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Directional Data analysis Multivariate Statistics Chapter 15 Mardia, Kent and Bibby

Directional Data analysis Multivariate Statistics Chapter 15 Mardia, Kent and Bibby. Presented by Steven Brown. Directional Data. Considering the direction of an object in 360 degrees as the main variable Cannot take the mean and variance according to normal assumptions

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Directional Data analysis Multivariate Statistics Chapter 15 Mardia, Kent and Bibby

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  1. Directional Data analysisMultivariate StatisticsChapter 15Mardia, Kent and Bibby Presented by Steven Brown

  2. Directional Data • Considering the direction of an object in 360 degrees as the main variable • Cannot take the mean and variance according to normal assumptions • Example-degrees on a compass (10o and 350o ) Both are close to north, but the mean is 180o which is south. • Have to use polar coordinates and the angle that the direction makes according to different axes.

  3. x2 (cos , sin )  x1 Circular direction data

  4. x1 P  0 x3 φ N x2 Spherical direction data

  5. Descriptive measures • I=() • ()=cos(i)  i-1 sin(i) • Mean direction Io=Ī / Ř • Ī = ¹/n Σn Ii • Distance from the center = Ř=(Ī´ Ī)½ i=0 i=1

  6. 1-Ř Ř Ii Io 0o

  7. Uniform Distribution • E(I)=0 •  not defined • Second moment equal to 1/p • P.d.f.=Cp=(½p)/(2p/2)

  8. Von Mises Distribution • Named after Richard Von Mises • His brother was a famous economist Ludwig Von Mises • He developed the axiom of randomness and convergence for the field of probability • He introduced the birthday problem in 1939 • The distribution is a Circular normal distribution

  9. Von Mises Distribution • Von mises and Fisher Distribution p.e.=Cp(k)e kldSp • P=2 in Von Mises distribution • k=concentration parameter • k=0 then uniform distribution • Mean =  • Variance = 1 − I1(κ)2 / I0(κ)2 (circular) • There is a Fisher distribution that has similar distribution with p=3 • There is modified Bessel function with a similar distribution

  10. Von Mises Distribution Wikipedia-Von Mises distribution

  11. Rayleigh Test of uniformity • Ho: k=0 vs Ha k0 • Use Likelihood ratio test log=-n(log(Cp(k)/Cp) + kA(k) • Critical region Ř > K • pn Ř2~2 for large n • Compare to a 2 distribution p p

  12. Orbits of nine planets(Watson 1970)

  13. Results • Direction (sin Ώ sin i - cos Ώ sin i, cos i) • Ř =0.79 • Reject null hypothesis for the Rayleigh test

  14. Watson and William Test for mean direction • Ho: = o vs Ha  o • Use Likelihood ratio test • F p-1,(n-1)(p-1) = (n-1)(R-r1o)/(n-R) • Compared to a F distribution

  15. Pluto’s origin • The ninth planet • An asteroid- largest of Kuiper Belt objects • Comet • A dwarf planet-Pluto and any other round object that "has not cleared the neighborhood around its orbit, and is not a satellite.“ • A solar system body-All other objects orbiting the Sun. Pluto Demoted: No Longer a Planet in Highly Controversial Definition By Robert Roy BrittSenior Science Writer posted: 24 August 2006 • Aug 16, 2006 Pluto demoted to dwarf planet

  16. Solar Nebula theory http://csep10.phys.utk.edu/astr161/lect/solarsys/nebular.html

  17. Solar Nebula theory The collapsing, spinning nebula begins to flatten into a rotating pancake http://csep10.phys.utk.edu/astr161/lect/solarsys/nebular.html

  18. Solar Nebula theory As the nebula collapses further, local regions begin to contract gravitationally on their own because of instabilities in the collapsing, rotating cloud http://csep10.phys.utk.edu/astr161/lect/solarsys/nebular.html

  19. The normals to the orbital of nine planets(Patrangenaru &Mardia 2002)

  20. Parametric Results • Direction (sin Ώ sin i - cos Ώ sin i, cos i) • Test the mean direction with Watson William • Ř =0.9987-highly significant according to Rayleigh • Reject null hypothesis using the Watson William F-test

  21. Non-Parametric Results • Extrinsic mean = Ř / || Ř|| • Assume that the asymptotic distribution is normal • Bootstrap approximation for the distribution T(Q)=T(X1,…, Xn, Q) with the data • Substitute the bootstrap samples of (X1*,…, Xn*) for (X1,…, Xn) • This is a better approximation than standard normal distribution

  22. DDSTAP • Software developed to analyze directional data • Developed by Ashis Sengupta

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