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# «By Nathan B. Goodale A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE ...»

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Summary While admitting that several variables in use here may be influenced by taphonomic bias or the natural destruction of archaeological deposits, “this is not to say that examining such datasets is a worthless endeavor” (Surovell and Brantingham 2007:1874). Instead, under a cautious approach utilizing several variables as well as analytical techniques to compare the resulting datasets, I hope to circumvent most of the problems of taphonomic bias. The next section provides a detailed overview of the technique, cross-correlation analysis, which will provide a graphical summary to test a series of expectations if the datasets are reliable indicators of population. As noted above, we expect that if each variable is a relatively accurate proxy of the same phenomenon (population growth), we should expect each variable to monitor and track that phenomenon in a similar manner.

Cross-correlation Analysis Cross-correlation is an alignment algorithm used to find patterns between two datasets (Rockwood et al. 2005). In every case, the researcher takes a pattern (for example DNA sequences, ink chromatograms, or a portion of two images) and then slides it over another pattern that is suspected to match up with it in some manner

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correlation provides one number; and when the numbers are plotted, there will generally be a few peaks; the highest peak demonstrates where the datasets best align.

Cross-correlation Example 1 One way the cross-correlation can be thought of is going down the aisles in a shoe store, trying on a bunch of pairs of shoes sizes 4 through 16+ and giving each a rating. Those you cannot put on, or fall right off, get very low scores; and those that fit your feet generally get a high rating. Producing a plot of rated score versus where you are in the shoe store will provide a view of the topographical landscape; the valleys correspond to shoes of the wrong size or shape, and the peaks (say in the right-sized sandals section) where size and fit are just right for you. In this case, your feet (a) are the sliding pattern, the shoes (b) in the shoe store are the fixed pattern you're sliding over, and however you rate the comfort of the shoes is the scoring criterion(N) for forming the cross-correlation (Figure 6.1 (A)).

131 Figure 6.1. A graphic illustration of the cross-correlation procedure where (A) represents the two functions for comparison, (B) represents the shift required to match up the two functions and (C) represents the cross-correlation where the peak is the best fit between the functions.

Thus, wherever the peaks are in the abN numbers, they tell us how far the analyst has to shift, say, b to get it to line up the best with a (Figure 6.1 (B)); the height of the peak (and in some sense, the narrowness of its width) demonstrates how good the match is between a and b; and so on. In other words, if ab9.5 is by far the highest number in all of the sets that are cross correlated, it indicates that a and b match up the best at shoe size 9.5 (Figure 6.1 (C)). Furthermore, cross-correlation

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different types of shoes fit the best. Thus, we can examine if sandals (c) fit the best in the same size as dress shoes (d), athletic shoes (e) etc. The two types of shoes that fit the best in the same range will score the highest (N) and thus, have the highest crosscorrelation.

Cross-correlation Example 2 For humans, picking patterns out from images is remarkably easy, sometimes too easy. We can recognize the faces of long-lost relatives in photographs and paintings, even though we may not have ever seen the person depicted before. We can pick our native written language out of examples of thousands of typeset or even handwritten words, and we can match pieces to gaps in a jigsaw puzzle, despite not being sure of which position or orientation the piece should have. Before computers took over the task of matching fingerprints found at a scene to those residing in a catalog, the task was done by having human investigators look at the found print (called here the target; see Figure 6.2) and turn them this way and that, running them down a row of suspect prints (the template), mentally filling in gaps and deleting inkblots, and rescaling the target print until they made the best match possible with the template prints.

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Having a computer do this is by no means a difficult feat. The heart of almost every pattern-matching program is cross-correlation analysis, which will be illustrated here in example 2 with the task of matching a fingerprint against a catalog of several others (potentially millions). One way in which to have a computer match a finger print against another is to digitize the print in strips, as shown in Figure 6.2.

Wherever a ridge (dark on a print) makes a prominent appearance in a given strip, a high number is given to the strip at that location; lower numbers indicate a valley of the print, or no print at all. The computer’s task is then to rapidly match each strip of a target print against the millions present in a large repository of template prints.

Figure 6.3b shows the blue trace from 6.

3a, and a copy of it that has been shifted to the right, and changed by adding noise to it. This simulates what might happen when the print is lifted under different situations. The challenge is to figure out how much the gray trace (the target) must be moved to line up best with the blue trace (the template). Additionally, we want to come up with a way of quantifying the

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this naturally.

Figure 6.3.

(a) A fingerprint, digitized in strips. The blue trace shows the pixel density in a given strip; higher pixel density indicates a ridge at that point in the strip.

Sampling the fingerprint under different conditions can lead to changes; (b)shows the original strip and a shifted, noisy version.

To start, the target trace is shifted all the way to the left, and both traces are zero-padded (Figure 6.4). Each trace is divided into bins which are one point wide, and the numbers in each corresponding bin are multiplied together. All of the resulting numbers are then added, to give one number (the correlation) corresponding to this fully leftward-shifted position of the target. Because of the zero-padding, those portions of the traces that do not overlap automatically get a value of zero and add nothing to the total.

Table 1 illustrates three steps in a cross-correlation of two small datasets. The resulting data structure can be expressed as

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where N is the length of the target and template (tem) datasets, and j represents the amount of the shift applied to the target (tar) set ( j can run from -N when the target set is shifted all the way to the left, to +N when the target set is shifted all the way to the right).

Figure 6.4 A zero-padded template (blue) and a fully left-shifted, zero-padded target (gray) at three different shifts in the cross-correlation process.

Once the set of numbers is compiled for all of the shifts, the computer can look for peaks. These peaks occur when the two datasets have numbers that match up well, so the sums of the products add to large numbers. Note that the sums can be negative if the multiplications give more negative numbers than positive numbers.

Negative sums imply that for those shifts, the datasets are generally matching peaks with valleys. The cross-correlation between the two traces in 6.3b is shown in Figure 6.5.

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Note: the template dataset is held constant, while the target dataset is slid alongside it, by one position during each step. The numbers at the same position in the template and target are multiplied, and then the entire result is added up to give a total number for each shift. The resulting cross-correlation would have the structure {…,{2,-0.09}, {3, -0.67}, {4, -0.65},…}.

Figure 6.5.

The result of the cross-correlation of the two traces in Figure 6.3b. The large peak at a shift of+61 indicates that the target trace (gray) should be shifted 61 positions to the left (alternatively, the blue trace should be shifted 61 places to the right) to match up best.

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There have been a number of ways in which archaeologists have attempted to model past population fluctuations. In this chapter I have presented the variables used to interpret past population growth through the transition to agriculture in the southern Levant. Importantly, each line of evidence to build the cultural model is a standard line of evidence that researchers normally present in site reports as well as in larger discussion pieces regarding changing settlement patterns and mobility strategies (Kuijt and Bar-Yosef 1994). Additionally, each variable has been adopted by archaeologists and paleodemographers as proxies of human population densities.

Through utilizing multiple variables in concert with regression and cross-correlation analysis, a number of expectations will tested in order to determine if the data are fairly accurate measures of population and not overly skewed by taphonomic bias. In Chapter Seven, the results of the data analysis are presented as well as the paleodemographic model of the NDT in the southern Levant.

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## LEVANT: A POPULATION GROWTH RATE MODEL

Although a daunting task, archaeologists largely remain fascinated by population size and the remains in the archaeological record that may allow us to estimate past population growth rates. The Neolithic demographic transition, defined as a substantial increase in human numbers, should logically be quantifiable based on archaeological data, apart from paleoanthropological markers in cemeteries (BocquetAppel 2002; Bocquet-Appel and Naji 2006). This chapter is dedicated to building a model of population growth rates based on the variables presented in Chapter Six.

I argued in previous chapters that the NDT was a caused by 1) an increase in fertility as a result of certain foods being available on a longer than seasonal basis, and 2) population growth as a result of labor reorganization to in incorporate younger age classes that helped feed larger families. While these points were argued, it is beyond my means to quantify these aspects of the model at this point. Instead, my focus in this study is to present the population growth rate portion of the model and bring all other components of the model together in the concluding chapter.

In order to avoid imposing specific numbers of people in the past for a certain unit of space, we can instead examine proxies of human populations or the intensity of human occupation in some defined space. The population proxy model discussed here attempts to demonstrate 1) that the data should not be overly influenced by taphonomic bias, 2) that each variable demonstrates a statistically significant positive

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good cross-correlation with the others, which would indicate that each variable is tracking the same phenomenon (population growth) in a relatively consistent manner, and 4) because results 1-3 indicate that the data should be fairly reliable indicators that are tracking population growth, we should be able to employ the data to examine population growth rates. Finally, the population growth model is compared to a simulation of how population growth should pattern if 1) there is a reasonably consistent growth of population in comparison to 2) differing rates of taphonomic processes that are influencing the destruction of the archaeological record at different rates. Surovell and Brantingham (2007) develop a similar simulation, yet mine differs in that it considers both the rate of addition and subtraction of sites from the record. The simulation demonstrates very similar patterns to increasing population through time with minimal destruction of the archaeological record.

The Data As covered in Chapter Six, the data utilized here to model population growth rates include the total depth of deposits, the total site area in hectares, the frequency of sites occupied and the frequency of 14C dates occurring in each 50-year increment from 20,000 to 8,000 cal BP. The data analyzed here can be found by site in Appendix A and by time interval in Appendix B. Additionally, all of the references to where the data were obtained are documented in Appendix C. All data are from the southern Levant and gathered from the published literature. Sites were included

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investigator agreed with the dates. In total, I recovered 109 archaeological sites with affiliated and reliable 14C dates spanning this period. I was able to recover information on the depth of deposits and site size for many of the sites, although not all of them as indicated in Appendix A. In some cases below, the raw data was utilized in the analysis, such as with the regression analysis, but were transformed for normality as required by the technique. However, in the population growth rate model, the raw data have been converted to the percentage of the total of each variable occurring in each 50-year increment. This was in order to put each variable on the same scale for comparison.

Decaying Exponentials and Taphonomic Bias Discussed in Chapter Six, taphonomic bias (Surovell and Brantingham 2007), or the destruction of sites over time, could present problems for accurately estimating past populations. In short, there are three rudimentary issues of taphonomic bias in using archaeological data to reconstruct population size, 1) the visibility of older versus younger sites lending to identification and dating, 2) the validity of older versus younger archaeological deposits as they become more disturbed through time and 3) the destructive processes that have removed sites from the archaeological record consistently through time, thus making older materials less abundant than they originally were.

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