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Simulated Grouped Hyper Data Frame

Tingting Zhan

Introduction

This vignette of package groupedHyperframe.random (Github, RPubs) documents the simulation of superimposed ppp.object and the groupedHyperframe object.

Note to Users

Examples in this vignette require that the search path has

library(groupedHyperframe.random)
#> Loading required package: spatstat.random
#> Loading required package: spatstat.data
#> Loading required package: spatstat.univar
#> spatstat.univar 3.1-2
#> Loading required package: spatstat.geom
#> spatstat.geom 3.3-6.001
#> spatstat.random 3.3-3
#> Loading required package: groupedHyperframe

Terms and Abbreviations

Term / Abbreviation	Description	Reference
	Forward pipe operator	`?base::pipeOp` introduced in `R` 4.1.0
`CRAN`, `R`	The Comprehensive R Archive Network	https://cran.r-project.org
`coords`	\(x\)- and \(y\)-coordinates	`spatstat.geom:::ppp`
`diag`	Diagonal matrix	`base::diag`
`groupedHyperframe`	Grouped hyper data frame	`groupedHyperframe::as.groupedHyperframe`
`hypercolumns`, `hyperframe`	(Hyper columns of) hyper data frame	`spatstat.geom::hyperframe`
`marks`, `marked`	(Having) mark values	`spatstat.geom::is.marked`
`pmax`	Parallel maxima	`base::pmax`
`ppp`, `ppp.object`	Point pattern	`spatstat.geom::ppp.object`
`recycle`	Recycling	https://r4ds.had.co.nz/vectors.html#scalars-and-recycling-rules
`rlnorm`	Log normal random variable	`stats::rlnorm`
`rMatClust`	Matern’s cluster process	`spatstat.random::rMatClust`
`rmvnorm_`	Multivariate normal random variable	`groupedHyperframe.random::rmvnorm_`; `MASS::mvrnorm`
`rnbinom`	Negative binomial random variable	`stats::rnbinom`
`rpoispp`	Poisson point pattern	`spatstat.random::rpoispp`
`superimpose`	Superimpose	`spatstat.geom::superimpose`
`var`, `cor`, `cov`	Variance, correlation, covariance	`stats::var`, `stats::cor`, `stats::cov`

Acknowledgement

This work is supported by NCI R01CA222847 (I. Chervoneva, T. Zhan, and H. Rui) and R01CA253977 (H. Rui and I. Chervoneva).

Simulated Point Pattern

Function .rppp() simulates superimposed ppp.objects with vectorized parameterization of random point pattern and distribution of marks.

Simulated un`marked` Point Pattern

Example below simulates a coords-only, unmarked, two superimposed Matern’s cluster processes \((\kappa, \mu, s) = (10,8,.15)\) and \((5,4,.06)\).

set.seed(125); r = .rppp(rMatClust(kappa = c(10, 5), mu = c(8, 4), scale = c(.15, .06)))
#> Point-pattern simulated by `spatstat.random::rMatClust()`
#> 
# plot(r) # suppressed for aesthetics

Simulated `marked` Point Pattern

Example below simulates two superimposed marked ppps,

Matern’s cluster process \((\kappa,\mu,s) = (10,8,.15)\), attached with a log-normal mark \((\mu,\sigma)=(3,.4)\), and a negative-binomial mark \((r,p)=(4,.3)\).
Matern’s cluster process \((\kappa,\mu,s) = (5,4,.06)\), attached with a log-normal mark \((\mu,\sigma)=(5,.2)\), and a negative-binomial mark \((r,p)=(4,.3)\).

set.seed(125); 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 parameter recycled
)
#> Point-pattern simulated by `spatstat.random::rMatClust()`
#> Marks simulated by `stats::rlnorm()`
#> Marks simulated by `stats::rnbinom()`

Example below simulates two superimposed marked ppps,

Poisson point pattern \(\lambda=3\), attached with a log-normal mark \((\mu,\sigma)=(3,.4)\), and a negative-binomial mark \((r,p)=(4,.3)\).
Poisson point pattern \(\lambda=6\), attached with a log-normal mark \((\mu,\sigma)=(5,.2)\), and a negative-binomial mark \((r,p)=(6,.1)\).

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()`
#> Marks simulated by `stats::rlnorm()`
#> Marks simulated by `stats::rnbinom()`

In the foreseeable future we will not support simulating more than one type of point patterns in a single call to function .rppp(). End user may manually superimpose different (marked) point patterns after simulating each of them separately.

spatstat.geom::superimpose(r1, r2)
#> Marked planar point pattern: 176 points
#> Mark variables: m1, m2 
#> window: rectangle = [0, 1] x [0, 1] units

Simulated `groupedHyperframe`

Now consider two superimposed Matern’s cluster processes attached with a log-normal mark. The population parameters are

(p = data.frame(kappa = c(3,2), scale = c(.4,.2), mu = c(10,5), 
                meanlog = c(3,5), sdlog = c(.4,.2)))
#>   kappa scale mu meanlog sdlog
#> 1     3   0.4 10       3   0.4
#> 2     2   0.2  5       5   0.2

We simulate for 3 subjects (e.g., patients). The subject-specific parameters deviate from the population parameters under a multivariate normal distribution with variance-covariance matrix \(\Sigma\). The matrix \(\Sigma\) may be specified by a numeric scalar, indicating all-equal diagonal variances and zero correlations/covariances. We also make sure that all subject-specific parameters satisfy that \(\kappa>1\), \(\mu>1\), \(s>0\) for Matern’s cluster processes, and \(\sigma>0\) for log-normal distribution. Each matrix of the subject-specific parameters has the subjects on the rows, and the parameters of the ppps to be superimposed on the columns.

set.seed(39); (p. = rmvnorm_(n = 3L, mu = p, Sigma = list(
  kappa = .2^2, scale = .05^2, mu = .5^2, 
  meanlog = .1^2, sdlog = .01^2)) |> 
    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)
    }))
#> $kappa
#>          [,1]     [,2]
#> [1,] 3.119196 1.962887
#> [2,] 2.906535 1.754151
#> [3,] 2.915672 1.914559
#> 
#> $scale
#>           [,1]      [,2]
#> [1,] 0.3519187 0.1489174
#> [2,] 0.3947885 0.1688761
#> [3,] 0.4029897 0.2418501
#> 
#> $mu
#>           [,1]     [,2]
#> [1,] 10.227306 5.351866
#> [2,] 10.058303 4.622515
#> [3,]  9.726093 4.727232
#> 
#> $meanlog
#>          [,1]     [,2]
#> [1,] 2.982251 5.007964
#> [2,] 2.878996 4.976754
#> [3,] 3.034019 4.852771
#> 
#> $sdlog
#>           [,1]      [,2]
#> [1,] 0.4055664 0.1975062
#> [2,] 0.4172841 0.2115214
#> [3,] 0.4076479 0.1970090

We simulate one to four ppps (e.g., medical images) per subject.

set.seed(37); (n = sample.int(n = 4L, size = 3L, replace = TRUE)) 
#> [1] 2 3 4

Function grouped_rppp() simulates a groupedHyperframe with a ppp-hypercolumn, and one-or-more columns of the grouping structure.

set.seed(76); (r = p. |> 
  with.default(expr = {
    grouped_rppp(
      rMatClust(kappa = kappa, scale = scale, mu = mu), 
      rlnorm(meanlog = meanlog, sdlog = sdlog),
      n = n
    )
  }))
#> Grouped Hyperframe: ~g1/g2
#> 
#> 9 g2 nested in
#> 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

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
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