Maximum likelihood method for evaluating correlation dimension

The c o r r e l a t i o n d imens ion [ I ] , [ 2 ] has been used w i d e l y to c h a r a c t e r i z e t i m e s e r i e s or d a t a s t r i n g s . I t is one among seve ra l p o s s i b l e g e o m e t r i c a l i n d i c a t o r s o f the t i m e a s y m p t o t i c p o r t i o n of phase space v i s i t e d by system o r b i t s of a dynamica l sys tem; o the rs i n c l u d e [ 3 ] the p o i n t w i s e d imens ion , c a p a c i t y , i n f o r m a t i o n d i m e n s i o n , and a lso the h i g h e r o rde r c o r r e l a t i o n and Renyi e n t r o p y b a s e d d imens ions [ 4 ] , [ 5 ] . H i s t o r i c a l l y , the c o r r e l a t i o n d imens ion was welcomed as an a l t e r n a t i v e to the c a p a c i t y as a " p h y s i c a l o b s e r v a b l e " s i nce the l a t t e r t y p i c a l l y has i n v o l v e d c o m p u t a t i o n a l e f f o r t which i nc reases e x p o n e n t i a l l y w i t h embedding d imens ion [ 6 ] , w h i l e the fo rmer does not s u f f e r from t h i s d i s e a s e . That c o r r e l a t i o n d imens ion measurement has been w i d e l y accepted by the p h y s i c s community is no doubt a l so due in pa r t to f a m i l i a r i t y of the t h e o r e t i c a l concept f o r p h y s i c i s t s accustomed to ana logous ideas from many-body t h e o r y ; and i n t e r e s t i n g f u r t h e r g e n e r a l i z a t i o n s m o t i v a t e d by these p a r a l l e l s to s t a t i s t i c a l mechanics have a l so been i n t r o d u c e d [ 4 ] , [ 5 ] , [ 7 ] . As a r e s u l t however , the o l d e r n o t i o n of p o i n t w i s e d imens ion , use fu l in dynamica l systems t h e o r y and b e t t e r known among m a t h e m a t i c i a n s , has been l a r g e l y n e g l e c t e d by p h y s i c i s t s even though i t a lso does not i n v o l v e e x p o n e n t i a l comput ing cost i n c r e a s e w i t h embedding d imens ion . But the dream of p a i n l e s s data a n a l y s i s to e x t r a c t d imens ion va lues encoun te rs i m p o r t a n t numer i ca l problems a s s o c i a t e d w i t h the f a c t t h a t the number of p o i n t s in a t i m e s e r i e s i s i n h e r e n t l y f i n i t e . There are some i n d i c a t i o n s t h a t these p r a c t i c a l ma t t e r s may cause less t r o u b l e in ma t t e r s o f accuracy f o r the p o i n t w i s e d imens ion than f o r the c o r r e l a t i o n d imens ion [ 8 ] .