Archaeological sequence diagrams and Bayesian chronological models

Thomas S. Dye

, Caitlin E. Buck

University of Hawai'i at M



anoa, 735 Bishop St., Suite 315, Honolulu, HI 96813, USA

School of Mathematics and Statistics, University of Shefﬁeld, UK

article info

Article history:

Received 13 November 2014

Received in revised form

9 August 2015

Accepted 10 August 2015

Available online 12 August 2015

Keywords:

Sequence diagram

Chronology

Directed graph

Bayesian radiocarbon calibration

abstract

This paper develops directed graph representations for a class of archaeological sequence diagrams, such

as the Harris Matrix, that do not include information on duration. These “stratigraphic directed graphs”

differ from previous software implementations of the Harris Matrix, which employ a mix of directed

graph and other data structures and algorithms. A “chronological directed graph” to represent the re-

lationships in a Bayesian chronological model that correspond to the possibilities inherent in a sequence

diagram, and an algorithm to map a stratigraphic directed graph to a chronological directed graph are

proposed and illustrated with an example. These results are intended to be a proof of concept for the

design of a front-end for Bayesian calibration software that is based directly on the archaeological

stratigrapher's identiﬁcation of contexts, observations of stratigraphic relationships, inferences con-

cerning parts of once-whole contexts, and selection of materials for radiocarbon dating.

1. Introduction

Advances in the methods and practice of radiocarbon dating in

archaeology, sometimes characterized as revolutionary (Bayliss,

2009; Taylor, 1995; Linick et al., 1989), have worked generally to

increase the precision of age estimates for archaeological events. A

recent phase of this radiocarbon revolution has as its focus Bayesian

calibration (Buck et al., 1996), which highlights the role of strati-

graphic interpretation in the development of radiocarbon-based

site chronologies. A key innovation of Bayesian calibration is its

ability to integrate ancillary sources of chronological information

with the information returned by the radiocarbon dating labora-

tory. In a typical archaeological application having to do with site

chronology, records of the stratigraphic relationships of deposits

and interfaces are a primary source of this ancillary information.

Common sense indicates that a site chronology based on “the

dates” and “the archaeology” is bound to be more reliable than one

that relies only on one or the other (Bayliss, 2009, 127). The

improvement yielded by Bayesian calibration has been demon-

strated, perhaps most convincingly for the early Neolithic period of

Southern Britain and Ireland where time-scales with resolutions

that approach a human generation have been achieved (Bayliss

et al., 2011). At Çatalh

€

oyük, a Neolithic village in Anatolia, a basic

goal of the Bayesian calibration is to provide “calendar date esti-

mates for the construction, use, and disuse of the excavated

buildings, in order to infer a structural narrative between buildings

that are not stratigraphically related” (Bayliss et al., 2014, 69). Given

that a typical house at Çatalh

€

oyük was constructed, used, and dis-

used over a period on the order of 60e145 years (Bayliss et al., 2014,

89), the ambitious goal of identifying contemporary houses from

spatially separate parts of the site without the aid of dendrochro-

nology (Towner, 2002) would have been wildly unrealistic prior to

the development of AMS dating and Bayesian calibration.

The data requirements to achieve high precision estimates are

sufﬁciently stringent that often specialists are sought to select

samples for radiocarbon dating. The specialist works with a list of

potential dating samples and a model of relative chronological re-

lations yielded by stratigraphy, sometimes in the form of a

sequence diagram such as the Harris Matrix (Harris, 1989) but more

often in the form of proﬁle drawings and excavation notes, to

develop a chronological model that maximizes the value of the

calibration results for interpretation. In effect, the specialist trans-

forms one relative chronological model into another, moving from

the stratigrapher's model expressed in terms of units of stratiﬁca-

tion, or contexts (Carver, 2005, 107), into the statistician's model

expressed in terms of formal algebraic relationships between

chronological phases.

This paper describes a transformation algorithm based on the

theory of directed graphs that takes as its input a suitably struc-

tured sequence diagram and information on potential dating

* Corresponding author.

E-mail address: tsd@tsdye.com (T.S. Dye).

Contents lists available at ScienceDirect

Journal of Archaeological Science

journal homepage: http://www.elsevier.com/locate/jas

http://dx.doi.org/10.1016/j.jas.2015.08.008

Journal of Archaeological Science 63 (2015) 84e 93

samples to produce a chronological model for use in Bayesian

calibration. To demonstrate its utility in automating the creation of

Bayesian chronological models, we apply the algorithm to Buildings

1 and 5 in the North Area at Çatalh

€

oyük (Cessford, 2007d, c, b, a).

This example represents a relatively rare situation where a detailed

sequence diagram is published (Bayliss et al., 2014, Fig. 3.17) and

dating specialists have carried out several Bayesian calibrations

(Cessford et al., 2005; Bayliss et al., 2014).

2. Computing the sequence diagram

In archaeology, the term sequence diagram refers to a family of

graphic displays designed to represent stratigraphic relationships

(Carver, 2009, 276). Perhaps the most widely used sequence dia-

gram is produced by the Harris Matrix, which is described by its

creator as a method by which the order of the deposition of the

layers and the creation of feature interfaces through the course of

time on an archaeological site can be diagrammatically expressed

in very simple terms (Harris, 1989, 34). This focus on the order of

deposition to the exclusion of other attributes distinguishes the

Harris Matrix from sequence diagrams which augment the order of

deposition with information about duration (Dalland, 1984; Carver,

1979), and it is this sense in which sequence diagram is used here.

Since the transformation algorithm we propose is based on the

theory of directed graphs, the sequence diagram used as input must

be capable of representation as a directed acyclic graph, or DAG,

which can be manipulated programatically. A DAG conceptualizes

the stratigraphic structure of an archaeological sequence as chro-

nological relationships on a set of depositional and interfacial

contexts. A directed graph consists of one or more of a ﬁnite set of

nodes and zero or more connections between ordered pairs of

distinct nodes, each of which deﬁnes an arc (Harary et al., 1965). In

the case of archaeological stratigraphy, an archaeological context is

represented as a node and a stratigraphic relationship between two

contexts is represented by an arc.

Available Harris Matrix software packages are closed-source and

do not permit programmatic access to the DAG representation, so it

proved necessary to develop the open-source software package, hm,

to achieve this goal (provided as supplementary material).

Although computer programmers quickly recognized that the

sequence of observed stratigraphic relationships at the heart of the

sequence diagram can be represented as a DAG (Ryan, 1988;

Herzog, 1993; Herzog and Scollar, 1991), the display conventions

of the Harris Matrix are tied to the layout of paper forms developed

in the 1970s (Harris, 1989, 34) and these conventions introduce

complexities that can not be represented by a DAG. Thus, the hm

software abandons certain display conventions of the Harris Matrix

in order to preserve a pure DAG representation of the sequence

diagram.

The following sections compare and contrast DAG and Harris

Matrix representations of the sequence diagram and present the

data inputs to the hm software as tables that deﬁne entities in a

relational database (Fig. 1). The ﬁrst three sections consider the

relationships between contexts recognized by the Harris Matrixdi)

no direct stratigraphic relationship, or context identity, ii) an

observed relationship of superposition, and iii) parts of a once-

whole contextdin turn, as steps in the construction of a

sequence diagram. This is followed by a consideration of periods

and phases, which are conceptually similar interpretive constructs.

2.1. Identiﬁcation of contexts

Archaeologists commonly identify ﬁ

ve types of context: de-

posits, horizontal feature interfaces, vertical feature interfaces,

upstanding layer interfaces, and horizontal layer interfaces. The

Harris Matrix was designed, in part, to ensure that all of the con-

texts identiﬁed at a site are included in the sequence (Roskams,

2001, 157) and to replace the previous archaeological practice of

recording contexts and their relationships with section drawings,

which typically take in only some small fraction of the contexts

identiﬁed at a site (Bibby, 1993, 108).

In practice, the archaeologist working with a printed Harris

Matrix sheet draws up a list of identiﬁed depositional and feature

interface contexts, then writes each context identiﬁer in a rectan-

gular box on the grid. Contexts close to one another in space are

placed in rectangular boxes close to one another on the grid and the

vertical position is chosen to reﬂect the context's position in the

stratigraphic sequence, with surﬁ cial contexts placed near the top

of the diagram and basal contexts placed near the bottom. At this

stage the Harris Matrix consists of rectangular boxes with context

identiﬁers within them, and the rectangular boxes are not yet

connected to one another (Fig. 2, center).

By convention, horizontal layer interfaces are not represented in

the Harris Matrix because they are considered to have “the same

stratigraphic relationships as the deposits and are recorded as an

integral part of the layers” (Harris, 1989, 54). This practice appears

to be deeply ingrained in the archaeological community, but it is

problematic from the point of view of relative chronology (Clark,

2000, 103). Treating the layer interface as an integral part of the

depositional context beneath it ignores the possibility that it rep-

resents a unit of time, either because the surface it represents was

deﬂated by erosion, exposing old deposits, or because the surface

itself was open for some time. The failure to record layer interfaces

potentially introduces hiatuses into the chronological model. A

hiatus-free sequence diagram (and thus the associated directed

graph) exhibits a particular structure with alternating interfacial

and depositional contexts. In contrast, conventional stratigraphic

practice places deposits in a relationship of direct superposition

across unrecorded layer interfaces. Of course, archaeologists who

use the Harris Matrix recognize the unrecorded layer interfaces and

these are brought back into the analysis at a later stage, when pe-

riods are identiﬁed (Harris, 1989, Fig. 25). It is at this late analytic

stage that the deﬁnition of a period boundary as an interface and its

speciﬁcation in the Harris Matrix as a mix of interfaces and deposits

is reconciled (Harris, 1989,67e68).

Because the representation of a directed graph is not con-

strained by the conventions of the Harris Matrix, the shapes of

nodes can express the fundamental distinction between deposi-

tional and interfacial contexts. The convention adopted here uses a

rectangular box, similar to the symbol used in a Harris Matrix,

when unit-type is set to deposit and a trapezium when unit-

type is set to interface

(Fig. 2, right).

2.2. Observed stratigraphic relationships

The next step in construction of the sequence diagram is to

indicate observed stratigraphic relationships. In practice, the

stratigrapher records observed relationships in a two-column table,

where one column contains the identiﬁers of the younger contexts

that assume a superior position in the observed stratigraphic

relationship and the other column contains the identi ﬁers of the

older contexts that assume an inferior position in the observed

stratigraphic relationship (Fig. 1). For each row of the table, the

stratigrapher identiﬁes on the sequence diagram the rectangular

box that represents the younger context and searches below it for

the rectangular box that represents the older context. An orthog-

onal line is then drawn from the bottom of the rectangular box

representing the younger context to the top of the rectangular box

representing the older context (Fig. 3, center).

The directed graph uses the same table of observed stratigraphic

T.S. Dye, C.E. Buck / Journal of Archaeological Science 63 (2015) 84e93 85

relationships that the stratigrapher uses to draw the Harris Matrix.

It is easy to see that each row of table (Fig. 3, left) represents an

ordered pair of nodes, which in the theory of directed graphs de-

ﬁnes an arc. The ordering is given by the stratigraphic relationship

of the nodes; the younger context is by convention designated the

start node of the arc and the older context the end node. It is

customary to represent the arcs in a directed graph as arrows, with

an arrowhead at the end of each arc to indicate direction. However,

the Harris Matrix convention that uses a plain line and indicates

direction by vertical position, such that a younger context appears

above an older context with which it shares a stratigraphic

relationship, is appreciated by archaeologists who see in it the

physical relationship of the contexts when viewed in section. Thus,

the directed graphs presented here adopt this convention and draw

arcs as lines rather than arrows (Fig. 3, right).

At this stage in its construction, the Harris Matrix is a partial

order, or poset (Orton, 1980, 67). The stratigraphic relationships

that it records are irreﬂexive, because an archaeological context

cannot be stratigraphically superior or inferior to itself, asymmet-

rical because a context that is stratigraphically superior to another

context cannot be stratigraphically inferior to it, and transitive

because, given three contexts, 1, 2, and 3,if1 is stratigraphically

superior to 2, and 2 is stratigraphically superior to 3, then 1 is

stratigraphically superior to 3.

2.3. Parts of once-whole contexts

In the Harris Matrix, pairs of contexts inferred to have been part

of a once-whole context are connected with two horizontal lines to

indicate this relationship (Fig. 4, bottom left). The information

needed for this step is a table with two columns, where each row

represents an inference that the two contexts in it are parts of a

once-whole context (Fig. 1). Parts of a once-whole context describe

a symmetrical relation that is transitive; this type of relation is

outside the theory of directed graphs. Parts of a once-whole context

can be treated in two ways by a directed graph. In the ﬁrst, the

directed graph is used to model only observations of stratigraphic

relationships; inferred parts of a once-whole context can be plotted

at the same vertical level of the sequence diagram, but stratigraphic

relationships implied by the inference of once-wholeness are not

taken into account (Fig. 4, top right). In the second, the inference of

once-wholeness is assumed to be true and parts of a once-whole

context are treated as a single context (Fig. 4, bottom right). Thus,

the Harris Matrix displays in a single sequence diagram observa-

tions of stratigraphic relationships and inferences about parts of

once-whole contexts; two directed graphs are required to show the

same information.

2.4. Stratigraphic periods and phases

The terms “period” and “phase” are de

ﬁned variously and

Fig. 1. Relational database design for the seven tables of information used to construct stratigraphic and chronological directed graphs. Note that table names are uppercase, column

names are lowercase, and divided entries deﬁne the domain of the column whose name is directly above, e.g., the unit-type column in the context table contains one of the two

values deposit and interface.

Fig. 2. Initial stage in construction of a sequence diagram consisting of an interface,

context 1, and a deposit, context 2: left,aﬁve-column context table that records in-

formation about contexts (see Fig. 1); center, Harris Matrix; right, directed graph.

Fig. 3. The sequence diagram after stratigraphic relationships are indicated with

vertical lines: left, a two-column observation table that records the stratigraphic

relationship between contexts 1 and 2 (see Fig. 1); center, a Harris Matrix showing a

younger interface, context 1, overlying an older deposit, context 2; right, a directed

graph showing a younger interface, context 1, overlying an older deposit, context 2.

T.S. Dye, C.E. Buck / Journal of Archaeological Science 63 (2015) 84e9386

sometimes interchangeably by archaeologists. For the Harris Ma-

trix, a “phase” groups contexts of similar age, and a “period” groups

phases of similar age, yielding a nested series of time intervals

(Harris, 1989, 158). Deﬁned in this way, both phases and periods are

interpretive constructs that are typically formulated with both

stratigraphic and non-stratigraphic information. Because “phase” is

also used to describe Bayesian chronological models, here we use

the term “stratigraphic phase” to refer to a group of contexts, and

the term “ chronological phase” to refer to a time period in a

chronological model.

Alternative ways to represent periods and stratigraphic phases

can be illustrated using a stratigraphic proﬁle drawing developed

by Harris (1989, Fig. 12a) and adapted here (Fig. 5). The Harris

Matrix displays periods and stratigraphic phases in the same way,

by horizontal lines drawn across the diagram (Fig. 6, left). In

contrast, the directed graph convention displays periods and

stratigraphic phases by altering the graphic attributes of nodes

(Fig. 6, right).

3. Structure of a Bayesian chronological model

The chronological model now widely used in Bayesian chro-

nology construction comprises entities different than those of an

archaeological sequence diagram. The basic entity of a sequence

diagram is a stratigraphic context; a Bayesian chronological model

comprises directly-dated events and the start and end dates of one

or more chronological phases. The start and end dates of a chro-

nological phase typically map directly to an archaeological context,

and so in this paper we will assume that no additional information

is needed to represent them beyond that which is available from

the stratigraphic directed graph.

Within software such as hm, it is convenient to capture the in-

formation about dated events in two tables. An “event table” as-

sociates a directly-dated archaeological event with its

archaeological context (Fig. 1) and indicates whether the event is

directly associated with the context, is older than the context and

thus disjunct, or is younger than the context and thus disparate

(Dean, 1978). An “event order table” records information on the

relative ages of archaeological events associated with the same

context (Fig. 1).

One difference between a Bayesian chronological model and an

archaeological sequence diagram is that the Bayesian chronological

model may include relationships that cannot be expressed by

stratigraphy. An illustration recognizes three possible relationships

between two chronological phases where one is older than the

other (Fig. 7). Only two of these relationships can be represented

stratigraphically.



One chronological phase can be older than the other such that

the end date for the older chronological phase is older than the

start date for the younger chronological phase (Fig. 7, left). This

relationship, where a time interval separates two chronological

phases, arises in archaeological stratigraphy when two contexts

are found on the same line of a (possibly multi-linear) sequence

but are separated by one or more contexts. This relationship is

relatively common in practical Bayesian chronological models.

Contexts that lack dating material are typically ignored in a

Bayesian chronological model.

 One chronological phase can be older than the other such that

the end date for the older chronological phase is the same age as

the start date for the younger chronological phase (Fig. 7, mid-

dle). This abutting relationship describes the relationship of

superposition that archaeologists typically observe in the ﬁeld.

 One chronological phase can be older than the other such that

the end date for the older chronological phase is younger than

the start date for the younger chronological phase (Fig. 7, right).

This overlapping relationship cannot be determined solely on

stratigraphic grounds because the two contexts must be from

different lines of a multi-linear stratigraphic sequence. Other

information, perhaps having to do with the content of the

contexts, is required to posit this kind of relationship (Triggs,

1993).

Another difference between a Bayesian chronological model and

an archaeological sequence diagram is that the archaeological

sequence diagram is concerned only with relationships between

archaeological contexts, but the chronological model includes re-

lationships among a variety of different entities, including early

phase boundaries, late phase boundaries, and dated events. In

addition, the notation for recording relationships between phase

boundaries must distinguish between phase boundaries that share

the same calendar age and phase boundaries that are separated in

time. For example, depositional context i, within which a single

event, e, was identiﬁed and dated might be represented by the

chronological model as a

> q

> b

, where a

and b

are the start and

end dates, respectively, of chronological phase i, q

represents the

calendar age of event e, and > means “is older than”. Alternatively,

this simple chronological model can be represented as a directed

Fig. 4. Three graphical representations of parts of a once-whole context: top left, two-

column once-whole table recording the inference that contexts 2 and 3 are parts of a

once-whole context (see Fig. 1); bottom left, the Harris Matrix connects contexts 2 and

3 with two horizontal lines; top right, a directed graph representation of the observed

relationships of superposition places contexts 2 and 3 at the same level, but does not

make explicit the inferred stratigraphic relationship between contexts 1 and 3; bottom

right, a directed graph representation of the sequence diagram where the inferred

relationship between contexts 2 and 3 as parts of a once-whole context is assumed to

be true and the contexts have been merged and labeled “2 ¼ 3”.

Fig. 5. Illustrative stratigraphic proﬁle drawing. Adapted from Harris (1989, Fig. 12a).

T.S. Dye, C.E. Buck / Journal of Archaeological Science 63 (2015) 84e93 87

graph (Fig. 8, left), where vertical position represents relative age,

similar to the convention used in directed graphs of archaeological

sequences.

4. Mapping a sequence diagram to a chronological model

Given a directed graph of a hiatus-free archaeological sequence

from which transitive relationships have been removed, it is

possible to construct a Bayesian chronological model by combining

the relative chronological information in the directed graph of the

archaeological sequence diagram with the potentially dated events.

Recall that a directed graph consists of a ﬁnite set of nodes and a

collection of ordered pairs of distinct nodes, the connection be-

tween any pair of which is called an arc. Two nodes connected by an

arc are said to be adjacent; the start node of the arc is adjacent to the

end node, and the end node is adjacent from the start node. The

outdegree of a node is the number of nodes adjacent from it, and the

indegree of the node is the number of nodes adjacent to it. A walk in

a directed graph is an alternating sequence of nodes and arcs, and a

path is a walk in which all nodes are distinct. If there is a path from

node u to node v, then node v is reachable from node u. The directed

graph concept of reachability can be used to determine whether

two contexts are on the same line of a possibly multilinear

sequence diagram. If, for two archaeological contexts, x and y, x is

reachable from y or y is reachable from x, then x and y are on the

same line of the sequence diagram. Conversely, if x is not reachable

from y and y is not reachable from x , then x and y are on different

lines of a multi-linear sequence diagram.

For two archaeological contexts, x and y, on the same line of an

hiatus-free sequence diagram such that y is reachable from x, the

directed graph concept of adjacency can be used to distinguish an

abutting chronological relationship, where x is adjacent to y, from a

separated relationship, where x is not adjacent to y

. These re-

lationships are illustrated in Fig. 9, which categorizes contexts ac-

cording to their chronological relationship to Context 4 using a

directed graph that includes contexts and their observed strati-

graphic relationships (Fig. 9, center) and one that augments this

Fig. 6. Hypothetical phasing of an example sequence developed by Harris (see Fig. 5): left, the Harris Matrix representation, after Harris (1989, Fig. 12c); right, directed graph

representation with nodes shaded to indicate phases.

Fig. 7. Schematic representation of three possible relationships between older and

younger chronological phases: left, chronological phases 1 and 2 separated, b

> a

;

middle, chronological phases 1 and 2 abutting, b

¼ a

; right, chronological phases 1

and 2 overlapping, b

< a

. Adapted from Buck et al. (1996, Fig. 9.8).

T.S. Dye, C.E. Buck / Journal of Archaeological Science 63 (2015) 84e9388

information with inferences about once-whole contexts (Fig. 9,

right). These graphs indicate that directed graph representations of

an archaeological sequence contain the information needed to

construct a Bayesian chronological model.

The maximal chronological directed graph is obtained by adding

to the stratigraphic directed graph extra nodes and arcs to repre-

sent the information in the event table and the event order table.

Since the number of contexts with potentially dated events is

Fig. 8. Entities and relationships of Bayesian chronological models represented as directed graphs: left, a chronological phase with a single dated event; middle, relationship

between boundary parameters of separated chronological phases; right, relationship between boundary parameters of abutting chronological phases.

Fig. 9. A hiatus-free sequence diagram with contexts shaded according to their chronological relationship to Context 4: left, the stratigraphic proﬁle after Harris (1989, Fig. 12), with

layer interfaces numbered 10e18 (cf. Fig. 5); center, a directed graph representation of the sequence diagram depicting observed relationships of superposition; right, a directed

graph representation of the sequence diagram in which inferences of once-whole contexts are assumed to be true.

T.S. Dye, C.E. Buck / Journal of Archaeological Science 63 (2015) 84e93 89

typically much smaller than the number of undated ones, however,

an algorithmic version of this approach would not closely mirror

what those constructing Bayesian models do at present. A six step

algorithm can, however, be used to construct the minimal chro-

nological directed graph (and hence chronological model) from the

directed graph of the archaeological sequence and the two tables of

potentially dated event information, as follows.

Suppose the set of all contexts in our stratigraphic directed

graph is C and that the subset of those with potentially dated events

is D. The number of elements in D, #D, is typically much smaller

than the number in C since relatively few contexts from the exca-

vation contain potentially dated ﬁnds. The set of potentially dated

events (i.e. events in the event table) is then E with individual el-

ements

; e

; /; e

, where E ¼ #E . Each member of E was

excavated from a context and so is associated with one and only one

member of D ¼

; d

; /; d

, where D ¼ #D.

1. For each member of D, d

, add two nodes to the chronological

directed graph, one for the early boundary date, a

, and the

other for the late boundary date, b

2. For each member, e

,ofE add one node, q

, to the chronological

directed graph to represent its absolute date.

3. For each row in the event order table add an arc from the

younger node to the older node.

4. For each row, j ¼ 1; 2; /; E, of the event table (associated with

archaeological context d

and event with absolute date q

Þ:

a) if the indegree of q

is 0 (and association is not equal to

“disjunct”) add an arc from a

to q

and assign it a value of 0;

b) if the outdegree of q

is 0 (and association is not equal to

“disparate”) add an arc from q

to b

and assign it a value of

5. For each pair ðd

; d

Þ of archaeological contexts in the event

table:

a) if d

is reachable from d

in the directed graph of the

archaeological sequence, add an arc from b

to a

in the

chronological directed graph;

b) if context d

is adjacent to context d

in the directed graph of

the archaeological sequence, assign the arc from b

to a

the chronological directed graph a value of 1, else assign it a

value of 2;

c) If context d

is reachable from context d

in the directed

graph of the archaeological sequence, add an arc from b

in the chronological directed graph;

d) if context d

is adjacent to context d

in the directed graph of

the archaeological sequence, assign the arc from b

to a

the chronological directed graph a value of 1, else assign it a

value of 2.

6. Perform transitive reduction.

5. Discussion

At present, it appears to be the case that no archaeologists build

their chronological models using formal algorithms. Instead they

apply their expert judgment, selecting features from the strati-

graphic record to include in the model on whatever basis they

choose and justify their decisions in prose in the resulting publi-

cation. Such an approach may well lead archaeologists to learn all

they wish to from the chronological evidence available, but it

would be hard to demonstrate that and few authors at present even

discuss the impact of their choice of chronological model on the

results obtained.

An example where the authors do discuss the impact of model

choice is the work undertaken to establish the chronology of

Buildings 1 and 5 in the North Area excavations at Çatalh

€

oyük

(Cessford et al., 2005; Bayliss et al., 2014). The initial work was

exploratory in nature, with one goal “to determine which types of

material and/or context provide good dating evidence” (Cessford

et al., 2005, 84). The reliability of each dated sample was ranked

as “low” where “there is a direct stratigraphic relationship between

determinations that contradicts the relationship between the ages

of the two determinations” (Cessford et al., 2005, 76), “high” where

the sample comes from “a consistently dated stratigraphic

sequence” (Cessford et al., 2005, 76) or where it is “short lived

material from a context with a low probability of residuality”

(Cessford et al., 2005, 76), or “medium” otherwise (Supplementary

Material Table S1). Where possible, contradictions were resolved

with reference to four of the ﬁve age determinations from Context

1332þ

in Building 1, a “deliberately-placed deposit of lentils which

represents a single year's harvest of a short-lived species that was

purposefully burnt” (Cessford et al., 2005, 86). Context 1332þ has a

direct stratigraphic relationship with all of the contexts excavated

from Building 5, which underlies Building 1, but its age relative to

most of the contexts from Building 1 cannot be determined (Fig. 10).

Since the full sequence diagram for Buildings 1 and 5 is too large

to reproduce here and given its pivotal role in the interpretation of

the chronology of both buildings, we focus our illustration on

Context 1332þ and those closest to it stratigraphically. However,

the full sequence diagram and the chronological models derived via

our algorithm are provided in the Supplementary Material.

A directed graph representation of the chronological model

implied by the exploratory analysis accepts the assumption that

each dated sample is associated with the context from which it was

collected (Fig. 11). The chronological model indicates that none of

the related contexts superior to Context 1332þ in Building 1 were

dated. Of the six dated contexts that are stratigraphically related to

Context 1332þ, ﬁve are from Building 1 and one, Context 3810þ,is

from Building 5. Thus, potential contradictions could be worked out

with direct reference to the lentil deposit for a small subset of the

dated contexts.

Carrying through the exploratory approach, Cessford et al.

rejected the age determination for one of the lentils, q

,as

inconsistent with the other four age determinations on lentils from

Context 1332þ, q

, q

, and q

. Two dates on animal bone, q

from Context 1295aþ and q

from Context 1456, were assigned

medium reliability because they were older than botanical material

from the same deposits and the four lentils (Cessford et al., 2005,

88). As can be seen in Supplementary Material Figure S1, these

comparisons with the lentils are not based on stratigraphic re-

lationships; Contexts 1295aþ and 1456 are not reachable from

Context 1332þ and their relative ages cannot be determined on

stratigraphic grounds. Instead, the comparison appears to be made

on the basis of “the division of the site into phases” (Cessford et al.,

2005, 65), and thus on inferences rather than direct observations.

Similarly, six dates on human bone were considered to be “in

agreement with the stratigraphic sequence and the determinations

from the lentils” (Cessford et al., 2005, 87), however ﬁve of these

dates, q

4953

, have no stratigraphic relationship to the lentils, and

these comparisons also appear to be a result of phasing. One age

determination, q

from Context 2519, is stratigraphically inferior

to the lentils and so directly comparable.

Subsequently, dates on human bone and antler processed at the

It was frequently the case that a single context was assigned two or more ﬁeld

numbers. These ﬁeld numbers were carried through the analysis and appear on the

published Harris matrix for the excavation (Bayliss et al., 2014, Figure 3.17). The

convention adopted here typically uses the ﬁrst ﬁeld number assigned to a context

and indicates multiple ﬁeld numbers for a single context by appending a “þ” to the

ﬁeld number.

T.S. Dye, C.E. Buck / Journal of Archaeological Science 63 (2015) 84e9390

Oxford Radiocarbon Accelerator Unit between 2000 and 2002 were

shown to be incorrect due to a technical problem. When re-dated,

the bone and antler samples from Çatalh

€

oyük, including the six

dates on human bone, were determined to be 50e150 BP younger

than the original measurements (Bayliss et al., 2014, 79). In

particular, q

, which replaced q

from Context 2529, strati-

graphically inferior to the lentils, returned a date younger than the

four lentils, but older than the lentil that was previously rejected.

Fig. 10. A portion of the sequence diagram for Buildings 1 and 5 of the North Area excavations at Çatalh

€

oyük showing Context 1332þ in Building 1, adjacent and reachable contexts

whose ages relative to Context 1332þ are known, and unreachable contexts whose ages relative to Context 1332þ can not be determined stratigraphically. Note that the majority of

the contexts shown on the diagram are deposits and that interfacial contexts are comparatively rare. The full sequence diagram, of which this is a part, is available as Supplementary

Material Figure S1.

Fig. 11. Representation of the dated lentils from Context 1332þ on a chronological model for determining which types of material and/or context provide good dating evidence

using the dated samples reported by Cessford et al. (2005, Table 4.10) and the sequence diagram for the North Area excavations (Fig. 10). The full chronological model is available as

Supplementary Material Figure S2.

T.S. Dye, C.E. Buck / Journal of Archaeological Science 63 (2015) 84e93 91

Accordingly, the four lentils previously determined to represent the

true age of the lentil deposits were interpreted as residual, and the

lentil previously believed to be a statistical outlier was accepted as

dating the true age of the deposit. This circumstance, and a

comprehensive reevaluation of the suitability of the dated sample

materials based largely on experience gained subsequent to the

original exploratory dating project (Bayliss et al., 2014,81e88),

resulted in a different chronological model, one in which a large

proportion of the dated samples are termini post quem for the end

date of the context from which they were collected but have no

relationship to the start date (Fig. 12). These “dangling

's” graph-

ically illustrate the substantial challenges posed by residuality for

the ambitious dating project at Çatalh

€

oyük.

6. Conclusions

Directed acyclic graphs are already in widespread use in a

number of disciplines in which, for reasons of practicality or logic, a

collection of tasks or ideas must be ordered into a sequence. Many

well established algorithms now exist for performing inference on

ideas that are represented as DAGs including, for example, the

Markov chain Monte Carlo (MCMC) algorithms now so widely used

in Bayesian inference in general and in Bayesian chronological

modelling in particular.

Like many other statistical models, Bayesian chronological

models are hierarchical in nature, with calendar ages of individual

samples, linked sequentially to those for contexts, phases, struc-

tures, and so on. Such models have for many years been repre-

sented as DAGs both in publications (Parent and Rivot, 2013; King

et al., 2010) and in software tools. Of the latter, the general pur-

pose Bayesian inference environment known as WinBUGS (Lunn

et al., 2000) e one of the ﬁrst to become widely used e allows

users the choice to deﬁne their model via a DAG from which the

software generates the Bayesian model automatically.

One natural future use of the construction of chronological

directed graphs from stratigraphic ones would thus be as a front-

end to Bayesian chronological modelling software. Users could

then develop a plethora of chronological directed graphs (based on

automated algorithms, expert judgment, or both), estimate the

parameters of the resulting models given real or simulated data,

compare the resulting chronologies and even conduct formal

model choice to establish which model best ﬁts the currently

available data.

Prototype software for creating and illustrating both strati-

graphic and chronological directed graphs was developed to carry

out the analyses in this paper.

The software establishes that the

conversion from archaeological sequence diagram to a Bayesian

chronological model can be made entirely rule-based and thus

relatively straightforward. However, if others wish to beneﬁt from

these developments, and particularly if the automated generation

of chronological directed graphs from stratigraphic ones is seen as

beneﬁcial, then more work is needed. The next phase of this project

will thus involve close collaboration with those who code Bayesian

chronological modeling software with a view to providing a

directed graph front-end that will offer a more intuitive way for

archaeologists to build chronological models than such software

offers at present and, ultimately, allow systematic exploration of

the impact of different models on the chronological inferences

made.

Acknowledgments

The authors thank Alex Bayliss for suggesting the example of

Çatalh

€

oyük Buildings 1 and 5 and for providing guidance during our

analysis; Craig Cessford for clarifying conventions used in the

representation of the Harris Matrix for Buildings 1 and 5 at Çata-

€

oyük; Julian Richards, Keith May, and Kieron Niven for advice on

data standards and assistance searching the ADS archives; Eric

Schulte for the graph.lisp library and for patient help during

development of the hm software; and two anonymous reviewers for

recommending that the paper include a real-world example. Any

errors are the authors'.

Appendix A. Supplementary data

Supplementary material related to this article can be found at

http://dx.doi.org/10.1016/j.jas.2015.08.008.

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