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#
# kernels.R
#
# rXXX, dXXX, pXXX and qXXX for kernels
#
# $Revision: 1.19 $ $Date: 2018/06/07 05:42:54 $
#
match.kernel <- function(kernel) {
kernel.map <- c(Gaussian ="gaussian",
gaussian ="gaussian",
Normal ="gaussian",
normal ="gaussian",
rectangular ="rectangular",
triangular ="triangular",
Epanechnikov="epanechnikov",
epanechnikov="epanechnikov",
biweight ="biweight",
cosine ="cosine",
optcosine ="optcosine"
)
ker <- pickoption("kernel", kernel, kernel.map)
return(ker)
}
kernel.factor <- function(kernel="gaussian") {
# This function returns the factor c such that
# h = c * sigma
# where sigma is the standard deviation of the kernel, and
# h is the corresponding bandwidth parameter as conventionally defined.
# Conventionally h is defined as a scale factor
# relative to the `standard form' of the kernel, namely the
# form with support [-1,1], except in the Gaussian case where
# the standard form is N(0,1).
# Thus the standard form of the kernel (h=1) has standard deviation 1/c.
# The kernel with standard deviation 1 has support [-c,c]
# except for gaussian case.
kernel <- match.kernel(kernel)
switch(kernel,
gaussian = 1,
rectangular = sqrt(3),
triangular = sqrt(6),
epanechnikov = sqrt(5),
biweight = sqrt(7),
cosine = 1/sqrt(1/3 - 2/pi^2),
optcosine = 1/sqrt(1 - 8/pi^2))
}
rkernel <- function(n, kernel="gaussian", mean=0, sd=1) {
kernel <- match.kernel(kernel)
if(kernel == "gaussian")
return(rnorm(n, mean=mean, sd=sd))
# inverse cdf transformation
u <- runif(n)
qkernel(u, kernel, mean=mean, sd=sd)
}
dkernel <- function(x, kernel="gaussian", mean=0, sd=1) {
kernel <- match.kernel(kernel)
stopifnot(is.numeric(x))
stopifnot(is.numeric(sd) && length(sd) == 1 && sd > 0)
a <- sd * kernel.factor(kernel)
y <- abs(x-mean)/a
dens <-
switch(kernel,
gaussian = { dnorm(y) },
rectangular = { ifelse(y < 1, 1/2, 0) },
triangular = { ifelse(y < 1, (1 - y), 0) },
epanechnikov = { ifelse(y < 1, (3/4) * (1 - y^2), 0) },
biweight = { ifelse(y < 1, (15/16) * (1 - y^2)^2, 0) },
cosine = { ifelse(y < 1, (1 + cos(pi * y))/2, 0) },
optcosine = { ifelse(y < 1, (pi/4) * cos(pi * y/2), 0) }
)
dens/a
}
pkernel <- function(q, kernel="gaussian", mean=0, sd=1, lower.tail=TRUE){
kernel <- match.kernel(kernel)
stopifnot(is.numeric(q))
stopifnot(is.numeric(sd) && length(sd) == 1 && sd > 0)
a <- sd * kernel.factor(kernel)
y <- (q-mean)/a
switch(kernel,
gaussian = {
pnorm(y, lower.tail=lower.tail)
},
rectangular = {
punif(y, min=-1, max=1, lower.tail=lower.tail)
},
triangular = {
p <- ifelse(y < -1, 0, ifelse(y > 1, 1,
ifelse(y < 0, y + y^2/2 + 1/2,
y - y^2/2 + 1/2)))
if(lower.tail) p else (1 - p)
},
epanechnikov = {
p <- ifelse(y < -1, 0, ifelse(y > 1, 1,
(2 + 3 * y - y^3)/4))
if(lower.tail) p else (1 - p)
},
biweight = {
p <- ifelse(y < -1, 0, ifelse(y > 1, 1,
(15 * y - 10 * y^3 + 3 * y^5 + 8)/16))
if(lower.tail) p else (1 - p)
},
cosine = {
p <- ifelse(y < -1, 0, ifelse(y > 1, 1,
(y + sin(pi * y)/pi + 1)/2))
if(lower.tail) p else (1 - p)
},
optcosine = {
p <- ifelse(y < -1, 0, ifelse(y > 1, 1,
(sin(pi * y/2) + 1)/2))
if(lower.tail) p else (1 - p)
})
}
qkernel <- function(p, kernel="gaussian", mean=0, sd=1, lower.tail=TRUE) {
kernel <- match.kernel(kernel)
stopifnot(is.numeric(p))
stopifnot(is.numeric(sd) && length(sd) == 1 && sd > 0)
a <- sd * kernel.factor(kernel)
if(!lower.tail)
p <- 1 - p
y <-
switch(kernel,
gaussian = {
qnorm(p, lower.tail=lower.tail)
},
rectangular = {
qunif(p, min=-1, max=1, lower.tail=lower.tail)
},
triangular = {
ifelse(p < 1/2, sqrt(2 * p) - 1, 1 - sqrt(2 * (1-p)))
},
epanechnikov = {
# solve using `polyroot'
yy <- numeric(n <- length(p))
yy[p == 0] <- -1
yy[p == 1] <- 1
inside <- (p != 0) & (p != 1)
# coefficients of polynomial (2 + 3 y - y^3)/4
z <- c(2, 3, 0, -1)/4
for(i in seq(n)[inside]) {
sol <- polyroot(z - c(p[i], 0, 0, 0))
ok <- abs(Im(sol)) < 1e-6
realpart <- Re(sol)
ok <- ok & (abs(realpart) <= 1)
if(sum(ok) != 1)
stop(paste("Internal error:", sum(ok), "roots of polynomial"))
yy[i] <- realpart[ok]
}
yy
},
biweight = {
# solve using `polyroot'
yy <- numeric(n <- length(p))
yy[p == 0] <- -1
yy[p == 1] <- 1
inside <- (p != 0) & (p != 1)
# coefficients of polynomial (8 + 15 * y - 10 * y^3 + 3 * y^5)/16
z <- c(8, 15, 0, -10, 0, 3)/16
for(i in seq(n)[inside]) {
sol <- polyroot(z - c(p[i], 0, 0, 0, 0, 0))
ok <- abs(Im(sol)) < 1e-6
realpart <- Re(sol)
ok <- ok & (abs(realpart) <= 1)
if(sum(ok) != 1)
stop(paste("Internal error:", sum(ok), "roots of polynomial"))
yy[i] <- realpart[ok]
}
yy
},
cosine = {
# solve using `uniroot'
g <- function(y, pval) { (y + sin(pi * y)/pi + 1)/2 - pval }
yy <- numeric(n <- length(p))
yy[p == 0] <- -1
yy[p == 1] <- 1
inside <- (p != 0) & (p != 1)
for(i in seq(n)[inside])
yy[i] <- uniroot(g, c(-1,1), pval=p[i])$root
yy
},
optcosine = {
(2/pi) * asin(2 * p - 1)
})
return(mean + a * y)
}
#' integral of t^m k(t) dt from -Inf to r
#' where k(t) is the standard kernel with support [-1,1]
#' was: nukernel(r, m, kernel)
kernel.moment <- local({
kernel.moment <- function(m, r, kernel="gaussian") {
ker <- match.kernel(kernel)
check.1.integer(m)
#' restrict to support
if(ker != "gaussian") {
r <- pmin(r, 1)
r <- pmax(r, -1)
}
if(!(m %in% c(0,1,2)) || (ker %in% c("cosine", "optcosine"))) {
## use generic integration
neginf <- if(ker == "gaussian") -10 else -1
result <- numeric(length(r))
for(i in seq_along(r))
result[i] <- integralvalue(kintegrand,
lower=neginf, upper=r[i],
m=m, ker=ker)
return(result)
}
switch(ker,
gaussian={
if(m == 0) return(pnorm(r)) else
if(m == 1) return(-dnorm(r)) else
return(pnorm(r) - r * dnorm(r))
},
rectangular = {
if(m == 0) return((r + 1)/2) else
if(m == 1) return((r^2 - 1)/4) else
return((r^3 + 1)/6)
},
triangular={
m1 <- m+1
m2 <- m+2
const <- ((-1)^m1)/m1 + ((-1)^m2)/m2
answer <- (r^m1)/m1 + ifelse(r < 0, 1, -1) * (r^m2)/m2 - const
return(answer)
},
epanechnikov = {
if(m == 0)
return((2 + 3*r - r^3)/4)
else if(m == 1)
return((-3 + 6*r^2 - 3*r^4)/16)
else
return(( 2 + 5*r^3 - 3* r^5)/20)
},
biweight = {
if(m == 0)
return((3*r^5 - 10*r^3 + 15*r + 8)/16)
else if(m == 1)
return((5*r^6 - 15*r^4 + 15*r^2 -5)/32)
else
return((15*r^7 - 42*r^5 + 35*r^3 + 8)/112)
},
# never reached!
cosine={stop("Sorry, not yet implemented for cosine kernel")},
optcosine={stop("Sorry, not yet implemented for optcosine kernel")}
)
}
integralvalue <- function(...) integrate(...)$value
kintegrand <- function(x, m, ker) {
(x^m) * dkernel(x, ker, mean=0, sd=1/kernel.factor(ker))
}
kernel.moment
})
kernel.squint <- function(kernel="gaussian", bw=1) {
kernel <- match.kernel(kernel)
check.1.real(bw)
RK <- switch(kernel,
gaussian = 1/(2 * sqrt(pi)),
rectangular = sqrt(3)/6,
triangular = sqrt(6)/9,
epanechnikov = 3/(5 * sqrt(5)),
biweight = 5 * sqrt(7)/49,
cosine = 3/4 * sqrt(1/3 - 2/pi^2),
optcosine = sqrt(1 - 8/pi^2) * pi^2/16)
return(RK/bw)
}
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