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\name{LatticeDistribution-class}
\docType{class}
\alias{AffLinLatticeDistribution-class}
\alias{LatticeDistribution-class}
\alias{lattice}
\alias{lattice-method}
\alias{lattice,LatticeDistribution-method}
\alias{initialize,LatticeDistribution-method}
\alias{initialize,AffLinLatticeDistribution-method}
\alias{sqrt,LatticeDistribution-method}
\alias{coerce,LatticeDistribution,DiscreteDistribution-method}
\alias{coerce,AffLinLatticeDistribution,AffLinDiscreteDistribution-method}
\title{Class "LatticeDistribution"}
\description{The \code{LatticeDistribution}-class is the mother-class of the
classes \code{Binom}, \code{Dirac}, \code{Geom}, \code{Hyper}, \code{Nbinom} and
\code{Poisson}. It formalizes a distribution on a regular affine
linear lattice.}
\section{Objects from the Class}{
The usual way to generate objects of class \code{LatticeDistribution} is to call
the generating function \code{\link{LatticeDistribution}}. \cr
Somewhat more flexible, but also proner to inconsistencies is a call to
\code{new("LatticeDistribution")}, where you may explicitly specify random
number generator, (counting) density, cumulative distribution and quantile
functions. For conveniance, in this call to \code{new("LatticeDistribution")},
an additional possibility is to only specify the random number generator. The
function \code{RtoDPQ.d} then approximates the three remaining slots \code{d},
\code{p} and \code{q} by random sampling.
}
\section{Slots}{
\describe{
\item{\code{img}}{Object of class \code{"Reals"}: the space of the image
of this distribution which has dimension 1 and the name "Real Space" }
\item{\code{param}}{Object of class \code{"Parameter"}: the parameter of
this distribution, having only the slot name
"Parameter of a discrete distribution" }
\item{\code{r}}{Object of class \code{"function"}:
generates random numbers}
\item{\code{d}}{Object of class \code{"function"}:
(counting) density/probability function}
\item{\code{p}}{Object of class \code{"function"}:
cumulative distribution function}
\item{\code{q}}{Object of class \code{"function"}:
quantile function}
\item{\code{support}}{Object of class \code{"numeric"}: a (sorted) vector
containing the support of the discrete
density function}
\item{\code{lattice}}{Object of class \code{"Lattice"}: the lattice
generating the support.}
\item{\code{.withArith}}{logical: used internally to issue warnings as to
interpretation of arithmetics}
\item{\code{.withSim}}{logical: used internally to issue warnings as to
accuracy}
\item{\code{.logExact}}{logical: used internally to flag the case where
there are explicit formulae for the log version of density, cdf, and
quantile function}
\item{\code{.lowerExact}}{logical: used internally to flag the case where
there are explicit formulae for the lower tail version of cdf and quantile
function}
\item{\code{Symmetry}}{object of class \code{"DistributionSymmetry"};
used internally to avoid unnecessary calculations.}
}
}
\section{Extends}{
Class \code{"UnivariateDistribution"}, directly.\cr
Class \code{"Distribution"}, by class \code{"UnivariateDistribution"}.
}
\section{Methods}{
\describe{
\item{\code{initialize}}{\code{signature(.Object = "LatticeDistribution")}:
initialize method }
\item{-}{\code{signature(e1 = "LatticeDistribution")}:
application of `-' to this lattice distribution}
\item{*}{\code{signature(e1 = "LatticeDistribution", e2 = "numeric")}:
multiplication of this lattice distribution
by an object of class `numeric'}
\item{/}{\code{signature(e1 = "LatticeDistribution", e2 = "numeric")}:
division of this lattice distribution by an object of class `numeric'}
\item{+}{\code{signature(e1 = "LatticeDistribution", e2 = "numeric")}:
addition of this lattice distribution to an object of class `numeric'}
\item{-}{\code{signature(e1 = "LatticeDistribution", e2 = "numeric")}:
subtraction of an object of class `numeric' from this lattice
distribution }
\item{*}{\code{signature(e1 = "numeric", e2 = "LatticeDistribution")}:
multiplication of this lattice distribution by an object of class `numeric'}
\item{+}{\code{signature(e1 = "numeric", e2 = "LatticeDistribution")}:
addition of this lattice distribution to an object of class `numeric'}
\item{-}{\code{signature(e1 = "numeric", e2 = "LatticeDistribution")}:
subtraction of this lattice distribution from an object of class `numeric'}
\item{+}{\code{signature(e1 = "LatticeDistribution",
e2 = "LatticeDistribution")}: Convolution of two lattice distributions.
Slots p, d and q are approximated by grids.}
\item{-}{\code{signature(e1 = "LatticeDistribution",
e2 = "LatticeDistribution")}: Convolution of two lattice
distributions. The slots p, d and q are approximated by grids.}
\item{\code{sqrt}}{\code{signature(x = "LatticeDistribution")}: exact
image distribution of \code{sqrt(x)}.}
\item{\code{lattice}}{accessor method to the corresponding slot.}
\item{\code{coerce}}{\code{signature(from = "LatticeDistribution",
to = "DiscreteDistribution")}: coerces an object from
\code{"LatticeDistribution"} to \code{"DiscreteDistribution"}
thereby cancelling out support points with probability 0.}
}
}
\section{Internal subclass "AffLinLatticeDistribution"}{
To enhance accuracy of several functionals on distributions,
mainly from package \pkg{distrEx}, there is an internally used
(but exported) subclass \code{"AffLinLatticeDistribution"} which has extra slots
\code{a}, \code{b} (both of class \code{"numeric"}), and \code{X0}
(of class \code{"LatticeDistribution"}), to capture the fact
that the object has the same distribution as \code{a * X0 + b}. This is
the class of the return value of methods
\describe{
\item{-}{\code{signature(e1 = "LatticeDistribution")}}
\item{*}{\code{signature(e1 = "LatticeDistribution", e2 = "numeric")}}
\item{/}{\code{signature(e1 = "LatticeDistribution", e2 = "numeric")}}
\item{+}{\code{signature(e1 = "LatticeDistribution", e2 = "numeric")}}
\item{-}{\code{signature(e1 = "LatticeDistribution", e2 = "numeric")}}
\item{*}{\code{signature(e1 = "numeric", e2 = "LatticeDistribution")}}
\item{+}{\code{signature(e1 = "numeric", e2 = "LatticeDistribution")}}
\item{-}{\code{signature(e1 = "numeric", e2 = "LatticeDistribution")}}
\item{-}{\code{signature(e1 = "AffLinLatticeDistribution")}}
\item{*}{\code{signature(e1 = "AffLinLatticeDistribution", e2 = "numeric")}}
\item{/}{\code{signature(e1 = "AffLinLatticeDistribution", e2 = "numeric")}}
\item{+}{\code{signature(e1 = "AffLinLatticeDistribution", e2 = "numeric")}}
\item{-}{\code{signature(e1 = "AffLinLatticeDistribution", e2 = "numeric")}}
\item{*}{\code{signature(e1 = "numeric", e2 = "AffLinLatticeDistribution")}}
\item{+}{\code{signature(e1 = "numeric", e2 = "AffLinLatticeDistribution")}}
\item{-}{\code{signature(e1 = "numeric", e2 = "AffLinLatticeDistribution")}}
}
There is also an explicit \code{coerce}-method from class
\code{"AffLinLatticeDistribution"} to class \code{"AffLinDiscreteDistribution"}
which cancels out support points with probability 0.
}
\author{Peter Ruckdeschel \email{peter.ruckdeschel@uni-oldenburg.de}}
\note{ Working with a computer, we use a finite interval as support which
carries at least mass \code{1-getdistrOption("TruncQuantile")}. }
\seealso{
\code{\link{LatticeDistribution}}
\code{\link{Parameter-class}}
\code{\link{Lattice-class}}
\code{\link{UnivariateDistribution-class}}
\code{\link{DiscreteDistribution-class}}
\code{\link{Binom-class}}
\code{\link{Dirac-class}}
\code{\link{Geom-class}}
\code{\link{Hyper-class}}
\code{\link{Nbinom-class}}
\code{\link{Pois-class}}
\code{\link{AbscontDistribution-class}}
\code{\link{Reals-class}}
\code{\link{RtoDPQ.d}}
}
\examples{
B <- Binom(prob = 0.1,size = 10) # B is a Binomial distribution w/ prob=0.1 and size=10.
P <- Pois(lambda = 1) # P is a Poisson distribution with lambda = 1.
D1 <- B+1 # a new Lattice distributions with exact slots d, p, q
D2 <- D1*3 # a new Lattice distributions with exact slots d, p, q
D3 <- B+P # a new Lattice distributions with approximated slots d, p, q
D4 <- D1+P # a new Lattice distributions with approximated slots d, p, q
support(D4) # the (approximated) support of this distribution is 1, 2, ..., 21
r(D4)(1) # one random number generated from this distribution, e.g. 4
d(D4)(1) # The (approximated) density for x=1 is 0.1282716.
p(D4)(1) # The (approximated) probability that x<=1 is 0.1282716.
q(D4)(.5) # The (approximated) 50 percent quantile is 3.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
}
\keyword{distribution}
\concept{discrete distribution}
\concept{lattice distribution}
\concept{lattice of a distribution}
\concept{S4 distribution class}
\concept{generating function}
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