Positive Semi-Definite Correlation Matrices: Recursive Algorithmic Generation. and Volume Measure
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1 Pue Matheatcal Scece Vol. 0 o Potve Se-Defte Coelato Matce: Recuve Algothc Geeato ad Volue Meaue Wee Hüla FRSGlobal Stelad Seefeldtae 69 CH-8008 Züch Stelad ee.huela@fglobal.co hula@blue.ch Abtact aed o a oto of d-caled deteat caled patal coelato ad to deteatal dette fo coelato atce e deve eplct ecuve cloed fo coelato boud fo potve e-defte coelato atce. A a applcato e obta a e ple poduct foula fo the volue eaue of the pace of potve-e defte coelato atce. Matheatc Subect Clafcato: 5A H0 65F0 Keod: coelato at patal coelato potve e-defte popet deteatal dett ecuve algoth. Itoducto The algothc geeato of vald coelato atce ha bee up to qute ecetl a challegg poble. A ft cotuctve ecuve optato algoth fo th decbed udde et al. []. A potat poveet [] Coolla 4. ho obta ecuve teatve eplct cloed fo boud fo coelato coeffcet. Sce the do ot eque a po a optato pocedue thee boud offe a oe taghtfoad pleetato tha the etoed optato algoth. Though geealato of the dea epeed [] Eaple. t
2 8 W. Hüla poble to get d of the pevou teatve ecuo ad htheto obta ve eplct ecuvel defed geec cloed fo coelato boud ou a Theoe.. The devato of th eult baed o to deteatal dette fo coelato atce hch ae peeted ad poved Secto. The ft oe a ecuve elato fo a d-caled deteat caled patal coelato hch obtaed b tafog the patal coelato appopatel. The ecod dett a ecuve elato fo deteat of coelato atce te of the quaed d-caled patal coelato. A a applcato e obta a e ple poduct foula fo the volue eaue of the pace of potve-e defte coelato atce. We ote that Joe [] ha obtaed pevoul a equvalet volue foula a pobabltc a a a b-poduct of oe e ethod to geeate ado coelato atce. The e poof ue ol ft-ea calculu ad o-pobabltc atue.. Deteatal dette fo coelato atce Fo fed a coelato at R.e. a etc at th ut dagoal ete uquel deteed b the uppe dagoal eleet <... ad a de et. Fo each { }... th... e code the ubcoelato at R hch uquel deteed b <. It coveet to code o pecal quatte. Defto. Deteat patal coelato ad d-caled patal coelato The deteat of the at R deoted b... det R.. Fo ad a de et the patal coelato of th epect to... ecuvel defed ad deoted b hee fo the quatte ued o the ght had de of. ae b coveto the coelato. The tafoed patal coelato defed ad deoted b
3 Potve e-defte coelato atce called d-caled deteat caled patal coelato. coveto e et. Rea. Though ot elevat the peet cotet e ote the follog tochatc tepetato. Fo a et of ado vaable X X... X th fte ea ad vaace the quatt... othg ele tha the patal lea coelato of X X th epect to X... X hch ca be tepeted a the coelato betee the poecto of X X o the plae othogoal to the pace paed b X... X. Th quatt atfe the defg ecuve foula. ee e.g. Yule ad Kedall [6]. Recall that the deteat of a coelato at atfe the poduct epeetato e.g. [] foula.0 o Kuoca ad Cooe [4] Theoe 4.5 fo a oe geeal patal coelato ve pecfcato hee a ept poduct equal oe b coveto. Ou ft goal the devato of oe ueful fudaetal ecuve dette that l deteat to d-caled patal coelato. Wthout lo of geealt t uffce to develop foula fo the caocal de et... obtaed b ettg.... Geeal foula ae obtaed b ug abta de et ad tag to accout the peutato popete of the volved quatte. I the pecal cae e have the dett..5 Popoto. Recuve elato fo d-caled patal coelato Aue that 4. The e have the dett th the coveto cae 4. Poof Th ho b ducto. Fo 4 e have b the ecuo. that
4 40 W. Hüla th Ug thee elato ad the defg equato. e have o fo the veo of.4 fo th de et 4 e ee that 4 4. Ieted to the pecedg elato ho the deed eult fo 4. It ea to ho that f the elato.6 hold fo the deo the t hold fo the deo. Poceedg lal e obeve that th Ietg thee elato to. e obta that equal to o Popoto. belo ea tue he eplacg the caocal de et b a othe de et. I patcula.8 vald fo the de et...
5 Potve e-defte coelato atce 4 th { }... hch eld tag to accout. ad the vaace of deteat ude peutato of dce Ieted to the pecedg elato ho.6 fo the deo. Though ucceve teato of.6 e obta a eplct fte epeetato. Coolla. See epao fo d-caled patal coelato Aue that 4. The e have the fte ee epeetato... / Poof. Th left a eece to the teeted eade. Whle.6 allo ecuve evaluato of the d-caled patal coelato the et dett pefo the ae fo deteat. Popoto. Recuve elato fo deteat of coelato atce Aue that. The e have the dett th the coveto cae. Poof The deo thee veo of the poduct epeetato.4 eld. Ug. ho.8 fo. o e have to ho that f.8 hold fo the deo the t hold fo the deo. ducto aupto ad th appopate choce of the de et e have fo the elato Wth th the ght had de of.8 fo the deo ca be ette a 4...
6 4 W. Hüla O the othe had fo the veo of.4 th abta de et e obta th the coveto that a ept poduct equal oe. Ieted to the pecedg elato ad ug aga the epeetato.4 e ee that th detfe th hch ho the dett.8 fo the deo..... Algothc ecuve geeato of coelato atce We ae ead fo ou a eult o potve e-defte coelato atce. Theoe. Recuve geeato of vald coelato atce A coelato at paaeteed b < potve e-defte f ad ol f the follog codto ae fulflled: [ ].... [ ] [ ]
7 Potve e-defte coelato atce 4 [ ] / Poof A a tatg pot e ecall Coolla 4. []. The foulato deped upo the ue of the ultvaate teated fucto defed ecuvel though [ ] [ ] The the coelato at potve e-defte f ad ol f the follog codto ae fulflled: [ ]....6 [ ]....7 [ ] [ ] It uffce to ho that the boud.7-.9 fo cocde th the eplct boud tated.-.4. Ft t clea that the boud
8 44 W. Hüla.7 cocde th thoe. b defto of the fucto ad the Defto. of the atheatcal bol. o let The epao of.8-.9 a fte ee eld the boud Cleal the ft te.0 cocde th the coepodg ft te.-.4. The detfcato of the othe te ho b ducto o. Fo e have b defto. the patal coelato foula.... hch ple that.... Slal th chaged de et the elatohp.8 eld.... hch ple that.....4
9 Potve e-defte coelato atce 45 Togethe th ho the foula. fo abta 4. o let... 5 ad aue that.0 th eplaced b cocde th the coepodg te.4 fo abta 5. We ut ho that.0 cocde th.4. We obeve that the uad fo... the u of.0 cocde b ducto aupto th the coepodg te.-.4 of the boud hch obvoul cocde th the coepodg te of the boud.... Theefoe t ea to ho that the uad th the u of.0 ad the lat te.0 cocde th the coepodg te.4 fo abta 5. Fo the uad the u of.0 e have b defto. that Ug the ducto aupto o the lat te of the ee epao fo e ee that Ug.5-.6 t follo that the uad the u of.0 equal ad cocde th the coepodg te.4. O the othe had th chaged de et the elatohp.8 eld
10 46 W. Hüla Ug th ad the elatohp.6 above e ee that the lat te.0 cocde th the lat te.4. The eult ho. 4. Volue eaue fo the cove et of vald coelato atce Code the et of potve e-defte coelato atce paaeteed b the et S { < } hch o to be a cove ubet of the -deoal cube [ ] th. The eplct ecuve geeato of the coelato boud ued to calculate the volue V of th et. The devato deped upo the deteatal dette toduced Secto. Theoe 4. Volue eaue fo the pace of vald coelato atce The volue of the pace gve b S V Vol S J J t dt. 4. π Poof Sce V J V J J e aue that 4. A cloe loo at the ecuve boud of Theoe. ho that the volue obtaed b tegatg bacad uccevel ove the pacage of vaable... th ode. Ft of all e ote that... d A et tep the tegal of 4. th epect to the vaable ol deped upo the quae oot epeo ad epaate. Theefoe e have to evaluate the tegal defed b I :... d. 4.
11 Potve e-defte coelato atce 47 Wth appopate choce of the de et the dett.8 eld the epeo Moeove the ue of the d-caled patal coelato epao.7 eld the follog epeo fo the quae bacet quatt 4.4 th chaged de et hee fo 4 the u obvoul ept ad et equal to eo. Ug.5 ad the coveto e ote that Settg.4 e ee though copao th 4.5 that Sce the lat epeo 4.6 doe ot deped o the tegatg vaable ad 4.5 deped o t ol though t ft te 4 t atual to ae the chage of vaable 4... / 4... d d. 4.7 Ug 4.4 the tegal 4. tafo to c I : 4... c 4... c c d 4... / Settg c t oe get I c J. Ug th ad 4. e ee that the e tegad ove the eag pace of tegato I / 4... J J 4... / 4... I. 4.9
12 48 W. Hüla We poceed o the ae a. I geeal the calculato of the foula 4. fo the deo 4 eque tep ubeed.... We cla that the ft tep e have to calculate tegal defed b I :... d ad that the eult of th tegato lead the et tep to the e tegad J... / hch fo the lat tep educe to the tegad J / 4. becaue b coveto. A fal tegato eld the deed foula a V J J t d dt J. 4. The eag aeto poved b ducto hee the ducto tep fo ha bee ho above. Shog that the aeto hold fo he t hold fo pefoed lal ad left to the eade. Rea 4. uecal value fo V V4 had bee etoed [5]. Moe ecetl Joe [] deve a volue foula a pobabltc a a a b-poduct of oe e ethod to geeate ado coelato atce. The e poof eque ol eleeta ft-ea calculu ad o-pobabltc atue. It ot dffcult to ecove Joe foula fo the follog ecuo ue patal tegato π J J J J J J
13 Potve e-defte coelato atce 49 Refeece [] M. udde P. Hadava ad L. Hoffa O the geeato of coelato atce Appled Matheatc E-ote [] W. Hüla Copatblt codto fo the ultvaate oal copula th gve a coelato at Poee Joual of Theoetcal ad Appled Stattc 0 to appea. [] H. Joe Geeatg ado coelato atce baed o patal coelato Joual of Multvaate Aal [4] D. Kuoca ad M.R. Cooe Ucetat aal th hgh deoal depedece odellg J. Wle Chchete 006. [5] P.J. Roueu ad G. Molebegh The hape of coelato atce The Aeca Stattca [6] G. Yule ad M. Kedall A Itoducto to the Theo of Stattc 4 th ed. Chale Gff & Copa elot Calfoa 965. Receved: Mach 0
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