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Parallel simulations in FPT problems for Gaussian processes
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for Gaussian pro esses
E.Di Nardo
Dip. diMatemati a,Univ. dellaBasili ata,ContradaMa hiaRomana,Potenza, e-mail: dinardounibas.it
A.G.Nobile
Dip. di Matemati a e Informati a, Univ. di Salerno, Via S. Allende, Salerno, e-mail: nobileunisa.it
E.Pirozzi
Dip. di Informati a,Matemati a, Elettroni a eTrasporti,Univ. di Reggio Ca-labria,ViaGraziella,Reggio Calabria,e-mail: pirozziing.unir .it
L.M.Ri iardi
Dip. di Matemati a e Appli azioni, Univ. di Napoli \Federi o II",Via Cintia, Napoli,e-mail: Luigi.Ri iardiunina.it
Abstra t
Themainresultsofourresear hrelatedto rstpassagetime(FPT)problemsfor stationaryGaussian pro essesaresyntheti allyoutlined. Theve torizedandparallel algorithm, eÆ iently implemented on CRAY-T3E in FORTRAN90-MPI, allows to simulatealargenumberofsamplepathsofGaussiansto hasti pro essesinorderto obtainreliableestimatesofprobabilitydensityfun tions(pdf) of rstpassage times throughpre-assigned boundaries. The lass of Gaussian pro esses hara terized by damped os illatory ovarian e fun tions and by Butterworth-type ovarian eshave been extensively analyzed in the presen e of onstant and/or periodi boundaries. Theanalysis basedon our simulationpro edure has been parti ularly pro table as ithasproved toprovidean eÆ ientresear h tool in all asesof interestto us when losed-formresultsoranalyti evaluationswerenotavailable. Lastbut notleast,in some asesithas allowedus to onje ture ertain generalfeatures ofFPT densities thatsu essivelyhavebeenrigorouslyproved.
Siillustranosinteti amenteglisviluppidellanostrari er ainmeritoalproblema del tempodi primo passaggio per pro essi gaussiani stazionari. L'algoritmo vetto-riale e parallelo, eÆ entemente implementatosul CRAY-T3E in FORTRAN90 e in ambiente MPI, ha resopossibilesimulareunelevatonumerodi realizzazionidi pro- essi sto asti i gaussiani no a poter disporre di aÆdabili stime delle densita del tempodi primopassaggio. Le lassidi pro essigaussiani onfunzione di ovarianza ados illazionismorzatee on funzione di ovarianzadi tipoButterworthsono state estensivamente analizzatein presenzadi barriere ostanti e/operiodi he. Lo studio basatosulle simulazionisi e os rivelatoparti olarmente utile an he per heha o-stituito uno strumento di indagineeÆ iente in tuttiquei asi nei qualisoluzioni in forma hiusasono risultate ina essibiliogli strumenti analiti i inadeguati; inoltre, talorahalas iato intuirenuove ongetturela ui validitae poi statarigorosamente dimostrata.
The rstpassagetime(FPT)probleminvolvesthedeterminationofthe prob-abilitydistributionfun tionoftherandomvariableT representingtheinstant when for the rst time a dynami system, modeled by a sto hasti pro ess, enters a pre-assigned riti al region of the state spa e. Numerous examples stressingtherelevan eofFPTproblemsareo eredbyvarious elds: in theo-reti alneurobiologytheneuronal ringmaysometimesbeviewedasaFPTof thepotentialdi eren ea rosstheneuronalmembranethroughsomethreshold value;inastrophysi sone an thinkofthetimene essary forastar toes ape fromagalaxy;otherexamplesareo eredby eldsasdiversi edase onomyor psy hologyand,of ourse,me hani alengineeringinwhi h rst passagetime problems bear relevan e in the ontext of stability and integrity of systems subje tto randomvibrations( f,forinstan e, [6 ℄).
In the neurobiologi al ontext a lassi al approa h to model FPT prob-lems onsistsofinvokingdi usionpro essesasresponsibleforthe u tuations of the sto hasti system under the assumption of numerous simultaneously andindependentlya tinginputpro esses (see[8℄ and referen estherein). All these modelsrest on thestrongMarkov assumption,whi h impliesthe possi-bilityof making useof various analyti methodsfor theFPTpdf evaluation. However, it is on eivable that, parti ularly if the des ribed system is sub-je ttostrongly orrelatedinputs,theMarkovassumptionisinappropriate,so thatmodelsbasedonnon-Markovsto hasti pro essesoughttobe onsidered. ByanalogywithGernstein-Mandelbrotand Ornstein-Uhlenbe kmodels([8 ℄), one an thus hallengetheuseof orrelatedGaussianpro esses. However, the diÆ ulty stems out of the la k of e e tive analyti al methods for obtaining manageable losed-formexpressions fortheFPTprobabilitydensity. Indeed, let fX(t);t0g be a one-dimensionalnon-singularstationary Gaussian pro- esswithmeanE[X(t)℄=0and ovarian eE[X(t)X( )℄= (t )= ( t) su h that (0)=1; _(0)=0 and (0) <0. Furthermore, let S(t)2C
1 [0;1) be the threshold, with S(0) > x
0
. The FPT random variable is de ned as follows: T =inf t0 ft:X(t)>S(t)g withX(0)=x 0 ;
and the FPT probability density fun tion
P(Tt) t g[S(t);tjx 0 ℄ of X(t) throughS(t),is givenby([9 ℄) g[S(t);tjx 0 ℄=W 1 (tjx 0 )+ 1 X i=1 ( 1) i Z t 0 dt 1 Z t t 1 dt 2 Z t t i 1 dt i W i+1 (t 1 ;:::;t i ;tjx 0 ); (1) with W n (t 1 ;:::;t n jx 0 )= (2) Z 1 _ S(t1) dz 1 Z 1 _ S(tn) dz n n Y i=1 [z i _ S(t i )℄p 2n [S(t 1 );:::;S(t n );z 1 ;:::;z n jx 0 ℄;
2n 1 n 1 n 0 1 n 1 _ X(t 1 );:::,Z(t n )= _ X(t n ) onditional uponX(0)=x 0 .
The omputational omplexityoftheaboveequationsindi atesthat alter-native pro edures shouldbe usedin order to obtaininformationon the FPT distributionfun tions.
A simulation pro edure,indetailsdes ribed in[3 ℄,hasbeenimplemented in order to dis lose the essential features of the FPT densities for a lass of Gaussian pro esses and spe i ed boundaries. Our approa h relies on a simulation pro edure by whi h sample paths of the sto hasti pro ess are onstru ted and their rst rossinginstantsthrough assigned boundariesare re orded. The underlyingidea anbeappliedtoanyGaussianpro esshaving spe tral densitiesof a rationaltype. Sin e thesample paths ofthe simulated pro essaregeneratedindependentlyofone another,thesimulationpro edure isparti ularlysuitedforimplementationon ve tor and super omputers.
1.1 Butterworth ovarian e fun tion
Extensive omputations have been performed by us to explore the possible di erent shapes of theFPT densities asindu ed by theos illatory behaviors of ovarian es and thresholds. In[3 ℄ we refer to thefX(t);t0g stationary, zero-meannormal pro ess withtheos illatory ovarian e
(t):=E[X(t+ )X( )℄ p 2e t sin( t+=4 ) t0; 2R + ; (3)
known as Butterworth-type ovarian e, whi h is the simplesttype of ovari-an e arryinga on reteengineeringsigni an e([10 ℄). We onsiderseparately the ase ofthevarying boundaryof theform
S(t)=S 0
+Asin(2t=Q ); (initiallyin reasing), (4)
and thatof thevaryingboundaryofthe form
S(t)=S 0
+A os(2t=Q ); (initiallyde reasing),
whereS 0
;A and Q arepositive onstants. Su h investigations have led usto formulate onje turesonthein uen eoftheratio
Q P
;whereQisthethreshold period and P is the ovarian e period, on the behaviour of g[S(t);tjx
0 ℄. In addition,wegave numeri alevaluationof W
1 (tjx
0
),the rsttermintheright hand of (1), whi h we used to validate the reliabilityof estimates obtained bysimulationsin orresponden e of smallvaluesof timet.
1.2 Damped os illatory ovarian efun tionanddouble
bound-ary ase
Motivatedbytheirrelevan efornumerousappli ations,wehaveimplemented the simulation pro edure forstationary normal pro esses f X(t);t0g har-a terizedbyamoregeneraldampedos illatory ovarian efun tiondepending ontwoparameters ([5 ℄)
0 throughpairsofsmoothboundaries,a onstant and aperiodi one:
8 < : 1 (t) = A; 2 (t) = A+sin 2t Q ; t0; A;Q2R + : (6)
Theresultsofsomesimulationsareshownand on lusionsaredrawnonthe e e tsoftheperiodi omponentsof ovarian esandboundariesonqualitative and quantitativefeatures of thefollowing rst passagetimedensities
g + (tjx 0 ):= t P inf t0 [t:X(t)> 2 (t);X( ) > 1 (t);8 2(0;t)℄ ; (7)
denotingtheFPTdensitythroughtheupperboundary,
g (tjx 0 ):= t P inf t0 [t:X(t)< 1 (t);X( ) < 2 (t);8 2(0;t)℄ ; (8)
denotingtheFPTdensitythroughthelowerboundary,g(tjx 0 )=g + ( tjx 0 ) + g (tjx 0
), denoting the FPT density of X(t) througheither boundariesand, -nally,theg
s (tjx
0
)providingtheFPTdensityofX(t)throughonlyone bound-aryof that in(6)(see Fig.1).
2 The up rossing FPT problem
The up rossing rst passage time problem through varying boundaries, in whi h the X(0) does not possess a delta-type probability density fun tion, is onsidered. At rst, fo using the attention on the problem of single neu-ron'sa tivitymodeling,we have onsideredsomeearlier ontributionsbyA.I. Kostyukov et al. ([7 ℄) in whi h a non-Markov pro ess of a Gaussian type is assumedto des ribethe time ourseoftheneuralmembrane potential. After re-formulating the problemin a rigorousframework, de ningthe up rossing FPT density g
u
[S(t);t℄, and pinpointing the limits of validity of the model we ompared in[4 ℄ Kostyukov'sresults on the ringprobabilitydensity with thoseobtainedbyusbymeansofanadho numeri alalgorithmimplemented for the leaky integrator di usion ring model with a variety of data on-stru tedbythesimulationpro edure ofnon-Markov Gaussianpro esses with pre-assigned ovarian es. AsshowninFig.2, fordi erent valuesof orrelation time :=
R 1 0
j ( )jd ;the parallelpro edure allowed us to simulate the be-havior of a lassof Gaussian pro esses with ovarian e (5) inthe presen eof de reasingboundaryS(t)=5 t: See[4℄ fordetails.
Along the lines indi ated in [9 ℄, we have provided a series expansion of theup rossingprobabilitydensityfun tionof the rst passage time. Spe i - ally,along the paradigmof a more extensive analysis of Gauss-Markov pro- esses ([1 ℄), we introdu ed, in orrisponden e of a xed real number " > 0,
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Figure 1: ForP (2= ) =2; = 0:1;A = 1and Q = 5densities g s (tjx 0 ) and g + (tjx 0
) are plotted respe tively in (a) and in (d). For P = 2; = 0:1;A = 1 andQ=7densitiesg s (tjx 0 ) andg + (tjx 0
) areplotted respe tivelyin (b)andin (e). For P = 3; = 0:1;A = 2 and Q = 4 densities g s (tjx 0 ) and g + (tjx 0 ) are plotted respe tivelyin ( )andin(f). the " up rossing FPT pdf g (") u
[S(t);t℄ related to the onditioned FPT pdf g[S(t);tjx 0 ℄ asfollows: g (") u [S(t);t℄= Z S(0) " 1 g[S(t);tjx 0 ℄ " (x 0 )dx 0 ; (t0);
underthe hypothesis thata subsetof sample paths of X(t) areoriginated at astate X
0
thatis ar.v. withpre-assignedpdf
" (x 0 ) 8 > < > : f 1 (x 0 ) " Z S(0) " 1 f 1 (z)dz # 1 ; x 0 <S(0) " 0; x 0 S(0) "; (9) and wheref 1 (x 0 ) denotesthepdfof X(0): 1 x 2 0 !
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Figure 2: Plotofg u[S(t);t℄forthevariousvaluesof.
Byusingtheparallelpro edure,suitablymodi ed([2 ℄)forup rossingFPT problem, estimates of the up rossingprobability density fun tionof the rst passagetime ~g
(") u
[S(t);t℄ have been onstru ted. In the ontext of up rossing
FPTproblemwe also worked outanumeri alpro edureto evaluate W (u) 1
(t); thatplaystheanalogous role ofW
1 (tjx
0
) inthe onditional FPTproblem. A omparison isthen provided withthe numeri al results obtainedby approxi-matingthesimulatedup rossingprobabilitydensitybythe rstorder partial sumof theseriesexpansion(see Fig.3).
3 Asymptoti results
Finally, by making use of a Ri e-like series expansion, we have investigated theasymptoti behaviorofFPTpdfthrough ertaintime-varyingboundaries, in ludingperiodi boundaries. In[2 ℄ wegave suÆ ient onditionsforthe ase of a single asymptoti ally onstant boundary, under whi h the FPT densi-ties asymptoti allyexhibit exponential behaviors. The parallel pro edure to simulate the sample paths allows to evaluate the order of magnitude of the parameters hara terizingthe exponentialbehaviorof FPTpdf's(see Fig.4).
A knowledgements
This work has beenperformed in part within a joint ooperation agreement between Japan S ien e and Te hnology Corporation (JST) and Universita di Napoli \Federi o II", under partial support of National Resear h Coun- il (CNR) and of Ministry of University and of S ienti and Te hnologi al
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Figure 3: Plot of W (u) 1 (t) and of g~ (") u[S(t);t℄ as fun tion of t (top to bottom) for a stationary Gaussian pro ess with zero mean and ovarian e fun tion (3) for the boundary (4)with thefollowingparametervalues: (a) S
0 =0:5, A=0, =1and "=0:1; (b)S 0 =1,A =0, =1and "=0:1; ( ) S 0 =1,A =1, Q=3, =1, "=0:1;(d)S 0 =1,A=1,Q=1, =1,"=0:1.
Resear h(MURST). AgrantbyCINECA hasmade possiblesomeof the ne -essary omputations.
Referen es
[1℄ E. Di Nardo, A.G. Nobile, E. Pirozzi and L.M. Ri iardi, A ompu-tational approa h to rst-passage-time problem for Gauss-Markov pro- esses. Preprint no.5, UniversitadegliStudidellaBasili ata(1998)
[2℄ E. Di Nardo, A.G. Nobile, E. Pirozzi and L.M. Ri iardi, Evaluation of up rossing rst passage time densities for Gaussian pro esses via a simulationpro edure.AttidellaConferenzaAnnualedellaItalianSo iety forComputer Simulation.pp.95-102 (1999)
[3℄ E. Di Nardo, A.G. Nobile, E. Pirozziand L.M. Ri iardi, Simulation of Gaussian pro esses and rst passage time densities evaluation. Le ture Notes inComputer S ien e. vol. 1798,pp.319-333 (2000)
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Figure4: Plotof~g (") u(S;t)with"=0:1andexponentialpdfg(t)=e t
whereis spe i edin[2℄,relatedtoaButterworth ovarian efun tionwith =1and onstant boundariesS=1in (a),S=1:5in(b)andS=2in( ).
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[7℄ A. I. Kostyukov, Yu.N. Ivanov and M.V. Kryzhanovsky, Probability of Neuronal Spike Initiation as a Curve-Crossing Problem for Gaussian Sto hasti Pro esses.Biologi alCyberneti s. 39, 157-163 (1981)
[8℄ L.M. Ri iardi,Di usionmodelsof neurona tivity. InTheHandbookof Brain Theory and Neural Networks (M.A. Arbib, ed.). The MIT Press, Cambridge,299-304 (1995)
[9℄ L.M. Ri iardi and S. Sato, Onthe evaluation of rst passagetime den-sitiesforGaussian pro esses.SignalPro essing11, 339{357 (1986)
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