On Selecting Models for Nonlinear Time Series

Co~utn~cting models frarn time scrics wirh nontrivial dynamics involvcs tlic problen~ of Ilow to clioosc lilt bcsr rncvlcl fi.o~n wilhin n class QF ~nodels, or to C ~ O O S C belwcc~i colilpcling classes. This papcr discirsscs a ~nelhorl of builcli~ig rlonlinciir ~nodcls of possibly chnolic sysrcms from data, rvllilc ~naintainiog good robus~rlcss ngainst noisc. 'Thc tnodels rhnt arc buill nrc clasc to thc sirnplcu possible nccoriling lo a dcscriplion Ie~lgth crilerion. ' I ' t~c lnctllod will clclivcr a lincnr LnodcI i f tl la~ liar sIio17er description lcngtli [ban a nonli~lear motlcl. Wc show how our modcls ciut bc uscd fclr prcdiclioo, smooltling and intcrpolnrion in ~ l l e usual way. Wc also show how to apply OIC rcsulls to irlc.ntilication of chiles by detccting ~ h c presence o l homoclinic orb~ts direc~ly from time series. 1. 'l'hc !tiode1 sclcctiorl problcol As o u ~ undcrstanctiag OF chaotic and o r l ~ e ~ nonlincar phenotncnn has grown. i t has bccolnc ~ p l ~ a r e n t lhnt linear modcls arc inadocluatc to modcl most dy nn~nical p~uccsses. Ncvcrtl-lcless, lincar modcls rcrn,~in nllrncrivc becausc of thc great powcr derived f ~ o m the clegancc and co~nplc~cncss of' rhc lheory. Thc art of consirucling ~ ~ o n l i ~ l c a r motlcls is by comparison i t ) its ~tlfancy, anti is unlikcly in lhc near rutuic to dcvelop anythirig like the compictcness that l incar r~ lodel i r~g posscsscs. I-Iowcvcr, hc rc arc steps i n the process of buildill:: a lincar mocicl that we would bc w tsc to m u [bat ttlcrc is always mor-c tl1i111 one possiblc lnotlcl -~rldccd thcrc arc infinitely many so orle must somehow dccidc wkicl~ to use beforc the fitting cvct~ starls. Onc con~rnonly-uscd cri(cr1011 for choosing thc bcst rnotlcl is that i t shoultl capturc Lhe csscntial d y r ~ a ~ l ~ i c s o l (he time scrics withot1 t "ovcr-fi tiing", ~v l i i ch l,es~lis i n including in rllc motlcl aspects o f ltlc li lnc scrics thai should bc ;ttlributed to noise. We call this higher. lcvel of t l~c 1110dcl bui lrling process, thc t ~ ~ o d ~ i se1t.uriot1 problet!~. 'I'hcrc has bccn a grcat cjeal o f rcccnt work on aigo~irhms to construct nooliacnr modcls, hut rnucll of this work tlns ignored the i~nportant sclectiorl