Nonlinear Time Series Analysis in a Nutshell

Transcription

1 9 Nonlnear Tme Seres Analyss n a Nutshell Ralph Gregor Andrzejak Contents 9. Introducton Dynamcs and Tme Seres Nonlnear Determnstc Dynamcs Lorenz Dynamcs Nosy Determnstc Dynamcs Nosy Lorenz Dynamcs Stochastc Processes Lnear Autoregressve Models Determnstc versus Stochastc Dynamcs Delay Coordnates Nonlnear Predcton Error The Concept of Surrogate Tme Seres Influencng Parameters What Can and Cannot be Concluded? References Introducton Nonlnear tme seres analyss s a practcal spnoff from complex dynamcal systems theory and chaos theory. It allows one to characterze dynamcal systems n whch nonlneartes gve rse to a complex temporal evoluton. Importantly, ths concept allows extractng nformaton that cannot be resolved usng classcal lnear technques such as the power spectrum or spectral coherence. Applcatons of nonlnear tme seres analyss to sgnals measured from the bran contrbute to our understandng of bran functons and malfunctons and thereby help to advance cogntve neuroscence and neurology. In ths chapter, we show how a combnaton of a nonlnear predcton error and the Monte Carlo concept of surrogate tme seres can be used to attempt to dstngush between purely stochastc, purely determnstc, and determnstc dynamcs supermposed wth nose. The framework of nonlnear tme seres analyss comprses a wde varety of measures that allow one to extract dfferent characterstc features of a dynamcal system underlyng some measured sgnal (Kantz and Schreber 997). These nclude the correlaton dmenson as an estmate of the number of ndependent degrees of freedom, the Lyapunov exponent as a measure for the dvergence of smlar system states n tme, predcton errors as detectors for characterstc trats of determnstc dynamcs, or dfferent nformaton theory measures. The aforementoned nonlnear tme seres measures are unvarate,.e., they are appled to sngle sgnals measured from ndvdual dynamcs. In contrast, bvarate measures are used to analyze pars of sgnals measured smultaneously from two dynamcs. Such bvarate tme seres analyss measures am to dstngush whether the two dynamcs are ndependent or nteractng through some couplng. Some of these bvarate measures am to extract not only the strength, but also the drecton of these couplngs. The Monte Carlo concept of surrogates allows one to test the results of the dfferent nonlnear measures aganst well-specfed null hypotheses. 25 K766_C9.ndd 25 /24/2 :35:58 PM

2 26 Eplepsy: The Intersecton of Neuroscences, Bology, Mathematcs, Engneerng and Physcs We should brefly defne the dfferent types of dynamcs. Let x(t) denote a q-dmensonal vector that fully defnes the state of the dynamcs at tme t. For purely determnstc dynamcs, the future temporal evoluton of the state s unambguously defned as a functon f of the present state: x() t = f( x( t)). (9.) Here x(t) denotes the tme dervatve of the state x(t). We further suppose that a unvarate tme seres s s measured at nteger multples of a samplng tme Δt from the dynamcs usng some measurement functon h: s = h(x(δt)). (9.2) The combnaton of Equaton 9. and Equaton 9.2 results n the purely determnstc case. When some nose ψ(t) enters the measurement: s = h(x(δt),ψ(δt)), (9.3) we obtan the case of determnstc dynamcs supermposed wth nose. Here t s mportant to note that the nose only enters at the measurement. The determnstc evoluton of the system state x(t) va Equaton 9. s not dstorted. In contrast to such measurement nose stands dynamcal nose ξ(t) for whch we have: x() t = f( x( t), ξ ()) t. (9.4) Due to the nose term, the future temporal evoluton of x(t) s not unambguously determned by the present state. Accordngly, Equaton 9.4 represents stochastc dynamcs. We agan suppose that a measurement functon, wth or wthout measurement nose (Equatons 9.3 and 9.2, respectvely), s used to derve a unvarate tme seres s from ths stochastc dynamcs. As stated above, we am to dstngush between purely stochastc, purely determnstc, and determnstc dynamcs supermposed wth nose. Accordngly, we want to desgn an algorthm A to be appled to a tme seres s that fulflls the followng crtera:. A(s ) should attan ts hghest values for tme seres s measured from any purely stochastc process.. A(s ) should attan ts lowest values for tme seres s measured from a nose-free determnstc dynamcs. *,. A(s ) should attan some value n the mddle of ts scale for tme seres s measured from nosy determnstc dynamcs. In partcular, the ranges obtaned for. and. should not overlap wth the one of. For., A(s ) should deally vary wth the relatve nose ampltude. In order to study our problem under controlled condtons, we use a tme seres of mathematcal model systems that represent these dfferent types of dynamcs. Whle ths chapter s restrcted to these model systems, the underlyng lecture ncluded results for exemplary ntracranal electroencephalographc (EEG) recordngs of eplepsy patents (Andrzejak et al. 2). These EEG tmes * The followng text contans a number of footnotes. These are meant to provde background nformaton. The reader mght safely gnore them wthout rsk of losng the essentals of the man text. The polarty of A does not matter. It could lkewse attan lowest values for. and hghest values for. K766_C9.ndd 26 /24/2 :35:58 PM

3 Nonlnear Tme Seres Analyss n a Nutshell 27 seres as well as the source codes to generate the dfferent tme seres studed here, the generaton of surrogates, and the nonlnear predcton error are freely avalable. * 9.2 dynamcs and Tme Seres 9.2. Nonlnear Determnstc Dynamcs Lorenz Dynamcs As nonlnear determnstc dynamcs we use the followng Lorenz dynamcs. x() t = ( yt ( ) xt ( )) (9.5) y() t = 39 xt () yt () xtzt () () (9.6) z() t = xtyt () () ( 83 / ) zt (). (9.7) Ths set of frst-order dfferental equatons can be ntegrated usng, for example, a fourth order Runge Kutta algorthm. To obtan tme seres x, y, z for =,,N = 248, we sample the output of the numercal ntegraton at an nterval of Δt =.3 tme unts. We denote the resultng temporal sequence of three-dmensonal vectors as L = (x, y, z ). In our analyss, we restrct ourselves to the frst component x (see Table 9. and Fgure 9.). Hence, we have an example of purely determnstc dynamcs (Equaton 9.). The dscretzaton n tme s obtaned from the numercal ntegraton of the dynamcs. Ths ntegraton can be consdered as part of the measurement functon (Equaton 9.2), whch furthermore conssts of the projecton to the frst component Nosy Determnstc Dynamcs Nosy Lorenz Dynamcs By addng whte nose to the frst component of our Lorenz tme seres x, we obtan nosy determnstc dynamcs: n = x + 8ψ, (9.8) where ψ denotes uncorrelated nose wth unform ampltude dstrbuton between and (see Table 9. and Fgure 9.). Our tme seres n s an example of a sgnal generated by determnstc dynamcs (Equaton 9.) where nose s supermposed on the measurement functon Equaton 9.3. To roughly assess the strength of the nose, we have to know that the standard devaton of the nose term ψ and the Lorenz tme seres x s approxmately.29 and 9.6, respectvely. Wth the prefactor of 8, the standard devaton of the nose amounts to approxmately 5.2. * Equatons 9.5 through 9.7 result n chaotc dynamcs. However, ths s not of our concern here. Rather, we use the Lorenz dynamcs as a representatve of some aperodc determnstc dynamcs. Here we use an algorthm wth fxed step sze of.5 tme unts. Startng at random ntal condton we use some arbtrary but hgh number (e.g., 6 ) of preteratons to dmnsh transents. In the sgnal analyss lterature, one often fnds 24, 248, 496, etc. for the number of samples. The reason s that these numbers are nteger powers of two: N* = 2 n wth n beng an nteger. For these numbers of samples, one can use the very effcent fast Fourer transform algorthm. For other numbers of samples, one has to use algorthms that dvde the number of samples n prme factors and that are slower. In assessng the relatve strength of nose supermposed to the Lorenz dynamcs, t s mportant to note that a major contrbuton to the overall standard devaton orgnates from the swtchng behavor of ths dynamcs. The standard devaton of the actual aperodc oscllatons n the two wngs s smaller (see Fgures ). K766_C9.ndd 27 /24/2 :35:58 PM

4 28 Eplepsy: The Intersecton of Neuroscences, Bology, Mathematcs, Engneerng and Physcs Table 9. Overvew of Dfferent Tme Seres Studed Type Orgn and Parameter a Stochastc AR process, Equaton 9.9 wth r = b Stochastc AR process, Equaton 9.9 wth r =.95 c Stochastc AR process, Equaton 9.9 wth r =.98 x Determnstc x-component of Lorenz dynamcs n Nosy determnstc x supermposed wth whte nose 5 x n c b a Fgure 9. Plots of the dfferent tme seres studed here. K766_C9.ndd 28 /24/2 :35:59 PM

5 Nonlnear Tme Seres Analyss n a Nutshell stochastc Processes Lnear Autoregressve Models As a stochastc process, we use a smple but stll very nstructve example a frst-order lnear autoregressve process wth varyng degrees of autocorrelaton: ζ + = r ζ + ξ, (9.9) Please verfy meanng as ntended for =,...,N = 248, where ξ denotes uncorrelated Gaussan nose wth mean and unt varance. * We can readly see that for r =, Equaton 9.9 results n whte,.e., uncorrelated nose. Upon the ncrease of r, wthn the nterval < r <, the strength of the autocorrelaton ncreases. For ths chapter, we study three dfferent tme seres derved from Equaton 9.9 usng dfferent values of r (see Table 9. and Fgure 9.). In contrast to the tme-contnuous stochastc dynamcs expressed n Equaton 9.4, the autoregressve process of Equaton 9.9 s defned for dscrete tme. What s common to both formulatons, and what s the mportant pont here, s that the nose s ntrnsc to the dynamcs; no addtonal nose enters through the measurement. We should note that the fact that we can wrte down a rule to generate an autoregressve process (as n Equaton 9.9) does not conflct wth ts stochastc nature n any way. The pont s that due to the nose term, ξ, an ntal condton does not unambguously determne the future temporal evoluton of the dynamcs. Furthermore, wthout ths nose term, the autoregressve process would not result n sustaned dynamcs but rather decay to. 9.3 determnstc versus Stochastc Dynamcs Plottng the three components of the Lorenz dynamcs L n a three-dmensonal state space, we obtan the trajectory of the dynamcs that form the famous Lorenz attractor (Fgure 9.2, left). Once we specfy some state of the Lorenz dynamcs x(), y(), z() at t = as an ntal condton the future temporal evoluton of the dynamcs s unambguously determned by Equatons As a result of ths determnstc nature of the Lorenz dynamcs, ts trajectory cannot ntersect wth tself. A trajectory crossng would mply that the state at the crossng would have two dstnct future evolutons, whch would contradct wth the unqueness of the soluton of Equatons Inspectng the trajectory n the left panel of Fgure 9.2, we can apprecate that neghborng trajectory segments are algned. That means that smlar present states of the determnstc Lorenz dynamcs result n smlar states n the near future. In opposton to determnstc dynamcal systems are stochastc dynamcs. When we start Equaton 9.9 twce for an dentcal ntal condton, ζ, and the same r value, we get two completely dfferent solutons. Moreover, smlar present states of stochastc dynamcs wll typcally not result n smlar future states. Accordngly, trajectores of stochastc dynamcs can ntersect, and neghborng segments are typcally not algned. Ths s llustrated n the rght panel of Fgure 9.2, where we plot the trajectory of smple stochastc dynamcs derved from Equaton 9.9) n a three dmensonal state space. 9.4 delay Coordnates In the prevous secton, we saw that the algnment of neghborng trajectory segments can be used as a crteron to try to dstngush determnstc from stochastc dynamcs (See the two panels of Fgure 9.2). Before we can quantfy ths crteron we have to realze, however, that n expermental measurements from real-world dynamcs we typcally cannot access all varables of a dynamcal * To avod transents, we agan run some arbtrary but hgh number of preteratons (e.g., 4 ). That s, we start Equaton 9.9 at ζ = before recordng the samples of the process startng from ζ. The edgness at the outer corners of the wngs results from the fxed nonzero step sze of the numercal ntegraton. K766_C9.ndd 29 /24/2 :35:59 PM

6 3 Eplepsy: The Intersecton of Neuroscences, Bology, Mathematcs, Engneerng and Physcs 5 z [a.u.] c3 [a.u.] x [a.u.] y [a.u.] c [a.u.] 2 2 c2 [a.u.] 2 Fgure 9.2 (left) The three components of the Lorenz dynamcs x, y, and z plotted n a three-dmensonal state space. (rght) Three ndependent realzatons of an autoregressve process (Equaton 9.9 for r = :98) plotted n a three-dmensonal state space. system. Suppose for example that we study a real-world dynamcal system that s fully descrbed by the Lorenz dfferental Equatons Suppose, however, that we cannot measure all three components of L, but rather only s = x s assessed by our measurement functon. In ths case, we can use the method of delay coordnates to obtan an estmate of the underlyng dynamcs (Takens 98): v = (s,,s (m ) ), (9.) wth embeddng dmenson m and delay for = + (m ),...,N = η,...,n. Where we defne η = + (m ) to smplfy the notaton below. An ntroducton to the embeddng theorem underlyng the reconstructon of dynamcs usng such delay coordnates can be found n (Kantz and Schreber 997). We restrct ourselves here to a hand-wavng argument that Equaton 9. allows us to obtan a structure that s topologcally smlar to the orgnal dynamcs. In Fgure 9.3, we contrast the full Lorenz dynamcs L wth delay coordnate reconstructons obtaned from ts frst component x usng m = 3 and dfferent values of. Comparng these structures, we fnd a strkng smlarty between the orgnal dynamcs and ts reconstructon for not too hgh values of. For low values of, the dynamcs s somewhat stretched out along the man dagonal and s not properly unfolded. At an ntermedate value of, the two-wng structure of the Lorenz dynamcs s clearly dscernble from the reconstructon. For too hgh a value of, the dynamcs s overfolded. K766_C9.ndd 3 /24/2 :35:59 PM

7 Nonlnear Tme Seres Analyss n a Nutshell z y 2 x j x 2 j 2 2 x 2 2 x x 6 j 2 2 x x x 6 j 2 2 x x Fgure 9.3 (frst row, left) The three components of the Lorenz dynamcs L = (x, y, z ) plotted n a threedmensonal state space. Hence, ths panel s the same as the left panel of Fgure 9.2, but vewed from a dfferent angle. (frst row, rght) Eucldean dstance matrx calculated for the Lorenz dynamcs L. (second row, left) Delay reconstructon of the Lorenz dynamcs based on the frst component, x. Here we use a tme delay of =. For obvous reasons, the maxmal embeddng dmenson that we can use for ths llustraton s m = 3. (second row, rght) Eucldean dstance matrx calculated from such an embeddng, but for m = 6. (thrd and fourth rows) Same as second row, but for = 6 and = 3, respectvely. K766_C9.ndd 3 /24/2 :36: PM

8 32 Eplepsy: The Intersecton of Neuroscences, Bology, Mathematcs, Engneerng and Physcs We can apprecate ths smlarty further by nspectng Eucldean dstance matrces derved from the full Lorenz dynamcs and ts delay coordnate reconstructon. * For a set of q-component vectors w for = η,...,n, such a matrx contans n ts element Dj the Eucldean dstance between the vectors w and w j. The Eucldean dstance between two q-component vectors w and w 2 s defned as q 2 dw (, w2) = ( w, l w2, l). (9.) l = Fgure 9.3 shows the dstance matrx for the full Lorenz dynamcs L, whch can also be seen as a set of N vectors wth q = 3 components. Ths s contrasted to the dstance matrx derved from delay coordnate reconstructons v, whch s a set of N vectors wth q = m components. Whle the overall scales of the dfferent matrces are dfferent, we can observe an evdent smlarty between the structures found n these matrces for the orgnal and ts delay coordnate reconstructon for not too hgh values of. Importantly, ths smlarty mples that close and dstant states n the full Lorenz dynamcs are mapped to close and dstant states n the reconstructed dynamcs, respectvely. Ths topologcal smlarty mples that our crteron of dstncton between determnstc and stochastc dynamcs, namely the algnment of neghborng trajectory segments, carres over from the orgnal dynamcs to the reconstructed dynamcs obtaned by means of delay coordnates. The dfferent dstance matrces derved from the delay reconstructons are, however, not dentcal to the orgnal dstance matrx and the degree of ther smlarty depends on the embeddng parameters m and. These aspects wll be dscussed n more detal below. To get a thorough understandng of the qualty of the embeddng requres playng wth dfferent parameters and vewng the result from dfferent angles. Ths can be done usng the source code referred to n the ntroducton. 9.5 nonlnear Predcton Error Above we have dentfed the algnment of neghborng trajectory segments as a sgnature of determnstc dynamcs. We wll now ntroduce the nonlnear predcton error as a straghtforward way to quantfy the degree of ths algnment (see Kantz and Schreber (997) and references theren). Before we begn, we frst normalze our tme seres s to have zero mean and unt standard devaton and varance. Subsequently, we form a delay reconstructon v from our tme seres for = η,...,n. To calculate the nonlnear predcton error wth horzon h we carry out the followng steps for = η,...,n h. We take v as reference pont and look up the tme ndces of the k nearest neghbors of the reference pont: { g }( g =,..., k ). These k nearest neghbors are smply those ponts that have the k smallest dstances to our reference pont n the reconstructed state space. In other words, { g }( g =,...,k ) are the ndces of the k smallest entres n row of the dstance matrx derved from v. Now we use the future states of these nearest neghbors to predct the future states of our reference pont and quantfy the error that we make n dong so by k ε = s + h s h g k +. (9.2) g= * The choce of the Eucldean dstance s arbtrary. The same arguments hold when other defntons of dstances are used. In ths way we do not have to normalze the predcton error tself and the subsequent notatons become smpler. Snce dstance matrces are symmetrc, these ndces concde wth the k smallest entres n the column of the matrx. Note however that f v s among the k nearest neghbors of v 2, ths does not mply that v 2 s among the k nearest neghbors of v. K766_C9.ndd 32 /24/2 :36: PM

9 Nonlnear Tme Seres Analyss n a Nutshell 33 Fnally, we take the root-mean-square over all reference ponts: E = N h η + = N h 2 ε = η. (9.3) Note that whle the nearest neghbors are determned accordng to ther dstance n the reconstructed state space, the actual predcton n Equaton 9.2 s made usng the scalar tme seres values. Regardng Equaton 9.2, t also becomes clear that we must exclude nearest neghbors wth ndces r > N h. Furthermore, we should am to base our predcton on neghborng trajectory segments rather than on the precedng and subsequent pece of the trajectory on whch our reference pont v s located on. For ths purpose we apply the so-called Theler correcton wth wndow length W (Theler 986), by ncludng only ponts to the set of k nearest neghbors that fulfll g > W. For the parameters, we use the followng values. We fx the embeddng dmenson used for the delay coordnates to m = 6, the number of nearest neghbors to k = 5, the predcton horzon to h = 5 samples, and the length of the Theler correcton to W = 3. The delay used for the delay coordnates s vared accordng to =, 2,..., 29. Fgure 9.4 shows results obtaned for our set of exemplary tme seres. The whte nose tme seres a results n E values around.. For ths stochastc process, ponts close to some reference ponts do not carry any predctve nformaton about the future evoluton of ths reference pont. Neghborng state space trajectory segments of ths stochastc process are not algned. Hence, for whte nose the predctons s r + h used n Equaton 9.2 are no better than guessng a set of k ponts randomly drawn from s. For the nose-free determnstc Lorenz tme seres x, neghborng trajectores segments are well algned. Ponts close to some reference ponts have a smlar future evoluton as ths reference pont. The predctons s r + h used n Equaton 9.2 are good predctons for s + h. Therefore, we get small values of E. For ncreasng the predcton error ncreases, reflectng the overfoldng of the reconstructed dynamcs (see Fgure 9.3). Nonetheless, as of now, E seems to fulfl the crtera. and. Nonlnear predcton error E a b c n x Fgure 9.4 Dependence of the nonlnear predcton error, E, on the tme delay,, used for the state space reconstructon. Here, results are shown only for the orgnal tme seres. K766_C9.ndd 33 /24/2 :36: PM

10 34 Eplepsy: The Intersecton of Neuroscences, Bology, Mathematcs, Engneerng and Physcs formulated n the Introducton. It allows us to clearly tell apart our determnstc Lorenz tme seres from whte nose. It can be objected that ths dstncton s not terrbly dffcult, and could readly be acheved by vsual nspecton of our data. We should therefore proceed to our nosy determnstc tme seres n and tme seres of autoregressve processes wth stronger autocorrelaton (b, c ). When we look at ther results n Fgure 9.4, we must realze that E s n fact not a good canddate for Fgure 9.4A. For the tme seres n t s clear that the nose must degrade the qualty of the predctons of Equaton 9.2 as compared to the nosefree case of x. Ths ncrease n E by tself would not dsqualfy E, as t s compatble wth crteron. The problem s that E values for b overlap wth those for n. Results for c are even lower than those for n and for hgher, even smlar to those of x. The reason why these autocorrelated tme seres exhbt a hgher predctablty as opposed to whte nose s the followng: v beng a close neghbor of v 2 mples that the scalar s s typcally smlar to the scalar s 2. Furthermore, s beng smlar to s 2 mples, to a certan degree, that s + h s smlar to s 2+ h. It s nstructve to convnce oneself that t s just the strength of the autocorrelaton that determnes the strength of the to a certan degree n the prevous statement. As a consequence, stochastc tme seres wth stronger autocorrelaton and nosy determnstc dynamcs cannot be told apart by our nonlnear tme seres algorthm E. We are stll mssng one very substantal ngredent to reach our am, whch wll we ntroduce n Secton the Concept of Surrogate Tme Seres Nonlnear tme seres analyss was ntally developed studyng complex low-dmensonal determnstc model systems such as the Lorenz dynamcs. The bran s certanly not a low-dmensonal determnstc dynamcs and wll typcally not behave as one. Rather, t mght be regarded as a partcularly challengng hgh-dmensonal and complcated dynamcal system that mght have both determnstc and stochastc features. Problems that arse n applcatons to sgnals measured from the nervous system contnue to trgger the refnement of exstng algorthms and development of novel approaches and, thereby, help to advance nonlnear tme seres analyss. One mportant concept n nonlnear tme seres analyss, whch has some of ts roots n EEG analyss (Pjn et al. 99), s the Monte Carlo framework of surrogate tme seres (Theler et al. 992; Schreber and Schmtz 2). Suppose that we have calculated the nonlnear predcton error for some expermental tmes seres and consder the followng scenaros. Frst, we obtan a relatvely low E value but have doubts that ths s due to some determnstc structure n our tme seres. Second, we obtan a relatvely hgh E value but we suspect that our tme seres should have some determnstc structure that s possbly dstorted by some measurement nose. In any case, we want to know what a relatvely hgh or a relatvely low value of the nonlnear predcton error means. A way to address ths problem s to scrutnze our tme seres further by testng dfferent null hypotheses about t by means of surrogate tme seres. Such surrogate tme seres are generated from a constraned randomzaton of an orgnal tme seres. * The surrogates are desgned to share specfc propertes wth the orgnal tme seres, but are otherwse random. In partcular, the propertes that the surrogates share wth the orgnal tme seres are selected such that the surrogates represent sgnals that are consstent wth a well-specfed null hypothess. Here, we use the smple but llustratve example of phase-randomzed surrogates (Theler et al. 992). To construct these surrogates, frst calculate the complex-valued Fourer transform of the orgnal tme seres. In the next step, randomze all phases of the complex-valued Fourer coeffcents between and 2π. Fnally, the nverse Fourer transform of these modfed Fourer * See Schreber and Schmtz (2) for the alternatve to usng a typcal randomzaton to generate surrogates. In ths randomzaton t s very mportant to preserve the antsymmetry between Fourer coeffcents representng postve and negatve frequences. Ths antsymmetry holds for real-valued sgnals. If t s not preserved the nverse Fourer transform wll not result n a real-valued tme seres. K766_C9.ndd 34 /24/2 :36: PM

11 Nonlnear Tme Seres Analyss n a Nutshell 35 coeffcents results n a surrogate tme seres. An ensemble of surrogates s generated usng dfferent randomzaton of the phases. Randomzng the phase of a complex number does not affect ts absolute value, that s, the length of the correspondng vector n the complex plane s preserved. Therefore, the power of the resultng sgnal and ts dstrbuton over all frequences, as measured va the perodogram, s dentcal to the one of the orgnal tme seres. Consequently, the surrogates also have the same autocorrelaton functon as the orgnal tme seres. However, regardless of the nature of the orgnal tme seres, the surrogate tme seres s a statonary lnear stochastc Gaussan process. Whether the orgnal was determnstc or stochastc, statonary or nonstatonary, Gaussan or nongaussan, the surrogate s stochastc, statonary, and Gaussan. In that sense, the surrogate algorthm projects from the space of all possble tme seres wth a certan perodogram to the space of all statonary lnear stochastc tme Gaussan tme seres wth ths same perodogram. To see ths pont, t s nstructve to convnce oneself that a phase-randomzed surrogate of a phase-randomzed surrogate cannot be dstngushed from a phase-randomzed surrogate of the orgnal tme seres. In Fgure 9.5, we agan show values of the nonlnear predcton error for our exemplary tme seres (see agan Fgure 9.4). Here, however, we not only show results for the orgnal tme seres but also contrast these to results obtaned for an ensemble of 9 phase-randomzed surrogate tme seres generated for each orgnal tme seres. For all tme seres from autoregressve processes a, b, and c, the results for the surrogates match those of the orgnal tme seres. The autoregressve processes generated by Equaton 9.9 are statonary lnear stochastc Gaussan processes. The r value determnes the strength of the autocorrelaton of ths process and thereby the shape of the perodogram. A phase-randomzed surrogate of a statonary lnear stochastc Gaussan process wth any perodogram s a statonary lnear stochastc Gaussan process wth ths very perodogram. Accordngly, phase-randomzed surrogates constructed from a, b, and c cannot be dstngushed from ther correspondng orgnal tme seres. The null hypothess s correctly accepted for all three cases. * For both the nose-free Lorenz tme seres x and the nosy Lorenz tme seres n, we fnd that the orgnal tme seres results n lower E values as compared to ther correspondng ensemble of surrogates. Here, we deal wth a purely determnstc tme seres and a nosy determnstc one. When we construct surrogates from them, ther determnstc structures are destroyed. In partcular, the randomzaton destroys the local algnment of neghborng trajectory segments. Regardng the sgnfcant dfference between the E values for the orgnal tme seres and the mean values obtaned for the surrogates, we can conclude that the null hypothess s correctly rejected for both x and n. From these results, t s only a small step for us to arrve at our am by defnng A = E surr E org, (9.4) where the brackets denote the average taken across the ensemble of 9 surrogates. The resultng surrogate-corrected nonlnear predcton error values A are shown n Fgure 9.5, and we can see that the defnton of Equaton 9.4 fulflls crtera.,., and. that we defned as the am of ths chapter. 9.7 Influencng Parameters In ths chapter, we have plotted the nonlnear predcton error only n dependence on the tme delay used for the delay coordnates. To fully understand how nonlnear technques and surrogates work, t s always mportant to study the nfluence of all parameters n detal. In our case, these parameters are the delay and embeddng dmenson m used for the delay coordnates, the number of nearest * A good check of whether one s stll on track n understandng ths chapter s to fnd an explanaton of why our dfferent autoregressve processes result n dfferent E values, and why the surrogates match closely wth ther correspondng orgnal tme seres n these E values, rather than just undscrmnatngly coverng the entre range of E values obtaned for a, b, and c. K766_C9.ndd 35 /24/2 :36: PM

12 36 Eplepsy: The Intersecton of Neuroscences, Bology, Mathematcs, Engneerng and Physcs E a A a E b A b E c A c E n A n E x A x Fgure 9.5 (left) Dependence of the nonlnear predcton error, E, on the tme delay,, used for the state space reconstructon. Here, results are shown for the orgnal tme seres (grey lnes wth symbols) and 9 surrogate tme seres (black lnes wth error bars). The error-bar denotes the range obtaned from the surrogates. (rght) Dependence of the surrogate-corrected nonlnear predcton error, A, on the tme delay,. Each row corresponds to results of one specfc tme seres (see legends). neghbors k and the predcton horzon h, and, fnally, the length of the Theler-correcton W. In general, the nfluence of these dfferent parameters wll not be ndependent from another. For example, t s often the so-called embeddng wndow (m ) that matters, rather then the two parameters and m ndependently. There exst very useful gudelnes and recpes for an adequate choce of these dfferent parameters (see Kantz and Schreber (997) and references theren). However, whenever one s analyzng some type of expermental data, one should play wth these parameters and see K766_C9.ndd 36 /24/2 :36:2 PM

13 Nonlnear Tme Seres Analyss n a Nutshell 37 how they nfluence results. Certanly, f one fnds some effect n expermental data usng nonlnear tme seres analyss technques, the sgnfcance of ths effect should not crtcally depend on the choce of the dfferent parameters. We have studed our problem based on only sngle realzatons of our tme seres. However, whenever we have controlled condtons, we should make use of them and generate many ndependent realzatons of our tme seres. In ths way, we can get more relable results and also learn more about our measures by regardng ther varablty across realzatons. Another mportant ssue that we have neglected here s that t can be very nformatve to study the nfluence of nose for a whole range of nose ampltudes and for dfferent types of nose. 9.8 What Can and Cannot be Concluded? Frst, we can draw a very postve concluson: We reached the am that we stated n the Introducton. Usng a combnaton of a nonlnear predcton error and surrogates, we were able to dstngush between purely stochastc, purely determnstc, and determnstc dynamcs supermposed wth nose. We demonstrated ths dstncton under controlled condtons usng mathematcal model systems wth well-defned propertes. However, how do these results carry over once we leave the secure envronment of controlled condtons? Under controlled condtons, we know what dynamcs underle our tme seres. Here, we can draw conclusons such as the null hypothess s correctly accepted or rejected, but what conclusons can be drawn when facng real-world expermental tme seres from some unknown dynamcal system? Let us suppose we want to test the followng workng hypothess consstng of two parts. Frst, the dynamcs exhbted by an epleptc focus s nonlnear determnstc. Second, the dynamcs exhbted by healthy bran tssue s lnear stochastc. Suppose that to test ths hypothess we compare EEG tme seres measured from wthn an epleptc focus aganst EEG tme seres measured outsde the focal area. We calculate the nonlnear predcton error for these EEG tme seres and phase-randomzed surrogates constructed from them. Suppose further that for the focal EEG tme seres we fnd sgnfcant dfferences between the results for orgnal EEG tme seres and those of the surrogates, whereas for the nonfocal EEG we fnd that the results for the orgnal are not sgnfcantly dfferent from the surrogates. (In the lecture underlyng ths chapter we looked at such results for real EEG tme seres.) Have we proven our workng hypothess? The answer s no, we have not nether ts frst nor ts second part. Frst, we need to recall that the rejecton of a null hypothess at say p <.5 n no way proves that the null hypothess s wrong. It only mples that f the null hypothess was true, the probablty to get our result (or a more extreme result) s less than 5%. Puttng that asde, let us assume that the dfference between the focal EEG and surrogates s so pronounced that we deem the underlyng null hypothess as ndeed very mplausble. Stll, ths does not prove that the dynamcs are nonlnear determnstc. Recall that we used phase-randomzed surrogates representng the null hypothess of a lnear stochastc Gaussan statonary process. We have to keep n mnd that the complement to our null hypothess s the world of all dynamcs that do not comprse all the propertes: lnear, stochastc, Gaussan, and statonary. Hence, f we deal, for example, wth a statonary lnear stochastc nongaussan dynamcs, our null hypothess s wrong. Our test should reject t. Lkewse, we mght have nonstatonary lnear stochastc Gaussan dynamcs or nonlnear determnstc process. We mght also deal wth statonary lnear stochastc Gaussan dynamcs measured by some nonlnear measurement functon. We have to realze that the dynamcs exhbted by an epleptc focus s nonlnear determnstc and s among these alternatves. However, t s only one of many explanatons of our fndngs. Gven only the results of our test, our workng hypothess s not more lkely than any of the other explanatons of the rejecton of our null hypothess. We should furthermore note that our test cannot prove the second part of our null hypothess ether. The fact that we could not reject the null hypothess for our nonfocal EEG does not prove the correctness of ths null hypothess. We cannot rule out that our nonfocal EEG orgnates from some K766_C9.ndd 37 /24/2 :36:2 PM

14 38 Eplepsy: The Intersecton of Neuroscences, Bology, Mathematcs, Engneerng and Physcs nonlnear determnstc dynamcs, but our test s not suffcently senstve to detect ths feature. Our tme seres mght be too short or too nosy for the predcton error to detect the algnment of neghborng trajectores, or we mght have used nadequate values of our parameters m,, and k. It s mportant to keep these lmtatons n mnd when nterpretng results derved from nonlnear measures n combnaton wth surrogates. * Nonetheless, nonlnear measures n combnaton wth surrogates can stll be very useful n characterzng sgnals measured from the bran. Andrzejak et al. studed the dscrmnatve power of dfferent tme seres analyss measures to lateralze the sezure-generatng hemsphere n patents wth medcally ntractable mesal temporal lobe eplepsy (Andrzejak et al. 26). The measures that were tested comprsed dfferent lnear tme seres analyss measures, dfferent nonlnear tme seres analyss measures, and a combnaton of these nonlnear tme seres analyss measures wth surrogates. Subject to the analyss were ntracranal electroencephalographc recordngs from the sezure-free nterval of 29 patents. The performance of both lnear and nonlnear measures was weak, f not nsgnfcant. A very hgh performance n correctly lateralzng the sezure-generatng hemsphere was, however, obtaned by the combnaton of nonlnear measures wth surrogates. Hence, the very strategy that brought us closest to the am formulated n the ntroducton of ths chapter to relably dstngush between stochastc and determnstc dynamcs n mathematcal model systems also seems key to a successful characterzaton of the spatal dstrbuton of the epleptc process. The degree to whch such fndngs carry over to the study of the predctablty of sezures was among the topcs dscussed at the meetng n Kansas. References Andrzejak, R. G., K. Lehnertz, F. Mormann, C. Reke, P. Davd, and C. E. Elger. 2. Indcatons of nonlnear determnstc and fnte-dmensonal structures n tme seres of bran electrcal actvty: Dependence on recordng regon and bran state. Phys. Rev. E 64:697. Andrzejak, R. G., G. Wdman, F. Mormann, T. Kreuz, C. E. Elger, and K. Lehnertz. 26. Improved spatal characterzaton of the epleptc bran by focusng on nonlnearty. Eplepsy Res. 69:3 44. Kantz, H., and T. Schreber Nonlnear tme seres analyss. Cambrdge: Cambrdge Unv. Press. Pjn, J. P., J. van Neerven, A. Noes, and F. Lopes da Slva. 99. Chaos or nose n EEG sgnals; dependence on state and bran ste. Cln. Neurophysol. 79: Schreber, T., and A. Schmtz. 2. Surrogate tme seres. Physca D 42: Takens, F. 98. Detectng strange attractors n turbulence. In Dynamcal systems and turbulence, Vol. 898 of Lecture notes n mathematcs, eds. D. A. Rand and L.-S. Young, Berln: Sprnger-Verlag. Theler, J Spurous dmensons from correlaton algorthms appled to lmted tme-seres data. Phys. Rev. A 34: Theler, J., S. Eubank, A. Longtn, B. Galdrkan, and J. D. Farmer Testng for nonlnearty n tme seres: The method of surrogate data. Physca D 58: * Consder as a further prncple lmtaton, a perodc process wth a very long perod that exhbts no regulartes wthn each perod. We would need to observe at least two perods to apprecate the perodcty of the process. In prncple, we need an nfnte amount of data to rule out that we msnterpret perodc, and hence determnstc, processes as stochastc processes. K766_C9.ndd 38 /24/2 :36:2 PM