Aligning Relative Sequences
Introduction
This vignette outlines functions that check and align multiple
relative sequences of events using eratosthenes. Not all
sequences are however of the same informational value. Some sequences
may be based on ideal or optimal theoretical assumptions, as with
frequency or contextual seriation, and others they may be based on
physical relationships, as with soil stratigraphy. It is clear that such
theoretical sequences should yield to those based on physical
relationships, but this raises the problem of how to coerce one or more
theoretical sequences which may contain events that also have physical
relationships. For example, one can produce an ideal seriation of
contexts that includes discrete, single deposits on the one hand (say a
number of separate pits with no overlap) and on the other stratified
deposits. In such an optimal seriation, it is possible for those
stratified deposits to be seriated “out of order” with respect to their
stratified sequence (i.e., the stratigraphy may have been perturbed at
one more moments so that their finds assemblages are not well
stratified). Accordingly, it is desirable to take the optimal order
achieved via seriation and constrain it back to agree with the soil
stratigraphy.
Checking Sequence Agreement
Events are structured within two classes of objects, produced by the
following functions: events() comprise a single, ordered
sequence of events, which must be unique character elements, while
sequences() comprise a list of events objects,
which must not conflict with one another in their ordering across all
events objects. Basically, events() is a more
restrictive class of c(), while sequences() is
a more restrictive class of list(). The operation of the
seq_check() function is run automatically when creating a
sequences() object, which will throw an error if any
events objects disagree in their ordering.
Both sequences x and y contain the same
events in the same order:
x <- events("A", "B", "C", "D", "E")
y <- events("B", "D", "F", "E")
a <- sequences(x, y)
Merging Sequences
The synth_rank() function will use recursion in order to
produce a single, “merged” or “synthesized” sequence from two or more
sequences. This is accomplished by counting the total number of elements
after running a recursive trace through all partial sequences (via the
quae_postea() function, on which see below). If partial
sequences are inconsistent in their rankings, a NULL value
is returned.
x <- events("A", "B", "C", "D", "H", "E")
y <- events("B", "D", "F", "G", "E")
a <- sequences(x, y)
synth_rank(a)
#> Events object of 8 elements:
#> A, B, C, D, F, H, G, E
Producing a single merged or synthesized sequence is a matter of
procedural convenience for the gibbs_ad() function (see the
vignette on Gibbs Sampling for Archaeological Dates).
To be sure, events missing from one sequence or another could occur at
different points in the merged sequence. In the example above,
"H" could occur at any point after "D" and
before "E", but the synth_rank() function has
situated it in between "F" and "G".
If sequences disagree, a NULL value is returned for the
synth_rank() function.
Adjusting Sequences
As mentioned above, one may have sequences which are derived via
theoretical considerations (e.g., frequency or contextual seriation),
and some which are known (e.g., soil stratigraphy or historical
documentation). The seq_adj() function will take an “input”
sequence and adjust its ordering to fit with another “target” sequence
of smaller size.
For example, the input sequence might be an ordering obtained from a
contextual seriation which is based on artifact types found in both tomb
assemblages as well as stratified deposits, from one or more sites. And
the target sequence might be a known stratigraphic sequence. One wants
to maintain the ordering of the target sequence, adjusting the input
into agreement:
# input
seriated <- events("S1", "T1", "T2", "S3", "S4", "T4", "T5", "T6", "S5", "T7", "S2")
# target
stratigraphic <- events("S1", "S2", "S3", "S4", "S5")
# input adjusted to agree with the target
seq_adj(seriated, stratigraphic)
#> Events object of 11 elements:
#> S1, T1, S2, T2, S3, T7, S4, T4, T5, T6, S5
To achieve this the seq_adj() function performs a linear
interpolation between jointly attested events, placing the input
sequence along the \(x\) axis and the
target sequence along the \(y\) axis,
coercing the order of all elements to the \(y\) axis:
For multiple stratigraphic sequences, the seq_adj()
function can be re-run, taking a new target sequence and using the
previous result as the new input sequence. If both input and target
sequences agree, the input sequence will be returned.
All Earlier Events, All Later Events
A core need of the gibbs_ad() function is to determine,
for each event, all events which come later and earlier than
that event. The functions quae_postea() and
quae_antea() achieve this need for later events and earlier
events respectively. Hence, there is no need to determine only one,
single sequence or ordering as an input. Instead, a list
object which contains multiple (partial or incomplete) sequences. The
output is a list indexed with each element, containing the
vector of contexts which precede or follow that element.
For quae_antea(), a dummy element of
"alpha" is included in all vectors, and for
quae_postea(), a dummy element of "omega" is
included. The elements of "alpha" and "omega"
are necesary as they constitute the fixed lower and upper limits in
which estimates are made in gibbs_ad().
x <- events("A", "B", "C", "D", "H", "E")
y <- events("B", "D", "F", "G", "E")
quae_postea(x, y)
#> $A
#> [1] "B" "C" "D" "H" "E" "F" "G" "omega"
#>
#> $B
#> [1] "C" "D" "H" "E" "F" "G" "omega"
#>
#> $C
#> [1] "D" "H" "E" "F" "G" "omega"
#>
#> $D
#> [1] "H" "E" "F" "G" "omega"
#>
#> $H
#> [1] "E" "omega"
#>
#> $E
#> [1] "omega"
#>
#> $F
#> [1] "E" "G" "omega"
#>
#> $G
#> [1] "E" "omega"
quae_antea(x, y)
#> $A
#> [1] "alpha"
#>
#> $B
#> [1] "A" "alpha"
#>
#> $C
#> [1] "A" "B" "alpha"
#>
#> $D
#> [1] "A" "B" "C" "alpha"
#>
#> $H
#> [1] "A" "B" "C" "D" "alpha"
#>
#> $E
#> [1] "A" "B" "C" "D" "H" "F" "G" "alpha"
#>
#> $F
#> [1] "A" "B" "C" "D" "alpha"
#>
#> $G
#> [1] "A" "B" "C" "D" "F" "alpha"
Using events as input into quae_postea()
and quae_antea() automatically check of the agreement of
events’ ordering. A sequences object can however also be
used as input:
x <- events("A", "B", "C", "D", "H", "E")
y <- events("B", "D", "F", "G", "E")
a <- sequences(x, y)
quae_postea(a)
#> $A
#> [1] "B" "C" "D" "H" "E" "F" "G" "omega"
#>
#> $B
#> [1] "C" "D" "H" "E" "F" "G" "omega"
#>
#> $C
#> [1] "D" "H" "E" "F" "G" "omega"
#>
#> $D
#> [1] "H" "E" "F" "G" "omega"
#>
#> $H
#> [1] "E" "omega"
#>
#> $E
#> [1] "omega"
#>
#> $F
#> [1] "E" "G" "omega"
#>
#> $G
#> [1] "E" "omega"
quae_antea(a)
#> $A
#> [1] "alpha"
#>
#> $B
#> [1] "A" "alpha"
#>
#> $C
#> [1] "A" "B" "alpha"
#>
#> $D
#> [1] "A" "B" "C" "alpha"
#>
#> $H
#> [1] "A" "B" "C" "D" "alpha"
#>
#> $E
#> [1] "A" "B" "C" "D" "H" "F" "G" "alpha"
#>
#> $F
#> [1] "A" "B" "C" "D" "alpha"
#>
#> $G
#> [1] "A" "B" "C" "D" "F" "alpha"