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Directional Statistics Data in degrees or coordinates.

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Presentation on theme: "Directional Statistics Data in degrees or coordinates."— Presentation transcript:

1 Directional Statistics Data in degrees or coordinates

2 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

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

4 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

5 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

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

7 Allen’s Creek Cemetery Mean = 126.1° Circular Mean = 43.6°

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

9 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

10 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

11 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

12 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)

13 > 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 = Rotation = clock [1] > mean(ErnestWitte$Direction, na.rm=TRUE) [1]

14 > mean(look, na.rm=TRUE) Circular Data: Type = angles Units = degrees Template = geographics Modulo = asis Zero = Rotation = clock [1] > mean(ErnestWitte$Look, na.rm=TRUE) [1] > 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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17 Rose Plot with Circular Plot and Kernel Density Plot

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


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