Allen's interval algebra for time vectors

Allen's interval algebra has thirteen base relations for time intervals. == is defined in mt_linear-compare/mt_cyclical-compare; the twelve operators below define the rest, computed from the same interval endpoints (start/end) that comparison derives. Together, they cover all thirteen relations:

Operator	Relation	e1 %op% e2 holds iff
%p%	precedes	e1 ends before e2 begins (a gap)
%m%	meets	e1 ends exactly where e2 begins (no gap, no overlap)
%o%	overlaps	e1 begins before e2, e1 ends between the start and end of e2
%s%	starts	e1 and e2 begin together, and e1 ends first
%d%	during	e1 is strictly within e2's span
%f%	finishes	e1 and e2 end together, and e1 begins later
==	equals	e1 and e2 share both endpoints
%pi%	preceded by	e2 %p% e1
%mi%	met by	e2 %m% e1
%oi%	overlapped by	e2 %o% e1
%si%	started by	e2 %s% e1
%di%	contains	e2 %d% e1
%fi%	finished by	e2 %f% e1

mt_linear-compare/mt_cyclical-compare's </> are a deliberate deviations from their use as 'precedes' in Allen's interval algebra (%p%/%pi% operators). The important difference is that </> include 'meets' relations, while %p%/%pi% requires gaps, so in the examples below jan < feb is TRUE in mixtime but the equivalent symbol in Allen's interval algebra (jan %p% feb) is FALSE because there is no gap.

The implementation here also extends to continuous time instants, which are treated as degenerate zero-width intervals. This allows testing how a specific time instant relates to time spans, for example does January 2020 contain 30% through the day 2020-01-24 is yearmonth("2020-01") %di% date(datetime("2020-01-24 07:12:00"), discrete = FALSE). This is a technical deviation from Allen's interval algebra, which assumes that intervals are non-degenerate (i.e. end > start, not end >= start).

Usage

e1 %p% e2

e1 %pi% e2

e1 %m% e2

e1 %mi% e2

e1 %o% e2

e1 %oi% e2

e1 %s% e2

e1 %si% e2

e1 %d% e2

e1 %di% e2

e1 %f% e2

e1 %fi% e2

Arguments

e1, e2

mt_linear or mt_cyclical vectors sharing a mode of time (and, for mt_cyclical, a cycle) - or values castable to one, such as plain numeric vectors in the other operand's chronon, read from its epoch as mixtime() reads them. mt_duration has no interval to compare and so is not supported.

Value

A logical vector.

References

Allen, J. F. (1983). Maintaining knowledge about temporal intervals. Communications of the ACM, 26(11), 832-843.

See also

mt_linear-compare and mt_cyclical-compare for ==, <, >, <=, >= - the three of Allen's relations that also form the package's ordering (</> in a looser, adjacency-inclusive form than %p%/%pi%).

Examples

jan <- yearmonth("2020 Jan")
feb <- yearmonth("2020 Feb")
mar <- yearmonth("2020 Mar")
q1 <- yearquarter("2020 Q1")

jan == jan    # January equals itself
#> [1] TRUE

jan %p% mar   # January precedes March: February leaves a genuine gap
#> [1] TRUE
mar %pi% jan  # ... equivalently, March is preceded by January
#> [1] TRUE

jan %m% feb   # January meets February: adjacent, no gap
#> [1] TRUE
feb %mi% jan  # ... equivalently, February is met by January
#> [1] TRUE
jan < feb     # `<` also holds for adjacent pairs, unlike `%p%`
#> [1] TRUE
jan %p% feb   # FALSE: no gap between them, so they don't (strictly) precede
#> [1] FALSE

jan %s% q1    # January and Q1 start together, but January finishes first
#> [1] TRUE
q1 %si% jan   # ... equivalently, Q1 is started by January
#> [1] TRUE

feb %d% q1    # February is strictly within Q1
#> [1] TRUE
q1 %di% feb   # ... equivalently, Q1 contains February
#> [1] TRUE

mar %f% q1    # March and Q1 finish together, but March starts later
#> [1] TRUE
q1 %fi% mar   # ... equivalently, Q1 is finished by March
#> [1] TRUE