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```
#' Overlap calculation for uniform niche regions.
#'
#' @details The overlap between niche regions \eqn{A} and \eqn{B} is defined as \eqn{vol(A \cap B)/vol(A \cup B)}, where the hypervolume of an \eqn{n}-dimensional region \eqn{S} is \eqn{vol(S) = \int_S dx}. For elliptical niche regions, there are simple formulas for \eqn{vol(A)} and \eqn{vol(B)}. Thus, we need only determine the volume of the intersection \eqn{vol(A \cap B)}, as the volume of the union is given by the formula \eqn{vol(A \cup B) = vol(A) + vol(B) - vol(A \cap B)}.
#'
#' For spherical niche regions, \eqn{vol(A \cap B)} has a closed-form expression (see 'References'). For elliptical regions, no such formula exists and a Monte Carlo method is used instead. That is, \eqn{vol(A \cap B)} is calculated by sampling uniformly from \eqn{A}, then multiplying \eqn{vol(A)} by the fraction of sampled points which fall into \eqn{B}.
#'
#' While the uniform overlap metric is invariant to permutation of niche regions \eqn{A} and \eqn{B}, the accuracy of the Monte Carlo calculation of \eqn{vol(A \cap B)} is not: higher accuracy is obtained when a higher fraction of sampled points are in the opposite niche region. [overlap.unif()] does not attempt to determine for which region this is the case, though the choice can be informed by plotting the niche regions, e.g., with [niche.plot()].
#'
#' @param muA,muB Mean of niche regions.
#' @param SigmaA,SigmaB Variance matrix of elliptical niche regions.
#' @param alphaA,alphaB Probabilistic size of niche regions.
#' @param nprob Number of uniform draws from niche region `A`.
#' @return A Monte Carlo estimate of the niche overlap for [overlap.unif()], and an analytic calculation for [overlap.sphere()].
#' @name overlap.unif
#' @example examples/overlap.unif.R
#' @export
overlap.unif <- function(muA, SigmaA, muB, SigmaB, alphaA = .95, alphaB = .95,
nprob) {
# niche sizes
volA <- niche.size(Sigma = SigmaA, alpha = alphaA)
volB <- niche.size(Sigma = SigmaB, alpha = alphaB)
# intersection: simulate data from one niche
simA <- niche.runif(nprob, mu = muA, Sigma = SigmaA, alpha = alphaA)
# fraction of pts in the other
Z <- backsolve(chol(SigmaB), t(simA) - muB, transpose = TRUE)
propAB <- mean(colSums(Z*Z) < qchisq(p = alphaB, df = length(muB)))
# overlap calculation
volAB <- propAB * volA
volAB / (volA + volB - volAB)
}
#' @rdname overlap.unif
#' @param sigmaA,sigmaB standard deviations (scalars) of spherical niche regions.
#' @references Li, S. "Concise formulas for the area and volume of a hyperspherical cap." *Asian Journal of Mathematics & Statistics* 4.1 (2011): 66-70. \doi{10.3923/ajms.2011.66.70}.
#' @export
overlap.sphere <- function(muA, sigmaA, muB, sigmaB,
alphaA = .95, alphaB = .95) {
n <- length(muA) # number of dimensions
# niche sizes
volA <- niche.size(Sigma = sigmaA^2 * diag(n), alpha = alphaA)
volB <- niche.size(Sigma = sigmaB^2 * diag(n), alpha = alphaB)
# volume of intersection
rA <- sigmaA * sqrt(qchisq(alphaA, df = n)) # sphere radii
rB <- sigmaB * sqrt(qchisq(alphaB, df = n))
dd <- sqrt(sum((muA - muB)^2)) # distance between sphere centers
if(dd > rA + rB) {
volAB <- 0 # no overlap
} else if(dd <= abs(rA-rB)) {
volAB <- min(volA, volB) # one sphere fully inside other
} else {
cA <- .5 * (dd^2 + rA^2 - rB^2)/dd
cB <- .5 * (dd^2 - rA^2 + rB^2)/dd
volAB <- vol.cap(rA, cA, n) + vol.cap(rB, cB, n)
}
volAB / (volA + volB - volAB)
}
# volume of an n-dimensional sphere cap
vol.cap <- function(r, a, n) {
vol <- .5 * pbeta((a/r)^2, .5, .5*(n+1), lower.tail = FALSE)
if(a < 0) vol <- 1 - vol
(sqrt(pi) * r)^n/gamma(1+.5*n) * vol
}
# cseg <- function(R, d) R^2 * acos(d/R) - d*sqrt(R^2 - d^2)
## # 1D example
## d <- 1
## mA <- rnorm(d)
## mB <- rnorm(d)
## sigA <- rexp(1)
## SigA <- sigA^2 * diag(d)
## sigB <- rexp(1)
## SigB <- sigB^2 * diag(d)
## nicheA <- mA + c(-1,1) * sigA * sqrt(qchisq(.95, 1))
## nicheB <- mB + c(-1,1) * sigB * sqrt(qchisq(.95, 1))
## plot(0, type = "n", xlim = range(nicheA, nicheB), ylim = c(0,3),
## xlab = "x", ylab = "y")
## lines(nicheA, c(1,1), col = "red", lwd = 3)
## lines(nicheB, c(2,2), col = "blue", lwd = 3)
## points(c(mA, mB), c(1,2), pch = 16, cex = 2)
## overlap.sphere(mA, sigA, mB, sigB)
## overlap.unif(mA, SigA, mB, SigB, nprob = 1e5)
## # 3d example
## require(rgl)
## open3d()
## d <- 3
## mA <- rnorm(d)
## ## mB <- mA + .1
## mB <- rnorm(d)
## sigA <- rexp(1)
## SigA <- sigA^2 * diag(d)
## ## sigB <- sigA
## sigB <- rexp(1)
## SigB <- sigB^2 * diag(d)
## clear3d()
## # open3d()
## shade3d(ellipse3d(x = SigA, centre = mA), col = "yellow")
## shade3d(ellipse3d(x = SigB, centre = mB), col = "green")
## overlap.sphere(mA, sigA, mB, sigB)
## overlap.unif(mA, SigA, mB, SigB, nprob = 1e5)
```

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