This paper explores several approaches to assess temporal trends within archaeological and historical datasets containing records marked with signi昀椀cant extent of uncertainty accompanying their dating. We evaluate the strengths and pitfalls of these methodologies by employing two datasets: one comprising ancient shipwrecks and the other ancient Greek inscriptions. While these objects can, in principle, be precisely dated to speci昀椀c years, they are o昀琀en assigned broader date ranges, spanning centuries or longer historical periods. We propose that the most promising approaches involve using these date ranges as de昀椀ning probabilities. By randomly assigning speci昀椀c dates based on these probabilities, we enable hypothesis testing for temporal trends. As we want to encourage other scholars to employ the methods we propose, we o昀er a detailed description of the implementation of these methods using functions from the Pythontempun package.
In this paper, we introduce three computational approaches to deal with temporal uncertainty
in historical datasets for the purposes of exploratory data analysis and hypothesis testing. By
historical datasets we mean datasets consisting of data points representing objects or events
from the past, such as collections of material artifacts, corpora of textual sources, or lists of
historical events studied by archaeologists or historians. Such datasets are o昀琀en marked by
some level of uncertainty accompanying dating of individual data points, as creators or curators
of the datasets are not able to say precisely when an object was produced or when an event
occurred. In what follows, we will focus on one particular form of such uncertainty, which is
calledwithin-phase uncertainty [
Our aim is to demonstrate the strengths and pitfalls of the three most commonly applied
approaches and to show how they can be employed in a straightforward manner using the Python
programming language and thetempun package [
The shipwreck database is available in a spreadsheet format with 1,784 records and 16 attributes. These attributes include details about the provenance, destination, and cargo of the ships, among other information. For our topic, two attributes are of crucial importance: ”post_0” and ”ante_0,” which represent the starting and ending points of the estimated temporal interval during which the shipwreck occurred.
The GIST (Greek Inscriptions in Space and Time) database of ancient Greek inscriptions is also available in a tabular format and covers 217,863 records with 29 column attributes. The GIST dataset is primarily based on an online collection of ancient Greek inscriptions published by the Packard Humanities Institute (PHI) and available via the Ithaca project as I.PH1I5[]. A key feature of the dataset is that, wherever available, the spatial and temporal metadata are expressed in a machine-readable form. Thus, in the case of temporal information, the estimated date of production is again expressed by means of two attributes, called here ‘not_before’ and ‘not_a昀琀er’, having the same meaning as ‘post_0’ and ‘ante_0’ in the case of the shipwrecks; both delineate the dating interval. For instance, the date of origin of an inscription which has been in the original collection dated to the 5th century BCE (e.g. PH174) is now in a machinereadable form expressed by means of an interval with limit points -500 and -401.
While in the case of shipwrecks the interval expresses a time range within which it is estimated that the wreckage occurred, in the case of inscriptions it is an interval within which the inscription was erected. In both cases, the interval captures a one time event, which might be in principle dated to a singular year, or even a day. Ultimately, the annual temporal resolution is probably the most appropriate one for working with data from the ancient Mediterranean context.
A 昀椀rst issue to explore in the case of a dataset with dating information encoded by means of an interval is the distribution of durations of these intervals. Fig1urinetroduces histograms of durations for both datasets. In the case of inscriptions, we see that there is a higher ratio of inscriptions within the two le昀琀most bins reserved to the shortest durations up to 15 years. These are the most precisely dated inscriptions in the dataset. In the case of shipwrecks, we see that such precise dates are rather exceptional.
It has been proposed that regardless of the issues with temporal uncertainty, the datasets of shipwrecks and inscriptions o昀er valuable information concerning economic and cultural development of the ancient Mediterranean world. Both have been used as proxies for economic trends, as a higher amount of shipwrecks or inscriptions dated to a certain period are generally considered to indicate higher intensity of trade or higher overall economic performance of the area under scrutiny in this period in comparison to a period from which we have a smaller number of records within these databases.
Thus, for instance, Hopkins relied on an older version of the shipwrecks’ dataset to argue that ”in the period of Roman imperial expansion and in the High Empire (200 BC – AD 200), there was more sea-borne trade in the Mediterranean than ever before, and more than there was for the next thousand years”6[, pp. 105-106] and concluded ”that trade in the third century A.D. declined” [6, pp. 115] (for the same argument, see also1[]). Similarly, Kaše and Glomb [kase_a昀툀uence_2022 ] argued that the temporal distribution of ancient Greek inscriptions can serve as a useful proxy for the economic development in the eastern Mediterranean region. But how do we evaluate such claims using a dataset with dates expressed by means of dating ranges?
All the below introduced analyses and visualizations were implemented using the Python 3
programming language1[
Perhaps the most straightforward approach to analyze temporal trends in historical data characterized by a substantial extent of within-phase temporal uncertainty is the one based on midpoint values, i.e. arithmetic mean of the two numeric values de昀椀ning the range. Thus, an inscription dated to the 3rd century BCE, which is expressed by means of an interval delimited by the values -300 and -201, is treated as being dated to the year 250 BCE. Such dates are 昀椀nally used within some temporal distribution plots using stable bins such as centuries, half centuries, quarter centuries etc.
This approach has at least one serious issue: In case that a dataset contains a lot of data points dated on the century basis, you will bias the dataset by overestimating the number of objects dated to the middle of centuries. Further, the dataset might also contain a substantial amount of data points dated with reference to historical periods such as ”Roman Imperial period”, conventionally delimited by an interval from 31 BCE to 410 CE and covering a period longer than four hundred years with mid-point at 190 CE. Again, using the mid-range values will substantially bias the data; in this case it will cause overestimation of the number of records dated to the end of the 2nd century CE.
One way to diminish the impact of the bias caused by extensive and imprecise ranges is to exclude all data points with temporal range (duration) greater than a certain threshold (e.g. 100 or 50 years). This approach might be useful for speci昀椀c applications, but it implies a substantial information loss, as broadly dated records get completely omitted, while we should expect that they still somehow contribute to the general pattern.
Figure2 demonstrates this approach while clearly revealing its limitations. The 昀椀gure comprises four histograms showing midpoint dates of records from the GIST dataset. Subplots (A) and (B) are based on a bin width of 50 years, while subplots (C) and (D) use a narrower bin width of 25 years. In subplot (A), we depict all records within the dataset that have a properly delimited dating range (i.e., both ”not_before” and ”not_a昀琀er” values are available). On the other hand, subplot (B) employs the same dataset but includes only data points with dating ranges shorter than 100 years, excluding inscriptions dated on a centurial or longer basis. As a result, this 昀椀ltering reduces the size of the GIST dataset to approximately half of its original size.
Subplots (C) and (D) o昀er a more granular overview of the temporal distribution of the data due to their narrower histogram bins of 25 years each. Subplot (D) further restricts the data by excluding inscriptions with intervals equal to or longer than 50 years, keeping only about 40 % of all dated inscriptions. Across the subplots, temporal trends show notable di昀erences.
By comparing subplots (A) and (B) and subplots (C) and (D), we can identify entirely distinct patterns in the data from the 2nd and 3rd centuries CE. During this period, ancient Greek inscriptions were primarily produced within the Eastern part of the Roman Empire, which experienced its peak economic performance, followed by a decline starting at some point during the third century, exhibiting substantial regional di昀erences. It is, therefore, not surprising that the 2nd century CE contains the highest number of dated epigraphic records. However, while subplot (A) suggests a signi昀椀cantly higher number of inscriptions from the second half of that century, subplot (B) presents a reverse trend. This indicates that the peak in (A) was, to some extent, an arti昀椀cial product of midpoints derived from widely dated inscriptions. Drawing on the higher granularity of subplot (C), one might easily form a false impression that a more nuanced picture of the period is obtained. However, this picture becomes entirely blurred when compared with subplot (D).
An alternative, which overcomes some of the aforementioned limitations, is the Aoristic Sum (AS) based approach, developed originally in criminology and geograp1h3y] a[nd later on adopted in archaeology (see7[]). AS converts date ranges into probabilities, providing a more nuanced representation of temporal distribution. For instance, an inscription dated to the 3rd and 4th centuries CE (i.e., numerically ranging from 201 to 400) would have a 0.5 probability of being erected in either of the two centuries, a 0.005 probability of being produced within any singular year within this range, and a 0 probability of being produced outside of this range. Essentially, for each record within the dataset, we generate its unique probability distribution based on the dating range. We then plot the overall cumulative distribution by combining the probability distributions from all individual records. In other words, 昀椀rst, for each time bin we sum up probabilities of all records within it and, second, plot these sums for all bins in form of a histogram.
In Figure3 we see that the temporal distribution is now rather smooth. Focusing on the 2nd century CE, we see much more balanced values across the bins on both subplots than in the case of the midpoints. Thus, by avoiding the biases listed above, a plot based on AS is a valuable contribution to the visual exploration of the data. But how do we draw on this approach when proceeding further to cross-data comparisons or hypothesis testing? This is where the Monte Carlo Simulation approach deserves our attention.
The Monte Carlo Simulation (MCS) approach to address within-phase temporal uncertainty in historical datasets2[] relies on the same assumptions as the Aoristic Sum method, treating date ranges as expressing probabilistic distributions. However, MCS derives these distributions as secondary outcomes of random number generation, rather than directly calculating them from the data. This technical distinction requires a more detailed explanation. In the following sections, we will demonstrate the e昀케cacy of the MCS approach by exploring our two example datasets using speci昀椀c functions from the tempun package, as tempun provides a range of useful functions for both the AS and MCS methodologies.
In the case of MCS, the initial step involves drawing on the dating intervals to generate random dates within them in the format of singular years. This task is accomplished using the functionmodel_date(), which requires two mandatory parameters: ”start” to specify the beginning of the dating interval, and ”stop” to indicate the end of the interval. Additionally, there are several optional parameters that can be utilized for speci昀椀c requirements, such as ”size”, specifying the number of the random dates we attempt to generate, or ”seed”, guaranteeing that we will obtain identical results once we again execute the command.
By default, the random number generator underlying tmhoedel_date() function follows a uniform distribution. This means that all numbers within the interval have an equal probability of being generated, which aligns with one of the shared assumptions of the AS approach. This characteristic becomes evident when we set the size parameter to a high value (e.g., 10,000) and subsequently plot the distribution of the resulting data. In a standard setting, we generate a 昀椀xed number of random dates for all records within a dataset in a single step. In the code snippet Listing1, we apply themodel_date() function to all records from the GIST dataset. The GIST dataset is loaded into the Python environment as a dataframe object, with the ”not_before” and ”not_a昀琀er” column attributes de昀椀ning the dating interval for each record. Thmeodel_date () function processes the values of these attributes for each row record and assigns to each of them 100 random dates using the ”size” parameter. To ensure reproducibility of our analyses, we also specify the seed for each row using their numerical ID inherited from PHI.
Listing 1: Assignment of 100 random dates to each record within the GIST dataset The resulting data is saved into a new dataframe column called ”random_dates”, where each row entry is represented by a list of 100 numbers. Collectively, the data in this column can be viewed as a matrix. In this object-date matrix, row vectors correspond to individual observations within the source dataset, while columns correspond to individual simulated date variants. For example, in Tabl1e, we display a few of the 100 modeled dates from a random sample of 5 dated inscriptions from the GIST dataset. Additionally, for reference, we include the PHI identi昀椀er of each record, its original verbal dating, and the ”not_before” and ”not_a昀琀er” attributes.
Now, focusing exclusively on the random dates data from the entire dataset and treating them as a matrix, we can proceed to various forms of analysis of temporal trends within the data. First and foremost, each of the 100 column vectors in the matrix can be regarded as
PHI_ID raw_date not_before
not_a昀琀er random_dates a time series on its own, with its unique temporal distribution. Our primary question is to what extent the individual temporal distributions are similar to each other, as this will re昀氀ect the impact of temporal uncertainty accompanying dating of individual records on the overall temporal trends. To address this question, we can plot all the histograms cumulatively and observe di昀erences within individual time blocks. However, bar-style histograms, commonly used for this type of visualization, may not be the most suitable choice. Instead, line-style histograms appear to be more useful in this regard.
With the help of thetempun package, you can easily generate a line-style cumulative histogram plot directly from the ”random_dates” attribute of a dataframe dataset. This can be achieved by utilizing thetimeblocksplot_from_randoms() function. Unlike plotting separate line histograms for each time series, this function consolidates all data within each time bin. Within each bin, the 2.5 % upper and lower margins are visually indicated using gray coloring, while all values within the 95 % interval are represented by a user-de昀椀ned color value.
Figure4 allows a straightforward visual comparison of the three aforementioned techniques
for exploring the diachronic distribution of historical datasets with range-de昀椀ned temporal
information, using the Ancient Shipwrecks dataset as an example. In Figur1e(A), we have
already demonstrated that this dataset exhibits a signi昀椀cant extent of temporal uncertainty,
with a relatively high number of records lacking precise dating. In Fig4u(rAe) we apply the
midpoints-based approach without any 昀椀ltering threshold for vaguely dated wreckage events.
Therefore, we can expect that these records signi昀椀cantly bias the overall distribution. This
becomes visible when we apply the Aoristic Sum methodology, as we do in Figu4r(Be). This
reveals an overall decline in the number of documented shipwrecks starting in the 1st half of
the 3rd century CE rather than its later half, as suggested by the un昀椀ltered midpoints data. This
observation echoes those made by Wilson1[
In Figure 4(C), we move a step forward, as we adopt the MCS timeblocks visualization method described above, putting together temporal distributions of 100 simulated time series. Unlike the AS methodology, this approach allows us to directly detect the periods most a昀ected by temporal uncertainty in the underlying data. For instance, in Figu4r(Be), there appears to be a higher number of shipwrecks from the 1st half of the 1st century CE than from its later half. This trend is apparent in subplot Figur4e(C) as well, but here the fat lines with gray margins also show that at least in case of some simulated time series data, the trend is in fact reversed. As we show below, this might be subjected to hypothesis testing.
Finally, on Figure4(D), we employ yet another visualization function of the simulated time
series data implemented in thetempun package: the kdeplot_from_randoms() function. This
visualization produces a cumulative kernel density estimate plot with each time series
represented by one line. The bandwidth of the KDE is calculated automatically on the basis of Scott’s
rule [
The visualizations above fall under the umbrella of exploratory data analysis, but perhaps the most promising possibility of employing the MCS approach is in the context of hypothesis testing. For instance, we can ask whether the temporal distribution of the simulated time series from one subset of the data di昀ers signi昀椀cantly from temporal distribution characterizing the rest of the dataset or another subset of the data. Thus, applying the MCS approach on a dataset of Latin Inscriptions of the Roman Empire (LIRE)1[0], Glomb et al (4[] focused on the distribution of a subset of inscriptions naming the god Asclepius while testing the hypothesis that these inscriptions (and hence the religious healing cult they are related to) was especially thriving during the period of the so called Antonine Plague (dated approximately to 165 - 180 CE), as proposed by a number of scholars (e.g.12[]). First, Glomb at el. tested whether the distribution of the inscriptions naming Asclepius di昀ers signi昀椀cantly from a distribution of a random control sample of inscriptions of the same size and with the same proportions of individual inscription types (e.g. epitaphs, votive inscriptions etc.). For that purpose, they employed the two-sample Kolmogorov-Smirnov test (2sKS5)] [to compare against each other 1,000 pairs of simulated time series data from the both groups of inscriptions and calculated the averageKS statistic and p values. The test revealed that the temporal distribution of inscriptions naming Asclepius was not signi昀椀cantly di昀erent from the distribution of its control sample.
Another option is to test statistically - in face of the temporal uncertainty within the underlying data - whether a number of records from a certain period is higher than a number of records from another period. On the most basic level, we can calculate the proportion of cases in which the trend goes in one direction. Thus, returning back to the GIST dataset, comparing statistically the data from the bins for the the 昀椀rst half and the second half of the 1st century CE, we could 昀椀nd that the number of inscriptions assigned to the earlier bin is higher in case of 68 out of 100 simulated time series. In other words, there is a non-negligible probability equal to 0.32 that the trend went in the opposite direction (c.f. Figur5e(A)). Accordingly, we should hesitate to formulate any strong claims concerning the distributional trend within this period. But, once we move one century forward and repeat the same test, we see that the number of inscriptions from the earlier half of the 2nd century is higher in 100 % of cases. Thus, there is a discernible decline starting at this point.
Finally, this logic might be further elaborated while focusing on some internal properties
of the data, such as word frequencies. It is in this respect where the advantage of MCS over
AS is most apparent. Thus, using the LIRE dataset of ancient Latin inscriptions from the
Roman Empire, Kaše et al. [
In this paper, we introduced three most commonly applied approaches for dealing with withinphase temporal uncertainty in historical datasets. The 昀椀rst introduced approach, based on the midpoint values, is the most straightforward one, but it is also associated with some substantial issues, as it tends to bias the data. The approach based on Aoristic Sum is free of the biggest problems of the midpoints’ based approach, but has its own limitation, as it does not allow us to identify in a straightforward fashion the periods most a昀ected by temporal uncertainty. Finally, the Monte Carlo approach allows us to move forward with hypothesis testing, even on the level of secondary features within a dataset (such as word frequencies in the case of inscriptions). Our implementation of the Aoristic Sum and MCS approach, including production of the 昀椀gures, relied heavily on thetempun Python package. Furthermore, we demonstrate the usefulness of thetempun package for the purposes of hypothesis testing in the studies referred to above. All our examples were elaborated within one Jupyter notebook script which we also share publicly for future reuse by other scholars.
This research was funded by the Aarhus University Forskningsfond Starting grant no. AUFFE-2018-7-22 awarded to the ‘Social Complexity in the Ancient Mediterranean‘ (SDAM) project.
A. Wilson. “Approaches to Quantifying Roman Trade”. InQ:uantifying the Roman Economy: Methods and Problems. Ed. by A. Bowman and A. Wilson. Oxford - New York: Oxford University Press, 2009, pp. 213–249.