On the relationship between the theory of cointegration and the theory of phase synchronization

0.67 MB·English

by  Rainer Dahlhaus

On the relationship between the theory of cointegration
and the theory of phase synchronization
Rainer Dahlhaus and Jan C. Neddermeyer
University ofHeidelbergandDZBANKAG
Version: January 4, 2012
2
1
0
2
Summary. The theory of cointegration has been the leading theory in econometrics with powerful

n
a applicationstomacroeconomicsduringthelastdecades. Ontheotherhandthetheoryofphasesyn-
J
4 chronizationfor weakly coupled complex oscillators has been oneof the leading theories in physics

] forsome15yearswithmanyapplicationstodifferentareasofscience. Forexampleinneuroscience
O
A phase synchronization is regarded as essential for functional coupling of different brain regions. In
.
n an abstract sense both theories describe the dynamic fluctuation around some equilibrium. In this
i
nl paper we point out that there exists a very close connection between both theories. We show that
[
there exists a system of stochastic difference equations which can be viewed as close to the Ku-

2
v ramotoequations,whosesolutionisacointegratedsystem. Asoneconsequencetherichtheoryon
1
5 statistical inference for cointegrated systems can immediately be applied for statistical inference on
6
0 phasesynchronizationbasedonempiricaldata. Thisincludestestsforphasesynchronization,tests
.
1
0 for unidirectional coupling and the identification of the equilibrium from data including phase shifts.
2
1 We give an example where an unidirectionally coupled Rössler-Lorenz system is identified with the
:
v methodsfromcointegration. Themethodsfromcointegrationmayalsobeusedtoinvestigatephase
i
X
synchronization in complex networks. On the contrary there are many interesting results on phase
r
a
synchronizationwhichmayinspirenewresearchoncointegration.
Keywords. Cointegration, phase synchronization, weakly coupled oscillators; driver response rela-
tionship;Rössler-Lorenzsystem.
Addressforcorrespondence: InstituteofAppliedMathematics, Universityof Heidelberg,ImNeuenheimer
Feld294,D-69120Heidelberg,Germany(e-mail:[email protected]).
Contents
1 Introduction 1
2 PhasesynchronizationandtheKuramotomodelfor interactingoscillators 4
2.1 Phasesynchronization ofweaklycoupledoscillators . . . . . . . . . . . . . . 4
2.2 Cointegrationasa stochastic Kuramoto-typemodel. . . . . . . . . . . . . . . 6
3 Theconceptofcointegration 8
3.1 Cointegrationandthe Johansen-Grangerrepresentationtheorem . . . . . . . 8
3.2 CointegrationTesting andModelFitting . . . . . . . . . . . . . . . . . . . . . 11
4 Thecointegrationapproachto phasesynchronization 15
5 Example: A coupledRössler-Lorenzsystem 19
6 Conclusionandoutlook 28
7 Appendix1: VEC -state oscillators 29
8 Appendix2: Computationalandmodelingaspects 32
8.1 Computationalaspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
8.2 Modelingaspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
0
1 Introduction
PhaseSynchronizationhasalonghistorydatingbackto1665wherethemathematicianand
physicist C. Huygens discovered some antiphase synchronization of two pendulum clocks
suspended close to each other on the same wooden beam. From that time on, the phe-
nomenon got more and more into the focus of scientists. During the last decades the be-
havior of two or several interacting oscillators has been intensively studied in the physics
literature - in particular in the context of nonlineardynamical systems (cf. [59, 42, 70, 9, 8];
- see also the monographs [63, 53]. In contrast to other types of synchronization, phase
synchronization purely dependson the phases of self-sustained weakly-coupled oscillators
while the amplitudes may even be empirically uncorrelated. Weak coupling means that the
phases of all oscillators may be subject to individual disturbances and the whole system
adjusts itself afterwards. Thus phase synchronization is rather regarded as a complex dy-
namicalprocess thana fixed state.
Thephenomenaof phasesynchronization has beenfoundexperimentallyin manyfields,
e.g. electrical circuits [66, 6], lasers [14], electrochemistry [40, 41], biological systems [18,
75],populationdynamics[7],betweenElNinoandtheIndianmonsoon[50]andeventennis
[57]. Since phase synchronization purely depends on the phases of the oscillators it can
also be detected between oscillators which are of a different type. A practical example of
this is the phase synchronization of the cardiac and respiratory system [47, 73] or between
maternalbreathingandthefetal-maternalheartrate coordination[76].
In neurosciencephasesynchronizationis regardedasessentialforfunctionalcouplingof
differentbrainregions. Singleneuronalactivity, phasecoupledtoanensembleofoscillatory
neuronal activity, enables the cells to transmit their information content long range across
differentcorticalareas. TocitefromtheabstractofFellandAxmacher[22]: “Inrecentyears,
studies ranging from single-unit recordings in animals to EEG and MEG studies in humans
have demonstrated the pivotal role of phase synchronization in memory processes. Phase
synchronization - here referring to the synchronization of oscillatory phases between differ-
entbrainregions- supportsboth working memoryandlong-term memoryandacts by facil-
itating neural communication and by promoting neural plasticity.” [10] discuss mechanisms
of gamma oscillations in the hippocampus and the functional role of the synchronization of
such oscillations in several key hippocampal operations, including cell grouping, dynamic
routing, and memory. In [79] synchronization in the gamma band is investigated, and it is
discoveredthatthemutualinfluenceamongneuronalgroupsdependsonthephaserelation
1
between rhythmic activities within the groups. Furthermore, the pattern of synchronization
is related to the pattern of neuronalinteractions. Furtherreviews on phase synchronization
in neuroscienceare [19, 77, 13].
To study the synchronization of a larger population of interacting units theoretically - for
example fireflies flashing at the same time or crickets chirping at the same time - Winfree
[78] studied the nonlineardynamics of a large populationof weakly coupled oscillators with
intrinsic frequenciesthatwere distributed according to some prescribed probabilitydistribu-
tion. He also ignored the amplitude and considered phase oscillators and worked with a
mean field model. Kuramoto [44, 45] introduced a more sound mathematical model to de-
scribethisphenomenologyleadingtoatheoreticaltreatmentofthemeanfieldapproach. He
also studied the limit behavior when the number of oscillators tends to infinity. This model
has been used since then in different forms to discuss theoretically phase synchronization
in a population of weakly coupled oscillators. For example the onset of synchronization is
discussed in this framework[74].
Although some statistical methods are used to analyze phase synchronization (e.g. the
spectral coherence) there hardly exist any stochastic models for synchronized phase pro-
cesses. Such stochastic phase models would be an important task to estimate the dynam-
icsofthephaseprocessesfrom observeddata. Specificoutcomescouldbetests forphase
synchronization,testsforunidirectionalcouplingandtheidentificationoftheequilibriumfrom
dataincludingphaseshifts.
In this paper we argue that the famous theory of cointegration provides also a good
stochastic model for phase synchronization and therefore is a good framework for inves-
tigating these questions. The trick is to use cointegration not directly as a model for the
oscillators but as a model for the phase processes driving the oscillators. Using the phase
processes as the key element for describing oscillators has been standard in physics for
many years. On the other hand this approach has never been used to our knowledge in
statistics or econometrics to model oscillators. Here oscillators have always been investi-
gated by means of Fourier transforms (spectral methods) or wavelet transforms. Further-
more, statisticians have always tried to build models directly for the oscillating processes
insteadfor thephaseprocesses.
The theory of cointegration may be used in phase synchronization both for theoretical
studies and for identifying unknown systems based on empirical data. The link may also
be of importance in the other direction in that the known results on phase synchronization
2
mayleadtonewinsightsorstimulateresearchincointegration. Thecommongroundofboth
theoriesisthattwo orseveral processesfluctuate randomlyaroundsome equilibrium.
The concept of cointegration was introduced by Granger 1981 [24] for the joint modeling
of several macroeconomic variables over time. In 2003 he received the Nobel Prize for his
discovery jointly with Robert Engle. The aim of the theory is to model common stochastic
trends, for example of income and consumption where the short-run dynamics is affected
by random disturbances and the long-run dynamics is restricted by economic equilibrium
relationships. Other examples are exchange rates and price levels, short and long-term
interestrates,andthe prices onspotandfuture markets.
Sinceitsintroductionthetheoryofcointegrationhasbeendevelopedbymanyresearchers
and it has become the leadingtheory in econometrics with many applicationsin macroeco-
nomics and beyond - for references see the monographs [5, 35, 21, 49, 36] among many
others. [61,39] havedevelopedthetheoryfor continoustime diffusion processes.
We now give a heuristic and elementary argument why the setting of cointegration is of
(i) (i) (i)
importance for phase synchronization: Suppose we observe y = A cos(φ ) + ε for
t i t t
(i)
i = 1,2 and t = 1,...,T with phase processes φ (alternatively we may just have the
t
(i)
phases φ from some oscillators calculated by means of the Hilbert transform or some
t
other method). A naive model for a random phase could be φ = ωt+ φ + δ with some
t 0 t
random error δ which could be a white noise process or some stationary process. This
t
modelhadthedrawbackthatit fluctuatesaroundthedeterministiclinearphaseωt+φ and
0
wouldthereforeonlybeadequateifsomeexternalforce orconstraintwouldkeepthephase
close to this linear phase. Instead a better model for most situations is the corresponding
model for the phase increments ∆φt = φt −φt−1 = ω +δt where δt is a stationary process
with mean zero and positive correlation. In the simplest case where δ is iid Gaussian this
t
means that φ = ωt + φ + t δ has the same distribution as a Brownian motion with
t 0 s=1 s
drift. In this case var(φt) ∼Pt which means that the phase may depart substantially from
φ = ωt+ φ for large t. This feature remains if the process δ is stationary with mean 0.
t 0 t
Sucha processφ is called anintegratedprocess.
t
Ifwelookattwoprocesseswithsynchronizedphasesthenbothphasesmayevolvelikea
(1) (2)
Brownianmotionwithdrift-howeverthedifferenceφ −φ shouldstayrelativelysmall;in
t t
particular it should not explode. A proper model therefore is that these differences follow a
stationary process. This is exactly the concept of cointegration: both phase-processes are
integratedbutthedifferenceremainsstationary. Theheuristicspresentedhereisformalized
3
in Appendix1.
The paper is organized as follows. Section 2 provides a brief introduction to phase syn-
chronization with focus on testing for phase synchronization and the Kuramoto model for
interacting oscillators. We point out that a cointegrated system arises as the solution of a
system of stochastic difference equations which can be viewed as close to the Kuramoto
equations. InSection3,wegiveabriefintroductiontothetheoryofcointegrationandreview
some tests for cointegration. Section 4 describes the use of cointegration for the statis-
tical analysis of phase synchronization. In particular we give our definition of stochastic
phase synchronization. In Section 5 we present a simulation showing that the proposed
methodworks for a coupled Rössler-Lorenz system. Conclusions and an outlookare given
inSection6followedbythedefinitionofVEC-stateoscillatorsinAppendix1andsomecom-
putationalandmodelingaspectsin Appendix2.
The article wants to address physicists, econometricians and statisticians, as well as
scientists whojustwantto applythemethod.
AquickreadingguideforphysicistsmaybetostartwiththediscussionoftheKuramoto-
model in Section 2.2, to see how (8), (9), (14), and (15) are related to the Kuramoto-model
andto proceedwith Sections4 and5.
Aquickreadingguideforeconometriciansandstatisticiansfamiliarwithcointegration
may be to start with the model for a VEC-state oscillator in Appendix 1, continue with the
exampleinSection5andtopassontotheKuramoto-modelinSection2.2andthedefinition
anddiscussion in Section 4.
Scientists mainly interested in the method may start directly with the example in Sec-
tion 5 to geta first impression oncointegration andon the impact ofcointegration for phase
synchronization.
2 Phase synchronization and the Kuramoto model for interact-
ing oscillators
2.1 Phase synchronization of weakly coupled oscillators
The starting point of phase synchronization is the definition of the phase φ of an oscilla-
t
tor. Traditionally phase synchronization has been mainly considered for chaotic oscillators
where the phase often is defined via the Hilbert transform of the signal. If the projection
of the attractor on some plane has only one rotation center then the phase can be defined
alternativelybyusingaPoincarémaporacertainrotationangle[64]. Alternatively,onemay
4
lookatstochastic systems, e.g. ofthe form
y = a cos(φ )+ε , t ∈ Z (1)
t t t t
orevenatsystems like
y = a f(φ )+b +ε , t ∈ Z (2)
t t t t t
wheref isa2π periodicrealvaluedfunctionrepresentingtheoscillationpattern,anda and
t
b are the amplitude and the baseline respectively [12]. The latter model may for example
t
beusedfor ECGsignals.
Based on discrete data of the oscillator, the phases φ have mainly been determined via
t
the Hilbert transform [70]. The Hilbert phase of a signal y is defined through the analytic
t
signalζ given by ζ = y +iyH = a exp(iφ ). The imaginarypartofζ is obtainedusing the
t t t t t t t
Hilberttransform which isdefinedthrough
1 y
yH = P s ds
t π t−s
Z
with P being the Cauchy principal value. a is the amplitude and the phase φ is computed
t t
through
yH
φ = arctan t .
t
y
t
Other methods for phase estimation are based on the Wavelet transform [26], on the peri-
odogram [29], on a local periodogram [58] or on a general state-space model with particle
filtering[12]. Thelastmethodhasinparticularbeenusedincombinationwiththemodel(2).
(1) (2)
In the next step two weakly coupledoscillators with phasesφ and φ are called m : n
t t
phasesynchronizedifthereexistssomeconstantΦ(phaseshift)andsomesmallδ > 0such
that
(1) (2)
mφ −nφ −Φ mod2π < δ (3)
t t
(cid:12)(cid:0) (cid:1) (cid:12)
holdsfor all t. m and n are in(cid:12)tegers which are usually kno(cid:12)wn in a practical application. The
idea behind this definition is that the (rescaled) phases “stay together” and do not move
arbitrarily far from each other. A standard measure for testing for phase synchronization is
thephasesynchronization index[51, 67]definedthrough
T 2
1
R2 = exp i mφ(1) −nφ(2) .
T t t
(cid:12) t=1 (cid:12)
(cid:12) X (cid:8) (cid:0) (cid:1)(cid:9)(cid:12)
(cid:12) (cid:12)
The synchronization index is an(cid:12)estimate of the correspondin(cid:12)g theoretical quantity called
mean phase coherence. It is normalized between 0 and 1 where values close to one in-
dicate phase synchronization. A value close to 1 is obtained for an almost constant phase
5
(1) (2)
differenceof mφ −nφ whichhasalsobeenobservedbetweentwononidenticalchaotic
t t
oscillators [70(cid:12)]. R2 has oft(cid:12)en been used for testing phase synchronization and there exist
(cid:12) (cid:12)
several articles where the distribution of R2 or other test statistics is approximated by dif-
ferent methods - e.g. with surrogate data. The most advanced approach is [72] where the
distribution of R2 under the null hypothesis R2 = 0 is approximated by using two indepen-
dent stochastic processes. In [2] different tests are presented by using a two sample test
setupandby utilizing insightsandmethodsfrom directionalstatistics andbootstrap theory.
Comprehensive reviews and comparisons of various phase synchronization measures
andtests can befound,in [67], [2], and[72].
Another important aspect of phase synchronization is directional dependency and there
exist approaches based on information theory with surrogate data [55] and mutual pre-
dictability [69] to detect the direction of coupling. A practical example of a phase syn-
chronized system with unidirectional coupling is the cardiac and respiratory system. [69]
providedevidencethattherespirationisdrivingthecardiovascularsystem. Wementionthat
manyotherdirectionalmeasuresexistwhichareassociatedwithothertypesofsynchroniza-
tion(cf. [56]andthe referencesgiventhere).
Despite of these results a good stochastic model for the phase processes themselves in
which questions like synchronization and unidirectional coupling can be investigated is still
missing. We showin this paperthatcointegratedsystems providesuch a framework.
2.2 Cointegration as a stochastic Kuramoto-type model
As mentioned in the introduction Kuramoto [44, 45] introduced a mathematical model to
describe the synchronization of a larger population of interacting units. A simple version of
theKuramotomodelforthe phasesofd oscillators is
d
K
φ˙(i) = ω + sin φ(j)−φ(i) (i =1,...,d). (4)
i
d
j=1
X (cid:0) (cid:1)
AllefeldandKurths[1]discuss thefollowingstochastic generalizationofthe model
d
φ˙(i) = ω + k sin φ(j)−φ(i) +η (i = 1,...,d) (5)
i ij i
j=1
X (cid:0) (cid:1)
wherethe η aretaken to beindependentGaussianwhite noisevariables.
i
There is a striking similarity of this model to cointegration: For φ(j)−φ(i) small we can
approximate sin(φ(j)−φ(i)) by φ(j)−φ(i) leading to the discretized approximation for the
6
(1) (d) ′ ′
vectorφ = φ ,...,φ ,ω = ω ,...,ω
t t t 1 d
(cid:0) (cid:1) (cid:0) (cid:1)
∆φt = ω+Πφt−1+ηt (6)
with Π = K−diag d k ,..., d k where K = (k ) . This is exactly the
j=1 1j j=1 dj ij i,j=1,...,d
representation (8) fr(cid:0)oPm below withPp = 1 a(cid:1)nd ω = µ, i.e. the cointegrated system (10), (9)
withp = 1 is exactly the solutionofthe stochastic difference equations(6). Wementionthat
cointegrationrequiresthe matrix Π to be of reduced rank which is fulfilled in this case since
thecolumnssum upto 0.
Being a bit more specific at this point gives us a deeper insight into the relevance of
cointegration theory for phase synchronization in networks: The above Π can be written in
′
the form Π = αβ with d×r - matrices α and β of full rank r where (in this case) r = d−1,
namely
1 0 ··· 0 −1
′
β
1 0 1 ··· 0 −1
β′ = ...  = .. .. .. 
. . .
′
β  
 r 0 0 ··· 1 −1
   
  ′
and α consisting of the first d − 1 columns of Π (only for this specific β). The β φ = 0
j t
(1) (d)
are obviously the d−1 phase synchronization relations φ = ··· = φ . The matrix α is
t t
sometimescalled adjustmentorloadingmatrix.
In the following we apply the concept of cointegration to phase synchronization. For the
model(6)thismeanstestingforcointegration(phasesynchronization)viadetermining(test-
′
ing)r = rankΠ. ThisresultsinthereducedrankrepresentationΠ=αβ withd×r-matrices
′
α and β of full rank r. After the parameters α and β have been estimated β φ are the r
t
cointegration(phasesynchronization)relationsandαgivestheintensitieswithwhichdevia-
tionsfromthe equilibriumleadto corrections. In particularαcan betested forunidirectional
coupling.
In addition it is very interesting to see that the vector ω splits under cointegration to a
lineardrift term ω (basicfrequency)anda constantterm intheequilibriumβ (meanphase
0 0
shift) transforming (6)finally to
′
∆φt = ω0+α β φt−1−β0 +ηt
(cid:0) (cid:1)
′
meaningthat the “true phase synchronization relations” are β φt−1 −β0 = 0. Of course the
correspondingKuramotomodel(4)and(5)shouldthencontain sin(φ(j)−φ(i)−β(ij))instead
0
of sin(φ(j)−φ(i)).
7
In the example of Section 5 we apply this analysis with the more general cointegration
model(14), (15)in the case d = 2 to datacomingfrom a coupledRössler-Lorenzsystem.
There is some connection to the work [4] where also the linearized dynamics of the Ku-
ramoto model is investigated (however in a deterministic framework). As in that work the
phase synchronization relations come out as the eigenvectors of some eigenvalueproblem
withthestrengthofthesynchronizationbeingdeterminedbythemagnitudeoftheeigenval-
ues. Dueto the stochastic structure theeigenvalueproblemhoweveris different.
Summarizing: in the case of phase synchronization, where (say) φ(j) −φ(i) is small, the
model (6) is almost equivalent to a stochastic Kuramoto model; in the case of non phase
synchronization the model (6) (or better its generalization (8)) is also a proper stochastic
modelfor the phase processes (in this case rankΠ = 0 or d). As a consequencethe model
(6) can be fitted in both situations (phase synchronization / non phase synchronization )
to the data and the hypothesis of phase synchronization can be tested on the matrix Π -
e.g. by looking at the fitted model. In addition one is able to identify the parameters of
the “Kuramoto-type” model from real data, and to conclude to the phase synchronization
relations,undirectionalcouplingetc.
Under the hypothesis of phase synchronization the difference between the two models
(5),(6) (resultingfromthereplacementofsin(φ(j)−φ(i))withφ(j)−φ(i))issmallbutofcourse
notnegligible. In the Kuramoto modelthe phasedifferences are employed modulo2π while
in the cointegration model the ordinary differences between the phases are employed. In
practisethismeansforexamplethattwoprocesseswhosephasedifferenceisclose tozero
atthebeginningbutmovesatacertaintimepoint(quickly)to2π maybephasesynchronized
intheclassicalsensebutnotphasesynchronizedinthecointegrationbaseddefinitiongiven
below(this happensexactly in the exampleofSection 5for ǫ = 9.6,10.2 –see Figure4).
3 The concept of cointegration
3.1 Cointegration and the Johansen-Granger representation theorem
We now briefly review a small part of the theory of cointegration (for more details see e.g
the monographs[35, 49, 36] - the last onecontaininga more appliedview). As it is usualin
manypaperswerestrict ourselvesto vectorautoregressive (VAR-)processes
Xt = A1Xt−1+...+AkXt−p +µ+ηt (7)
8