Specify the Support of a Distribution

A support says where a distribution's probability lives, and in what form. Probability comes in two forms: mass, which sits on single points, and density, which is spread over regions. A support records both — the points carrying mass, its atoms, and the regions carrying density — and discrete(), continuous() and mixed() build one from those pieces.

Usage

discrete(atoms = numeric(0))

continuous(...)

mixed(discrete = numeric(0), continuous = numeric(0))

Arguments

atoms: For discrete(), the points carrying mass: a discretes object (see the discretes package, e.g. discretes::natural0()), a numeric vector of finitely many atoms, or a purely discrete support. A bare numeric vector is unambiguous here because the argument names the intent (contrast with passing one to .support, which is rejected).
...: For continuous(), one or more regions, each given as a length-2 numeric c(lower, upper). With no arguments, continuous() defaults to the whole real line, c(-Inf, Inf). Overlapping or touching regions are merged and sorted into a canonical form.
discrete, continuous: For mixed(), the two halves. Each takes the same things its own constructor takes, or a support already built by it: discrete as for atoms above, continuous as for ... below.

Value

A support object (class "support").

Details

The variable type (vtype()) is derived from the support: a support with only atoms is "discrete", only a continuous part is "continuous", both is "mixed", and neither is "empty".

Because the type is derived, none of the three insists on being handed something non-empty. Each builds whatever the parts describe, and describing nothing gives empty_support(). So mixed(continuous = continuous()) is the whole real line, discrete(numeric(0)) is empty, and mixed() is empty too. This is what makes them usable when the parts are computed rather than typed and may come out empty; mixed() is then the general constructor, with discrete() and continuous() the direct way to say one kind on its own.

A region is written as a closed interval, but its endpoints carry no probability either way, a single point having no width, so open against closed makes no difference there. An atom that happens to sit on a region's boundary is simply tracked as an atom.

Recording where the mass is and where the density is are two pieces of information, not one. Knowing which values are possible is not enough: continuous(c(0, 1)) and mixed(discrete = 0, continuous = c(0, 1)) cover the same values, but they are different supports and the distributions over them differ: one has P(X = 0) = 0, the other does not. This is why an atom lying inside a region is kept rather than absorbed into it.

The third kind

Strictly, a measure on the real line splits into three parts, not two: mass on points, density over regions, and a third kind with neither — all of its probability on a set of zero total length, none of it sitting on any point. The Cantor distribution is the usual example. This is the Lebesgue decomposition, and the third part is called singular continuous. A support here has no way to describe one, so such distributions are out of reach.

Examples

discrete(discretes::natural0())   # e.g. the support of a Poisson
#> <support: discrete>
#> -- atoms --
#> Integer series of length Inf:
#> 0, 1, 2, 3, 4, 5, ...
discrete(c(3.5, 1.2, 6.7))        # finitely many atoms
#> <support: discrete>
#> -- atoms --
#> Numeric vector series of length 3:
#> 1.2, 3.5, 6.7
continuous(c(0, Inf))             # e.g. the support of a Gamma
#> <support: continuous>
#> -- continuous --
#> [0, Inf]
continuous(c(0, 1), c(3, 4))      # a union of regions
#> <support: continuous>
#> -- continuous --
#> [0, 1] U [3, 4]
mixed(discrete = 0, continuous = c(0, Inf))  # an atom, plus a tail
#> <support: mixed>
#> -- atoms --
#> Numeric vector series of length 1:
#> 0
#> -- continuous --
#> [0, Inf]