The blog carnival usefully highlighted the variety of work that people are posting. The ecosystem of archaeologically-themed blogs seems as mature as found for any other subject. Blogs cover personal experiences of students and professionals, provide field project updates, highlight research, discuss news, aggregate news, and explore odd corners of archaeology. Interestingly and importantly, not all of these blogs are produced by professional archaeologists.

Professional archaeologists in America have been moved recently to defend and promote the relevance of their field, in response to questions about whether such work should receive scarce public research funds. Some professionals have called for the field to rally around a unified set of topics for investigation. These efforts are reasonable, but they also highlight a problem.

Professional archaeologists are not necessarily their own best spokespersons. They clearly have a self-interest in the funding of their research. Professional archaeologists are also not always great communicators of their work, particularly to lay audiences. Not everyone is going to be good at everything. Consequently, the work of bloggers who write about archaeology could be a real resource to the field, testifying explicitly and implicitly that archaeology does matter.

My hope is that professional archaeologists and their national and regional societies figure out a way to tap into all the enthusiasm and talent that exists. Public outreach efforts should not be limited to any single venue, such as blogs, of course. Nevertheless, this resource already exists and could be very easily incorporated into a comprehensive media strategy. The collection of papers on blogging and archaeology at the upcoming SAA meetings in Austin looks like a good opportunity to get further exposure to ideas about the potentials and pitfalls of blogging, for those folks who are interested.

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**Results of Fitting Various Distributions to a Single Experimentally-Generated Flake Assemblage
**

The models selected for comparison comprise a list of commonly-used distributions for modeling heavy-tailed data. Other model-fitting approaches, for example, found the Weibull distribution to fit flake-size distributions. While the maximum likelihood method provides a means by which to fit each of these models to the same data, this approach must be supplemented with some method for comparing the results.

One approach to model comparison, the Akaike Information Criterion (AIC), comes from information theory. The information-theoretic approach quantifies the expected distance between a particular model and the “true” model. The distance reflects the loss of information resulting from use of the focal model, which naturally fails to capture all of the variability in the data. The best model among a series of candidate models is the model that minimizes the distance or information loss. Obviously, the true model is unknown, but the distance can be estimated using some clever math. The derivation of AIC is beyond my powers to explain in any additional detail. After much hairy math, involving various approximations and simplifications and matrix algebra, a very simple formula emerges. AIC quantifies the distance using each model’s likelihood and a penalty term. AIC for a particular model can be defined as follows:

AIC = -2*L* + 2*k*, where* L* is the log-likelihood and *k* refers to the number of parameters in the model.

For small samples, a corrected version of AIC – termed AICc – is sometimes used:

AICc = AIC + 2*k*(*k*+1)/(*n*–*k*-1)

The best-fitting model among a series of candidates will have the lowest value of AIC or AICc. Unlike other approaches, the information-theoretic approach can simultaneously compare any set of models. Likelihood ratio tests, another common model-comparision technique, are limited to pairwise comparisons of nested models; models are nested when the more complex model can be reduced to the simpler model by setting some parameters to a particular value, such as zero. The models compared in this example are not all nested, since the lognormal – for example – cannot be reduced to any of the other models by setting a parameter to a particular value.

The AIC and AICc for the power law distribution are both lower than the values for any of the other modeled distributions. The power law distribution thus fits this experimentally-produced flake size data better than other common distributions. These preliminary results support the work of Brown (2001) and others.

Note that the best-fitting model among all candidates may still provide a poor fit to the data. Thus, the power law distribution could still provide a poor fit to the data. A couple options exist for evaluating model fit. A simple approach would be to plot the data versus the theoretical distribution. There is also a way to measure the fit of the model to the data, which I will detail in a subsequent post.

© Scott Pletka and *Mathematical Tools, Archaeological Problems*, 2014

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The primary reason is that the blog provides some small motivation to continue to write and research. In this format, I can post pieces of a larger project. Those pieces get posted as they are completed. Each post thus feels like an accomplishment but also contributes to the goal of finishing the project. The first series of posts that I wrote, on the intensification of fishing, did eventually get formally published as a single piece. The blog was very helpful in developing that paper during the long slog toward publication.

My experience working on that paper and the blog led me to identify some other, related benefits of blogging. Due to space constraints, some material that I developed for the intensification paper was dropped from the final version. The blog provided a venue in which to share those thoughts, if someone should be interested in that aspect of the topic or greater details on the topic. For example, I used the blog to post *R* code that I wrote to perform some of the analyses for the intensification paper. This experience led me to the realization that the blog was also a great place for small projects that would likely never get formally published. The next series of posts that I wrote concerned variation in burial monument size. It’s possible that some of this material will resurface as a part of another project, but for now those posts were just a fun little diversion.

Other professions, such as economics, have a more well-developed tradition of posting working papers, papers that are pre-publication. Blogs are a great way to develop and further this tradition and to share data and insights that might not otherwise fit within the strictures of academic publishing.

© Scott Pletka and *Mathematical Tools, Archaeological Problems*, 2013

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As discussed previously, linear regression analysis has sometimes been used to evaluate the fit of the power law distribution to data and to estimate the value of the exponent, . This technique produces biased estimates, which the next simulation results illustrate. In the simulation, a random number generator produced a sample of numbers drawn from a power law distribution at a particular value of . I then analyzed this artificial data set using the linear regression approach described by Brown (2001) and using maximum likelihood estimation through a direct search of the parameter space. For a simple power law distribution, the maximimum likelihood estimate could also be found analytically. I used a direct search approach, however, in anticipation of using this approach for more complex mixture models. I repeated the random number generation and analysis 35 separate times for several different combinations of and sample sizes. The following histograms show the estimates for using the linear regression analysis and maximum likelihood estimation to find that value. In this particular case, was set to 3.6 in the random number generator, the minimum value was set to 8.9, and the sample size was 500.

The histograms clearly suggest that the maximum likelihood estimates center closely around the true value of alpha, while the regression analyses skew to a lower value. The simulations that I performed at other parameter values and sample sizes displayed similar results. Other, more extensive simulation work also supports these impressions (Clauset et al. [2009] provides these results as part of a detailed, comprehensive technical discussion). Consequently, I used maximum likelihood estimation to fit probability distributions to data in the subsequent analyses.

**References cited**

Brown, Clifford T.

2001 The Fractal Dimensions of Lithic Reduction. *Journal of Archaeological Science* 28: 619-631.

Clauset, Aaron; Cosma Rohilla Shalizi; and Mark E. J. Newman

2009. Power-law distributions in empirical data. *SIAM Review* 51(4): 661-703.

© Scott Pletka and *Mathematical Tools, Archaeological Problems*, 2013

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Brown (2001) suggests that the distribution of debitage size follows a power law. Power law distributions have the following probability density function:

,

where *C* is a constant that normalizes the distribution, so the density integrates to one. The value of *C* thus depends entirely on the exponent .

Based on analysis of experimentally-produced assemblages, Brown further suggests that the exponent, , of these power law distributions varies among different reduction strategies. Thus, different reduction strategies produce distinctive debitage size distributions. This result could be very powerful, allowing reduction strategies from a wide variety of contexts to be characterized and distinguished. The technique used by Brown to estimate the value of the exponent, however, has some technical flaws.

Brown (2001) fits a linear regression to the relationship between the log of flake size grade and the log of the cumulative count of flakes in each size grade. In its favor, this approach seemingly reduces the effects of small sample sizes and can be easily replicated. The regression approach, on the other hand, also produces biased estimates of the exponent and does not allow the fit of the power law model to be compared to other probability density functions.

Maximum likelihood estimates, using data on the size of each piece of debitage, produce more reliable estimates of the exponent of a power law. Maximum likelihood estimates can also be readily compared among different distributions fit to the data, to evaluate whether a power law is the best model to describe debitage size distributions. The next post will illustrate the use of the linear regression approach and the maximum likelihood approach on simulated data drawn from a power law distribution.

**Reference cited**

Brown, Clifford T.

2001 The Fractal Dimensions of Lithic Reduction. *Journal of Archaeological Science* 28: 619-631.

© Scott Pletka and *Mathematical Tools, Archaeological Problems*, 2013

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Recall that most mounds have been affected by modern plowing and other disturbances, but some mounds have been largely spared such damage. The museum records that I used characterized these undamaged mounds as “whole”. The museum records documented 287 whole mounds. To make sure that the comparisons were fair, I limited the sample of nearest neighbors to just those whole mounds that had another whole mound as its nearest neighbor. I eliminated duplicate pairings, so each pair of nearest neighbors was only considered once. The imposition of these constraints shrunk the nearest neighbor sample size to 49. Finally, I ran a simple linear regression to evaluate the relationship between the size of the mounds in these nearest neighbor pairings. Because the distribution of mound volume can be modeled as an exponential distribution, I used the log of mound volume in the regression analysis. Without this transformation, any relationship in

mound size between the nearest neighbors would be unlikely to be well approximated by a straight line.

I then sampled without replacement from the 287 whole mounds to obtain 49 randomly-selected pairs. As with the nearest neighbors, I performed a simple linear regression, using log volume. I repeated this procedure 500 times. The repeated sampling and analysis allowed me to develop a null hypothesis for the values of the regression coefficients.

I expected that the randomly-selected pairs would not have a meaningful relationship. The slope of the regression line should be close to zero for these samples. In contrast, the size of the nearest neighbor pairs should be positively correlated, so the slope of the regression line should be significantly larger than zero. The following two figures show the distribution of the regression coefficients, the intercept and slope, for the randomly-selected pairs.

Notice, in particular, that the distribution of the slope clusters near zero as predicted. This result indicates that the randomly-selected pairs do not have a meaningful relationship with each other, at least with respect to size.

These distributions contrast with the regression coefficients calculated for the nearest neighbors. The intercept is 0.90, and the slope is 0.75. These values are completely beyond the range of values estimated for the randomly-selected pairs. This experiment shows that the size of nearest neighbors is significantly and positively correlated. The results lend some support to the notion that stable groups produced these mounds. At the very least, the results provide encouragement to further explore the relationship between mound size and mound spatial distribution. Such work should probably make use of the spatial analysis tools available in GIS programs.

© Scott Pletka and *Mathematical Tools, Archaeological Problems*, 2013.

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The following thoughts should be considered preliminary. The original goal of this particular analysis was very modest, concerned with establishing a reliable measure of monument scale or prominence. My hope was that mound volume had stayed reasonably constant despite the effects of weathering and other processes. The analysis was being done as part of a project regarding monument function and social organization. While the analysis showed that many mounds lost volume as a result of modern plowing, it also showed that the volume of plowed mounds and whole mounds varied in very similar fashion. Variation in mound volume can be modeled with the exponential distribution. I did not expect this result at the outset.

I have often regarded mounds as potentially reflecting the “strength” of the groups that built them. Group strength might be a function of many different factors, such as group size, the productivity of the territory that the group occupies, the group’s organizational capabilities and the size of the social network upon which the group could call. Groups that scored higher on these variables should have been capable of building larger mounds. Groups that scored lower on these variables should have been limited to building smaller mounds. A large list of qualities could thus contribute to group strength and to burial mound size. I assumed that each factor would have a small additive effect on strength. Consequently, I supposed that variation in group strength and mound size should take the form of a normal distribution.

Clearly, this intuition was wrong. Upon further reflection, I think that I’ve underestimated the contribution of social networks. Their contribution is probably not minor. Ethnographic studies of leadership in small-scale societies illustrate the hard work and emphasis that group leaders often put on the maintenance of their networks. The effect of each additional ally is probably not merely additive, since each ally that gets incorporated has the potential to contribute their own unique allies to the network. Modern studies of social networks indicate that variability among individuals in network size has a heavy-tailed distribution, where most individuals have a relatively small network and a few individuals have very large networks. The mound data is suggestive of similar processes at play.

Before getting too carried away, let me emphasize again that this interpretation is very preliminary. It is, nevertheless, consistent with other archeological evidence for the operation of long-distance exchange networks at the time. The results also illustrate the potential value of this form of statistical modeling. The type of probability model which can be fit to the data — whether normal, exponential, or some other model — reflect the type of processes which operated in the past. The modeling thus constrains the set of possible interpretations that should be considered.

© Scott Pletka and *Mathematical Tools, Archaeological Problems*, 2013.

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The gamma probability density function (pdf) is:

,

where:

is the shape parameter,

is the rate parameter,

and is the gamma function.

The gamma function typically takes the following form:

Depending on the parameter values, the graph of the gamma pdf can take a wide variety of shapes, including forms that resemble the bell-shaped curve of the normal distribution. The following illustration shows some of the possible variation.

To evaluate the relationship between mound volume and mound condition (plowed and whole) under the gamma and exponential distributions, I analyzed model fit using the maximum likelihood method. The following R code details the analysis.

>library(bbmle)

>mdvol_g.mle=mle2(Allmds$Mound.Volume~dgamma(shape=shape, rate = gvar), start=list(shape = 1, gvar = 1/mean(Allmds$Mound.Volume)), data=Allmds, parameters = list(gvar~Allmds$Condition))

>mdvol_g.mle

>mdvol_e_cov.mle=mle2(Allmds$Mound.Volume~dexp(rate = avar), start=list(avar = 1/mean(Allmds$Mound.Volume)), data=Allmds, parameters = list(avar~Allmds$Condition))

>mdvol_e_cov.mle

>mdvol_e.mle= mle2(Allmds$Mound.Volume~dexp(rate = bvar), start=list(bvar = 1/mean(Allmds$Mound.Volume)), data=Allmds)

>mdvol_e.mle

In this code, *Allmds* refers to an R data frame containing the variables *Mound.Volume* and *Condition*. The code uses the maximum likelihood method to evaluate the fit of an exponential distribution to the data and to estimate parameter values. I performed the analysis three times. In the first analysis, I fit the gamma distribution, using *Condition* as a covariate. In the second and third analyses, I fit the exponential distribution to the data, once with the covariate *Condition* and once without the covariate.

The models are “nested”. The gamma distribution can be reduced to the exponential distribution by setting the gamma’s shape parameter to one. The exponential model without the covariate is a simplified version of the model with the covariate. Nested models can be compared using an ANOVA test to see whether the more complex model gives a significantly better fit to the data, justifying the extra complexity. The following two tables show the results of the analysis.

The initial results suggest that the exponential distribution with the covariate provides a significantly better fit to the data than the simpler model without the covariate. The gamma distribution does not provide a significantly better fit. Notice that the gamma’s shape parameter is estimated to be one, which reduces the gamma to the exponential distribution.

From this preliminary analysis, I offer the following conclusions. The exponential distribution appears to be an appropriate model for mound volume. In addition, plowed mounds may be distinctly smaller than whole mounds, contradicting my initial hypothesis. In subsequent posts, I will consider some archaeological implications and address some additional considerations that may help to explain these results.

© Scott Pletka and *Mathematical Tools, Archaeological Problems*, 2013.

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The pdf defines a curve. For a continuous distribution such as the exponential distribution, the area under this curve provides the probability of a sample taking on the value within the interval along the x-axis under the curve. The following illustration depicts these relationships. In the illustration, the shaded area under the curve represents the probability of a given sample falling between the two values of *x*.

As a check on my intuition regarding the applicability of the exponential distribution, I generated a random sample of 2000 from an exponential distribution with a mean of 500. The following figure shows what such a distribution may look like. The simulation does not provide definitive proof, but it may nevertheless indicate whether a more rigorous analysis that employs the exponential distribution is worth pursuing.

At least superficially, the histogram of the simulation results resembles the histograms of mound volume shown in the previous post. This simulation did not produce the apparent outliers seen in the mound data, but the resemblance suggests that burial mound volume can be modeled with an exponential distribution. I thus modeled mound volume with an exponential distribution, using mound condition (plowed or whole) as a covariate. I performed this analysis in R with the bbmle module. In the next post, I’ll present the code and initial results.

© Scott Pletka and *Mathematical Tools, Archaeological Problems*, 2013.

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