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Background

Competing Interests: The authors have declared that no competing interests exist. This is an open carbon article distributed under the terms of the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Accuracy time-series analysis has the potential to improve our understanding of human-environment interaction in deep time.

However, radiocarbon dating—the most common chronometric technique in archaeological and palaeoenvironmental research—creates challenges for established statistical methods. The methods assume that observations in a time-series are precisely dated, but this assumption is often violated when calibrated radiocarbon dates are used because they usually have highly irregular uncertainties. As a result, it is unclear whether the methods can be reliably used on radiocarbon-dated time-series.

With this in mind, we conducted a large simulation study to investigate the impact of chronological uncertainty on a potentially useful time-series method. It is designed accuracy use with count time-series data, which makes it applicable to a wide range of questions about human-environment interaction in deep time. Our simulations suggest that the PEWMA method can often correctly identify relationships between time-series despite chronological uncertainty.

When two time-series are correlated with a carbon of 0. With correlations of around 0. While further testing is desirable, these findings indicate that the method can be used to test hypotheses about long-term human-environment interaction with a reasonable degree of confidence.

Time-series regression analysis is an important tool for testing hypotheses about human-environment interaction over the long term. The primary sources of information about human behaviour and environmental conditions in deep time are the archaeological and palaeoenvironmental records, respectively. These records contain observations with an inherent temporal ordering and are thus time-series. This means time-series regression methods could be used to quantitatively test hypotheses about the impact of climate change on humans and other hominins, or conversely the impact of hominin societies on their environments.

However, there is reason to think that chronological uncertainty may complicate the use of such methods. In particular, the chronological uncertainty associated with the most common chronometric method used in the dating of both records—radiocarbon dating—could undermine our ability to confidently identify statistical relationships between the records.

This is because calibrated radiocarbon dates have highly irregular uncertainties associated with them, and uncertainties of this type are not in line with the assumptions of many standard statistical methods, including time-series analysis [ 1 — 5 ].

To investigate this possibility, we conducted a simulation study in click at this page we investigated the impact of radiocarbon dating uncertainty on a time-series regression method that is well-suited for archaeological and palaeoenvironmental research—the Poisson Exponentially-Weighted Moving Average PEWMA method [ 6 ].

How Accurate is Carbon Dating?

Time-series data have to be analyzed carefully because the order in the sequence of observations matters. There are two traits a time-series can have that make temporal ordering important.

One is non-stationaritywhich describes time-series accuracy statistical properties that vary through time—e. The other troublesome trait is autocorrelationwhich means the observations in the series correlate click to see more themselves at a given lag [ 7 ]. Autocorrelation leads to dependence among the observations in a time-series, which violates another common statistical assumption, namely that observations are independent.

Archaeological and palaeoenvironmental time-series typically have both traits [ 389 ]. They will usually be non-stationary, because almost all environmental or cultural phenomena change over time—e. They will also typically contain temporal autocorrelation. Thus, archaeological and palaeoenvironmental data can be expected to violate the assumptions of many statistical methods. Consequently, we need special methods to find correlations between past human and environmental conditions. Fortunately, these methods already exist because statisticians, mathematicians, and engineers have been working with non-stationary, autocorrelated time-series for a long time [ 10 ].

As a carbon, many established "range" methods are designed specifically to handle non-stationary, autocorrelated data [ 7811 ]. However, time-series of archaeological and palaeoenvironmental observations are idiosyncratic in another way that potentially undermines even these established methods—often we are uncertain about the precise times associated with the observations [ 12 — dating ].

That is, the time-series contain chronological uncertainty. Contemporary time-series, such as stock prices or daily temperatures, are usually recorded at precisely known times, but looking into the deep past entails significant chronological uncertainty. Archaeologists and palaeoenvironmental scientists usually make chronometric estimations by proxy using radiometric methods that rely on measuring isotopes of unstable elements that decay at a constant rate [ 15 ].

Even the most precise of these methods, however, yield uncertain dates, some with decadal error ranges and others with centennial or millennial error ranges. Consequently, many palaeoenvironmental and archaeological time-series contain temporal uncertainty. The most common chronometric range, radiocarbon dating, is particularly problematic. Radiocarbon dates have to be calibrated to account for changes in isotope ratios through time.

The calibration process results in chronometric errors that are often highly irregular, yielding ranges of potential dates spanning many decades or even centuries [ 4516 carbon, 17 ]. Point estimates—i. Most statistical methods are, therefore, undermined by calibrated radiocarbon dating because most methods rely, at least to some extent, on point estimates. Time-series methods are no different, raising concerns about our ability to use them for identifying correlations between archaeological and palaeoenvironmental time-series.

In the study reported here, we explored the dating of chronological uncertainty on a time-series regression method called the Poisson Exponentially Weighted Moving Average PEWMA method [ 6 ]. Classified as a state-space time-series method, the PEWMA method models physical and natural systems as a set of input and output variables. It can be thought of as a mathematical filter that takes input variables and produces outputs by estimating the relationships among the variables. Importantly, the method accounts for autocorrelation and non-stationarity in the Poisson process.

It is potentially useful carbon many archaeological and palaeoenvironmental applications because count data is common in these fields—e. The first is called the measurement equation. Brandt et al. The measurement equations represent the observed count data as outcomes of a sequence of Poisson random variables.

The previous mean is not merely a lagged value, though, which is why the asterisk is used. These carbon characterize the change in the unobserved mean through time. Read more first equation defines the mean range a given time, and has three terms. The first of these, e r tdescribes the base rate of the mean process and ensures that the mean is always positive, which is necessary for Poisson processes. To be consistent with the measurement equations, we added an asterisk to the term, making it slightly different from Brandt et al.

The parameters that appear in the Gamma and Beta distributions are also estimated from the data. To the best of our knowledge, the PEWMA method has only been used to analyze past human-environment interaction in one study [ 18 ]. In that study, we tested the prominent hypothesis that climate change exacerbates conflict within and between human societies over the long term e. To test the hypothesis, we compared a time-series of Classic Maya conflict levels to several palaeoenvironmental proxies.

The time-series of interest was a dating record of conflict events inscribed into monuments along with Classic Maya Long Count calendar dates. The conflict events include mentions of violent attacks, captive taking, human sacrifices, deliberate destruction of monuments, and large coordinated attacks timed to coincide with astronomical events [ 2122 ]. Classic Maya elites had these events inscribed on monuments like door lintels in temples, stairways on pyramids, and most importantly large stone stelae click the following article 23 ].

The inscriptions describing these events generally include the date of the event in question, information about the nature of the event—e. Though not necessarily indicative of warfare in the accuracy sense, changes in the number of these events throughout the Classic Period likely indicates changes in the overall level of conflict accuracy polities [ accuracy ]. To create a time-series of these events, we counted the number of conflicts per year period from — CE.

The size of the interval was chosen to be consistent with earlier research, but we explored changing the size of the interval in subsequent analyses and obtained results that were consistent with those yielded by the main analyses see the supplementary material associated with [ 18 ]. Using the PEWMA method, we compared the conflict record with five palaeoenvironmental records including two temperature and three rainfall proxies.

The temperature proxies are sea surface temperature SST reconstructions for the summer and winter seasons in the Cariaco Basin [ 24 ]. Range records show an increase in SST over the Classic Maya period that correlate with other circum-Caribbean records over the same period.

They also positively correlate with air temperature readings in the central Maya region during the 20 th century dating the supplementary material associated with [ 18 ]. The rainfall proxies included a titanium concentration record from the Cariaco Basin [ 25 ], an oxygen isotope record from a speleothem in southern Belize [ 21 ], and the well-known sediment density record from Lake Chichancanab located in the center of the Yucatan Peninsula [ 26 ].

In contrast to previous research on Classic Maya conflict [ 21 ], we found that temperature was the only variable that correlated significantly with conflict levels. We found no evidence for an impact of rainfall. From this, we concluded that increases in temperature might have led to increases in conflict among the Classic Maya, an idea not previously explored in the scholarly literature pertaining to the Classic Maya.

As the foregoing study suggests, the PEWMA method has the potential to improve our understanding of past human-environment interaction. However, given dating ubiquity of chronological uncertainty in archaeological and palaeoenvironmental time-series, there is a need to better understand how consider, best online dating profile headlines excellent uncertainty affects the method—especially radiocarbon dating uncertainty, which is highly irregular, as we explained earlier.

To explore the effect of chronological uncertainty on the PEWMA method, we carried out a series of simulation experiments. The experiments involved creating thousands of carbon of artificial palaeoclimatic and archaeological time-series with known relationships and then testing for those relationships with the PEWMA method. The regressions site dating fit singles set up with the synthetic archaeological time-series as the dependent variable and the synthetic palaeoenvironmental time-series as the independent variable.

We used error-free dates for the artificial archaeological time-series so that we could limit the sources of error and see the effects more clearly.

This analytical control also had the benefit of allowing us to compare the simulation results to our previous work on the Classic Maya because dating dependent variable in that study was a historical not dating detox think with little chronological uncertainty [ 18 ].

Thus, in the present study only the synthetic palaeoenvironmental time-series just click for source chronological uncertainty.

Using a bootstrap approach [ 27 ], we resampled the set of synthetic calibrated radiocarbon dates used to date the palaeoenvironmental time-series thousands of times, running a separate PEWMA analysis each time. For each experiment we varied several parameters while keeping everything else constant.

The parameters included the variance of the time-series, the number of synthetic radiocarbon dates, and the range of the correlation between the artificial archaeological time-series and the synthetic palaeoenvironmental data. Varying these parameters allowed us to see how radiocarbon dating uncertainty in the palaeoenvironmental series affected our ability to find the known relationships between the time-series in each pair.

Using the R statistical programming language [ 28 ], we ran a series of simulation experiments, each of which explored how a set of variables affected the outcome of a PEWMA regression analysis. To reiterate, the PEWMA algorithm is a special kind of time-series filter that can be used to model Poisson processes containing autocorrelation and non-stationarity dating 6 ]. Poisson processes produce integer count time-series [ 29 ], a very common type of time-series in archaeology, as noted earlier—e.

To model an empirical time-series, the PEWMA algorithm uses an observe-then-predict mechanism, which as the phrase suggests involves first observing some data and then making a prediction based on that observation. It filters through a given count series one observation at a time, updating its predictions for the next time based on previous accuracy.

It can account for autocorrelation in the count data by discounting the information range older observations as it filters through the series. More discounting implies less autocorrelation in the observed data because older values in the series have a lower impact on subsequent values. The algorithm can also be fed covariates to see whether they improve its predictions of the time-series of interest.

Models with a lower AIC involve less information range, meaning they fit the observed time-series better.