The goal of seriation is to determine an ordering of a number of items (artifacts or assemblages) that replicates as closely as possible the temporal order of events they represent. Seriation relies on the assumption that the items to be seriated share characteristics or features whose measures evolve in a predictable fashion over time. The ordering of items that best conforms to that prediction for all features and all artifacts is considered to be the optimal seriation.
There are many predictions that may be made and OptiPath allows users a choice among a few, represented by objectives available in the Seriations table and on the Seriate menu. Given an objective, OptiPath seeks the seriation that optimizes that objective. Most of the predictions about the evolution of features or styles used in seriation boil down to two broad categories: unimodality and gradual change. The ordering of artifacts that produces the most gradual, or the most unimodal, evolution of all features for all artifacts is considered the optimal seriation. It can be shown that these two criteria are often equivalent (paper in preparation by the authors). OptiPath provides seven objectives to choose from: maximize unimodality and six others that all minimize some aspect of the rate of change of feature values over time.
All seriation techniques rely on the assumption that the artifacts to be seriated share characteristics or features whose measures evolve gradually over time. If they did not evolve gradually over time, styles would not be identifiable and the pattern of their evolution would not be predictable. Even unimodality depends upon recognizing styles that are relatively stable, or at least gradually evolving, over time. However, rather than focusing on unimodality, which is problematic in that perfectly unimodal seriations rarely exist, and often difficult to justify in any case, it is possible to optimize gradual change directly. Other than minimizing unimodality, all of the objectives in OptiPath are directed at finding seriations that maximize the gradualness of change in the features being analyzed. In this case the ordering of artifacts that produces the most gradual evolution of all features for all artifacts is considered the optimal seriation. However, there remain alternatives in defining what is meant by gradual change and how to resolve tradeoffs in gradualness in one feature over another, or in a little bit of fast change over a lot of slow change.
The first alternative is minimizing path length. If the user has not specified Earliest or Latest dates for individual items or has not chosen Use Input Dates from the Seriate menu, then it can be shown that minimizing path length is equivalent to the other gradualness objectives which follow (this is shown in a paper by Brett Shepardson and Fred Shepardson which is being prepared for publication). In other words, if the user has restricted the allowable date ranges for certain items, then it is generally not enough to minimize path length.
Each seriation involves a number of items (artifacts or assemblages) and a number of feaures. Mathematically, each feature can be considered as a separate dimension. In that case, each item is defined by its feature values which determine a unique point in feature space. A seriation determines a path through all of these points (items) in feature space. The distance from one point to the next in the seriation is a measure of the change in features from one item to the next. The path with the least overall change will be the shortest path through all the points. The ordering of artifacts that produces the most gradual evolution of all features for all artifacts is the shortest path.
Finding the shortest path is a variant of the well known traveling salesman problem known as the Hamiltonian path problem. This problem is notoriously difficult and there are no known algorithms that will solve large problems optimally in a reasonable amount of time. Solving the problem with restrictions on the dates for individual items is a variant known as the Hamiltonian path problem with time windows and is even more difficult to solve. OptiPath uses a heuristic technique that produces good answers with moderate effort.
The Shortest Path seriation Technique is specifically designed to minimize the overall path length of the seriation.
Maximizing the gradualness of a seriation is very similar to minimizing the path length, the difference being that maximizing gradualness maximizes the gradual index directly. The gradual index is a measure of the gradualness of a seriation or an ordered sequence of numbers. The gradual index is scaled from -1 to 1. A seriation with a Gradual Index of 1 cannot be made more gradual by reordering. A seriation with a Gradual Index of -1 could not be less gradual. The overall Gradual Index for the seriation is the weighted average (using the Weights in the Features table) of the individual feature indices. Gradualness, as opposed to the path length, allows us to optimize unimodality and gradualness simultaneously by taking an average value of the two.
Unimodality is the concept that a style appears in the archaeological record, gains in popularity over time and then loses popularity until it vanishes from the record. Seriation has a long history of using unimodality as the objective in choosing an optimal seriation. These efforts have been hampered by the fact that for a long time archaeologists had no objective measure of unimodality when a seriation failed to be perfectly unimodal (for example, a seriation in which three of the features considered were unimodal but the fourth one wasn't, or a seriation in which a style grew in popularity for many time periods then lost popularity for a few time periods, then regained its old popularity only to have it fade away once again). OptiPath uses an objective measure of unimodality, referred to as the Unimodal Index, developed by the authors.
The Unimodal Index is a measure of the unimodality of a seriation or an ordered sequence of numbers. The unimodal index is scaled from -1 to 1. If each feature's data values were plotted against their assigned dates, a graph showing a single peak would be perfectly unimodal and have a Unimodal Index of 1. A graph with a single valley would be perfectly anti-unimodal and would have a Unimodal Index of -1. A random ordering of the data would have an expected Unimodal Index of 0. The overall Unimodal Index for the seriation is the weighted average (using the Weights in the Features table) of the individual feature indices.
Maximize a Weighted Combination of Gradualness and Unimodality
It is possible to optimize unimodality and gradualness simultaneously by optimizing a weighted sum of their indices. The unimodal weight must be between 0 and 1. In seriating OptiPath will weight the unimodal index by the Unimodal Weight and the gradual index by 1 - the Unimodal Weight. The Unimodal Weight can be set in the Seriations table.
Minimize the Average Rate of Change
If the user has specified Earliest or Latest dates for individual items and has chosen Use Input Dates from the Seriate menu, then the dates that can be assigned to items are restricted (and, as a consequence, orderings can also be restricted).
If the user has specified Earliest or Latest dates for individual items and has chosen Use Input Dates from the Seriate menu, then the dates that can be assigned to items are restricted (and, as a consequence, orderings can also be restricted).
If the user has specified Earliest or Latest dates for individual items and has chosen Use Input Dates from the Seriate menu, then the dates that can be assigned to items are restricted (and, as a consequence, orderings can also be restricted). Without these restrictions, OptiPath simply assigns dates so that the rate of change between two consecutive items in the seriation is just equal to the average rate of change for the overall seriation, and there is no difference between minimizing the path length or minimizing the average rate of change or even maximum rate of change between consecutive items.
However, when the user has restricted the dates of individual items, then dates of the first and last items in the seriation will depend upon which items ar first and last in the current ordering, and therefore may depend upon user input, and even the average rate of change will be affected. In this case minimizing the path length is not necessarily equivalent to minimizing the average rate of change. Therefore, OptiPath allows the user to specify the average rate of changed should be minimized.
Minimize the Maximum Rate of Change
There may be times when the user feels it is more important to keep the largest rate of change between any two items as small as possible. OptiPath allows the user to specify the maximum rate of changed should be minimized.
Minimize the Sum of Squared Rates of Change
There may be times when the user feels that a few big rates of change are worse than numerous smaller ones. OptiPath allows the user to minimize the sum of squared rates of change. The result is that a rate of ten is penalized four times as much as a rate of five. This is equivalent to reducing the sum of squared errors in order to reduce the variance.