Transform a Support

Push a support through a strictly monotonic map, giving the support of the transformed variable.

Usage

support_transform(
  support,
  fun,
  inv,
  ...,
  increasing = TRUE,
  domain = c(-Inf, Inf),
  range = c(-Inf, Inf)
)

support_shift(support, by)

support_scale(support, by)

support_reciprocal(support)

Arguments

support

A support object, or a distribution.

fun, inv

The map and its inverse. Both must be vectorised, and fun must be strictly monotonic on the support.

...

Not used; must be empty. Present so that the arguments below are matched by name.

increasing

Whether fun is increasing. FALSE for a decreasing map, which reverses each interval's endpoints.

domain, range

The domain and range of fun, needed to transform an atomic part that is described rather than enumerated.

by

For support_shift() and support_scale(), the amount to shift or scale by.

Value

A support object.

Details

support_shift(), support_scale(), and support_reciprocal() are the common cases, and avoid having to supply an inverse, a domain, and a range by hand.

support_reciprocal() maps each side of zero separately, since 1 / x is monotonic on each side but not across the two. A support with an atom at zero has no reciprocal, and is an error. Zero lying inside a region is fine: a single point carries no probability there.

Scaling by zero sends every value to 0. Density that was spread over a region is compressed onto that single point, and density compressed onto a point is mass. So whatever the support was, the result has a mass at 0 and density nowhere: discrete(0). Only an empty support, having nothing to compress, stays empty.

It only works in that direction. A mass sits on one point and lands on one point, so mass stays mass.

A strictly monotonic map stretches and shifts regions but never squashes one down to a point, so density stays density and mass stays mass. That is why support_transform() asks for a monotonic map, and why scaling by zero — which is not one — is handled separately.

See also

Other Support algebra: support_add_atoms(), support_contains(), support_restrict(), support_union()

Examples

support_shift(continuous(c(0, 1)), by = 5)
#> <support: continuous>
#> -- continuous --
#> [5, 6]
support_scale(discrete(natural0()), by = 2)
#> <support: discrete>
#> -- atoms --
#> Transformed series of length Inf:
#> 0, 2, 4, 6, 8, 10, ...

# A decreasing map reverses the region.
support_scale(continuous(c(1, 2)), by = -1)
#> <support: continuous>
#> -- continuous --
#> [-2, -1]

# Scaling by zero collapses everything onto a single atom.
support_scale(continuous(c(1, 2)), by = 0)
#> <support: discrete>
#> -- atoms --
#> Numeric vector series of length 1:
#> 0

# Reciprocal of a support spanning zero.
support_reciprocal(continuous(c(-2, 4)))
#> <support: continuous>
#> -- continuous --
#> [-Inf, -0.5] U [0.25, Inf]

# The general form.
support_transform(
  continuous(c(0, Inf)),
  fun = exp, inv = log,
  domain = c(0, Inf), range = c(1, Inf)
)
#> <support: continuous>
#> -- continuous --
#> [1, Inf]