Estimation of dimension and order of time series

Abstract

Since t he beginning of th e eight ies there has been a n expand ing activity in a na-lyzing tim e series in terms of (reconstruct ed) at trac tors and th eir d imensions. For a sur veys and further references, see [ABST ,1993], [C,1991], and [GSS,l 991]. The notion of dim ension , which was based on considera t ions concerning determinis tic tim e series, describes roughly how many par a mete rs one needs to specify th e possible st ate s on the attractor of t he und erlyin g dyn amical system. Ano th er approach, prop osed by Cheng a nd Ton g, is to a nalyze a ti me series prim aril y in terms of th e order, which can be heur istically describ ed as th e number of successive elements of a tim e series which det ermine th e st ate of th e und erlyin g system [CT,l992]; thi s notion was introduced in th e context of non-d eterministic sys tems. th e state shou ld be interp reted as an (abst rac t) notion which su mmcrizes th e infor mation from th e past , as far as it is of influence on th e future: t his not ion of st ate makes sense for both det erminist ic and stoc has t ic syste ms. This same idea of order was also st udied by Savit and Green [SG ,1991]. T heir a pproa ch is closer to th e above mentioned conside rations concer ning deterministic syst ems. T he noti ons of ord er and d imension are different , bu t t here are clea r relat ions. e.g. the order should at least be equal to th e dimension. The main pu rpose of t his pap er is to give a survey of t hese different approa ches and to discuss in this contex t some new numer ical exa mples. Fir st we fix some notation a nd definitions. \\'e consider tim e series X = {X,,};;"=o' with tim e parametrized by t he int egers I I E N. a nd t akin g values in some finite dimensional vector space I V , which, when not exp licit ly stated ot herwise, is assum ed to be R. Time series of finite length , i.e. with 11 running from 0 to some N , will be considered as sa mples of (potc nt ially) infinite tim e series. From a time series one can obtain a new one by th e process of recon st ruct ion: for X as above, th e tim e series R k(X), obtained by an orde r k , or a k-di mens iona l. reconstruction of X, has as its nth element (RdX))1l = (X,"· ··. X,,+k-ll E R k-th ese elements ar e called k-dilll ensiona l reconstruction vectors. So each tim e series equa ls its order one reconstructio n.