\name{Heaviside}
\alias{Heaviside}
\alias{Sign}
\alias{Delta}
\alias{Boxcar}
\alias{Ramp}
\title{Haviside and Related Functions}
\description{
Functions which compute the Heaviside and
related functions. These include the sign
function, the delta function, the boxcar
function, and the ramp function.
\cr
The functions are:
\tabular{ll}{
\code{Heaviside} \tab Computes Heaviside unit step function, \cr
\code{Sign} \tab Just another signum function, \cr
\code{Delta} \tab Computes delta function, \cr
\code{Boxcar} \tab Computes boxcar function, \cr
\code{Ramp} \tab Computes 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{a}{
a numeric value, the location of the break.\cr
}
\item{x}{
a numeric vector.
}
}
\details{
The Heaviside step function \code{Heaviside} is 1 for \code{x>a},
\code{1/2} for \code{x=a}, and \code{0} for \code{x<a}.
\cr
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}.
\cr
The delta function \code{Delta} is defined as:
\code{Delta(x) = d/dx H(x-a)}.
\cr
The boxcar function \code{Boxcar} is defined as:
\code{Boxcar(x) = H(x+a) - H(x-a)}.
\cr
The ramp function is defined as:
\code{Ramp(x)= (x-a)*H(x-a)}.
}
\value{
returns the function values of the selected function.
}
\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{
## Heaviside -
x = sort(round(c(-1, -0.5, 0, 0.5, 1, 5*rnorm(5)), 2))
h = Heaviside(x)
## Sign -
s = Sign(x)
## Delta -
d = Delta(x)
## Boxcar -
Pi = Boxcar(x)
## Ramp -
r = Ramp(x)
cbind(x = x, Step = h, Signum = s, Delta = d, Pi = Pi, R = r)
}
\keyword{programming}