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################################################################################
### _L_agged power-law kernel f(s) = (||s||/sigma)^-d for ||s|| >= sigma, else 1
### Similar to the density of the Pareto distribution (but value 1 for < sigma)
###
### Copyright (C) 2013-2014,2017 Sebastian Meyer
###
### This file is part of the R package "surveillance",
### free software under the terms of the GNU General Public License, version 2,
### a copy of which is available at https://www.R-project.org/Licenses/.
################################################################################
siaf.powerlawL <- function (nTypes = 1, validpars = NULL, engine = "C")
{
nTypes <- as.integer(nTypes)
stopifnot(length(nTypes) == 1L, nTypes > 0L)
engine <- match.arg(engine, c("C", "R"))
## for the moment we don't make this type-specific
if (nTypes != 1) stop("type-specific shapes are not yet implemented")
## helper expression, note: logpars=c(logscale=logsigma, logd=logd)
tmp <- expression(
logsigma <- logpars[[1L]], # used "[[" to drop names
logd <- logpars[[2L]],
sigma <- exp(logsigma),
d <- exp(logd)
)
## spatial kernel
f <- function (s, logpars, types = NULL) {}
body(f) <- as.call(c(as.name("{"), tmp, expression(
sLength <- sqrt(.rowSums(s^2, L <- length(s)/2, 2L)),
fvals <- rep.int(1, L),
inPLrange <- which(sLength > sigma),
fvals[inPLrange] <- (sLength[inPLrange]/sigma)^-d,
fvals
)))
environment(f) <- baseenv()
## numerically integrate f over a polygonal domain
F <- siaf_F_polyCub_iso(intrfr_name = "intrfr.powerlawL", engine = engine)
## fast integration of f over a circular domain
Fcircle <- function (r, logpars, type = NULL) {}
body(Fcircle) <- as.call(c(as.name("{"),
tmp,
expression(
## trivial case: radius of integration domain < sigma (=> constant f)
if (r <= sigma) return(pi * r^2),
## otherwise, if r > sigma, integration via f^-1
fofr <- (r/sigma)^-d,
basevolume <- pi * r^2 * fofr, # cylinder volume up to height f(r)
intfinvsq <- sigma^2 * if (d == 2) -d*log(sigma/r) else {
d/(d-2) * (1 - (sigma/r)^(d-2))
},
basevolume + pi * intfinvsq
)
))
environment(Fcircle) <- baseenv()
## derivative of f wrt logpars
## CAVE: the derivative of f wrt logsigma is mathematically NaN at x=sigma
## this non-differentiability at the threshold causes false convergence
## warnings by nlminb but is otherwise not relevant (could use slow and
## robust Nelder-Mead instead)
deriv <- function (s, logpars, types = NULL) {}
body(deriv) <- as.call(c(as.name("{"), tmp, expression(
sLength <- sqrt(.rowSums(s^2, L <- length(s)/2, 2L)),
derivlogsigma <- derivlogd <- numeric(L),
inPLrange <- which(sLength > sigma),
fPL <- (sLength[inPLrange]/sigma)^-d,
derivlogsigma[inPLrange] <- d * fPL,
derivlogd[inPLrange] <- fPL * log(fPL),
cbind(derivlogsigma, derivlogd)
)))
environment(deriv) <- baseenv()
## Numerical integration of 'deriv' over a polygonal domain
Deriv <- siaf_Deriv_polyCub_iso(
intrfr_names = c("intrfr.powerlawL.dlogsigma", "intrfr.powerlawL.dlogd"),
engine = engine)
## simulate from the lagged power law (within a maximum distance 'ub')
##simulate <- siaf.simulatePC(intrfr.powerlawL) # <- generic simulator
##environment(simulate) <- getNamespace("surveillance")
## faster implementation taking advantage of the constant component:
simulate <- function (n, logpars, type, ub)
{
sigma <- exp(logpars[[1L]])
d <- exp(logpars[[2L]])
## Sampling via polar coordinates and inversion method
## random angle
theta <- runif(n, 0, 2*pi)
## sampling radius r
## trivial case u < sigma: p(r) \propto r on [0;u]
if (ub < sigma) {
r <- ub * sqrt(runif(n)) # inversion sampling
## now rotate each point by a random angle to cover all directions
return(r * cbind(cos(theta), sin(theta)))
}
## case u >= sigma: p(r) \propto r if r<sigma, r*(r/sigma)^-d otherwise
## sample hierarchically from mixture
## calculate probability for r < sigma (uniform short-range component)
mass1 <- sigma^2/2 # = int_0^sigma x dx
## mass2 = int_sigma^u x * (x/sigma)^-d dx; corresponding primitive:
## prim <- function (x) {
## sigma^d * if (d == 2) log(x) else x^(2-d) / (2-d)
## }
mass2 <- sigma^d *
if (d == 2) log(ub/sigma) else (ub^(2-d)-sigma^(2-d))/(2-d)
## probability for r < sigma is mass1/(mass1+mass2) => sample component
unir <- runif(n) <= mass1 / (mass1 + mass2)
## samples from the uniform short-range component:
n1 <- sum(unir)
r1 <- sigma * sqrt(runif(n1)) # similar to the case u < sigma
## samples from power-law component: p2(r) \propto r^(-d+1) on [sigma;u]
## For d>2 only, we could use VGAM::rpareto(n,sigma,d-2), d=1 is trivial
n2 <- n - n1
r2 <- if (d==1) runif(n2, sigma, ub) else { # inversion sampling
P2inv <- if (d == 2) { function (z) ub^z * sigma^(1-z) } else {
function (z) (z*ub^(2-d) + (1-z)*sigma^(2-d))^(1/(2-d))
}
P2inv(runif(n2))
}
## put samples from both components together
r <- c(r1, r2)
## now rotate each point by a random angle to cover all directions
r * cbind(cos(theta), sin(theta))
}
environment(simulate) <- getNamespace("stats")
## return the kernel specification
list(f=f, F=F, Fcircle=Fcircle, deriv=deriv, Deriv=Deriv,
simulate=simulate, npars=2L, validpars=validpars)
}
## integrate x*f(x) from 0 to R (vectorized)
intrfr.powerlawL <- function (R, logpars, types = NULL)
{
sigma <- exp(logpars[[1L]])
d <- exp(logpars[[2L]])
pl <- which(R > sigma)
upper <- R
upper[pl] <- sigma
res <- upper^2 / 2 # integral over x*constant part
xplint <- if (d == 2) log(R[pl]/sigma) else (R[pl]^(2-d)-sigma^(2-d))/(2-d)
res[pl] <- res[pl] + sigma^d * xplint
res
}
## integrate x * (df(x)/dlogsigma) from 0 to R (vectorized)
intrfr.powerlawL.dlogsigma <- function (R, logpars, types = NULL)
{
sigma <- exp(logpars[[1L]])
d <- exp(logpars[[2L]])
pl <- which(R > sigma)
res <- numeric(length(R))
xplint <- if (d == 2) log(R[pl]/sigma) else (R[pl]^(2-d)-sigma^(2-d))/(2-d)
res[pl] <- d * sigma^d * xplint
res
}
## local({ # validation via numerical integration -> tests/testthat/test-siafs.R
## p <- function (r, sigma, d)
## r * siaf.powerlawL()$deriv(cbind(r,0), log(c(sigma,d)))[,1L]
## Pnum <- function (r, sigma, d) sapply(r, function (.r) {
## integrate(p, 0, .r, sigma=sigma, d=d, rel.tol=1e-8)$value
## })
## r <- c(1,2,5,10,20,50,100)
## dev.null <- sapply(c(1,2,1.6), function(d) stopifnot(isTRUE(
## all.equal(intrfr.powerlawL.dlogsigma(r, log(c(3, d))), Pnum(r, 3, d)))))
## })
## integrate x * (df(x)/dlogd) from 0 to R (vectorized)
intrfr.powerlawL.dlogd <- function (R, logpars, types = NULL)
{
sigma <- exp(logpars[[1L]])
d <- exp(logpars[[2L]])
pl <- which(R > sigma)
res <- numeric(length(R))
res[pl] <- if (d == 2) -(sigma*log(R[pl]/sigma))^2 else
(sigma^d * R[pl]^(2-d) * (d-2)*d*log(R[pl]/sigma) -
d*(sigma^2 - R[pl]^(2-d)*sigma^d)) / (d-2)^2
res
}
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