The application of directional methods in p dimensions
Jorge Holguâin · Summit (Simon Fraser University) · 1980
I n r e c e n t years t h e r e have been many advances i n t h e a n a l y s i s of d i r e c t i o n a l d a t a expressed a s two and t h r e e dimensional u n i t vectors.The extension of these methods of a n a l y s i s t o p dimensions is presented i n t h i s t h e s i s .Although d i r e c t i o n s i n higher dimensions'do not have a physical i n t e r p r e t a t i o n , d a t a which can be recorded a s p-dimensional u n i t vectors a r i s e i n many a r e a s ; an example i s when t h e d a t a a r e i n terms of p continuous proportions.The von Mises and Fisher d i s t r i b u t i o n s have been widely used i n t h e a n a l y s i s of d i r e c t i o n a l d a t a i n two and t h r e e dimensions respectively; these a r e described i n Chapter 1.The extension of these d i s t r i b u t i o n s t o higher dimensions i s given i n Chapter 2. T e s t s a r e given f o r hypotheses of i n t e r e s t i n t h e a n a l y s i s of groups of p-dimensional u n i t vectors.I n d i r e c t i o n a l uses, a common technique i s a form of a n a l y s i s of variance introduced by Watson.It i s shown t h a t t h i s technique can be used with p-dimensional u n i t vectors.Further, it can be developed a l s o f o r a two way layout with a natural extension t o a multi-way layout.The a n a l y s i s of variance can a l s o be expressed i n terms of angles between t h e vectors and t h e i r r e s u l t a n t s , and t h i s i s a useful representation with many types of data.I t i s a l s o o f t e n of i n t e r e s t t o examine the c l u s t e r i n g of u n i t vectors; a simple method of c l u s t e r i n g i s given i n Chapter 3 .The r e s u l t s obtained i n the examples a r e compared t o those found by means of standard algorithms.* Chapter 4 i s a chapter of worked d a t a s e t s ; s e v e r a l examples a r e worked through t o demonstrate both t h e p-dimensional ANOVA techniques and t h e c l u s t e r i n g method.A s a r e s u l t o f working with t h e s e techniques, a number of problems have been i d e n t i f i e d .These a r e b r i e f l y discussed i n Chapter 5.