RECOGNIZING DIFFERENT TYPES OF STOCHASTIC PROCESSES

Transcription

1 RECOGNIZING DIFFERENT TYPES OF STOCHASTIC PROCESSES JONG U. KIM AND LASZLO B. KISH Department of Electrcal and Computer Engneerng, Texas A&M Unversty, College Staton, TX , USA Receved (receved date) Revsed (revsed date) Accepted (accepted date) Communcated by We propose a new cross-correlaton method that can recognze ndependent realzatons of the same type of stochastc processes and can be used as a new knd of pattern recognton tool n bometrcs, sensng, forensc, securty and mage processng applcatons. The method, whch we call bspectrum correlaton coeffcent method, makes use of the cross-correlaton of the bspectra. Three knds of cross-correlaton coeffcents are ntroduced. To demonstrate the new method, sx dfferent random telegraph sgnals are tested, where four of them have the same power densty spectrum. It s shown that the three coeffcents can map the dfferent stochastc processes to specfc sub-volumes n a cube. Keywords: Stochastc process recognton, Bspectra, Pattern recognton, Forensc, Securty, Bometrcs. 1. Introducton Identfcaton and pattern recognton technques are of crucal mportance n bometrc, sensng, forensc, securty and mage processng applcatons. In ths Letter, we wll ntroduce a new method, whch s a process recognton tool, recognzng dfferent types of stochastc processes. The method s based on the bspectra (a hgher order statstcal tool) whch have recently been appled to dentty gases by fluctuaton-enhanced gas sensng [1, ]. We name ths new process recognton tool bspectrum correlaton coeffcent (BCC) method because t utlzes normalzed cross-correlaton coeffcents based on the bspectra of the process realzatons. Conventonal cross-correlaton technques recognze only the same realzaton of a stochastc process, and they gve zero value for the ndependent realzatons of the same process or for two dfferent processes. Consequently, cross-correlaton technques cannot dstngush between the case of two ndependent realzatons of the same process and that of two dfferent processes. In ths letter, we wll show that the BCC method s useful for the dentfcaton of stochastc processes even though ther power densty spectra (PDS) or ampltude dstrbuton functons are ndstngushable.

2 Recognzng dfferent types of stochastc processes. Bspectrum correlaton coeffcent The bspectrum for a statonary sgnal x ( k) s defned as []: ( ω, ω ) = { E[ x( k) x( k + τ ) x( k + τ )] [ ( ω τ + ω τ )]} S, (1) 1 1 exp τ1= τ= where τ and ω are dscrete tme and the angular frequency, respectvely, and E [...] means ensemble average. The sgnal s supposed to be statonary wth zero mean. Due to the symmetry of the bspectrum [], the whole nformaton les n the non-redundant regon: ω ω 0 ω, and ( τ τ )( ω + ω ) π 1, () To calculate the bspectrum from a statonary sgnal of fnte length, we can use the so-called drect conventonal method [,4]: ( ω ω ) * ( ω ) X ( ω ) X ( ω + ω ) X 1 1 =, () N S 1, where N s the number of samples, the astersk represents complex conugate and X ( ω) s the Fourer transform: M 1 k = 0 ( ) = x( k ) exp( kω) X ω, (4) where M s the length of a sample. As mpled n Eq. (), the bspectrum for a realvalued sgnal s a two-dmensonal matrx wth complex number elements. To study the correlatons between two bspectra, we ntroduce the BCC whch has three dfferent types. The "real" bspectrum correlaton coeffcent (RBCC) s: RBCC = Re[ S ω )] Re[ S ω )] { Re[ S ω )]} { Re[ S ω )]} the "magnary" bspectrum correlaton coeffcent (IBCC) s: IBCC = Im[ S ω )] Im[ S ω )] { Im[ S ω )]} { Im[ S ω )]} and the "magntude" bspectrum correlaton coeffcent (MBCC) s: where S and MBCC = S S ω ) S ω ) ( ω, ω ) S ( ω, ω ) 1 1, (5a), (5b), (5c) S are bspectra of the -th realzaton and the -th realzaton. Here, Σ ω n the n Eqs. (5a) through (5c) represents the summaton over the frequences ω 1 and non-redundant regon. Trvally, each type of the above defned BCCs yelds 1 f we cross-correlate the same realzaton of the stochastc process wth tself.. Demonstraton by computer smulatons To demonstrate the effectveness of the BCC method n stochastc processes, we consder sx dfferent types of random telegraph sgnals (RTS) and generate seven realzatons of each type. The sx dfferent RTS types are as follows:

3 J. U. Km and L. B. Ksh Fg. 1. Probabltes or the ampltude densty functons g() x of the sx random telegraph sgnals. The length of each sample s 104 and the number of samples s RTS 1 has only two ampltudes, 1 and -1. The value s swtched at Posson tme events wth rate 0./step. RTS has only two ampltudes, -1 and 1. The ampltude of the sgnal s changed by the combnaton of two Posson rates dependng on the ampltude of the sg-

4 Recognzng dfferent types of stochastc processes nal. The Posson rate s 0./step at the ampltude 1 whle t s 0.18/step at the ampltude -1. RTS has contnuous ampltudes randomly selected at each Posson tme events. The ampltude values are unformly dstrbuted n the range [-1.74, 1.74]. The Posson rate s 0.4/step. RTS 4 has ntal ampltude chosen randomly from the range [0,.45]. At a Posson tme event, the ampltude s changed as follows: f the ampltude s postve, the next ampltude s the prevous one mnus a random number unformly dstrbuted n the range [0,.45]. Otherwse, the next ampltude s the prevous one plus a random number generated n the same way. Here, the Posson rate s 0.4/step. RTS 5 has only two ampltudes, 1 and -1. The -1 ampltude s the ground state and the 1 ampltude belongs to a "frng" event. At a Posson tme event, frng s ntated. After a unformly dstrbuted random perod, whch s selected from the range of 1-11 steps, the frng s termnated and the system s reset to the ground state. Then a new Posson tmng starts for the ntaton of the new frng event. The Posson rate s 0.1/step. RTS 6 has only two ampltudes, 1 and -1. The sgnal holds ts orgnal ampltude for a random tme perod. After a random perod unformly dstrbuted n the nterval 1-6 steps, the sgn s alternated. The evaluaton of the statstcal propertes of the RTS sgnals was obtaned by computer smulatons. The smulaton length of each sample s 104 and the number of samples s Fgure 1 shows the probablty or the ampltude densty functon (ADF) g(x) of the sx RTSs. Snce the ampltude of RTS 1, RTS, RTS 5, and RTS 6 s dscrete, we need to use probablty, not the ampltude densty functon. P 1 and P -1 are probabltes for ampltude to be found 1 and -1 at a step, respectvely. P 1 and P -1 n RTS 1 s the same snce the Posson rate s constant. In RTS, P -1 s bgger than P 1 snce the Posson rate depends on the ampltude. The ampltude densty functon of the RTS s unform snce the ampltude s selected randomly n a unform dstrbuton and the Posson rate s constant. The ampltude densty functon of the RTS 4 has a maxmum at the ampltude zero and zero at the ampltude.45 or -.45 snce the smaller absolute value of the ampltude happens more frequently than the bgger. In RTS 5 case, snce the condton for the ampltude 1 s dfferent from that for the ampltude -1, P 1 and P -1 s not the same. Snce the random tme perod s ndependent of the ampltude, P 1 and P -1 n RTS 6 s the same. These propertes of the RTSs are shown n Fgure 1 as expected. Fgure shows the PDSs of the sx RTSs. It can be seen that RTS 1 through RTS 4 have the same PDS, but the PDSs of RTS 5 and RTS 6 are dfferent. Let us call auto-bcc the BCC between dfferent realzatons of the same type of stochastc process and cross-bcc the BCC between realzatons of dfferent stochastc processes, respectvely. We generated 7 ndependent realzatons for each RTS and evaluated the bspectrum for each sgnal. Thus the smulatons provde 1 (=7*6/) auto-bccs for each RTS and 49 cross-bccs for each pars of RTSs. We use a -dmensonal plot (see Fgure ) so that the x-coordnate corresponds to the RBCC, the y-coordnate to the IBCC, and the z-coordnate to the MBCC, respectvely. Thus the BCC plot of any realzaton of any stochastc process wth tself would be a fxed pont wth coordnates (1,1,1). In Fgure, the pont groups labeled a through f, represent the cross-bccs whle the ponts nsde the ellpse represent the auto-bccs.

5 J. U. Km and L. B. Ksh The label a represents the RTS 4-5 par, the label b the RTS -5 par, the label c the RTS -4 par, the label d the RTS 1-4, 1-5, -4, -5, 4-6, and 5-6 pars, the label e the RTS 1-, - and -6 pars, and the label f the RTS 1-, 1-6 and -6 pars. The plots of almost all of the cross-bccs, except that of the -4 par, are around the z axs; that s, they have about zero RBCC and IBCC. The plots of the cross-bccs of the -4 par also have zero IBCC, but nonzero RBCC. Fgure 4 shows the proecton of the -dmensonal BCC plot onto the RBCC-IBCC plane. The features of the BCC locaton are as follows: The plots of the cross-bccs are around the z axs. The plots of the BCCs tend to form well dstngushable groups whch tend to form characterstc shapes. The plots of the auto-bccs tend to form lnes and the cross-bccs form - dmensonal bundles. Because ther locatons and shapes are strongly dfferent, they are clearly dstngushable from each other. Fg.. Power densty spectra of sx random telegraph sgnals. They were obtaned by drect conventonal method wthout wndow. Fg.. Three-dmensonal bspectrum correlaton coeffcent; the x-coordnate corresponds to the real-bcc, the y-coordnate to the magnary-bcc, and the z-coordnate to the magntude-bcc. The ponts nsde the ellpse represent the auto-bccs. The labeled a represents RTS 4-5 par, the labeled b RTS -5 par, the labeled c RTS -4 par, the label d RTS 1-4, 1-5, -4, -5, 4-6, and 5-6 pars, the labeled e RTS 1-, - and -6 pars, and the labeled f RTS 1-, 1-6 and -6 pars.

6 Recognzng dfferent types of stochastc processes Therefore, the BCC method s a new type of pattern recognton tools. It s useful to determne whch type of stochastc processes a realzaton belongs to. Conventonal correlaton technque s unable to do that. When we test f a certan realzaton x(t) of an unknown stochastc process s a realzaton of a known stochastc process K, we calculate and plot the BCCs of the x(t) process and several realzatons of the process K. Then we get a bundle of ponts n the -dmensonal space. The locaton and the shape of the bundle of the BCC ponts n Fgure and Fgure 4 can be regarded as a classfer. Fg. 4. Two-dmensonal bspectrum correlaton coeffcent plot; the x-coordnate corresponds to the real-bcc, and the y-coordnate to the magnary-bcc. The column at the rght sde shows the RTS pars. 4. Summary A new cross-correlaton technque, whch gves enhanced nformaton about stochastc patterns, the bspectrum correlaton coeffcent method, was ntroduced. We demonstrated the usefulness of the BCC method wth sx types of random telegraph sgnals. Ths new pattern recognton has potental applcatons n bometrcs, sensng, forensc, securty, and mage processng. Acknowledgement J. U. Km would lke to acknowledge the support of Ebensbeger/Fouraker Graduate Fellowshp. References [1] J. M. Smulko and L. B. Ksh, Hgher-Order Statstcs for Fluctuaton-Enhanced Gas-Sensng, Sensor. Mater. 16 (004) 91. [] G. Schmera, L.B. Ksh, J. Smulko, System and Method for Gas Recognton by Analyss of Bspectrum Functon, Patent pendng, Navy Case #9610 (March 004). [] C. L. Nkas and J. M. Mendel, Sgnal Processng wth Hgher-Order Spectra, IEEE Sgnal Proc. Mag. 10, (199) 10. [4] T. Subba Rao and M. M. Gabr, An Introducton to Bspectral Analyss and Blnear Tme Seres Models, Sprnger-Verlag, Berln, 1984.