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Directional Statistics Data in degrees or coordinates

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Directional Statistics North, Northeast, 30°, 215°, etc Standard descriptive statistics are misleading Anything measured in degrees Any cyclic data: daily, weekly, monthly, or yearly Mathematical vs Geographic standards

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Orientation North/South, East/West Orientation ignores directionality (e.g. “Texas Avenue runs both ways.”) Mostly roads, paths, and other transportation links

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Testing for Direction Rayleigh’s Test for directionality Chi-Square Test for even distribution For orientation data, we double the value and use the directionality tests

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Mean Direction Convert to rectangular form: –x = cos , y = sin [may need to convert degrees to radians] Sum coordinates: –C = (cos i ), S = (sin i ) Mean depends on sign of C, S

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Mean Direction (degrees) C=0, S>090° C=0, S<0270° C>0, S≥0arctan(S/C) C<0arctan(S/C) + 180 C>0, S<0arctan(S/C) + 360

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Allen’s Creek Cemetery 13 35 34 10 4 2 2 6 17 16 Mean = 126.1° Circular Mean = 43.6°

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Rayleigh’s Test Null hypothesis of uniformity (or bimodal opposing directions) Based on mean angle and Rayleigh’s spread:

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Chi Square Divide data into groups: 4, 8, etc Calculate simple Chi Square test This ignores circular nature of the data, but provides a rough guide to finding differences

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Package circular 1 Circular object class holds information about the kind of circular data being used: –units – radians, degrees, or hours –zero – location of zero point –rotation – clockwise or counterclockwise templates set both zero and rotation

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Package circular 2 Descriptive statistics adjusted for circular data Plotting circular plots, density plots, rose, and windrose plots Statistical tests for directionality, regression, and comparing samples

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Example Ernest Witte has two directional variables: –Direction (direction the spinal column head is pointing toward) –Looking (direction eyes are looking toward) They are stored as geographic data (degrees clockwise from north)

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> library(circular) > dir <- circular(ErnestWitte$Direction, units="degrees", template="geographics") > look <- circular(ErnestWitte$Looking, units="degrees", template="geographics") > mean(dir, na.rm=TRUE) Circular Data: Type = angles Units = degrees Template = geographics Modulo = asis Zero = 1.570796 Rotation = clock [1] 43.62221 > mean(ErnestWitte$Direction, na.rm=TRUE) [1] 126.0643

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> mean(look, na.rm=TRUE) Circular Data: Type = angles Units = degrees Template = geographics Modulo = asis Zero = 1.570796 Rotation = clock [1] 184.7019 > mean(ErnestWitte$Look, na.rm=TRUE) [1] 173.9577 > plot(dir) > plot(look) > plot(dir, main="Head/Body Direction") > plot(look, main="Direction Looking“) > rose.diag(dir, bins=6, prop=1.1, col="gray") > rose.diag(look, bins=6, prop=1.2, col="gray")

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Rose Plot with Circular Plot and Kernel Density Plot

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> rayleigh.test(dir) Rayleigh Test of Uniformity General Unimodal Alternative Test Statistic: 0.4806 P-value: 0 > rayleigh.test(look) Rayleigh Test of Uniformity General Unimodal Alternative Test Statistic: 0.07 P-value: 0.7062

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