## Posts Tagged ‘mathematical modeling in archaeology’

### Identifying and Explaining Intensification in Prehistoric Fishing Practices VI: Modeling the Fish Live Weight-Caudal Vertebra Size Relationship

September 23, 2009

In the previous post in this series, I suggested that existing theory could provide a model appropriate to the observed relationship between live fish weight and vertebra size. This relationship appears to be an instance of allometric scaling. Many animal species exhibit a power-law relationship between the scale of particular traits and overall body size. Such relationships take the following form. Let y = body size and x = the size of a particular trait. Then:

$y=ax^b\!$

where a and b are constants and the parameters to be estimated from the data.

Note that a log transform of this relationship would result in the following equation:

$\log(y) = \log(a) + b\log(x)\,$.

This equation produces a straight line with the y-intercept at log(a) and a slope of b. The log-log transform of the data illustrated in the previous post showed this kind of line, which is a necessary (but not sufficient) condition for demonstrating a power-law relationship.

For my sample of fish, I evaluated the relationship between fish live weight and caudal vertebra size using a linear regression analysis. This technique identifies the values of a and b that minimize the deviation between the observed values and the predicted values. The estimates for the sample are a=4.54 and b=2.77. The 95 percent confidence interval for a ranges from 3.05 to 6.76, while the 95 percent confidence interval for b ranges from 2.49 to 3.05. For this model, r2=0.93, and the p-values for both parameters are less than 0.001. The low p-values indicate that the sample size was sufficiently large.

The following plot illustrates the fit of this model to the data. In the plot, the dashed lines show the prediction interval around the model. This interval depicts the range within which 95 percent of live weights for new samples would be expected to fall for a given value of caudal vertebra height.

Log of Caudal Vertebra Height and Log of Live Fish Weight

The sample of fish for this analysis should be expanded. Nevertheless, it supports the common-sense notion that fish bone size reflects the overall size of fish. The data that corroborate this middle-level theory are not comprehensive, but they provide sufficient justification to proceed. Caudal vertebra height will be used as a measure of fish size.

© Scott Pletka and Mathematical Tools, Archaeological Problems, 2009.

### Software for Archaeological Analysis

September 22, 2009

Archeologists can be very particular when choosing equipment. We prefer Marshalltown trowels, Brunton compasses, Sharpies, Trimble GPS units, and Munsell color charts. We spend a lot of time using this equipment, so such preferences are understandable. I have also thought a bit about the software that I use to analyze data once I’ve gotten out of the field and have processed my finds. A good software package for graphics and analysis can greatly speed report production and open many possibilities for the rigorous documentation of variability in the data.

Among commercial statistical software, I am a big fan of Statistica. I particularly appreciate the graphical capabilities of Statistica. Statistica can produce a wide array of graph types out of the box, and these options are all easy to customize. I have gotten considerable mileage from its options for categorized histograms, for example. Statistica also offers a wide range of analyses, available in different optional packages. Lately, however, I have begun the transition to R.

R is a statistical computing program. As such, it is extremely flexible and powerful. R is command-line driven, which makes it very easy to document and replicate the steps that you undertook during an analysis. If you’ve ever had to conduct an analysis several times, you will understand the appeal of this feature. Graphics can be generated in R, and they can be extensively customized. A lot of packages for it have been developed, providing functions by which you can conduct an extremely broad range of analyses. R is also free, which is another compelling reason to adopt it. R has an avid group of users, so you can have confidence that the program will continue to be supported in the future. The downside, however, is the steep learning curve.

To aid novices, lots of guide books to R are available, and some of them are quite good. I recommend starting with Introductory Statistics with R. With this book, you should be able to jump right into generating basic graphics and running common statistical tests. Another really useful book is Ecological Models and Data in R. This book covers model-building and maximum likelihood methods for evaluating the fit of data to those models. While the book’s examples reflect the interests of an ecologist (obvs), the book focuses on developing skills that transcend any particular discipline. The book is quite well-written. The presentation is also sufficiently detailed that you can very easily use it as a point of departure for playing with your own data and models.

Whatever you choose, make sure that the program offers a lot of flexibility. The analyses that I have done varied quite a bit from one research project to the next. Any program that you obtain will require a considerable investment—both financial and intellectual—so you’ll want to have that investment pay off again and again.

© Scott Pletka and Mathematical Tools, Archaeological Problems, 2009.

### Identification and explanation of intensified prehistoric fishing practices I

September 13, 2009

In this series of posts, I will be exploring ways to identify and explain intensified prehistoric fishing practices. Intensification refers to the input of greater amounts of labor per unit capita to procure resources. As this formal definition implies, people are working harder at subsistence activities when they intensify their way of making a living. How do we determine when people are working harder from archaeological evidence? And what factors would induce people to intensify their efforts?

A lot of theories exist to address the latter question, but the former question is the more immediate problem. We need middle-level theory (sometimes labeled middle-range theory) appropriate to the nature of the archaeological evidence. Middle-level theory links archaeological data to phenomena of interest. It allows archaeologists to say with some confidence what happened in the past, based on that evidence.

My evidence comprises collections of fish bones and other artifacts from an organic-rich trash dump at a single archaeological site. Such trash dumps are often called middens. The occupation of the site spans several hundred years. The trash dump, however, has been sufficiently undisturbed since it was deposited that it could be excavated to recover evidence representative of much shorter spans of time. The mathematical tools that I found useful as I developed appropriate middle-level theory for this evidence included regression analysis, mixture models, and maximum likelihood models. In the next post, I will begin to develop the middle-level theory in detail, talking about the blind alleys down which I went, mistakes that I made, and solutions at which I arrived. Check back soon.

© Scott Pletka and Mathematical Tools, Archaeological Problems, 2009.

### Getting started…

September 12, 2009

I am in the process of getting the format and content into shape. Check back soon for more informative posts.