# extinction_risk_1.R, ver. 1.6 2015/6/24 # $Id: extinction_risk_1.R,v 1.5 2015/06/25 15:23:44 hako Exp $ # # Author: Hiroshi Hakoyama # Copyright (c) 2013-2015 Hiroshi Hakoyama , All rights reserved. # # Redistribution and use in source and binary forms, with or without # modification, are permitted provided that the following conditions # are met: # 1. Redistributions of source code must retain the above copyright # notice, this list of conditions and the following disclaimer. # 2. Redistributions in binary form must reproduce the above copyright # notice, this list of conditions and the following disclaimer in the # documentation and/or other materials provided with the distribution. # # THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY # EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, # THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A # PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE # AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, # EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT # NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; # LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER # CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, # STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) # ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF # ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. # # Description: # Estimates the demographic parameters and the probability of extinction within # a specific time period, t, from a time series of population size based on the # Wiener-drift process model (Dennis et al., 1991). # # Usage: # ext1(dat, t = 100, ne = 1, verbose = FALSE) # # Arguments: # dat data.frame of 2 variables: Time and Population size # t a time period of interest # ne a lower (extinction) threshold of population size # verbose If set to FALSE, give the ML estimate of the probability of extinction # within a specific time period t. If set to TRUE, give a more verbose output: # the ML estimate of the growth rate (mu), # the ML estimate of variance (s), # the unbiased estimate of variance (us), # the ML estimate of the probability attaining the threshold ne (psi), # the ML estimate of the conditional mean time to reach the threshold, given # the threshold is reached (theta), # the ML estimate of the conditional probability of extinction within a specific # time period t, given the threshold is reached (G), # the ML estimate of the probability of extinction within a specific time period t (G*psi), and # a lower 95 % confidence limit of parameter * (Low.CL.*), and # a higher 95 % confidence limit of parameter * (High.CL.*) # formatted If set to TRUE, give the result by formatted output. If set to FALSE, give a list of estimates. # # References: # R. Lande and S. H. Orzack. Extinction dynamics of age-structured populations # in a fluctuating environment. Proceedings of the National Academy of # Sciences, 85(19):7418-7421, 1988. # B. Dennis, P. L. Munholland, and J. M. Scott. Estimation of growth and # extinction parameters for endangered species. Ecological Monographs, # 61:115-143, 1991. # ext1 <- function(dat, t = 100, ne = 1, alpha = 0.05, verbose = FALSE, formatted = TRUE) { yr <- ts(dat[, 1], start = c(dat[, 1][1])) # Year ps <- ts(dat[, 2], start = c(dat[, 1][1])) # Population size complete <- complete.cases(yr, ps) yr <- yr[complete] ps <- ps[complete] tau <- diff(yr) # time intervals, \tau_i = t_i - t_{i-1} w <- diff(log(ps)) # W_i = \log(N(t_i)/N(t_{i-1})) = X(t_i) - X(t_{i-1}) q <- length(yr) - 1 # yr = \{t_0, t_1, \dots, t_q\} tq <- sum(tau) mu <- sum(w) / tq # ML estimate of growth rate s <- (1 / q) * sum((w - mu * tau)^2 / tau) # ML estimate of variance us <- q * s / (q - 1) # an unbiased estimate of variance xd <- log(ps[length(ps)] / ne) psi <- function(xd, mu, s) { # The probability that the process will attain the threshold ifelse (mu <= 0, 1, exp(- 2 * mu * xd / s)) } var.for.psi <- function(xd, mu, s) { ifelse (mu <= 0, 0, (4 * xd^2 / s) * (2 *(q -1) * mu^2 / (q^2 * s))) } Low.CL.psi <- function(xd, mu, s) { ifelse (mu <= 0, 1, exp(- (2 * mu * xd / s) - qnorm(1 - alpha / 2) * sqrt(var.for.psi(xd, mu, s)))) } High.CL.psi <- function(xd, mu, s) { ifelse (mu <= 0, 1, exp(- (2 * mu * xd / s) + qnorm(1 - alpha / 2) * sqrt(var.for.psi(xd, mu, s)))) } theta <- xd / abs(mu) Low.CL.mu <- mu - qt(1 - alpha / 2, q - 1) * sqrt(us / tq) High.CL.mu <- mu + qt(1 - alpha / 2, q - 1) * sqrt(us / tq) Low.CL.s <- q * s / qchisq(1 - alpha / 2, q - 1) High.CL.s <- q * s / qchisq(alpha / 2, q - 1) var.mu <- s / tq var.s <- 2 * s^2 * (q - 1) / q^2 erfc <- function (x) 2 * pnorm(-sqrt(2) * x) dGpsidmu <- - (xd / s) * exp(- (2 * mu * xd) / s) * erfc((- mu * t + xd) / sqrt(2 * s * t)) dGpsids <- (exp(- (mu * t + xd)^2 / (2 * s * t)) * t * xd) / (sqrt(2 * pi) * (s * t)^(3/2)) + ((mu * xd) / s^2) * exp(- (2 * mu * xd) / s) * erfc((- mu * t + xd) / sqrt(2 * s * t)) G <- function(t, xd, mu, s) { # G = Pr[T <= t]: See Eq. (16) and Appendix of Dennis et al. (1991). # Note that there is a typo in Eq. (16) of Dennis et al. (1991). a <- xd / sqrt(s) b <- abs(mu) / sqrt(s) y <- (b * t - a) / sqrt(t) z <- (b * t + a) / sqrt(t) d0 <- 0.2316419 d1 <- 0.319381530 d2 <- - 0.356563782 d3 <- 1.781477937 d4 <- - 1.821255978 d5 <- 1.330274429 qz <- 1 / (1 + d0 * z) g <- ifelse(z >= 4, pnorm(y) + dnorm(y) * (1 - (1 / z^2) + (1 * 3 / z^4) - (1 * 3 * 5 / z^6) + (1 * 3 * 5 * 7 / z^8) - (1 * 3 * 5 * 7 * 9 / z^10) + (1 * 3 * 5 * 7 * 9 * 11 / z^12) - (1 * 3 * 5 * 7 * 9 * 11 * 13 / z^14) ) / z, pnorm(y) + dnorm(y) * (d1 * qz + d2 * qz^2 + d3 * qz^3 + d4 * qz^4 + d5 * qz^5)) gg <- pnorm((-xd + abs(mu) * t) / sqrt(s * t)) + exp(2 * xd * abs(mu) / s) * pnorm((-xd - abs(mu) * t) / sqrt(s * t)) ifelse(is.nan(gg), g, gg) } H <- function(t, xd, mu, s) { log(G(t, xd, mu, s) * psi(xd, mu, s) / (1 - G(t, xd, mu, s) * psi(xd, mu, s))) } Gpsi <- G(t, xd, mu, s) * psi(xd, mu, s) var.H <- var.mu * (dGpsidmu / (Gpsi * (1 - Gpsi)))^2 + var.s * (dGpsids / (Gpsi * (1 - Gpsi)))^2 hh <- H(t, xd, mu, s) Low.CL.Gpsi <- 1 / (1 + exp(- hh + qnorm(1 - alpha / 2) * sqrt(var.H))) High.CL.Gpsi <- 1 / (1 + exp(- hh - qnorm(1 - alpha / 2) * sqrt(var.H))) if (verbose == TRUE) { results <- list(ne = ne, t = t, verbose = verbose, Growth.rate = mu, Low.CL.mu = Low.CL.mu, High.CL.mu = High.CL.mu, Variance = s, Low.CL.s = Low.CL.s, High.CL.s = High.CL.s, Low.CL.psi = Low.CL.psi(xd, mu, s), High.CL.psi = High.CL.psi(xd, mu, s), Unbiased.variance = us, Probability.attaining.the.threshold = psi(xd, mu, s), Conditional.mean.extinction.time = theta, Conditional.extinction.probability = G(t, xd, mu, s), Extinction.probability = G(t, xd, mu, s) * psi(xd, mu, s), Low.CL.Gpsi = Low.CL.Gpsi, High.CL.Gpsi = High.CL.Gpsi) if (formatted == TRUE) { class(results) <- "ext1" } return(results) } else { results <- list(ne = ne, t = t, verbose = verbose, Extinction.probability = G(t, xd, mu, s) * psi(xd, mu, s)) if (formatted == TRUE) { class(results) <- "ext1" } return(results) } } print.ext1 <- function(obj, digits = 5) { # cat("\nExtinction risk\n") if (obj$verbose == TRUE) { output.est <- data.frame( c(formatC(obj$Growth.rate, digits = digits), formatC(obj$Variance, digits = digits), formatC(obj$Unbiased.variance, digits = digits), formatC(obj$Probability.attaining.the.threshold, digits = digits), # formatC(obj$Conditional.mean.extinction.time, digits = digits), # formatC(obj$Conditional.extinction.probability, digits = digits), formatC(obj$Extinction.probability, digits = digits)), c(paste("(",formatC(obj$Low.CL.mu, digits = digits),", ", formatC(obj$High.CL.mu, digits = digits),")", sep = ""), paste("(",formatC(obj$Low.CL.s, digits = digits),", ", formatC(obj$High.CL.s, digits = digits),")", sep = ""), paste("(",formatC(obj$Low.CL.s, digits = digits),", ", formatC(obj$High.CL.s, digits = digits),")", sep = ""), paste("(",formatC(obj$Low.CL.psi, digits = digits),", ", formatC(obj$High.CL.psi, digits = digits),")", sep = ""), # "-", paste("(",formatC(obj$Low.CL.Gpsi, digits = digits),", ", formatC(obj$High.CL.Gpsi, digits = digits),")", sep = ""))) dimnames(output.est) <- list( c("Growth rate:", "Variance:", "Unbiased variance:", "Probability attaining the threshold:", # "Conditional mean extinction time:", # "Conditional extinction probability:", paste("Probability of decline to", obj$ne, "within", obj$t, "years:")), c("Estimate","95% confidence interval")) print(output.est) } else { output.est <- data.frame( c(formatC(obj$Extinction.probability, digits = digits))) dimnames(output.est) <- list( c(paste("Probability of decline to", obj$ne, "within", obj$t, "years:")), c("Estimate")) print(output.est) } } # # Examples # Yellowstone grizzly bears (from Dennis et al., 1991) # dat <- data.frame(Year = c(1959, 1960, 1961, 1962, 1963, 1964, 1965, 1966, 1967, 1968, 1969, 1970, 1971, 1972, 1973, 1974, 1975, 1976, 1977, 1978, 1979, 1980, 1981, 1982, 1983, 1984, 1985, 1986, 1987), # Population = c(44, 47, 46, 44, 46, 45, 46, 40, 39, 39, 42, 44, 41, 40, 33, 36, 34, 39, 35, 34, 38, 36, 37, 41, 39, 51, 47, 57, 47)) # The probability of extinction (of decline to population size 1) within 100 years # ext1(dat, 100) # The probability of decline to 10 individuals within 100 years # ext1(dat, 100, 10, verbose = TRUE)