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The offset observed in the weighted decadal averages of T-947 data can be traced to the existence of a difference between the actual path of past radiocarbon variability and the model that is the calibration curve. In general, the fit over the target date is good with the χ² test statistics between the curve and the decadal estimates following for the most part the expected distribution (Figure 5.24), but when plotted against the curve it becomes clear that although the decadal averages overlap with the 95.4%

confidence interval, they are placed for the most part above the mean of the calibration

curve, with a decreasing trend (Figure 5.25). Yet this decrease is misidentified by the wiggle-matching algorithm as the minor break in the calibration plateau around 500 cal BC and therefore drawn to it, leading to the observed discrepancy between the wiggle-match results and the target date.

Figure 5.24: Quantile-quantile plot of the χ² test statistics between decadal averages of the T-947 data and interpolated IntCal13 values at the dendrochronological point of fit. Despite minor deviations from this distribution for the low values of the test statistics, the overall fit of the data to the curve is within expectation, indicating that there is no significant difference between the decadal averages of T-947 single ring data and IntCal13 at the dendrochronological point of fit.

This discrepancy can be explained through autocorrelation of the calibration curve errors. Autocorrelation is a process whereby a value of a time series at a given instance is dependent on preceding values (Shumway and Stoffer 2011). Such behaviour can be expected in a radiocarbon time series because of the cyclical nature of¹⁴C production, and because of the impact of previous years concentrations on succeeding years ¹⁴C concentration. At the same time the calibration data sets are subject to measurement error. Even if this error is symmetrical, there is a chance that over a group of measure-ments a number of consecutive determinations will underestimate or overestimate the true values. Once these measurements are built into the calibration curve, the mean of the curve over that period will diverge from the past trend of radiocarbon. The

Figure 5.25: Decadal averages of the T-947 single-ring data plotted against the 95.4% confi-dence interval of the calibration curve. Bars represent two standard deviations.

errors on the calibration curve, understood as the differences between the curve mean and the actual past trend, will then auto-correlate, leading to a temporary asymmetry in the distribution of errors around the mean of the curve. As the wiggle-matching algorithm assumes that no such process is taking place, it will attempt to find a point in time where the measurements will have a symmetric distribution around the curve.

Hence, in short-span wiggle-matches there is a potential risk of an offset emerging on the basis of statistical uncertainties inherent in calibration.

This risk of an autocorrelation offset depends on the precision and frequency of mea-surement on the unknown sample and the calibration data. Decreasing the calibration measurement frequency and the precision should increase the duration and the magni-tude of potential autocorrelation based offsets (Figure 5.26). For the unknown sample the converse is true: because the wiggle-match is conditioned by the actual past trend, it will converge toward it as analytical uncertainty decreases and as the number of constituent measurements increases. As long as the uncertainty on the radiocarbon measurements is large enough relative to the uncertainty on the calibration curve, this is not a problem: the amount of information about the actual trend of radiocarbon will be insufficient to induce a significant bias in the model fit. However, as the mea-surement precision and sampling frequency increase, the description of the actual path improves and the measurements begin to cluster to one side of the curve. This can be observed through simulating wiggle-matches from T-947 decadal estimates, where in-creasing the precision leads to a clearer distribution to one side of the curve, drawing the wiggle-matches towards 480 BC (Figure 5.27), leading to a paradoxical situation where a greater number of measurements conducted to better precision produce a less accu-rate and therefore poorer end result. Of course, the results from the measurements on T-947 only indicate that autocorrelation-based offsets are a risk in wiggle-match dat-ing and need not necessarily happen in all cases where short-lived samples are dated – indeed, it would be surprising if such offsets were common enough to affect results on a regular basis. Nevertheless, the presence of the small-scale offset in the T-947 data does mean that it might be prudent to somehow mitigate for the risk of small-scale off-sets whenever wiggle-match dating short-lived sequences, in case the particular sample grew on an affected part of the calibration curve.

The interpretation presented here begs the question of why the autocorrelation effect has not been recognized earlier. This can be attributed to measurement precision, statistical models used and explanations alternative to autocorrelation. T-947 decadal estimates provide a much more precise determination of the actual path of the past radiocarbon variability than any published data set for their period. Furthermore, because of the short span of the series, the offset is much easier to recognize than it would have been in a long sequence where the effects of the intermittent biases would cancel each other out and the distribution of the measurements around the calibration curve would become more symmetrical. Most other series are either longer, have their measurements more spread out, or have lower precision and so identifying autocorrelation becomes difficult. For example, the data of Tyers et al. (2009) might be derived from a trend with alternating offsets from the calibration curve mean, but the

Figure 5.26: Curve precision and autocorrelation. As the precision of the curve is improved, its detail becomes more apparent and hence the probability of the actual trend of past radiocarbon persisting above or below the mean of the curve decreases.

Figure 5.27: Simulates of wiggle-matches from 50 year spans sampled into decadal blocks at overlapping five year intervals at analytical precision of 40, 20 and 10 radiocar-bon years at one standard deviation (left, middle and right, respectively). As the precision of individual measurements increases, so does their concentration above the mean of the calibration curve and although the target date might be contained within the 95.4% HPD area, the 68.2% HPD area now has a consistent position towards older dates, indicating bias. Simulations are based on T-947 combined decades. Vertical bar marks the target date. Error bars are at two standard deviations.

analytical precision of the data and their frequency is insufficient to choose between the two models (Figure 5.28). Furthermore, there are also other forms of offsets and because they are subject to more active pursuit, autocorrelations may have been confused with them. In the case of T-947 the confidence that autocorrelation is the causal factor hinges on geographic proximity to Ireland and comparable growth seasons to both Irish and German series, which precludes most offset based on climatic or physical factors (see Chapter 3.2). Nevertheless, these alternative sources of bias need attention whenever wiggle-match dating outwith areas close to the sources of the calibration data.

Figure 5.28: Radiocarbon offsets between early modern pine from Jermyn Street (London) and the mean of IntCal13. While for the most part the empirical data overlap with the calibration curve, they are insufficient to reject alternative models (e.g.

the one implied by the dashed line). The error bars represent two standard deviations. Data from Tyers et al. (2009).