On Monument Volume IV

This post evaluates burial mound volume, fitting various probability models to the data. As noted previously, the exponential distribution seems like an appropriate model to fit to the mound volume data. This model is not the only possibility, of course, so I will also consider an alternative, the gamma distribution. The exponential distribution is a simplified version of the gamma distribution.

The gamma probability density function (pdf) is: $f(x\vert \alpha , \lambda) = \frac{\lambda ^{\alpha} x^{\alpha - 1}e^{- \lambda x}}{\Gamma (\alpha)}$,

where: $\alpha$ is the shape parameter, $\lambda$ is the rate parameter,

and $\Gamma$ is the gamma function.

The gamma function typically takes the following form: $\Gamma (\alpha) = \int_{0}^{\infty} t^{\alpha -1} e^{-t} dt$

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.