CAPÍTULO II. MARCO CONCEPTUAL
2.3. MODELOS TEÓRICOS DE CREACIÓN DE EMPRESAS
There are two technical issues that affect each one of the wiggle-match dating case studies presented in the succeeding sections: wiggle-match date stability and potential for excessive shrinkage. Concerns over wiggle-match date stability are most pertinent during calibration plateaux, when a single outlying measurement can cause a major shift in the wiggle-match date range. Shrinkage is a process in which the hierarchy of a statistical model causes the contraction of parameters to a particular value; in general this is a welcome effect as it reduces the effect of noise in the data, but in some cases can lead to biased results. The following section explains these two issues in more detail to provide the necessary background for when they arise in the case studies.
6.2.1 Wiggle-match dating stability and systematic bias in outermost and decayed rings
Radiocarbon wiggle-match dates are conditional upon the relationship between the samples measured (in Bayesian wiggle-match dating this takes the form of a prior) and on the data in the form of calibrated ranges of the individual determinations. However, different determinations will have different impact on the distribution of the wiggle-matched date range; for example, if the wiggle-match contains a major feature of the calibration curve, such as the onset of the Hallstatt plateau. or the large wiggle around 675 cal BC, the measurements that represent these features will have a much greater impact on the final position of the wiggle-match than the measurements describing the surrounding flat parts of the curve. Hence, if a single measurement mimics one of these features either through a systematic bias in the underlying material, or due to measurement uncertainty, the wiggle-match result as a whole can become biased. To
ensure that this does not happen it becomes important to assess wiggle-match stability.
The easiest way of doing this is the removal of individual constituent determinations from the wiggle-match model and taking note on the effect this has on the posterior probability of the results. If these include the range of the unmodified results then the match is stable. If, on the other hand, the removal has a significant effect on the posterior distribution of the wiggle-match, further action has to be taken to ensure that the results are valid. In general, if such situations arise, the ideal course of action is to repeat the measurement in question. Given that this is often impossible, the more viable alternative is to run two alternatives of the site model, one with the key measurement included, and the other without it, so that the effects on the overall picture of the site can be assessed. With a sufficient number of measurements available to the project, this approach should be sufficient to identify any instability caused by measurement scatter, as the problematic determination ought to have poor agreement with the site model. There will be cases, however, where this approach is not enough due to systematic offsets.
One such case became clear in the archaeological case studies discussed below, as some timbers displayed a consistent shift towards older radiocarbon ages in decayed wood and outermost rings (Figure 6.18). This has been encountered in some form on all of the sites discussed and appears to be independent of the period in which the timbers were felled, thus excluding the possibility that a minor offset in the calibration curve is the underpinning cause. Common to all the cases was the species affected (alder) and exposure of the wood to the water saturating the site, either due to position on the outermost rings, or due to porosity caused by decay. This in turn allows an increased movement of sediments, as well as fulvic and humic substances that can have an adverse effect on the results of the radiocarbon dates unless the pre-treatment protocol is strict enough. The problem is only compounded by the possibility that in some cases dissolved cellulose might dissolve and then reconstitute (Zugenmaier 2008, 101-103), perhaps incorporating molecules of intrusive carbon into the new strands.
Such soluble cellulose can be removed with a strict enough pre-treatment protocol, but an experiment conducted on implementing such a stringent protocol showed that the sample loss rates would be unsustainable on a routine basis (see Chapter 4.1).
This excess variability has an adverse effect on both the quality assurance indices and the date ranges of the wiggle-matches. The problem with the quality indices emerge as the drift of some samples towards greater ages means that the wiggle-match resembles the underlying calibration curve to a lesser extent leading to an apparent mismatch between the model and the data. The bias in the date ranges has been observed most often with regard to the outermost rings from the timbers that have been felled in the 5th century BC. In these timbers a bias to older ages on the outermost rings of a short or medium span sequence will make the wiggle-match resemble the beginning of the large wiggle around 675 cal BC, or the rise to the middle part of the calibration plateau around 640 BC and hence a drift of results towards that part of the calibration curve.
Figure 6.18: The calibrated date ranges for the actual determinations on the affected outer-most rings and their estimates based on the remainder of the constituent wiggle-matches, by site. Note the systematic shift towards older dates in the actual measurements.
In all the case studies presented here, these biases could have been resolved through comparison of the results to other timbers within a particular feature, or through the emergence of stratigraphic incongruences. In situations where this kind of additional information is unavailable, the recommendation is to avoid decayed or outermost rings, at least in alder timbers. If this proves impossible either due to overall state of the site, or because a particular feature of interest did not preserve well, the recommendation is to check if the removal of the outermost rings affects the site models and, if so, publish both results, as long as they do not conflict with other archaeological information.
6.2.2 Shrinkage
Shrinkage is a process where individual parameters converge towards one another under the influence of an overarching distribution (Kruschke 2015b). In the context of the Bayesian analysis of radiocarbon determinations the simplest example of shrinkage is the narrowing of the modelled date ranges under the influence of the uniform phase prior (see chapter 3.3.2). In most cases shrinkage is a welcome phenomenon, as it means the reduction of statistical scatter and improved precision of model results. However, if the overarching distribution does not describe the model well, shrinkage may lead to the emergence of a systematic bias.
In the context of the current chapter, adverse instances of shrinkage arose during the Hallstatt calibration plateau when date ranges of different precision were placed under the same distribution based on the uniform prior. When this happens the distributions of the less precise determinations will shrink towards the more precise ones. This is not an issue if the model in question is a good description of the underpinning site formation process and in the vast majority of cases is welcome as a means of negotiating the effect of the calibration plateau. There does exist, however, a small number of possible situations where the dates from a specific part of the phase have a greater precision than those from the remainder of the dated activity. In these instances, the less precise dates will shrink towards the more precise ones even if their updated distributions do not provide a good description of the actual date (Figure 6.19). This applies for the most part to models which include both wiggle-match dates and single radiocarbon determinations, but can also affect some wiggle-match based models; for example, wiggle-matches that catch the break in the calibration plateau around 550 BC have the potential to bias wiggle-match dates from a later period that have lower precision as they failed to catch that particular feature of the calibration curve. It is important to stress that instances where such adverse shrinkage affects the 95.4% HPD areas are rare and often require conscious effort to develop. These issues are a part of the broader set of concerns regarding the effects of the calibration curve on the robustness of Bayesian modelling (Steier and Rom 2000).
For the most part instances of adverse shrinkage will become apparent in review of model results. The first sign is the presence of groups of skewed modelled date ranges of the data-based parameters (skewed distributions are expected for a number of inferred
Figure 6.19: Induction of adverse shrinkage in a uniform bound phase model through the con-centration of high-precision determinations on one edge of the modelled phase.
In this simulation a group of fixed high-precision distributions near the onset of a bound uniform phase draws in the posterior distributions on the simulated radiocarbon dates. Because of this, the modelled simulated radiocarbon dates display a systematic offset from their target dates (vertical bars), even though all the measurements follow a uniform distribution. While it is improbable that this magnitude of the adverse effect is to emerge in practice, wiggle-match dating research design ought to avoid clustering of very precise wiggle-matches to one side of a bound model that also contains dates of a lower precision.
parameters such as boundary events). If such groups display very narrow modelled date ranges (¡50 cal yrs within the 95.4% HPD area during the plateau) and if they display systematic divergence from their non-updated distribution, adverse shrinkage may have taken place. The easiest way to identify whether this is indeed the case is to assess if the model makes sense in archaeological terms; compression of 20 refurbishment events into a 10-year window indicates a high probability of adverse shrinkage. The best way to avoid such situations is to ensure that the site stratigraphy is well-understood and that this understanding is mirrored in the site model implemented.