The Patterns Are Unique
Modeling the PatternHere we develop a probabilistic model of the pattern of dots in a fisher footprint and use it to predict the odds that two prints with n matching pairs of dots are different by chance alone. In addition to demonstrating that the overall variation in foot-pad patterns is sufficient, the model is useful for estimating the rarity of an smaller portion of the pattern; important given the incomplete pattern typically available from field-collected tracks. 
Conceptualization of the Dot Pattern The model approximates the patterns in fisher metacarpal pads as a series of parallel rows of dots with variable distance between neighboring dots (measured from each dot's centroid) within any row. Assuming that the spacing between any pair of dots is independent of the spacing of other nearby pairs, the most common arrangement will be when all spacings are equal to the median value P(Ro). Because any deviation from the pattern where all dot spacings equal the median value represents a less likely combination, the probability of occurrence of the most common condition is an upper limit and serves as a conservative estimate of pattern rarity. When applied to an arbitrary number of dots, the maximum probability of occurrence for that portion of a footprint is given by the product of the probabilities of occurrence of each encountered spacing. Thus, the maximum probability of occurrence for a particular pattern of dots within a row, Pn, is given by: n Pn ≤ P(Ro) (Equation 1) where n is the number of spaces between neighboring dots of the same row. Equation 1 estimates how many corresponding dots one must find between two footprints in order to know with some degree of certainty that the same foot made those two prints. The possibility that two footprints made by different feet will match by chance alone can be set arbitrarily small by increasing the number of dot pairs matched. To determine Ro, we measured the distance between 1400 pairs of adjacent dots (dots found within the same row) in footprints made by 14 fisher forefeet (100 pairs per foot, 10 left and 4 right feet). Of 1400 measurements, 496 (approximately 35%) fell into the most common size bin, from 0.22 to 0.28 mm, with all other spacing being more rare. Thus, the most common pattern for any row of dots must be when each space falls within this range. If spacing between neighboring dot pairs within a row can be considered independent (see below), according to equation 1, the probability of encountering a row of dots with any given spacing configuration, Pn, is described by: n Pn ≤ 0.35 (Equation 2) With over 1000 dots typically present in a fully detailed metacarpal pad, the probability of occurrence of two similar pads by chance alone can be seen to be vanishingly small. Does the Independence Assumption Hold? To investigate the degree to which spacing between any pair of dots is independent of the spacing of other nearby pairs (i.e., spatial autocorrelation), we measured the spacing of 100 neighboring dot triplets (resulting in sets of two spacing measurements) and evaluated the correlation coefficient between these paired distances. This was found to be 0.30, a statistically significant but fairly mild correlation. As a further check on the magnitude of this effect, we identified 196 cases of the most common spacing (0.22-0.28 mm, see below) and measured 306 adjacent dot distances. We determined the percentage of these adjacent spaces that also fell within the range 0.22-0.28 mm and compared that figure to the percentage predicted by the overall distribution of spacings.One would expect by the results depicted in Figure C that 35%, or approximately 107, would fall into the most common range of 0.22-0.28 mm. In fact, with the sample examined there were 108, reflecting virtually no increased tendency for adjacent spaces to measure the same. Thus, it would appear that the spacing of any pair of dots is largely independent of the spacing of its neighbors within the same row and, therefore, equation 2 may be used to estimate the probability of occurrence of a dot pattern. Probability of Falsely Matching Tracks Due to factors already mentioned, one can expect to match only a portion of each dot pattern for any two tracks of the same foot. The analysis of the model suggests that the probability of a false match can be predicted by noting the number of dot features that match between the two tracks. More dots in the matching area imply greater confidence that two tracks were, indeed, made by the same foot. Due to the exponential nature of equation 2, the probability of a false match decreases rapidly as the number of dots (or, more strictly keeping with the analytical approach used here, the number of spaces between dots) increases. The figures in the table assume that the portions of the tracks being matched are extracted from exactly the same part of the metacarpal pad. The chance of a false match would increase if, say, a dot array from the upper left portion of the candidate track was matched against all similar sized arrays located throughout the reference track. While extreme cases are easily avoided, one must allow for such factors as incomplete tracks, distortion of the track due to the flexible nature of the foot-pad, and errors in alignment of the two tracks. A thorough attempt to match two tracks requires a certain degree of shifting of the relative position of the two images. The net result of this is an increase in the probability of a false match over what equation 3 indicates, albeit one that is difficult to quantify. There are further complications to be considered. Recall that several simplifying assumptions were made in arriving at equation 3, assumptions that tended to make the equation conservative in its prediction. These effects would tend to cancel the degradation described in the last paragraph. It is clear that a rigorous determination of the probability of falsely matching tracks made by two different individuals would be very complicated indeed, with many factors in common with the similar question regarding human fingerprints, a matter that has yet to be thoroughly resolved despite intense effort on the part of many researchers (Pakanti et al. 2002). Depending on the goals of the study, knowledge of the home range and other aspects of fisher autecology for the population in question might provide a useful input in determining how many dots need to correspond between two prints in order to safely conclude they were made by the same individual. Fishers are thought to exhibit a strong tendency for intrasexual territoriality throughout most of the year (Arthur et al. 1989, Powell 1979). This assumption, along with information about home range size (reported to be on the order of 20 to 50 km sq for males, 8 to 30 km sq for females, Powell and Zielinski 1994), recruitment, and an estimate of the likelihood of encountering transient or dispersing animals, could provide a useful upper limit on the number of individuals that might be encountered in a given study area. Considering these factors, the number of animals that could be responsible for a track collected from any individual track plate station might be on the order of 10. Thus, depending on the goals of the study, matching even as few as 10 dots between tracks may be suitable to conclude that two tracks were made by a single individual. Back to Basics
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