From Nuclear Reactions to High Frequency Trading: an R function Approach

Transcription

1 FromNuclearReactionstoHigh Frequency Trading:anR functionapproach FrankWKFirk TheHenryKoernerCenterforEmeritusFaculty, YaleUniversity,NewHaven,CT06520 Abstract. The R function theory of Thomas (1955) a practical version of the general R matrix theory of Wigner and Eisenbud(1947) is used to calculate the fine, intermediate and gross structure observed in the energy dependence of nuclearreactions,andinthetime dependenceofthedowjonesindustrialaverage, akeyeconomicindex.inthesetwodisparatefields,thethreebasicstructuresare characterizedbythevaluesofthefundamental strengthfunction,<γ>/<d>where <Γ>istheaveragewidth(lifetime)oftheunderlyingstates,and<D>istheaverage spacingbetweenadjacentstates.itisproposedthatthevaluesof<γ>/<d>forthe fineandintermediatestructureoftheindex,determinedinthefirsthouroftrading onagivenday,providevaluableinformationconcerningthelikelyperformanceof the index for the remainder of the trading day. The universality of fluctuations observedinresonatingsystemsisdiscussedintermsofthestatisticalpropertiesof complexbivariates. NUCLEARREACTIONS.TheR functiontheoryofthomasisusedtomodelneutron inelasticscatteringtoadefinitestateandtomodelthefine,intermediate,andgross structureobservedinthedowjonesindustrialaverageonatypicaltradingday. 1

2 1.INTRODUCTION Animportantparameterinsystemsthatexhibittime dependent structures(resonancesorfluctuationsofageneralnature),both classicalandquantum,istheratio,averagewidth(lifetime)ofthe states/averagespacingofthestates,denotedby<γ>/<d>.if<γ>/<d> <<1,thestatesareclearlyseparated,andtheparametersthat characterizeindividualstatescanbedeterminedwithanexacttheory (nuclearresonancetheoriesarediscussedinthestandardworkoflynn, 1968).Asthecomplexityofthesystemincreases,aregionisreachedin which<γ>/<d>>1,andnewchallengesforbothobserversandanalysts areencountered.thisisduetothefactthat,atagivenenergyortime, manyoverlappinglocalanddistantstatescontributetotheresponseof thesystem,andallthestatesmustbetreatedcoherently.ifthesystems arestudiedasafunctionofenergy,time,orfrequency,constructiveand destructiveinterferenceeffectsgenerate fluctuations intheobserved spectra.theoriginoffluctuationsdiffersinafundamentalwayfromthe originof collective states.theexacttheoryofstronglyoverlapping statesis,ingeneral,intractable;statisticalargumentsmustthenbeused. Astrikingsimilarityisobservedbetweentheformsofthefine, intermediate,andgrossstructureofphotonuclearstates,studiedasa functionofenergyinmedium massnuclei,andthefine,intermediate, andgrossstructureobservedinthefluctuationsinatypicaleconomic 2

3 indexoveraperiodoftime.examplesareshowninfigure1. Figure1.Thecrosssectionforthereaction 40 Ca(γ,n o) 39 Ca(Wu,Firk,andPhillips,1970)intheregionof thegiantresonance,andtheobserveddjiaina30 minuteinterval,displayedin10 secondintervals. 3

4 In nuclei, the gross, intermediate and fine structures are attributed to the evolution of nuclear reactions from the underlying single particle states to few particle few hole states to manyparticle many hole compound states, respectively (see Bohr and Mottelson, 1975). In the fluctuations of an economic index, the three structural forms are associated with long, medium, and short range time correlationsinthecomplextradingoptionsthattakeplaceamong all elements that contribute to the index. The fine structure is a characteristicfeatureofhigh FrequencyTrading. Themarketstrategyofbuyinglowandsellinghighisareflection of a lack of a sophisticated model of the structure associated with a typicaleconomicindex.itisnecessarytounderstandthemathematical form of the time dependence of the fluctuations; the R function theory describedhereprovidesananalyticalmeansforstudyingtheindividual andtheaveragepropertiesofthestates.thetheoryincludes,explicitly, theall important lifetime ofeachstate, Inthenuclearcase,theoriginoftheintermediatestructureisnot alwaysclear.theobservedformcanresultnotonlyfromfew particlefew holestatesbutalsofromericsonfluctuations(ericson,1960,1963). InthemodeloftheDJIApresentedhere,theoriginoftheintermediate structure is not of primary importance; it is the mathematical method usedtodescribethestructuralformthatmatters. Inaseminalpaper,Thomas(1955),developedtheexacttheoryof resonant quantum states (Wigner and Eisenbud (1947)) to include cases in which <Γ>/<D> > 1. Using arguments that were of practical importanceinthestudyofnuclearresonances,hedevelopedthetheory 4

5 neededtoanalyzeobservedspectra.hismethodhasformedthebasis of a great deal of the subsequent work in the field of nuclear spectroscopy (Vogt (1958), Reich and Moore (1958), Firk, Lynn and Moxon (1963)). Thomas also discussed the forms of nuclear cross sectionswhenaveragedoverlargeenergyintervals. A major development in dealing with energy averaged nuclear cross sections was made by Ericson (1960,1963). He showed that in regions in which <Γ>/<D> > 1, fluctuations naturally occur. Ericson averaged the scattering amplitude given by Feshbach (1962), and Moldauer(1964)refinedthemethodbyusingthestatisticalarguments ofporterandthomas(1956)todealwiththeaveragepropertiesofthe widths(lifetimes)oftheoverlappingresonances. Thomas R function theory is developed to describe the fine, intermediate, and gross structures observed in both inelastic neutron scattering,andthepricesofstocksreportedonatypicaltradingday. 2.ONTHEUNIVERSALITYOFFLUCTUATIONS Universality in physical systems is frequently discussed in terms ofacommonunderlyingmathematicalstructure.wearethereforeled to ask is there a mathematical structure associated with universal fluctuations,and,ifsowhatisit? Acluetoansweringthisquestionis obtainedbynotingthattheamplitudesofresonatingsystemshavethe complexforms f=a+ib. 5

6 (Recallthat,alightlydampedlinearoscillatorwitharesonantfrequency ωr,hasanamplitude,a,whendrivenwithadrivingfrequencyω,given by A=(Γ/2)/[(ω ωr)+i(γ/2)], andanintensity I(ω)=AA*=(Γ/2) 2 /[(ω ωr) 2 +(Γ/2) 2 ],alorentzian, whereγisthewidthoftheresonance). Wethereforetakethefollowingapproach:letasetofcomplexnumbers z=x+iy bestatisticallydistributed,andlettheirmeanvaluesandmeancomplex squaresbeequaltozero.fluctuationsofzaboutthemeanvalueoccur inarandomway.wehave <z>=<x>+i<y>=0 andtherefore <x>=<y>=0. Also, <z 2 > =<x 2 >+2i<xy> <y 2 >=0 andtherefore <x 2 >=<y 2 >=s 2 (say), <xy>=0, and, < z 2 >=<x 2 >+<y 2 > 0. Theprobabilitydensityfunctionofabivariatedistributionis P(x,y)=(1/2πsxsy(1 ρ 2 ))(exp{ G}) where G=(1/2(1 ρ 2 ))[(x µx) 2 /sx 2 2ρ(x µx)(y µy)/sxsy+(y µy) 2 /sy 2, 6

7 ρisthecorrelationcoefficient,µx,µyarethemeans,andsx,syarethe standarddeviationsofxandy,respectively. Ifthevariablesxandyhaveindependentnormaldistributions,andthey have the same variance, s 2, then their bivariate probability density distributionis P(x,y)=[1/(2πs 2 )]exp{ (x 2 +y 2 )/2s 2 }. z 2,givenbythesumofthesquaresoftwouncorrelatedvariableswith thesamevariance,s 2,hasaprobabilitydistributionthatbelongstothe χ 2 familywithtwodegreesoffreedom,anexponentialdistribution: P(w)=[exp{ w/2}]/2 where w= z 2 /< z 2 >. Significantfluctuationsofwaboutthemeanvaluearepredicted. Figure2.Thecalculateddistributionof100independent,randomlychosen,normalbivariates. 7

8 In this set, containing 100 members, the ratio of largest to least value ofwis~10. Theabovestatisticalapproachcanbeappliedtoallsystemsinwhichthe complex values (amplitudes) of z generate resonant like structures, and their real and imaginary parts belong to some probability distribution functions. Ericsonconsideredanautocorrelationfunction,C( n),definedas C( n)=<[z(n+ n) < z>][ z(n) <z>]>/< z> 2 where < z>=(1/n) n=1,nz(n). Thenormalizedvarianceisgivenwhen n=0: C(0)={< z 2 > < z> 2 }/< z> 2. Amplitudecorrelationsinvolvetheautocorrelationfunction,A( n): A( n)={<z(n+ n)z*(n)> <z> 2 }/< z> Ifthecorrelationshavenormaldistributionswithzeromeanthen C( E)= A( E) 2. Ifthescatteringamplitudez >f(e),intheregionofaresonanceatan energyer,withtotalwidthγ,hastheform f(e)~1/[(e Er)+i Γ/2] thentheamplitudeautocorrelationfunctioncanbewritten A( E)=1/[1 i( E/ Γ)]. Theenergyautocorrelationfunctionthenbecomes C( E)= A( E) 2 =Γ 2 /[( E) 2 +Γ 2 ]. ThisisakeyresultinEricson stheory. At sufficiently high excitation energy, the average width <Γ> becomesgreaterthantheaveragespacing<d>betweentheresonances. 8

9 The total width, made up of many partial widths, then becomes essentiallyconstant.atagivenenergy,thecontributionsfromthemany overlappingresonancesmustbetreatedcoherently.theaveragewidth <Γ>isthereforereferredtoasthe coherenceenergy. Inpractice,itisoftendifficulttostudyasetofresonancesofthe same spin and parity, and to know all the possible decay modes; the correlationsarethendampened.also,theeffectsoffiniteresolutionof the spectrometer used to measure the cross section can artificially reduce the correlation effects. If possible contributions to the cross sectionfromnon compoundnucleareffectsareincluded,theproblemof analysisbecomesevenmorechallenging. 3.AMICROSCOPICAPPROACHTOFLUCTUATIONS Thomas theory(1955) provides a method of dealing with many overlappingresonances(<γ>/<d>>1).aproblemofageneralnature concernstheinelasticscatteringofaparticlefromhighlyexcitedstates ofamany bodysystemtoadefinitefinalstate.theinelasticscattering is accompanied by elastic scattering, and by transitions to many alternative channels. Here, a specific problem is considered in which inelastic scattering of a neutron by an even even nucleus to a definite state is considered(see Firk(2010) for a detailed discussion). Elastic scattering is included explicitly, and inelastic scattering and radiative capture,toallotherallowedstates,aretreatedinanaverageway.each state,λ,withaspinandparityj π ischaracterizedbyanenergye λ anda total width Γ λ. An incident neutron interacts with a heavy nucleus to form a state that decays into many different channels. The two main 9

10 channels are inelastic scattering to a definite state (width Γn ), and elastic scattering (width Γn). All the subsidiary channels involve inelasticscattering(widthγn i)andradiativecapture(widthγ γ i). 3.1.TheThomasapproximation TheR matrixhastheenergy dependentform(laneandthomas, 1958) R(E)= λ γ λ c γ λ c /(E λ E) where the sum is over all levels λ of energy E λ, and the γ λ c s are the reduced width amplitudes associated with the channels c, c. If the signsoftheamplitudesaresufficientlyrandomtoensurethatthenondiagonalelementsofraresmallcomparedwiththediagonalelements, ThomasshowedthattheresultingcollisionmatrixUcanbewritten U= λ (a λ c a λ c )/(E λ E iγ λ /2),c c,(an amplitude inthe spirit of the earlier discussion) even for overlapping states. The quantities,a λ c saregivenby a λ c=γ λ c (2Pc), wherepcisthepenetrationfactor,andthewidthis Γ λ c=a λ c 2. Thewidththatoccursinthedenominatoris Γ λ = cγ λ c, inwhich Γ λ c= a λ c/(1 ipc(rc +iπρ<γ λ c> 2 ) 2, R istheeffectofallstatesoutsidetherangeofinterest,ρisthedensity ofstates,and< >denotedtheaveragevalueintheregionofe. Thecrosssectionforthereactionc >c is σc c =(π/kc ) 2 c,c Uc,c 2 10

11 wherekc isthewavenumberoftherelativemotionofthetwoparticles intheincidentchannel.(thespinweightingfactorhasbeenputequal tounity). In the present case that involves two main channels and many subsidiarychannels,thomas(1955)showedthatareducedformofthe R matrixisvalid;itis R(E)= λ (γ λ c γ λ c )/(E λ E iγ λe /2),theR function, where Γ λe is a suitable average of the widths of all the subsidiary, or eliminated channels. Here, the eliminated channels are all channels except theincident channel, and the inelastic scattering channel to the definitestate.thereducedr matrixisvalidifthemeansofthepartial widths for the eliminated channels are less than the spacings, and their reducedwidthamplitudesarerandominsign.itisseenthatthereduced R matrixcanbeobtainedfromthetraditionalr matrixbyevaluatingit atthecomplexenergye=e+iγ λe /2. 3.2Crosssectionforinelasticneutronscatteringtoadefinitestate The cross section for the inelastic scattering of a neutron to a definitestateinthepresenceofelastic,allotherinelasticchannels,and radiativecaptureiscalculatedusingthomas R function: λ (Γ λn/2) 1/2 (Γ λn /2) 1/2 /f λ (E) σ n,n (k n 2/4π)= 2 [1 i λ (Γ λn/2)/f λ (E)][1 i λ (Γ λn /2)/f λ (E)]+[ λ (Γ λn/2) 1/2 (Γ λn /2) 1/2 /f λ (E)] 2 where f λ (E)=E λ E iγ λe /2, and the sums are over all states λ of the same spin and parity. The widthoftheeliminatedchannelsis Γ λe = n Γ λ n + iγ λγ i=γ λ (Γ λ n+γ λ n ), 11

12 whereγ λ isthetotalwidth,γ λ nistheelasticscatteringwidth,γ λ n isthe inelastic scattering width to the definite state, Γ λ n is an inelastic scattering width to an eliminated state, and Γ λγ i is a partial radiation widthtoaneliminatedstate.theneutronwavenumberassociatedwith theincidentchanneliskn.thecrosssectionσnn iscalculatedforupto 1000interferingstatesatupto10000energies. 3.3Onsetoffluctuations Thecalculated inelastic neutron scattering cross section for two valuesofthestrengthfunction,<γ>/<d>=0.4,and7isshown: Figure3.Theinelasticneutronscatteringcrosssectiontoastateat1MeVforthetwovalues<Γ>/<D> =0.4and7.Theunderlyingfine structureresonanceenergiesarethesameinthetwocases. 12

13 ThespacingsarechosenrandomlyfromthesameWignerdistribution, using the same set of random numbers, and the neutron widths for elastic and inelastic scattering are chosen randomly from Porter Thomas distributions. For both values of the strength function, a constant, average value for the sum of the eliminated channels is assumed.theonsetoffluctuationsisclearlyseen. 4.ANR FUNCTIONMODELOFFLUCTUATINGECONOMICINDICES The R function theory of Thomas, described and used above to studytheonsetoffluctuationsinnuclearreactionsisadaptedtomodel thevariationsobservedinthedowjonesindustrialaverageonatypical trading day. Values of the fundamental parameters of the theory that bestdescribethethreebasicformsofthedailydjiaareobtained.the predictivefeaturesofthetheoryarediscussed. It is well known that the fluctuations observed in the timedependentdjiahaveanearest neighborspacingdistributionthatisofa WignerformassociatedwithrandommatricesbelongingtoaGaussian Orthogonal Ensemble(Plerou et al., 2000). Observed departures from the Wigner distribution were interpreted as a demonstration of longterm correlations in the market process. In the present analysis, a WignerspacingdistributionofadjacentstatesintheDJIAassumed.The widthsofthestatesareselectedfromχ 2 distributions,withdegreesof freedomthatdependonthenumberofmodesoftradingassociatedwith a given state. The importance of high resolution, time dependent information in the analysis of an economic index is necessary in determining the basic parameter (average width/average spacing) for 13

14 the fine structure states. A minimum time resolution of 10 seconds is requiredifareliablevalueof<γ>/<d>istobeobtainedfromthedata. Examplesofthethreestructuralmodes,calculatedusinganR function model,areshowninthefollowingfigure. Figure 4. The three basic components of the DJIA calculated using the R function theory of Thomas (1955). The key parameter in each case is the value of <Γ>/<D>. The underlying states are uncorrelated. 14

15 Anexampleofthecombinedfineandintermediatestructureforvalues of <Γ>/D> = 0.4 and 4 is shown in Figure 5. Figure5.Thecalculatedfineandintermediatestructuresarecombinedtogiveaformtypicalofthat observed on a given trading day. Here, the average spacing of the fine structure is chosen to be 4 minutes,andtheaveragewidthis1.6minutes.theunderlyingstatesofthetwostructuresareassumed tobeuncorrelated.therichformsofthefinestructurestates,characteristicofther functionmodel, areclearlyseen. 15

16 Thefinestructureobservedcanhaveanaveragespacingassmall as1minute,andanaveragewidthof30seconds.thevalueofthefine structurestrengthfunctiondependsupontheaverageload,bandwidth, processing capacity, and delays in buy and sell orders (the latency) associatedwithhigh FrequencyTradingonagivenday.Itisassumed that the underlying states of the intermediate structure are not correlated with the states of the fine structure; this is not necessarily thecase. It is proposed that the values of <Γ>/<D> for the fine and intermediatestructureofthestatesofthedjia,onanyday,beobtained byanalyzingthedatainthefirstone totwo hoursoftrading,andthat these values be used as predictors of the values for the remaining trading time on the given day. This procedure assumes that there is sufficientinertiainthefineandintermediatestructureassociatedwith tradesonthegivenday;thereisamplenumericalevidenceinsupportof thisassumption. Sophisticated non linear fitting programs, based on R matrix theory, are available in Nuclear and Atomic Physics, and recently in Astrophysics. Precise resonance parameters have been obtained from measured cross sections, and their values compared with theoretical predictions. 5.CONCLUSIONS The R function theory of Thomas, a practical variant of R matrix theory, was developed to analyze and understand nuclear and atomic reactioncross sections.here,itusedtoaccount,quantitatively,forthe 16

17 threemajorstructuresobservedinatypicaleconomicindex,suchasthe DJIA.Afundamentalparameterinthetheoryisthe strengthfunction, <Γ>/<D>. The method emphasizes not only the spacing between adjacentfluctuationsbutalsothewidths,orlifetimesofthefluctuations. Knowledge of the lifetimes of the three major components of an economic index is critically important in developing any successful investmentstrategy. References BohrAandMottelsonBR1975NuclearStructureVol.II(ReadingMass: Benjamin) EricsonTEO1960Phys.Rev.Lett Ann.Phys.(N.Y.)23390 FeshbachH1962Ann.Phys.(N.Y.)19,287 FirkFWK,LynnJEandMoxonMC1963Proc.Phys.Soc FirkFWK2010arXiv: v2 LaneAMandThomasRG1958Rev.Mod.Phys LynnJE1968TheTheoryofNeutronResonanceReactions(Oxford:The ClarendonPress) MoldauerPA1964aPhys.Rev B 1964bibid136949B 17

18 PlerouV,GopikrishnanP,RosenowB,AmaralLAN,andStanleyHE 2000PhysicaA PorterCEandThomasRG1956Phys.Rev ReichCWandMooreMS1958Phys.Rev ThomasRG1955Phys.Rev VogtE1958Phys.Rev ibid WignerEPandEisenbudL1947Phys.Rev.7229 WignerEP1957Proc.Conf.onNeutronPhysicsbyTime of Flight Methods(GatlinburgTN,Nov.1956)OakRidgeNationalLab.Report. ORNL 2309 WuC P,FirkFWKandPhillipsTW1970Nucl.Phys.A