Given that the TM is a nearly ubiquitous measurement across the site simply by chance, is there a way to determine what distances are most common?  In other words, instead of beginning a priori with a set distance and testing its ubiquity, is there a way to measure all possible distances between potential targets and then analyze this dataset for regularities?  Beginning again with GIS, this is possible with statistical frequency analysis through periodograms.  Periodograms are used to determine peak periodicities in serial data, usually applied to time-series variables (see Warner 1998 for periodogram and other time-series analysis methods).  A periodogram of electrical discharge from a person's heartbeat, for example, can show the regularity and timing of the heartbeats' main cycles and sub-cycles.  Stock market fluctuations are subjected to periodogram analysis in the search for cycles of rising and falling prices.  For this analysis, I apply the technique not to a temporal sequence, but to a spatial sequence: distances of all potential targets radiating from the edges and centers of the mounds. 

            In order to cover all potential distances within mounds, and to give a larger overall picture of distances across the site as a whole, the entire area covered by each mound is included as a potential target.  The datasets thus include distances from the centers and edges of each mound, measured to all raster grid cells covered by each of the other mounds.  Because the embankment is a very large target area and covers the ground in a qualitatively different way from the mounds (essentially surrounding them all), the analysis is conducted both with and without it as a target.  The analysis is also conducted from both the center points and edges of the mounds, resulting in a total of four distance determination methods (from mound edges with and without the embankment, and from mound center points with and without the embankment).

            Figure 7 illustrates the construction of one of these distance datasets.  The distance surface (Figure 7a), in this case created from the edge of Mound A, is used as a base layer masked by all other mounds, including the embankment.  The resulting raster (Figure 7b) contains values corresponding to all possible measurements from the edge of Mound A to all cells within all of the other mounds and the embankment.  The raw distance values were extracted from this raster and imported into statistical software (S-Plus, version 6.0).  This dataset contains all of the measurements in the raster, but in a tabular rather than spatial format. 

Figure 7.  (Left, a): Distance surface: all points are coded with their distance from the edge of Mound A.  (Right, b): Distances corresponding to all other mounds and the embankment have been extracted by masking out all other cells. 

Displaying the distances in this dataset as a histogram shows peaks and valleys in distance frequencies.  Figure 8 shows the Mound A histogram constructed in this way.  The shaded lines correspond to TM multiples.  A small mound is conjoined to Mound A, resulting in a few very short distances on the histogram.  At a distance of just over 100 m there are no other mounds from the edge of Mound A, and no measurements are therefore represented.  The most common distance measurements from the edge of Mound A are around 150 to 200 m, mostly corresponding to the mounds around the southern plaza.  Similar histograms were constructed for all mounds, in each of the four variations described above, for a total of 78 distance datasets.

Figure 8.  Histogram of distances from the edge of Mound A to all grid cells within all other mounds and the embankment.  The most common distance measurements are between about 150 and 200 m.  Vertical bars represent TM multiples. 

            Note that the most common distances expressed in the Mound A dataset do not generally correspond to TM multiples (shaded gray lines in Figure 8).  In fact, TM distances entirely miss the most prominent peaks.  The single co-occurrence of a TM multiple and a local peak in the data is at about 95 m (2 TM).  Mounds B, C, and D contribute the greatest number of measurements to this local peak in the data, and as Sherrod and Rolingson noted, some of the mounds ringing the southern plaza are close to 95 m from Mound A (1987:36).  The histograms from the other 77 iterations of this analysis are similar: there are peaks and valleys in the data, but few corresponding directly to TM multiples.  Figure 9 shows combined distance histograms from all mounds, computed with all four distance methods.  The shaded lines correspond to TM multiples.  Peaks in the measurements correspond fairly closely to TM multiples in some cases, but not at all in others.  Even if one of the methods were to generate sharp peaks in distance measurements which corresponded only with TM distances, citing this as evidence for the existence of the TM would be post hoc reasoning without prior theoretical justification for choosing one particular distance determination method over another.

Figure 9.  Histograms of all distances from the edges and centers of all mounds and the edge of the embankment, measured to all grid cells within the mounds and embankment.  Vertical bars represent TM multiples. 

            Another way to look at the distance histograms is to consider not the highest peaks they contain (which define the most common measurements from the feature in question), but the distances between the largest peaks, which define the most common measurements in the dataset as a whole.  In Figure 8, for example, the most common measurement expressed in the data is not the highest peak (at about 175 m).  The most commonly expressed distance in the dataset is actually represented by the most common distance between prominent peaks, each of which represents a large number of measurements from the target in question.

            This is where time series analysis is applied: viewing the datasets as spatial sequences, periodograms derive the most common distances between the peaks.  Figure 10 is the periodogram derived from the histogram in Figure 8 (distances to all mounds and the embankment from the edge of Mound A).  The horizontal axis is given in meters and represents potential frequencies within the data, from 2 m to infinity (because the histogram is composed of 1 m bins, two is the shortest frequency expressible).  The vertical axis represents the spectrum, or relative strength of each frequency. 

Figure 10.  Periodogram of the spatial-series data presented in Figure 8.  Frequencies of 2 and 4 m are weakly expressed.  These frequencies are an artifact of the square raster cells; distance measurements are "stepped" around the edges of mounds (see Figure 11).  The most significant frequency within the data occurs at 720 m.

            Two relatively short frequencies are weakly expressed in this periodogram, at two and four meters.  These frequencies are the result of square pixels within the GIS raster – the distances represented in the histogram do not sweep smoothly around the edges of the mounds, but step around them stair-wise at the distance of the edge of a pixel.  When such stepping around several mounds lines up in phase (rounded to the nearest meter in the histograms), these frequencies are expressed as significant peaks in the periodograms.

            The important frequencies for this analysis are the strongest ones expressed in the data, or the frequencies with the highest spectrum in each periodogram plot.  These represent the most common measurements in each dataset.  In the case of Figure 10, it is 720 m.  Figure 11 is a series of line graphs, showing each of the peak frequencies from all mounds, derived from each of the four iterations described above.  Not all of the periodicities shown here are statistically significant.  A frequency of 720 m from the edge of Mound A, for example (see Figure 10), is larger than the maximum distance in the histogram (about 500 m).  In this case, the periodogram analysis failed to derive a most significant frequency within the limits of the dataset, and we may consider the data to lack a clearly expressed frequency above 4 m.  Many of the distance datasets did contain significant frequencies within an appropriate range, however, mostly clustering between about 75 and 250 m (see Figure 11).  What this analysis demonstrates is at least one (quite experimental) way to quantify the most commonly occurring distances across a site as a whole.  Taking into account variations in the potential targets, in fact, there are at least four ways to do so.  In all four cases, there is a decided lack of correspondence between the most common distances and TM multiples. 

Figure 11.  Bar graphs of the most significant frequencies derived through all four iterations of periodogram analysis.

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