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CIRCULAR STATISTICS

CIRCULAR STATISTICS. Circular statistics are used to quantify the time of occurrence of hydrologic variables on a circle—typically on a yearly basis. Successive samples of circular statistic results The math :( Really comprehensive analysis. Circular Statistics — see BOX 4-3.

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CIRCULAR STATISTICS

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  1. CIRCULAR STATISTICS Circular statistics are used to quantify the time of occurrence of hydrologic variables on a circle—typically on a yearly basis. • Successive samples of circular statistic results • The math :( • Really comprehensive analysis

  2. Circular Statistics—see BOX 4-3 Circular statistics are used to quantify the time of occurrence of hydrologic variables on a circle—typically on a yearly basis. Two values require calculation: Average Time of Occurrence (Angle of the Mean) - analogous to the arithmetic mean Index of Seasonality - analogous to the standard deviation The average hydrologic quantity (say a monthly value) is considered to be a vector quantity. Length is proportional to the amount and direction (angle) of the time of the value.

  3. Circular Statistics • Average Time of Occurrence (Angle of the Mean) • Time through the year (or other interval) is represented on a circle with (usually) each month assigned an angle. Think of the sin/cos terms as weight factors. • Resultant Angle Prime: fR’ = atan(S/C) • Resultant Angle (deal with quadrant):fR = fR’ if(S > 0 and C > 0)fR = fR’+180 if(C < 0)fR = fR’+360 if(S < 0 and C > 0) But other conversions are sometimes needed depending upon the output of the atan function.

  4. Circular Statistics • Resultant Angle (deal with quadrant):$PHI = ( ($Sterm > 0 and $Cterm > 0) or • ($Sterm > 0 and $Cterm < 0) ) ? $PHIp : $PHIp+360;fR = fR’ fR = fR’+360 if[(S > 0 and C > 0) or (S < 0 and C < 0)] • 2. Index of Seasonality (IS) PR = sqrt(S2 + C2) IS = PR / (Total of Xm Values) In the Perl language

  5. Circular Statistics List of examples of hydrologic variables on which circular statistics would be useful: Example: Total Rainfall = 36 inches-------------------------------------------------Season Rainfall sin cos-------------------------------------------------Spring (Mar.31;DoY=90) 4.00 0.9998 0.0215Summer(Jun.30;DoY=181) 16.00 .0258 -.9997Fall (Sept.30;DoY=273) 11.00 -.9999 -.0129Winter(Dec.31;DoY=365) 5.00 .0000 1.0000-------------------------------------------------S = -6.587; C = -11.05; f’=atan(S/C)=> 30.8 degreesf = 30.8 + 180 = 211 degreesPR = 12.87; IS = 12.87/36 = 0.357

  6. Circular Statistics for 08155500 Barton Springs at Austin, Texas • 1978 to 2003 • Vector lengths are short • No definitive angle • Are these observations consistent with your expectation?

  7. Circular Statistics for 08158000 Colorado River at Austin, Texas • 1899 to 2003 • Vector lengths are moderately long. • Concentration of angle near end of September to (through?) November. • Are these observations consistent with your expectation?

  8. Circular Statistics for 08169000 Comal River at NewBraunfels, Texas • 1933 to 2002 • Vector lengths are short • No definitive angle--but perhaps more in January through March?

  9. Circular Statistics for 08169000 Comal River at NewBraunfels, Texas

  10. Circular Statistics for 08169000 Comal River at NewBraunfels, Texas

  11. ExtensiveCircularStatistics

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