Identifying and Explaining Intensification in Prehistoric Fishing Practices VIII: Establishing the Number of Fish Caught by Nets or Other Gear through Mixture Models

The previous post in this series provided the middle-level theory needed to quantify the number and size of fish represented in an archeological assemblage. Recall that I am interested in determining how many of these fish were caught by nets and how many were caught by other gear. These gear should differ in the size range of fish that they are likely to capture.

The next step is to look at histograms of the data, which show the number of fish bone that occur within a particular size interval. Modes, or peaks, in the fish size-frequency histograms should reflect the use of different fishing gear. The mode of smaller fish should represent fish taken by nets, and the mode of larger fish should reflect fish taken by hooks and line or by spears. The following histograms show data from my archeological site. As noted in an earlier post, the fish bone assemblages derive from different levels within a single site, where minimal mixing has occurred among the levels.

Histograms of Fish Caudal Vertebra Height by Level from an Archaeological Site

Histograms of Fish Caudal Vertebra Height by Level from an Archaeological Site

The histograms for some of the levels show distinct modes. In particular, two modes seem to be present in the 50-55 cm level, while three modes apparently occur in the 30-35 cm level. The other levels are harder to interpret.

Clearly, the identification of these modes is not straightforward. The lack of more clear-cut patterning likely results from a heavy reliance on nets. Nets may catch both large fish as well as small fish. Other gear like hook and line or spears is much more likely to catch large fish. Prehistoric fishers used spears tipped with large stone points or sharpened bone. They employed hooks made from shells. Hooks and line or spears may not be able to catch fish smaller than some threshold value of size. In assemblages where net-caught fish predominate, the prevalence of net-caught fish may obscure any mode in the fish size distribution formed by fish caught with other gear.

Fortunately, statistical techniques exist which may help to distinguish separate populations which are mixed together in a single distribution. Finite mixture distributions model such situations. Such distributions can be analyzed using the mixdist package for R. This package allows the parameters of the contributing populations to be estimated, including the proportion of each population represented in the distribution and the mean vertebra size in each separate population. The following graph illustrates the application of a mixture model to data from the 50-55 cm level at the site.

Example of the Mixture Distribution Fit to Data from the 50-55 cm Level

Example of the Mixture Distribution Fit to Data from the 50-55 cm Level

For the mixture model, I fit two lognormal distributions to the data. The histogram depicts the original data. Note that the histogram interval differs from the interval used in the previous graph. The two dotted lines show the separate lognormal distributions fit to the data, and the black triangles identify the means of those distributions. The solid line shows the mixture model prediction that results from combining the two individual lognormal distributions. The gray bars at the bottom of the graphic show the deviations of the model from the observed distribution. The scale of the deviations is depicted in relative terms. This model appears to fit the data reasonably well.

I have also been working on a more rigorous analysis of the mixture models and their fit. This analysis is ongoing and has been plagued by some problems that I may have finally resolved. I will present some the results and issues in the next post in this series.

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

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