# Chronometric dating and seriation math

Conceived and designed the experiments: Data are available along with an implementation of the algorithm described in the GitHub repository *Chronometric dating and seriation math* Frequency seriation played a key role in the formation of archaeology as a discipline due to its ability to generate chronologies.

Interest in its utility for exploring issues of contemporary interest beyond Chronometric dating and seriation math, however, has been limited. This limitation is partly due to a lack of quantitative algorithms that can be used to build deterministic seriation solutions. When the number of assemblages becomes greater than just a handful, the resources required for evaluation of possible permutations easily outstrips available computing capacity.

On the other hand, probabilistic approaches to creating seriations offer a computationally manageable alternative but rely upon a compressed description of the data to order assemblages.

This compression removes the ability to use all of the features of our data to fit to the seriation model, obscuring violations of the model, and thus lessens our ability to understand the degree to which the resulting order is chronological, spatial, or a mixture.

Recently, frequency seriation has been reconceived as a general method for studying the structure of cultural transmission through time and across space. The use of an evolution-based framework renews the potential for seriation but also calls for a computationally feasible algorithm that is capable of producing solutions under varying configurations, without manual trial and "Chronometric dating and seriation math" fitting.

Here, we introduce the Iterative Deterministic Seriation Solution IDSS for constructing frequency seriations, an algorithm that dramatically constrains the search for potential valid orders of assemblages. Our initial implementation of IDSS does not solve all the problems of seriation, but begins to moves towards a resolution of a long-standing problem in archaeology while opening up new avenues of research into the study of cultural relatedness.

The results "Chronometric dating and seriation math" favorably to previous analyses but add new details into the structure of cultural transmission of these late prehistoric populations. Frequency seriation is a technique that produces chronological sequences by arranging descriptions of assemblages so that the frequencies of artifact classes jointly form unimodal distributions.

Developed in the early 20th century, frequency seriation played an integral role in the emergence of archaeology as a coherent discipline [ 2 ] and enabled culture historians to construct regional chronologies of prehistory throughout the New World [ 3 — 12 ].

Yet, for the last 50 years, frequency seriation has been largely ignored due to its association Chronometric dating and seriation math relative chronology and the mistaken belief that radiometric dating techniques have replaced it. While there has been some interest in seriation for disciplines outside of archaeology [ 15 — 18 ], to the extent that methodological development has occurred in archaeology over the last 50 years, the focus has been largely on reducing the method to probabilistic similarity-ordering problems that can be attacked via multivariate statistical methods [ 19 — 23 ].

The roots of Chronometric dating and seriation math seriation, however, stem from a deterministic algorithm that identifies orders on the basis of occurrence and frequency criteria. Recently, deterministic frequency seriation hereafter, DFS received some attention due to the demonstration that the method can be theoretically rationalized using an evolutionary framework.

While the potential of this idea has been long recognized [ 24 — 26 ], the work of Neiman [ 27 ] firmly established an explanatory basis within cultural transmission models for the unimodal distributions that form the core of the frequency seriation algorithm.

With these advances, there remains substantial promise for DFS to again become a primary tool for archaeological analyses as it enables researchers to quantitatively track patterns of interaction, define social communities, and trace lineages among past populations, in addition to informing upon chronology.

In this way, frequency seriation could serve as a key method in the establishment of a fully evolution-based discipline. Despite its potential, the use of DFS as a productive tool for archaeological research remains difficult, and methods for constructing and evaluating solutions are incomplete.

While a handful of assemblages can be seriated using *Chronometric dating and seriation math* manipulation, sorting through all possible orderings for a set of assemblages is neither feasible nor systematic. When the numbers of assemblages grows, a combinatorial explosion sets in, first visible once 10 or more assemblages are analyzed.

The order of magnitude of numbers involved makes brute force approaches impossible even using modern computing power. This limitation was recognized early in the discipline. When archaeologists became concerned with the quantitative basis of their methods, probabilistic approaches were developed that could construct orders on Chronometric dating and seriation math

basis of similarity scores [ 41 — 49 ]. With probability-based seriation techniques one is guaranteed to *Chronometric dating and seriation math* a solution, but the order produced reflects sources of variability beyond time including the effects of sample size, biased transmission processes and spatial variation [ 1 ].

While one may suspect that the final order is largely chronological, it is not possible to ascertain the degree to which the order represents time or other possible factors. The order of any particular subset of assemblages might be explained as a consequence of several factors: Allowing a computational method to obscure the causal influence of these factors destroys the value that seriation can have in helping disentagle such factors in real data sets.

Here, we introduce a new quantitative seriation algorithm that addresses the computational barrier inherent in DFS while also Chronometric dating and seriation math upon the logical structure of the original method. The algorithm succeeds by iteratively constructing small seriation solutions and then using the successful solutions as the basis for creating larger ones. Significantly, the proposed algorithm produces the entire set of unique valid seriation solutions, and does not stop when a single valid solution has been located.

This is important because there are typically a number of valid orderings. Some are suboptimal solutions because they are subsets of larger, more complete ones. Others are simply valid alternative solutions, which point to the influence of multiple causal factors. By including all valid orders, one can use the distribution of Chronometric dating and seriation math as data regarding the structure of interaction between localities, and thus evidence about past cultural transmission.

Our algorithm also enables statistical assessment of the significance of solutions, given the sample sizes employed. Using an example from the Mississippi River Valley, we demonstrate how the new algorithm provides detailed insight into the temporal and spatial structure of inheritance. Suitably extended in this way, we argue that DFS has the potential to inspire new innovative approaches to the archaeological record as much as it did in the s as a critical tool for building chronology.

While not in common usage, seriate and seriation are English words that refer to arranging or occurring in one or more series [ 50 ]. The terms describe an archaeological method without defining it—there are many ways to order or arrange items in a series. The origins of the method are a bit opaque since variants were in used before it was given the name.

Identifying its history and understanding the scope of the method, therefore, requires tracing the components involved in seriation that emerge over time and under which contemporary seriation now exists.

Sir Flinders Petrie [ 51 ] is generally credited with inventing seriation. Working with predynastic Egyptian materials, Petrie used ceramics found in graves to develop a chronology. Since the history of Egyptian ceramics must have followed some particular course and thus presented an unique sequence of ceramic type replacements, the combinations of ceramic types found in grave lots allowed him to reconstruct both the history Chronometric dating and seriation math

ceramics and arrange the grave lots in chronological order.

As in all seriation, Chronometric dating and seriation math product was just an order; one had to determine independently usually through superposition which end of the order was most recent. Kroeber [ 52 ] is credited with stimulating the American development. Kidder and Nels C. Nelson all of whom were conducting stratigraphic excavations in the American Southwest [ 75052 — 54 ]. As powerful as seriation proved to be, these early formulations were entirely intuitive and based on the *Chronometric dating and seriation math* that greater temporal differences between assemblages caused larger differences between frequencies of decorated types.

The shape of the curves that led to the ability to order assemblages were not justified and even the terms used were ad hoc: Since knowledge of rates of change was impossible, all that one could say about the characteristic distributions were that they were unimodal in that they had a single peak frequency and decreased in value away from the peak in both directions.

Furthermore, there was little interest in figuring out why the characteristic distributions occurred.

It was enough that they did and could be used to order assemblages. Such statements are, of course, just descriptions of the observed frequencies and represent, moreover, the selection of simply one type of distribution that the popularity of styles can take. Seriation thus was based on an empirical generalization about the distribution of stylistic classes through time.

Almost all of the early work involved frequencies of stylistic historical pottery Chronometric dating and seriation math

used as attributes of assemblages, the assemblages being groups of artifacts, usually but not always, pottery. By the s, use of the method had spread from the Southwest to include the Eastern Chronometric dating and seriation math States and the Arctic and by the s even Peru and Amazonia had chronologies based on seriation [ 955 ].

Ford [ 5657 ] played a critical role in disseminating the method so widely and was the only scholar to take an interest in the theoretical aspects of seriation until the s [ 58 — 60 ].

Although Kroeber had been aware of potential problems derived from sample size effects, Ford brought these considerations to the fore, albeit in a highly intuitive, non-quantitative, and ultimately incorrect way.

More importantly, he deduced a series of conditions under which the empirical generalization driving seriation might be expected to hold: Ford, like his predecessor, arrived at the final arrangement by eyeballing trial and error orderings for conformance to the unimodal distribution model.

InGeorge Brainerd and Eugene Robinson proposed an entirely new technique for arriving at the order of groups [ 4361 ]. They devised *Chronometric dating and seriation math* measure of similarity, since termed the Brainerd and Robinson Index of Agreement or simply the Brainerd and Robinson Coefficient, with which pairs of assemblages could be compared in terms of type composition.

Thus described, they noted that in correct solutions the most similar assemblages were adjacent to one another; since this order was unique, groups could be chronologically ordered simply by arranging them so that the most similar units were adjacent.

Brainerd and Robinson did this by rearranging rows and columns in a square matrix each group is compared with every other group of similarity coefficients; in a perfect solution, the magnitude of the similarity coefficients would decrease uniformly monotonically away from the diagonal of the matrix the groups compared with themselves.

Cowgill [ 62 ] developed a similarity-based approach for occurrence descriptions paralleling the techniques developed by Brainerd and Robinson for frequency descriptions.

Thus, two kinds of seriation approaches emerged. Frequency seriation uses ratio level abundance information for historical classes *Chronometric dating and seriation math* 545657 ]. Like Ford, one could insist on an exact match with the unimodal model before regarding an order as chronological, a Chronometric dating and seriation math solution.

Each of these approaches to seriation can subsequently be built to utilize raw data identity information whether frequency or occurrence values or similarity coefficient e. Historical classes are those which *Chronometric dating and seriation math* more variability through time than through space. Frequency seriation uses ratio level abundance information in percentage for for historical classes [ 545764 ]. Frequency and occurrence seriation techniques can take the form of deterministic algorithms that require an exact match with the unimodal model or probabilistic algorithms that accept departures from an exact fit.

Identity approaches employ raw data whether frequency or occurrence to perform the ordering. Similarity approaches transform the raw data into a non-unique coefficient e.

Since Brainerd and Robinson [ 4361 ], the majority of efforts have focused on probabilistic approaches and researchers have brought increasingly sophisticated numerical approaches to bear on seriation [ 4665 — 76 ].

These probabilistic approaches generally seek to find approximate solutions by reducing the dimensionality of the data set. They will find a solution even when joint unimodality is not possible and most measure the departure from a perfect solution by calculating stress residuals "Chronometric dating and seriation math" by examining variability within higher dimensions.

Variability in the frequencies of classes beyond the generalization is treated as noise rather than information about violations to the model and much of the utility of deterministic solutions that can be created by hand ordering is lost.

Consequently, most of these quantitative approaches remain in the programmatic literature. Most practical work continues to be done pretty much as Ford did it in the s, hand creating orders using graphical representations of relative frequencies in order to establish deterministic solutions. To understand how to build an automated algorithm that is true to the seriation method, one must look in detail at its requirements.

Groups did not have to be of short duration time between the addition of the first and last element to the group in some absolute sense as Ford supposed, but group duration did have to be comparable among the included cases.

Groups did have to belong to the same tradition ancestor-descendant relationships. While there was no way to assess whether these "Chronometric dating and seriation math" were met a priori by a given set of assemblages, Dunnell showed that when deterministic-identity approaches were used, seriation could not be made to yield incorrect answers on these grounds, thus securing the chronological warrant for arrangements derived by those techniques.

The other techniques are not robust in this regard and the orders arrived by those means may or may not be chronological. Dunnell [ 1 ] showed that this condition did "Chronometric dating and seriation math" apply to the groups to be seriated as Ford had assumed. Rather it was a deficiency in the warranting generalization; the method was under determined. The generalization only spoke to temporal distributions of types, not their spatial distributions.

As Ford "Chronometric dating and seriation math" appreciated and others showed empirically in the s, frequencies of types varied in space and that variation could be mistaken for difference in age. To get rid of spatial variations would limit a seriation to a simple point in space; one would simply be doing superposition under a different name.

Using the different properties of space and time, Dunnell showed that the effect of spatial variation could be eliminated by multiple seriations of the same events using different materials e. Seriation thus became a more complicated and demanding dating method. Archaeological reaction to this was mixed. Two broad categories of dating or chronometric techniques that archaeologists use are Seriation, on the other hand, was a stroke of genius. as amino acid racemization, fission track dating, Chronometric dating and seriation math

dating, and point allows us to carry out the arithmetic operands of multiplication and.

It is difficult for today's students of archaeology to imagine an era when chronometric dating methods were unavailable. However, even a casual perusal of the.