordinates by Matérn (1986)’s cluster process with parameters \((\kappa_1,\mu_1,s_1) = (10,8,.15)\);
   • log-normal \((\log\mu, \log\sigma)\) marks with parameters \((\log\mu_1,\log\sigma_1)=(3,.4)\);
   • negative-binomial \((r, p)\) marks with parameters \((r,p)=(4,.3)\).
2. another marked point-pattern with
   • \(x\)- and \(y\)-coordinates by Matérn (1986)’s cluster process with parameters \((\kappa_2,\mu_2,s_2) = (5,4,.06)\);
   • log-normal marks with parameters \((\log\mu_2,\log\sigma_2)=(5,.2)\)
   • negative-binomial marks with parameters \((r,p)=(4,.3)\).

using a vectorized parameterization of Table 4.2.

Table 4.2: Vectorized parameterization of Listing 4.2 and parameter restriction

	kappa \(\kappa\)	mu \(\mu\)	scale \(s\)	meanlog \(\log\mu\)	sdlog \(\log\sigma\)	size \(r\)	prob \(p\)
Vec.	\((\kappa_1, \kappa_2)\)	\((\mu_1, \mu_2)\)	\((s_1, s_2)\)	\((\log\mu_1, \log\mu_2)\)	\((\log\sigma_1, \log\sigma_2)\)	\((r_1, r_2)\)	\((p_1, p_2)\)
Res.	\(\kappa_i>1\)	\(\mu_i>1\)	\(s_i>0\)	(none)	\(\log\sigma_i>0\)	\(r_i>0\)	\(0<p_i<1\)

Listing 4.2: Superimposed Matérn (1986)’s cluster processes with log-normal and negative-binomial marks

set.seed(25); r1 = .rppp(
  rMatClust(kappa = c(10, 5), mu = c(8, 4), scale = c(.15, .06)), 
  rlnorm(meanlog = c(3, 5), sdlog = c(.4, .2)),
  rnbinom(size = 4, prob = .3) # shorter arguments recycled
)

Point-pattern simulated by `spatstat.random::rMatClust()`

Mark simulated by `stats::rlnorm()`
Mark simulated by `stats::rnbinom()`

Listing 4.3 simulates two superimposed point-patterns,

1. one marked point-pattern with
   • \(x\)- and \(y\)-coordinates by Poisson point-pattern \(\lambda\) with parameter \(\lambda_1=3\),
   • log-normal marks with parameters \((\log\mu_1,\log\sigma_1)=(3,.4)\);
   • negative-binomial marks with parameters \((r_1,p_1)=(4,.3)\).
2. another marked point-pattern with
   • \(x\)- and \(y\)-coordinates by Poisson point-pattern with parameters \(\lambda_2=6\)
   • log-normal marks with parameters \((\log\mu_2,\log\sigma_2)=(5,.2)\)
   • negative-binomial marks with parameters \((r_2,p_2)=(6,.1)\).

using a vectorized parameterization of Table 4.3.

Table 4.3: Vectorized parameterization of Listing 4.3 and parameter restriction

	lambda \(\lambda\)	meanlog \(\log\mu\)	sdlog \(\log\sigma\)	size \(r\)	prob \(p\)
Vecterization	\((\lambda_1, \lambda_2)\)	\((\log\mu_1, \log\mu_2)\)	\((\log\sigma_1, \log\sigma_2)\)	\((r_1, r_2)\)	\((p_1, p_2)\)
Restriction	\(\lambda_i>0\)	(none)	\(\log\sigma_i>0\)	\(r_i>0\)	\(0<p_i<1\)

Listing 4.3: Superimposed Poisson point-patterns with log-normal and negative-binomial marks

set.seed(62); r2 = .rppp(
  rpoispp(lambda = c(3, 6)),
  rlnorm(meanlog = c(3, 5), sdlog = c(.4, .2)),
  rnbinom(size = c(4, 6), prob = c(.3, .1))
)

Point-pattern simulated by `spatstat.random::rpoispp()`

Mark simulated by `stats::rlnorm()`
Mark simulated by `stats::rnbinom()`

Listing 4.4 superimposes the simulated (marked) point-patterns r1 and r2 (Listing 4.2, Listing 4.3).

Listing 4.4: Superimpose simulated point-patterns (Listing 4.2, Listing 4.3)

spatstat.geom::superimpose.ppp(r1, r2)

Marked planar point pattern: 140 points
Mark variables: rlnorm, rnbinom 
window: rectangle = [0, 1] x [0, 1] units

4.2 Simulated Grouped Hyper Data Frame

The function grouped_rppp() (Chapter 32) simulates a grouped hyper data frame containing one-and-only-one point-pattern (ppp) hypercolumn (Chapter 3) via matrix parameterization, which is supported in the subject-specific parameter-specification of both the random point-pattern and the random marks.

4.2.1 with Superimposed Matérn (1986)’s Cluster Processes

Table 4.4 outlines the three steps described in Section 4.2.1,

Table 4.4: Steps of Section 4.2.1

	To Simulate	Primary Function
1.	subject-specific parameters of (marked) point-patterns, based on multivariate normal distribution, Listing 4.5	mvrnorm2() (Chapter 33)
2.	number of (marked) point-patterns per subject, Listing 4.7	sample()
3.	grouped hyper data frame with one-and-only-one ppp-hypercolumn, Listing 4.8	grouped_rppp() (Chapter 32)

Consider two superimposed Matérn (1986)’s cluster processes, each with one log-normal mark. The population parameters are

• \(x\)- and \(y\)-coordinates by Matérn (1986)’s cluster process \((\kappa_1,\mu_1,s_1) = (3,10,.4)\), with a log-normal mark \((\log\mu_1,\log\sigma_1)=(3,.4)\)
• \(x\)- and \(y\)-coordinates by Matérn (1986)’s cluster process \((\kappa_2,\mu_2,s_2) = (2,5,.2)\), with a log-normal mark \((\log\mu_2,\log\sigma_2)=(5,.2)\)

Listing 4.5 simulates for three (3L) subjects (e.g., patients). The subject-specific parameters deviate from the population parameters under a multivariate normal distribution specified by the standard deviation(s) \(\sigma\) of the subject-specific random effects, i.e., \(\sigma_\kappa=.2\), \(\sigma_\mu=.5\), \(\sigma_s=.05\), \(\sigma_{\log\mu}=.1\) and \(\sigma_{\log\sigma}=.01\), as well as the restrictions in Table 4.2.

Listing 4.5: Create subject-specific parameters p_Matern

set.seed(39); p_Matern = mapply(
  FUN = mvrnorm2, 
  mu = list(kappa = c(3,2), mu = c(10,5), scale = c(.4,.2), meanlog = c(3,5), sdlog = c(.4,.2)), 
  sd = list(kappa = .2, mu = .5, scale = .05, meanlog = .1, sdlog = .01), 
  MoreArgs = list(n = 3L, row.prefix = 'Subj', col.prefix = 'Pattern'), 
  SIMPLIFY = FALSE
) |>
  within.list(expr = {
    kappa = pmax(kappa, 1 + .Machine$double.eps)
    mu = pmax(mu, 1 + .Machine$double.eps)
    scale = pmax(scale, .Machine$double.eps)
    sdlog = pmax(sdlog, .Machine$double.eps)
  })

Listing 4.6 shows the subject-specific parameters p_Matern as a list of matrices, where each matrix represents one parameter used in generating the random marked point-patterns. In these matrices,

• Each row corresponds to a subject; i.e., each matrix in p_Matern contains three (3L) rows, one for each of the three (3L) subjects;
• Each column corresponds to a random generation parameter for each superimposed marked point-pattern; i.e., each matrix in p_Matern contains two (2L) columns, one for each of the two (2L) marked Matérn (1986)’s cluster processes to be superimposed.

Listing 4.6: Subject-specific parameters p_Matern (Listing 4.5)

p_Matern

$kappa
       Pattern 1 Pattern 2
Subj 1  3.119196  1.962887
Subj 2  2.906535  1.754151
Subj 3  2.915672  1.914559

$mu
       Pattern 1 Pattern 2
Subj 1  9.519187  4.489174
Subj 2  9.947885  4.688761
Subj 3 10.029897  5.418501

$scale
       Pattern 1 Pattern 2
Subj 1 0.4227306 0.2351866
Subj 2 0.4058303 0.1622515
Subj 3 0.3726093 0.1727232

$meanlog
       Pattern 1 Pattern 2
Subj 1  2.982251  5.007964
Subj 2  2.878996  4.976754
Subj 3  3.034019  4.852771

$sdlog
       Pattern 1 Pattern 2
Subj 1 0.4055664 0.1975062
Subj 2 0.4172841 0.2115214
Subj 3 0.4076479 0.1970090

Listing 4.7 simulates one (1L) to four (4L) point-patterns (e.g., medical images) for each of the three (3L) subjects (e.g., patients).

Listing 4.7: Simulate number of point-patterns per subject

set.seed(37); (n = sample(x = 1:4, size = 3L, replace = TRUE)) 

[1] 2 3 4

Listing 4.8 simulates a grouped hyper data frame with

• one point-pattern (ppp) hypercolumn, based on the subject-specific parameters in each matrix of p_Matern (Listing 4.6); and
• one-or-more regular columns representing the grouping structure.

Listing 4.8: Simulate grouped hyper data frame r_Matern (Listing 4.5, Listing 4.7)

set.seed(76); r_Matern = p_Matern |> 
  with.default(expr = {
    grouped_rppp(
      rMatClust(kappa = kappa, scale = scale, mu = mu), 
      rlnorm(meanlog = meanlog, sdlog = sdlog),
      n = n
    )
  })
r_Matern

Grouped Hyper Data Frame: ~g1

3 g1

    ppp g1 g2
1 (ppp)  1  1
2 (ppp)  1  2
3 (ppp)  2  1
4 (ppp)  2  2
5 (ppp)  2  3
6 (ppp)  3  1
7 (ppp)  3  2
8 (ppp)  3  3
9 (ppp)  3  4

4.2.2 with Superimposed Poisson Point-Pattern

Table 4.5 outlines the three steps described in Section 4.2.2,

Table 4.5: Steps of Section 4.2.2

	To Simulate	Primary Function
1.	subject-specific parameters of (marked) point-patterns, based on multivariate normal distribution, Listing 4.9	mvrnorm2() (Chapter 33)
2.	number of (marked) point-patterns per subject, Listing 4.7	sample()
3.	grouped hyper data frame with one-and-only-one ppp-hypercolumn, Listing 4.11	grouped_rppp() (Chapter 32)

Consider two superimposed Poisson point-pattern, each with one negative-binomial mark. The population parameters are

• \(x\)- and \(y\)-coords by Poisson point-pattern \(\lambda_1=3\), with a negative-binomial mark \((r_1,p_1)=(4,.3)\).
• \(x\)- and \(y\)-coords by Poisson point-pattern \(\lambda_2=6\), with a negative-binomial mark \((r_2,p_2)=(6,.1)\).

Listing 4.9 also simulate for three (3L) subjects (e.g., patients). The subject-specific parameters deviate from the population parameters under a multivariate normal distribution specified by the standard deviation(s) \(\sigma\) of the subject-specific random effects, i.e., \(\sigma_\lambda=.1\), \(\sigma_r=.2\) and \(\sigma_p=.05\), as well as the restrictions of Table 4.3.

Listing 4.9: Create subject-specific parameters p_Poisson

set.seed(39); p_Poisson = mapply(
  FUN = mvrnorm2, 
  mu = list(lambda = c(3, 6), size = c(4, 6), prob = c(.3, .1)), 
  sd = list(lambda = .1, size = .2, prob = .05), 
  MoreArgs = list(n = 3L, row.prefix = 'Subj', col.prefix = 'Pattern'), 
  SIMPLIFY = FALSE
) |>
  within.list(expr = {
    lambda = pmax(lambda, .Machine$double.eps)
    size = pmax(size, .Machine$double.eps)
    prob = pmin(pmax(prob, .Machine$double.eps), 1 - .Machine$double.eps)
  })

Listing 4.10: Subject-specific parameters p_Poisson (Listing 4.9)

p_Poisson

$lambda
       Pattern 1 Pattern 2
Subj 1  3.059598  5.981443
Subj 2  2.953268  5.877076
Subj 3  2.957836  5.957280

$size
       Pattern 1 Pattern 2
Subj 1  3.807675  5.795670
Subj 2  3.979154  5.875505
Subj 3  4.011959  6.167400

$prob
       Pattern 1  Pattern 2
Subj 1 0.3227306 0.13518657
Subj 2 0.3058303 0.06225150
Subj 3 0.2726093 0.07272321

Listing 4.11 simulates a grouped hyper data frame using the same grouping size as in Listing 4.7.

Listing 4.11: Simulate grouped hyper data frame r_Poisson (Listing 4.9, Listing 4.7)

set.seed(76); r_Poisson = p_Poisson |> 
  with.default(expr = {
    grouped_rppp(
      rpoispp(lambda = lambda),
      rnbinom(size = size, prob = prob),
      n = n
    )
  })
r_Poisson

Grouped Hyper Data Frame: ~g1

3 g1

    ppp g1 g2
1 (ppp)  1  1
2 (ppp)  1  2
3 (ppp)  2  1
4 (ppp)  2  2
5 (ppp)  2  3
6 (ppp)  3  1
7 (ppp)  3  2
8 (ppp)  3  3
9 (ppp)  3  4

4.2.3 Superimpose Grouped Hyper Data Frames

Listing 4.12 superimposes the simulated grouped hyper data frames r_Matern and r_Poisson (Listing 4.8, Listing 4.11),

Listing 4.12: Superimpose simulated grouped hyper data frames (Listing 4.8, Listing 4.11)

(r = spatstat.geom::superimpose(r_Matern, r_Poisson))

Grouped Hyper Data Frame: ~g1

3 g1

    ppp g1 g2
1 (ppp)  1  1
2 (ppp)  1  2
3 (ppp)  2  1
4 (ppp)  2  2
5 (ppp)  2  3
6 (ppp)  3  1
7 (ppp)  3  2
8 (ppp)  3  3
9 (ppp)  3  4

Listing 4.13 checks the return of superimposing (Listing 4.12). Note that the marks may be plotted on different scales.

Listing 4.13: A quick check of Listing 4.12

par(mar = c(0,0,0,0))
z = list(
  Mattern = r_Matern$ppp,
  Poisson = r_Poisson$ppp,
  Superimposed = r$ppp
) |>
  .mapply(FUN = spatstat.geom::solist, dots = _, MoreArgs = NULL) |>
  do.call(what = spatstat.geom::anylist, args = _)
z[[7L]] |>
  spatstat.geom::plot.solist(main = '')

Matérn, Bertil. 1986. Spatial Variation. Springer. https://doi.org/10.1007/978-1-4615-7892-5.