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\name{Heaviside}
\alias{Heaviside}
\alias{Sign}
\alias{Delta}
\alias{Boxcar}
\alias{Ramp}
\title{Heaviside and related functions}
\description{
Functions which compute the Heaviside and related functions. These
include the Heaviside function, the sign function, the delta
function, the boxcar function, and the ramp function.
}
\usage{
Heaviside(x, a = 0)
Sign(x, a = 0)
Delta(x, a = 0)
Boxcar(x, a = 0.5)
Ramp(x, a = 0)
}
\arguments{
\item{x}{
a numeric vector.
}
\item{a}{
a numeric value, the location of the break.
}
}
\details{
\code{Heaviside} computes the Heaviside unit step function.
\code{Heaviside} is 1 for \code{x > a},
\code{1/2} for \code{x = a}, and \code{0} for \code{x < a}.
\code{Sign} computes
the sign function. \code{Sign} is \code{1} for \code{x > a},
\code{0} for \code{x = a}, and \code{-1} for \code{x < a}.
\code{Delta} computes the delta function.
\code{Delta} is defined as: \code{Delta(x) = d/dx H(x-a)}.
\code{Boxcar} computes the boxcar function.
\code{Boxcar} is defined as: \code{Boxcar(x) = H(x+a) - H(x-a)}.
\code{Ramp} computes ramp function.
The ramp function is defined as: \code{Ramp(x) = (x-a) * H(x-a)}.
}
\value{
numeric vector
}
\note{
The Heaviside function is used in the implementation of the skew
Normal, Student-t, and Generalized Error distributions, distributions
functions which play an important role in modelling GARCH processes.
}
\seealso{
\code{GarchDistribution},
\code{GarchDistributionFits}
}
\references{
Weisstein W. (2004);
\emph{http://mathworld.wolfram.com/HeavisideStepFunction.html},
Mathworld.
}
\examples{
x <- sort(round(c(-1, -0.5, 0, 0.5, 1, 5*rnorm(5)), 2))
h <- Heaviside(x)
s <- Sign(x)
d <- Delta(x)
Pi <- Boxcar(x)
r <- Ramp(x)
cbind(x = x, Step = h, Signum = s, Delta = d, Pi = Pi, R = r)
}
\keyword{math}
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