## On Monument Volume III

For my study area, the distribution of burial mound volume for plowed and whole mounds looks similar. This distribution is also quite different from the normal distribution that characterizes so many traits in the natural world. The distribution of burial mound volume resembles the form of an exponential distribution. Exponential distributions have a peak at the extreme left end of the distribution and decline steadily and rapidly from that point. The exponential distribution has a single parameter, the rate, typically denoted by $\lambda$. The following function gives the probability density (sometimes called the pdf) of the exponential distribution.

$f(x\vert \lambda) = \lambda e^{-\lambda x}$

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.