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##########################################################
## Demo: n-fold convolution of absolutely continuous
## probability distributions
##########################################################
require(distr)
options("newDevice"=TRUE)
### from version 1.9 of distr on available in the package
### already
if(!isGeneric("convpow"))
setGeneric("convpow",
function(D1, ...) standardGeneric("convpow"))
##########################################################
## Function for n-fold convolution
## -- absolute continuous distribution --
##########################################################
##implentation of Algorithm 3.4. of
#P. Ruckdeschel, M. Kohl (2014): General Purpose Convolution Algorithm for
# Distributions in S4 Classes by Means of FFT. J. Statist. Softw. 59(4), 1-25.
setMethod("convpow",
signature(D1 = "AbscontDistribution"),
function(D1, N){
if((N<1)||(abs(floor(N)-N)>.Machine$double.eps))
stop("N has to be a natural greater than 0")
m <- getdistrOption("DefaultNrFFTGridPointsExponent")
##STEP 1
lower <- ifelse((q.l(D1)(0) > - Inf), q.l(D1)(0),
q.l(D1)(getdistrOption("TruncQuantile"))
)
upper <- ifelse((q.l(D1)(1) < Inf), q.l(D1)(1),
q.l(D1)(getdistrOption("TruncQuantile"),
lower.tail = FALSE)
)
##STEP 2
M <- 2^m
h <- (upper-lower)/M
if(h > 0.01)
warning(paste("Grid for approxfun too wide,",
"increase DefaultNrFFTGridPointsExponent"))
x <- seq(from = lower, to = upper, by = h)
p1 <- p(D1)(x)
##STEP 3
p1 <- p1[2:(M + 1)] - p1[1:M]
##STEP 4
## computation of DFT
pn <- c(p1, numeric((N-1)*M))
fftpn <- fft(pn)
##STEP 5
## convolution theorem for DFTs
pn <- Re(fft(fftpn^N, inverse = TRUE)) / (N*M)
pn <- (abs(pn) >= .Machine$double.eps)*pn
i.max <- N*M-(N-2)
pn <- c(0,pn[1:i.max])
dn <- pn / h
pn <- cumsum(pn)
##STEP 6(density)
## density
x <- seq(from = N*lower+N/2*h,
to = N*upper-N/2*h, by=h)
x <- c(x[1]-h,x[1],x+h)
dnfun1 <- approxfun(x = x, y = dn, yleft = 0, yright = 0)
##STEP 7(density)
standardizer <- sum(dn) - (dn[1]+dn[i.max+1]) / 2
dnfun2 <- function(x) dnfun1(x) / standardizer
##STEP 6(cdf)
## cdf with continuity correction h/2
pnfun1 <- approxfun(x = x+0.5*h, y = pn,
yleft = 0, yright = pn[i.max+1])
##STEP 7(cdf)
pnfun2 <- function(x) pnfun1(x) / pn[i.max+1]
## quantile with continuity correction h/2
yleft <- ifelse(((q.l(D1)(0) == -Inf)|
(q.l(D1)(0) == -Inf)), -Inf, N*lower)
yright <- ifelse(((q.l(D1)(1) == Inf)|
(q.l(D1)(1) == Inf)), Inf, N*upper)
w0 <- options("warn")
options(warn = -1)
qnfun1 <- approxfun(x = pnfun2(x+0.5*h), y = x+0.5*h,
yleft = yleft, yright = yright)
qnfun2 <- function(x){
ind1 <- (x == 0)*(1:length(x))
ind2 <- (x == 1)*(1:length(x))
y <- qnfun1(x)
y <- replace(y, ind1[ind1 != 0], yleft)
y <- replace(y, ind2[ind2 != 0], yright)
return(y)
}
options(w0)
rnew = function(N) apply(matrix(r(e1)(n*N), ncol=N), 1, sum)
return(new("AbscontDistribution", r = rnew, d = dnfun1, p = pnfun2,
q = qnfun2, .withArith = TRUE,
.withSim = FALSE))
})
## initialize a normal distribution
A <- Norm(mean=0, sd=1)
## convolution power
N <- 10
## convolution via FFT
AN <- convpow(as(A,"AbscontDistribution"), N)
## ... for the normal distribution , 'convpow' has an "exact"
## method by version 1.9 so the as(.,.) is needed to
## see how the algorithm above works
## convolution exact
AN1 <- Norm(mean=0, sd=sqrt(N))
## plots of the results
eps <- getdistrOption("TruncQuantile")
par(mfrow=c(1,3))
low <- q.l(AN1)(eps)
upp <- q.l(AN1)(eps, lower.tail = FALSE)
x <- seq(from = low, to = upp, length = 10000)
## densities
plot(x, d(AN1)(x), type = "l", lwd = 5)
lines(x , d(AN)(x), col = "orange", lwd = 1)
title("Densities")
legend("topleft", legend=c("exact", "FFT"),
fill=c("black", "orange"))
## cdfs
plot(x, p(AN1)(x), type = "l", lwd = 5)
lines(x , p(AN)(x), col = "orange", lwd = 1)
title("Cumulative distribution functions")
legend("topleft", legend=c("exact", "FFT"),
fill=c("black", "orange"))
## quantile functions
x <- seq(from = eps, to = 1-eps, length = 1000)
plot(x, q.l(AN1)(x), type = "l", lwd = 5)
lines(x , q.l(AN)(x), col = "orange", lwd = 1)
title("Quantile functions")
legend("topleft",
legend = c("exact", "FFT"),
fill = c("black", "orange"))
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