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## This R package is free software; you can redistribute it and/or
## modify it under the terms of the GNU Library General Public
## License as published by the Free Software Foundation; either
## version 2 of the License, or (at your option) any later version.
##
## This R package is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU Library General Public License for more details.
##
## You should have received a copy of the GNU Library General
## Public License along with this library; if not, write to the
## Free Foundation, Inc., 59 Temple Place, Suite 330, Boston,
## MA 02111-1307 USA
################################################################################
## FUNCTIONS: DESCRIPTION:
## dstable Returns density for stable DF
## pstable Returns probabilities for stable DF
## qstable Returns quantiles for stable DF
## rstable Returns random variates for stable DF
## UTILITY FUNCTION DESCRIPTION:
## .integrate2 Integrates internal functions for *stable
################################################################################
##==============================================================================
### MM TODO:
## 0) All d, p, q, q -- have identical parts
## a) 'parametrization' (pm) check
## b) checking, alpha, beta,
## c) subdivisions etc {all but rstable}
## --- to do: "Fix" these in dstable(), then copy/paste to others
##==============================================================================
##' @title omega() according to Lambert & Lindsey (1999), p.412
##' @param gamma
##' @param alpha
##' @return
.om <- function(gamma,alpha)
if(alpha == 1) (2/pi)*log(gamma) else tan(pi*alpha/2)
dstable <- function(x, alpha, beta,
gamma = 1, delta = 0, pm = 0, log = FALSE,
tol = 16*.Machine$double.eps, subdivisions = 1000)
{
## Original implemented by Diethelm Wuertz;
## Changes for efficiency and accuracy by Martin Maechler
## Description:
## Returns density for stable DF
## Details:
## The function uses the approach of J.P. Nolan for general
## stable distributions. Nolan derived expressions in form
## of integrals based on the charcteristic function for
## standardized stable random variables. These integrals
## can be numerically evaluated.
## Arguments:
## alpha = index of stability, in the range (0,2]
## beta = skewness, in the range [-1, 1]
## gamma = scale, in the range (0, infinity)
## delta = location, in the range (-infinity, +infinity)
## param = type of parmeterization
## Note: S+ compatibility no longer considered (explicitly)
## Parameter Check:
## NB: (gamma, delta) can be *vector*s (vectorized along x)
stopifnot( 0 < alpha, alpha <= 2, length(alpha) == 1,
-1 <= beta, beta <= 1, length(beta) == 1,
0 <= gamma, length(pm) == 1, pm %in% 0:2,
tol > 0, subdivisions > 0)
## Parameterizations:
if (pm == 1) {
delta <- delta + beta*gamma * .om(gamma,alpha)
} else if (pm == 2) {
delta <- delta - alpha^(-1/alpha)*gamma*stableMode(alpha, beta)
gamma <- alpha^(-1/alpha) * gamma
} ## else pm == 0
## Shift and Scale:
x <- (x - delta) / gamma
ans <-
## Special Cases:
if (alpha == 2) {
dnorm(x, mean = 0, sd = sqrt(2))
} else if (alpha == 1 && beta == 0) {
dcauchy(x)
} else {
## General Case
if (alpha != 1) { ## 0 < alpha < 2 & |beta| <= 1 from above
tanpa2 <- tan(pi*alpha/2)
zeta <- -beta * tanpa2
theta0 <- atan(beta * tanpa2) / alpha
## Loop over all x values:
unlist(lapply(x, function(z)
.fct1(z, zeta, alpha=alpha, theta0 = theta0,
tol=tol, subdivisions=subdivisions)))
}
## Special Case alpha == 1 and -1 <= beta <= 1 (but not = 0) :
else { ## (alpha == 1) and 0 < |beta| <= 1 from above
## Loop over all x values:
unlist(lapply(x, function(z) {
if (z >= 0) {
.fct2( z , beta, tol=tol, subdivisions=subdivisions)
} else {
.fct2(-z, -beta, tol=tol, subdivisions=subdivisions)
}
}))
}
}
ans <- ans/gamma
## Return:
if (log) log(ans) else ans
}
## ------------------------------------------------------------------------------
.fct1 <- function(x, zeta, alpha, theta0, tol, subdivisions,
verbose = getOption("dstable.debug", default=FALSE))
{
## -- Integrand for dstable() --
x.m.zet <- abs(x - zeta)
f.zeta <- function()
gamma(1+1/alpha)*cos(theta0) / (pi*(1+zeta^2)^(1/(2*alpha)))
## Modified: originally was if (z == zeta),
## then (D.W.) if (x.m.zet < 2 * .Machine$double.eps)
## then (M.M.) if (x.m.zet <= 1e-5 * abs(x))
if(x.m.zet < 1e-10)
return(f.zeta())
## the real check should be about the feasibility of g() below, or its integration
if(x < zeta) theta0 <- -theta0
## constants ( independent of integrand g1(th) = g*exp(-g) ):
## zeta <- -beta * tan(pi*alpha/2)
## theta0 <- (1/alpha) * atan( beta * tan(pi*alpha/2))
## x.m.zet <- abs(x - zeta)
a_1 <- alpha - 1
cat0 <- cos(at0 <- alpha*theta0)
g <- function(th) (cat0 * cos(th) * (x.m.zet/sin(at0+alpha*th))^alpha)^(1/a_1) * cos(at0+ a_1*th)
## Function to Integrate:
g1 <- function(th) ## and (xarg, alpha, beta) => (zeta,at0,g0,.....)
{
## g1 = g(.) exp(-g(.))
v <- g(th)
## making sure that we don't get Inf * 0 => NaN
r <- v * exp(-v)
if(any(L <- v > 1000)) ## actually, v > -(.Machine$double.min.exp * log(2)) == 708.396...
r[L] <- 0
r
}
c2 <- ( alpha / (pi*abs(a_1)*x.m.zet) )
## Result = c2 * \int_{-t0}^{pi/2} g1(u) du
## where however, g1(.) may look to be (almost) zero almost everywhere and just have a small peak
## ==> Split the integral into two parts of two intervals (t0, t_max) + (t_max, pi/2)
## However, this may still be bad, e.g., for dstable(71.61531, alpha=1.001, beta=0.6),
## or dstable(1.205, 0.75, -0.5)
## the 2nd integral is "completely wrong" (basically zero, instead of ..e-5)
## FIXME --- Lindsey uses "Romberg" integration -- maybe we must split even more
g. <- if(alpha >= 1) g(-theta0+ 1e-6*abs(theta0)) else g(pi/2 -1e-6*(pi/2))
if(is.na(g.) || identical(g., 0))# g() is not usable
return(f.zeta())
## We know that the maximum of g1(.) is = exp(-1) = 0.3679 "at" g(.) == 1
## find that by uniroot :
ur <- uniroot(function(th) g(th) - 1, lower = -theta0, upper = pi/2, tol = tol)
theta2 <- ur$root
## now, because the peak may be extreme, we find two more intermediate values
## NB: Max = 0.3679 ==> 1e-4 is a very small fraction of that
## to the left:
th1 <- uniroot(function(th) g1(th) - 1e-4, lower = -theta0, upper = theta2,
tol = tol)$root
if((do4 <- g1(pi/2) < 1e-4))
## to the right:
th3 <- uniroot(function(th) g1(th) - 1e-4, lower = theta2, upper = pi/2,
tol = tol)$root
Int <- function(a,b)
.integrate2(g1, lower = a, upper = b,
subdivisions=subdivisions, rel.tol= tol, abs.tol= tol)
r1 <- Int(-theta0, th1)
r2 <- Int( th1, theta2)
if(do4) {
r3 <- Int( theta2, th3)
r4 <- Int( th3, pi/2)
} else {
r3 <- Int( theta2, pi/2)
r4 <- 0
}
if(verbose)
cat(sprintf(".fct1(%11g,%10g,..): c2*sum(r[1..4])= %11g*(%9.4g + %9.4g + %9.4g + %9.4g)= %g\n",
x,zeta, c2, r1,r2,r3,r4, c2*(r1+r2+r3+r4)))
c2*(r1+r2+r3+r4)
}
## ------------------------------------------------------------------------------
##' is only used when alpha == 1 (!)
.fct2 <- function(xarg, beta, tol, subdivisions)
{
## Integration:
i2b <- 1/(2*beta)
p2b <- pi*i2b # = pi/(2 beta)
pi2 <- pi/2
g. <- exp(-p2b*xarg)
## Function to Integrate; x is a non-sorted vector!
g2 <- function(x) { ## and (xarg, beta)
g <- p2b+ x # == g'/beta where g' := pi/2 + beta*x
v <- g / (p2b*cos(x)) * exp(g*tan(x))
g <- g. * v
gval <- g * exp(-g)
if(any(ina <- is.na(gval))) gval[ina] <- 0 ## replace NA at pi/2
gval
}
theta2 <- optimize(g2, lower = -pi2, upper = pi2,
maximum = TRUE, tol = tol)$maximum
r1 <- .integrate2(g2, lower = -pi2, upper = theta2,
subdivisions = subdivisions,
rel.tol = tol, abs.tol = tol)
r2 <- .integrate2(g2, lower = theta2, upper = pi2,
subdivisions = subdivisions,
rel.tol = tol, abs.tol = tol)
abs(i2b)*(r1 + r2)
}
### ------------------------------------------------------------------------------
pstable <- function(q, alpha, beta, gamma = 1, delta = 0, pm = 0,
tol = 16*.Machine$double.eps, subdivisions = 1000)
{
## A function implemented by Diethelm Wuertz
## Description:
## Returns probability for stable DF
x <- q
## Parameter Check:
## NB: (gamma, delta) can be *vector*s (vectorized along x)
stopifnot( 0 < alpha, alpha <= 2, length(alpha) == 1,
-1 <= beta, beta <= 1, length(beta) == 1,
0 <= gamma, length(pm) == 1, pm %in% 0:2,
tol > 0, subdivisions > 0)
## Parameterizations:
if (pm == 1) {
delta <- delta + beta*gamma * .om(gamma,alpha)
} else if (pm == 2) {
delta <- delta - alpha^(-1/alpha)*gamma*stableMode(alpha, beta)
gamma <- alpha^(-1/alpha) * gamma
} ## else pm == 0
## Shift and Scale:
x <- (x - delta) / gamma
## Return directly
## ------ first, special cases:
if (alpha == 2) {
pnorm(x, mean = 0, sd = sqrt(2))
} else if (alpha == 1 && beta == 0) {
pcauchy(x)
} else {
## General Case
if (alpha != 1) { ## 0 < alpha < 2 & |beta| <= 1 from above
tanpa2 <- tan(pi*alpha/2)
zeta <- -beta * tanpa2
theta0 <- atan(beta * tanpa2) / alpha
## Loop over all x values:
unlist(lapply(x, function(z) {
z.m.zet <- abs(z - zeta)
if (z.m.zet < 2 * .Machine$double.eps)
## FIXME? same problem as dstable
(1/2- theta0/pi)
else if (z > zeta)
.FCT1(z.m.zet, alpha=alpha, theta0= theta0,
tol = tol, subdivisions = subdivisions)
else ## (z < zeta)
1 - .FCT1(z.m.zet, alpha=alpha, theta0= -theta0,
tol = tol, subdivisions = subdivisions)
}))
}
## Special Case alpha == 1 and -1 <= beta <= 1 (but not = 0) :
else { ## (alpha == 1) and 0 < |beta| <= 1 from above
## Loop over all x values:
unlist(lapply(x, function(z) {
if (beta >= 0) {
.FCT2(xarg = z, beta = beta,
tol = tol, subdivisions = subdivisions)
} else {
1 - .FCT2(xarg = -z, beta = -beta,
tol = tol, subdivisions = subdivisions)
}
}))
}
}
}
## ------------------------------------------------------------------------------
.FCT1 <- function(x.m.zeta, alpha, theta0, tol, subdivisions)
{
a_1 <- alpha - 1
aa1 <- alpha/a_1
cat0 <- cos(at0 <- alpha*theta0)
g0 <- x.m.zeta^aa1
## Function to integrate:
G1 <- function(x) { ## (xarg, alpha, beta)
v <- (cat0*cos(x))^(1/a_1) * sin(at0+ alpha*x)^-aa1 * cos(at0+a_1*x)
exp(-(g0 * v))
}
theta2 <- optimize(G1, lower = -theta0, upper = pi/2,
maximum = TRUE, tol = tol)$maximum
c1 <- if(alpha < 1) 1/2 - theta0/pi else 1
c3 <- sign(1-alpha)/pi
r1 <- .integrate2(G1, lower = -theta0,
upper = theta2, subdivisions = subdivisions,
rel.tol = tol, abs.tol = tol)
r2 <- .integrate2(G1, lower = theta2,
upper = pi/2, subdivisions = subdivisions,
rel.tol = tol, abs.tol = tol)
c1 + c3*(r1+r2)
}
## ------------------------------------------------------------------------------
.FCT2 <- function(xarg, beta, tol, subdivisions)
{
i2b <- 1/(2*beta)
p2b <- pi*i2b # = pi/(2 beta)
pi2 <- pi/2
g. <- exp(-p2b*xarg)
## Function to Integrate; x is a non-sorted vector!
G2 <- function(x) { ## and (xarg, beta)
g <- p2b+ x # == g'/beta where g' := pi/2 + beta*x
v <- g / (p2b*cos(x)) * exp(g*tan(x))
g <- g. * v
gval <- exp(-g)
if(any(ina <- is.na(gval))) gval[ina] <- 0 ## replace NA at pi/2
gval
}
## Integration:
theta2 <- optimize(G2, lower = -pi2, upper = pi2,
maximum = TRUE, tol = tol)$maximum
r1 <- .integrate2(G2, lower = -pi2,
upper = theta2, subdivisions = subdivisions,
rel.tol = tol, abs.tol = tol)
r2 <- .integrate2(G2, lower = theta2,
upper = pi2, subdivisions = subdivisions,
rel.tol = tol, abs.tol = tol)
(r1+r2)/pi
}
### ------------------------------------------------------------------------------
qstable <- function(p, alpha, beta, gamma = 1, delta = 0, pm = 0,
tol = .Machine$double.eps^0.25, maxiter = 1000,
integ.tol = 1e-7, subdivisions = 200)
{
## A function implemented by Diethelm Wuertz
## Description:
## Returns quantiles for stable DF
## Parameter Check:
## NB: (gamma, delta) can be *vector*s (vectorized along x)
stopifnot( 0 < alpha, alpha <= 2, length(alpha) == 1,
-1 <= beta, beta <= 1, length(beta) == 1,
0 <= gamma, length(pm) == 1, pm %in% 0:2,
tol > 0, subdivisions > 0)
## Parameterizations:
if (pm == 1) {
delta <- delta + beta*gamma * .om(gamma,alpha)
} else if (pm == 2) {
delta <- delta - alpha^(-1/alpha)*gamma*stableMode(alpha, beta)
gamma <- alpha^(-1/alpha) * gamma
} ## else pm == 0
result <-
## Special Cases:
if (alpha == 2)
qnorm(p, mean = 0, sd = sqrt(2))
else if (alpha == 1 && beta == 0)
qcauchy(p)
else if (abs(alpha-1) < 1) { ## -------------- 0 < alpha < 2 ---------------
.froot <- function(x, p) {
pstable(q = x, alpha=alpha, beta=beta, pm = 0,
tol=integ.tol, subdivisions=subdivisions) - p
}
## Calculate:
unlist(lapply(p, function(pp) {
if (beta < 0) {
xmin <- -(1-pp)/pp
## xmax = pp/(1-pp)
if (pp < 0.5) {
xmax <- qnorm(pp, mean = 0, sd = sqrt(2))
} else {
xmax <- qcauchy(pp)
}
}
if (beta > 0 ) {
## xmin = -(1-pp)/pp
if (pp < 0.5) {
xmin <- qcauchy(pp)
} else {
xmin <- qnorm(pp, mean = 0, sd = sqrt(2))
}
xmax <- pp/(1-pp)
}
if (beta == 0 ) {
## xmin = -(1-pp)/pp
if (pp < 0.5) {
xmin <- qcauchy(pp)
} else {
xmin <- qnorm(pp, mean = 0, sd = sqrt(2))
}
## xmax = pp/(1-pp)
if (pp < 0.5) {
xmax <- qnorm(pp, mean = 0, sd = sqrt(2))
} else {
xmax <- qcauchy(pp)
}
}
root <- NA
counter <- 0
while (is.na(root)) {
root <- .unirootNA(.froot, interval = c(xmin, xmax), p = pp,
tol=tol, maxiter=maxiter)
counter <- counter + 1
xmin <- xmin-2^counter
xmax <- xmax+2^counter
}
root
}))
}
## Result:
result * gamma + delta
}
## ------------------------------------------------------------------------------
rstable <- function(n, alpha, beta, gamma = 1, delta = 0, pm = 0)
{
## Description:
## Returns random variates for stable DF
## slightly amended along nacopula::rstable1
## Parameter Check:
## NB: (gamma, delta) can be *vector*s (vectorized along x)
stopifnot( 0 < alpha, alpha <= 2, length(alpha) == 1,
-1 <= beta, beta <= 1, length(beta) == 1,
0 <= gamma, length(pm) == 1, pm %in% 0:2)
## Parameterizations:
if (pm == 1) {
delta <- delta + beta*gamma * .om(gamma,alpha)
} else if (pm == 2) {
delta <- delta - alpha^(-1/alpha)*gamma*stableMode(alpha, beta)
gamma <- alpha^(-1/alpha) * gamma
} ## else pm == 0
## Calculate uniform and exponential distributed random numbers:
theta <- pi * (runif(n)-1/2)
w <- -log(runif(n))
result <-
## If alpha is equal 1 then:
if (alpha == 1 & beta == 0) {
rcauchy(n)
## Otherwise, if alpha is different from 1:
} else {
## FIXME: learn from nacopula::rstable1R()
b.tan.pa <- beta*tan(pi*alpha/2)
theta0 <- atan(b.tan.pa) / alpha ## == \theta_0
c <- (1+b.tan.pa^2)^(1/(2*alpha))
a.tht <- alpha*(theta+theta0)
r <- ( c*sin(a.tht)/
(cos(theta))^(1/alpha) ) *
(cos(theta-a.tht)/w)^((1-alpha)/alpha)
## Use Parametrization 0:
r - b.tan.pa
}
## Result:
result * gamma + delta
}
## ------------------------------------------------------------------------------
##' Numerically Integrate -- basically the same as R's integrate()
##' --------------------- main difference: no errors, but warnings
.integrate2 <- function(f, lower, upper, subdivisions, rel.tol, abs.tol, ...)
{
## Originally implemented by Diethelm Wuertz -- without *any* warnings
if (class(version) != "Sversion") {
## R:
f <- match.fun(f)
ff <- function(x) f(x, ...)
wk <- .External("call_dqags", ff,
rho = environment(), as.double(lower),
as.double(upper), as.double(abs.tol),
as.double(rel.tol), limit = as.integer(subdivisions),
PACKAGE = "base")[c("value","ierr")]
iErr <- wk[["ierr"]]
if(iErr == 6) stop("the input is invalid")
if(iErr > 0)
## NB: "roundoff error ..." happens many times
warning(switch(iErr + 1, "OK",
"maximum number of subdivisions reached",
"roundoff error was detected",
"extremely bad integrand behaviour",
"roundoff error is detected in the extrapolation table",
"the integral is probably divergent"))
wk[[1]]
} else {
## SPlus:
integrate(f, lower, upper, subdivisions, rel.tol, abs.tol, ...)[[1]]
}
}
################################################################################
|