Information transfer at multiple scales

Max Lungarella* and Alex PittiERATO Asada Synergistic Intelligence Project, JST, The University of Tokyo, 113-8656 Tokyo, Japan

Yasuo KuniyoshiDepartment of Mechano-Informatics, The University of Tokyo, 113-8656 Tokyo, Japan

Received 20 April 2007; published 27 November 2007

In the study of complex systems a fundamental issue is the mapping of the networks of interaction betweenconstituent subsystems of a complex system or between multiple complex systems. Such networks define theweb of dependencies and patterns of continuous and dynamic coupling between the system’s elements char-acterized by directed flow of information spanning multiple spatial and temporal scales. Here, we propose awavelet-based extension of transfer entropy to measure directional transfer of information between coupledsystems at multiple time scales and demonstrate its effectiveness by studying a three artificial maps, bphysiological recordings, and c the time series recorded from a chaos-controlled simulated robot. Limitationsand potential extensions of the proposed method are discussed.

DOI: 10.1103/PhysRevE.76.056117 PACS numbers: 89.75.k, 05.45.Tp, 89.70.c

I. INTRODUCTION

The question often occurs of how to identify and charac-terize information structure and hidden dependencies be-tween the components of a complex dynamical system orbetween multiple complex systems, given only a set of si-multaneously recorded typically multivariate time series. Inparticular, it may be required to identify causal dependenciesor quantify flow of information. Such flow can beunidirectional—i.e., time series Xt affects time series Yt,while Yt has no influence on Xt— or in the general casebidirectional. Moreover, because complex systems operateacross multiple spatial and temporal scales, it is likely thatinformation flow will exhibit a multiscale structure.

Cross-correlation is a common way of evaluating the sta-tistical dependencies among data sets; it is not only effi-ciently computed but has also an appealing and natural inter-pretation. Cross-correlation, however, is a second-orderstatistics handling merely linear dependencies. Mutual infor-mation provides an attractive way of circumventing the re-strictions of cross-correlation because it is sensitive tohigher-order relationships, both linear and nonlinear ones.Yet, mutual information is a symmetric measure and fails todetect directional asymmetric flow of information unlessone of the time series is delayed 1. Measures of interdepen-dence aimed at overcoming the limitations of mutual infor-mation have recently been introduced, e.g., extendedGranger causality 2,3, nonlinear Granger causality 4,similarity index 5, predictability improvement 6, andtransfer entropy 7. All these measures, however, do notaccount for features related to structure on scales other thanthe shortest one.

Here, we introduce a multiscale extension of transfer en-tropy 7 as a tool for understanding the internal dynamics ofcomplex systems. By projecting a time series into the wave-let space, a new set of variables is obtained, whose statistics

allow us to extract causal dependencies between the vari-ables. In what follows, we show that our wavelet-based ap-proach augments “sample-based” embedding vector ap-proaches by disclosing previously undetectable causalstructure in potentially nonstationary and discontinuous sig-nals. Our analysis is applicable to a wide range of nonsta-tionary physical and biological signals, regardless of whetherthe underlying fluctuations have stochastic origins or arisefrom nonlinear dynamical processes.

The paper is organized as follows. We first introduce awavelet-based multiscale extension of transfer entropy.Transfer entropy builds upon conditional entropy, which incontrast to the standard form of mutual information, ordelayed-related measures, allows one to distinguish actually“transported” from shared information. We then illustrate theapplication of our method by studying three synthetic datasets, a physiological time series, and data collected using asimulated robot. Next, we discuss the results, point to someof the shortcomings of the extension, and eventually drawsome conclusions.

II. METHODS

In this section we give the mathematical background forthe proposed measure. We first outline some formalism andthen describe the method. The method was implemented inMATLAB R7.2 MATHWORKS, MA and the MATLAB wavelettoolbox using purpose-built script files.

A. Transfer entropy

We are interested in detecting directed exchange of infor-mation between two distinct dynamical systems X and Y andin comparing their mutual influence. A good approach toobtain knowledge about asymmetric dependencies betweencoupled systems or the direction and strength of the cou-pling between subsystems is to measure to which extent itsindividual components contribute to information productionand at what rate they exchange information among each*Corresponding author. lunga@ifi.uzh.ch

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other. Transfer entropy 7 is an appropriate starting pointbecause it is an information theoretic measure which notonly shares some of the properties of mutual information butalso takes the dynamics of information transport into ac-count. It is a specific version of the mutual information forconditional probabilities, but unlike mutual information it isdesigned to detect information transferred between two sys-tems, separate for both directions, and conditional to com-mon history and input signals.

Let Xt= xtt=1,. . .,N and Yt= ytt=1,. . .,N denote bivariate sta-tionary time series simultaneously measured from X and Y.We start by reconstructing the dynamics of the two timeseries by the method of time delay embedding 8,9. Thestate or embedding vector of dimension m of Xt is formedby delayed past scalar observations of the time series:

xtm,d = xt,xt−d,xt−2d, . . . ,xt−m−1dT, 1

where d is the time delay or lag between successive ele-ments of the state vector, and t=+1, . . . ,N with embeddingwindow = m−1d. If not otherwise stated in this paper d=1, and we use the following expression:

xtm = xt,xt−1,xt−2, . . . ,xt−m+1T. 2

Similarly, we can define ytm,d and yt

m. Note that the statevectors xt

m,d and ytm,d are points in the m-dimensional spaces

X and Y. To reconstruct the state spaces of the time series,the embedding dimension m and the time delay d have to bedetermined. The choice of both parameters depends on thedynamics of the underlying data.

Essentially, transfer entropy measures the deviation fromthe generalized Markov property

pxt+1xtk = pxt+1xt

k,ytl , 3

where p · · denotes the transition probability, and k and lare the dimensions of the two delay vectors. If the deviationfrom the generalized Markov process is small, i.e.,pxt+1xt

k pxt+1xtk ,yt

l, then it can be assumed that thestate vectors of space Y have no or little influence on thetransition probabilities of the state vectors of space X. If thedeviation is large, however, then the assumption of the Mar-kov process is not valid. The incorrectness of the assumptioncan be quantified by the transfer entropy which is formulatedas a special instance of the Kullback-Leibler entropy be-tween pxt+1xt

k and pxt+1xtk ,yt

l 7:

TY→X = xt+1,xt

k,ytl

pxt+1,xtk,yt

llogpxt+1xt

k,ytl

pxt+1xtk

, 4

where xt and yt represent the states of time series Xt and Yt attime t. An appealing feature of the transfer entropy is that—unlike mutual information or conditional entropy—it is ex-plicitly nonsymmetric under the exchange of Xt and Yt asimilar expression exists for TX→Y. It can thus be used todetect coupling and directional transport between two sys-tems. In other words, the transfer entropy represents the in-formation about a future observation xt+1 obtained from thesimultaneous observation of k past values of Xt and l past

values of Yt, after discarding the information about the futureof Xt obtained from k past values of Xt alone.

The transfer entropy is obtained by approximating thejoint probabilities pxt+1 ,xt

k ,ytl through kernel density esti-

mation and by coarse-graining the continuous state space at aresolution r. For computational reasons we set k= l=1 andobtain:

przt =1

Lt

r − zt − zt , 5

where zt= xt+1 ,xt ,ytT, · is the Euclidean distance, and Lis the number of pairs t , t which are selected to excludedynamically correlated pairs. As in 7, we use a step kernel:x0=1; x0=0 that is, the probabilities are evalu-ated as normalized histograms. Alternative kernels can befound in 10. The conditional probabilities required for Eq.4 are then calculated from the joint probabilities.

When the available data is limited number of samplesN1000 and the coupling between the time series is small,transfer entropy suffers from a finite sample effect in par-ticular for small resolutions, r0.05 which makes the as-sessment of the significance of the obtained values difficult11. To counter this problem, a slightly modified “finitesample”-corrected variant of transfer entropy, called effectivetransfer entropy, has been proposed 12. Note that we do notdeal with the finite sample problem here. For all our experi-ments N1000 and r0.05, and we can safely assume thatour results are affected by a finite sample problem just to asmall degree.

B. Wavelet-based extension

Multiresolution wavelet analysis is a useful technique foranalyzing signals at multiple scales, even in the presence ofnonstationarities which often obscure such signals 13,14.Wavelet analysis is known for its superiority to both conven-tional and short-time windowed Fourier analysis when itcomes to time and frequency localization 14,15, and thuslends itself particularly well as a tool for analyzing nonsta-tionary and discontinuous signals. Wavelet coefficients notonly provide a compact representation of the power distribu-tion in time and scale or frequency, but they also captureinformation about the higher frequency oscillatory propertiesand long-term trends.

A doubly indexed family of wavelet coefficients Vs, canbe extracted from Xt by convolving Xt with a scaled andtranslated version of a function called the mother wavelet orprototype function an analogous family Ws, can bedefined for Yt:

Vs, =1s

t=1

N

Xt t −

s , 6

where the * denotes the complex conjugate, s0 is thescale dilation parameter corresponding to the width of thewavelet, and R is the translation parameter indicating thelocation of the wavelet function as the prototype function isshifted through the signal the point of reference of the con-

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volution. The scale s determines the width of the waveletfunction and hence its resolution. Smaller scales correspondto more rapid variations and therefore to higher frequencies;that is, the larger the scale, the wider is the wavelet, and themore the details are smeared out. In this regard, the wavelettransform resembles a series of bandpass filters. Each filtermay be assigned an equivalent Fourier period which gives anidea of the frequency content of the signal analyzed. This isdone by substituting a cosine wave of a known frequencyinto Eq. 6 and by computing the scale that matches themaximum wavelet power spectrum.

Many choices for the mother wavelet functions exist thatdiffer according to symmetry, width of support, and orthogo-nality 13. Here, we use the complex MORLET wavelet as itis characterized by a good trade-off between time and fre-quency localization 36:

= −1/4ei0e−2/2, 7

where 0 is the normalized frequency and is here taken to be0 5,6. With reference to the MATLAB wavelet toolbox,the mother wavelets used are cmor2–0.9549 0=6; syn-thetic and physiological data; Secs. III A and III B andcmor2–0.7958 0=5; robot data; Sec. III C. For the MOR-

LET wavelet the relationship between scale and equivalentFourier period is =1.03 s.

We extend transfer entropy by mapping the informationtransfer in the space spanned by the wavelet coefficients. Theformulation is straightforward. First, the wavelet coefficientsVs, and Ws, are extracted from the time series Xt and Yt forthe scales s=s0 , . . . ,sn. The mth scale is sm=s02m/V, wheres0R+− 0 and V is a positive integer number correspond-ing to the number of scales for each octave. The smallestresolvable scale s0 m=0 should be chosen so that theequivalent Fourier period s02Ts where Ts is the sam-pling period. For V 4 the wavelets constructed are quasi-orthogonal and the scales quasi-independent 13,14 for allour analyses, V 8. In analogy to Eq. 2, we define two“embedding” vectors using the wavelet coefficients:

Vs, = Vs1,, . . . ,Vsn,T, 8

Ws, = Ws1,, . . . ,Wsn,T. 9

Assuming that Vs, and Ws, are “lag-stationary” the lag isdenoted by , one can express the multiscale extension ofEq. 4 for k= l=1 as

TY→Xs = pVs,+1,Vs,,Ws,logpVs,+1Vs,,Ws,

pVs,+1Vs,,

10

TX→Ys = pWs,+1,Ws,,Vs,logpWs,+1Ws,,Vs,

pWs,+1Ws,,

11

where s0 indicates the scale. If TY→Xs−TX→Ys0, in-formation flows from Y to X at scale s, and vice versa ifTX→Ys−TY→Xs0, information flows from X to Y.

One pitfall of this measure as well as of other measures16 is that it does not readily allow us to assess the signifi-cance of the result. This problem can be addressed with theconcept of surrogates 17. Here, we generate a set of surro-gates of Vs, ,Ws, by permuting the temporal indices ofWs, while keeping those of Vs, fixed. This operation effec-tively destroys the realizationwise pairings of the times se-ries while preserving their original statistical structure. Ap-plying transfer entropy to the original time series and to qsurrogate data sets—each obtained through a different ran-dom permutation—yields Ts ,q and allows testing the nullhypotheses that Xt and Yt are extracted from decoupled sys-tems. As in 18 or 12, however, our primary use of surro-gates is not to test this null hypothesis but rather for an offsetcorrection, which can be interpreted as a significance thresh-old. We finally define the offset-corrected wavelet-basedtransfer entropy as

Tcs = Ts − maxq

Ts,q . 12

In what follows, Tc,X→Y will indicate “offset-corrected” in-formation transferred from X to Y, whereas Tc,Y→X will indi-cate information transferred in the opposite direction. In allthe following numerical experiments, q=10.

III. EXPERIMENTS

We tested our method on five data sets: three unidirection-ally coupled chaotic maps, one physiological time series, anda data set generated using a robot.

A. Synthetic data

As a first experiment, for simplicity, we studied the sys-tem formed of two unidirectionally coupled autoregressiveprocesses of first order where one process contains a cosine:

xt+1 = 0.7xt + 0.7 cos0.3t + ntx0,2 , 13

yt+1 = 0.7yt + nty0,2 + ext, 14

where the parameter e 0,1 regulates the couplingstrength, and nt

x and nty are independent Gaussian random

processes with zero mean and variance 2=0.2. By construc-tion, the two time series are unidirectionally coupled withtime series Xt influencing time series Yt. For each coupling,we generated a bivariate time series Xt ,Yt random initialconditions extracted from a normal distribution with zeromean and unit variance; N=5104 samples, the first 2104 samples are discarded.

The parameters used for the analysis were r=0.125, 0=6, s0=0.5, V=10, and n=64 scales. Figure 1 displays theoffset-corrected directed information exchange between thetwo maps c=Tc,xt→yt

−Tc,yt→xtas a function of the coupling

strength e and scale s. Because for e=0 the two processes areindependent, c0 for any value of s. For increasing valuesof e the influence of Xt on Yt increases as demonstrated by

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the larger values of c. In particular, one can observe theeffect of the cosine on the dynamics of the coupled system: aslowly increasing ridge, rather well-localized in scale, in thescale interval 12,28. We also estimated the information ex-change as a function of the coupling e using the originalnonmultiscale definition of transfer entropy Fig. 4a; seealso 11. The comparison between the “traditional” and themultiscale approach reveals the advantage of the latter; thatis, a more fine-grained detection of information flow or,causal dependencies potentially allowing a better under-standing of the mechanisms underlying the dynamics of aparticular system.

In our second numerical experiment, we studied a one-dimensional ring lattice of 100 unidirectionally coupledUlam maps with periodic boundary conditions:

xt+1l = fext

l−1 + 1 − extl , 15

fu = 2 − u2, l = 1, . . . ,100, 16

where the parameter e 0,1 denotes the coupling strength,l is the lattice index, and t the time index. Note that byconstruction, information can be transported only in the di-rection of increasing l. For each coupling, we generated thetwo time series Xt

1l=1 and Xt2l=2 with a sample size of

N=2104, after 2104 steps of transients random initialconditions.

The analysis was conducted using the following param-eters: r=0.125, 0=6, s0=0.2, V=8, and n=64 scales. Figure2 shows the directed exchange of information c=Tc,xt

1→xt2

−Tc,xt2→xt

1 as a function of e and s. As evident from the figure,c correctly detects the gradual increase of coupling strengthfor increasing values of e, reflecting the causality of the sys-tem. This result is to be expected by the way the ring latticewas constructed, and for high values of the scale index isconsistent with what reported in the literature 6,7. For e=0.18 and 0.82, there is a sharp drop in the flow of informa-tion at all scales, and the estimated coupling is zero. Forthese two values of e, map l=1 and map l=2 are in a

generally synchronized state, that is, driver and responsesystem are indistinguishable from each other. As in the pre-vious numerical experiment, also in this case, we calculatedthe amount of information exchanged between the two se-lected neighboring maps using the original version of trans-fer entropy Fig. 4b and compared it to the proposed mul-tiscale approach. Although for coupling values e0.5 andfor scales s6.4 the trends are qualitatively similar, one canobserve a discrepancy for e0.5 and s6.4: at low scalesthe influence of a high coupling value is less pronounced.For s6.4, however, the relative influence of a higher cou-pling value is larger, hinting at a low frequency informationflow between the two maps.

In our third numerical experiment, we studied two unidi-rectionally coupled Henon maps:

xt+2 = 1.4 − xt+12 + 0.3xt, 17

yt+2 = 1.4 − ext+1 + 1 − eyt+1yt+1 + 0.3yt. 18

As in both previous systems, e 0,1 is the coupling pa-rameter. In this particular system, the maps synchronize fore0.7. For such values of the coupling, no quantifiable dif-ference between Xt and Yt exists, and the information trans-ferred from one map to the other is zero. The wavelet-basedtransfer entropy was calculated using N=2104 samples af-ter discarding the 2104 initial samples as transients.

The parameters for the analysis were r=0.125, 0=6, s0=0.2, V=10, and n=64 scales. The directed exchange of in-formation c=Tc,xt→yt

−Tc,yt→xtis shown in Fig. 3. The mea-

sure increases with the coupling strength e up to a criticalpoint e=0.7. For ee, the two maps synchronize at allscales and c0, as expected. As in the case of the coupledUlam maps, we note that the trends of both indices remainconsistent for all couplings and scales, i.e., Tc,xt→yt

0 andTc,yt→xt

0, reflecting the causality in the system. Our find-ings are consistent with the results reported in 2. As in thetwo previous numerical experiments, also here we comparedour multiscale approach with a “single scaled” one Fig.4c. There is a good match between the amount of ex-

12

48

1632 0

0.20.4

0.60.8

1

0

0.05

0.1

0.15

0.2

0.25

0.3

coupling

scale

Ωc

FIG. 1. Wavelet-based transfer entropy for coupled autoregres-sive processes xt

1 contains a cosine term. Displayed is c

=Tc,xt1→xt

2 −Txt2→xt

1 as a function of scale s and coupling strength e0=6, s0=0.5, V=10.

0.4

1.6

6.4

25.60

0.20.4

0.60.8

1

0

0.2

0.4

0.6

0.8

coupling

scale

Ωc

FIG. 2. Wavelet-based transfer entropy for coupled Ulam maps.Displayed is c=Txt

1→xt2 −Tc,xt

2→xt1 as a function of scale s and cou-

pling strength e 0=6, s0=0.2, V=8.

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changed information as a function of the coupling measuredby the wavelet-based multiscale extension at any singlescale s and the original version of transfer entropy.

B. Physiological data

For our fourth numerical experiment, we used a bivariatetime series extracted from a publicly available physiologicaldata set 19 consisting of the breath rate measured as lungpressure and the instantaneous heart rate of a sleeping hu-man patient suffering from a breathing disorder. Under nor-mal, physiological conditions, the heart rate is modulated byrespiration through a process known as respiratory sinus ar-rhythmia RSA. Sleep apnea a pathological condition char-acterized by brief interruptions of breathing during sleepaffects the normal process of RSA disturbing the usualphysiological patterns of interaction and feedback amongheart rate, respiration, and blood oxygen concentration. As aconsequence, the control of heart rate by respiration becomesunclear. We conjecture that the application of wavelet-basedtransfer entropy may provide additional insights into the na-ture of the interaction. In particular because it is likely thatthe dynamics of mutual interaction among physiologicalvariables act on different a priori unknown time horizons.

Figure 5 reproduces the data set that we analyzed. Thefigure indicates that bursts of breath and beat-to-beat alter-ations in heart rate heart rate variability HRV are interde-pendent samples no. 305–4305; approximately 32 min. Theparameters used for the analysis were r=0.16, 0=6,s0=0.9, V=60, and n=256 scales. Figure 6 shows that ourmultiscale approach finds bidirectional flow of informationfrom heart rate to breath rate, and vice versa. The graphalso illustrates that at lower scales s 0.0,1.463,corresponding to the frequency interval 0.321–2.0 Hz theflow of information from heart rate to breath rate Tc,H→B islarger than the one from breath rate to heart rate Tc,B→H.This result is consistent with the findings reportedin 2,7,20. Similarly, at higher scales s 5.5 ,8.0 ;0.058–0.086 Hz, the causal influence of heart rate onbreath rate is larger than vice versa Tc,H→BTc,B→H. For

s8.0 f 0.058 Hz, however, the direction of the causalinteraction is inverted, i.e., Tc,H→BTc,B→H. Such inversionof information flow is a documented phenomenon 21, butto our knowledge no conclusive evidence exists as to why itoccurs. Our result may point to a hidden low frequencymechanism f 0.06 Hz underlying the causal influencethat may be worthwhile investigating further.

Another interesting result is that Tc,B→H0 in the scaleinterval 1.21, 3.15 0.15–0.39 Hz; whereas Tc,H→B=0 inthe interval 1.46, 2.4 0.19–0.32 Hz and is 0.01 for 1.2

0.40.8

1.63.2

6.412.8 0

0.20.4

0.60.8

1

0

0.1

0.2

0.3

0.4

0.5

coupling

scale

Ωc

FIG. 3. Wavelet-based transfer entropy for coupled Henonmaps. Displayed is c=Tc,x→y −Ty→x as a function of scale s andcoupling strength e 0=6, s0=0.2, V=10.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

coupling

tran

sfer

entr

opy

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

coupling

tran

sfer

entr

opy

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

coupling

tran

sfer

entr

opy

FIG. 4. Transfer entropy as a function of the coupling e for acoupled autoregressive processes Txt→yt

: dotted, Tyt→xt: solid; b

coupled Ulam maps Txt1→xt

2: dotted, Txt2→xt

1: solid; and c coupledHenon maps Txt→yt

: dotted, Tyt→xt: solid.

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s4.0 which hints at a weak coupling. This finding isconsistent with the known fact that the high frequency com-ponent of HRV HF, 0.15–0.45 Hz is synchronous with therespiratory cycle and is identical to the RSA which is due tothe parasympathetic activity. Notable are also the high val-ues of Tc,B→H and Tc,H→B in the scale interval 5.5, 11.50.0418–0.0625 Hz which corresponds to the lower part ofthe low frequency component of the HRV spectrum LF,0.04–0.15 Hz. Note that all our findings are plausible underthe assumption that the processes underlying heart rate andbreath rate dynamics mutually causally influence each otherthat is, both can either act as driver or as response. Toconclude this forth application scenario, we point out that theproposed multiscale approach is rather specific and reason-ably sensitive even when applied to a realistic data set, with-out requiring too much parameter tuning.

C. Robot data

The goal of our last application is to identify and mapinformation flow in a simple robot composed of one sensorand one actuator; henceforth the “controlled system”coupled to a chaotic map henceforth the “controller”. We

opted for a chaotic controller not only because it representsone of the most exciting theoretical outcomes of contempo-rary dynamical systems theory 22,23, but also because sucha controller might provide an interesting alternative to con-ventional implementations. We emphasize that the goal ofthe present paper is not to discuss the chaos-based controlframework which will be explained in depth in another pa-per.

Specifically, we collected field data using a two-dimensional two-link model kinematically equivalent to adouble pendulum and vaguely resembling a human walkingon level ground Fig. 7. To simplify our analysis we re-stricted the motion of the biped to the sagittal plane, forcingthe robot to behave like a compass hence the namecompass-gait biped robot. The biped has two rigid knee-less legs of point mass m connected by a frictionless hingejoint at the hip. A third point mass mH representing the massof the upper body is concentrated at the same joint. We as-sume that when the nonsupporting foot hits the ground itsvelocity jumps instantaneously to zero. That foot then be-comes the supporting leg and stays on the ground acting likea hinge until the other foot touches the ground. The impact ofthe foot with the ground is modeled as an inelastic no-slip,no-bounce collision see below—energy is lost at each im-pact and in order for the system to keep walking withoutfalling, energy must be supplied externally.

The configuration of the compass-gait biped is determinedby

= ns,sT, 19

where ns and s are the angles spanned by the nonsupportswing and the support stance leg with the vertical, respec-

tively, and ns and s are their temporal derivatives see Fig.7. The equation of motion governing the dynamic behaviorof the biped is

4 8 12 16 20 24 28 32

−2

0

2

time (min)

hear

trat

e

4 8 12 16 20 24 28 32−5

0

5

time (min)

brea

thra

te

FIG. 5. Simultaneously recorded time series of heart rate topand lung pressure breath rate bottom. The data corresponds tosamples no. 305–4305 of the Santa Fe data set B archived in Phy-sioNet databank 19. The sampling rate is fs=2 Hz. Both tracesare normalized to zero mean and unit variance.

0.91 1.16 1.89 2.73 3.95 5.71 8.66 11.97 17.32

0.01

0.03

0.05

0.07

0.09

0.11

scale

Tc

TH−>B

TB−>H

FIG. 6. Wavelet-based transfer entropy Tc,B→H solid andTc,H→B dashed. B=breath, H=heart. The parameters used are: fs

=2 Hz, s0=0.9, V=60, n=256, and 0=6.

FIG. 7. Lumped model of compass-gait biped robot. The massof each leg is m; the mass of the hip is mH. The length of each legis l divided into a segment a and a segment b. s and ns are theangles formed by the supporting stance and the nonsupportingswing leg with the vertical counterclockwise positive. The inter-leg angle is ns−s and g is the gravity vector.

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M + C, + g = S 20

with

M = mb2 − mbl coss − ns− mbl coss − ns m + mHl2 + ma2 ,

21

C, = 0 − mbl sinns − ss

− mbl sinns − sns 0 ,

22

g = g mb sin ns

− mHl + ma + mlsin s , 23

where M is the 22 inertia matrix of the robot, C , isa 22 matrix containing centrifugal and Coriolis terms,g is a 21 vector of gravitational torques, S= −1,1T isa conversion vector, and is the torque applied to the hipjoint. The values of the parameters used in the simulation canbe found in Table I. The equations of motion were numeri-cally integrated using a fourth order Runge-Kutta with inte-gration step t=510−3 s.

Equation 20 adequately expresses the dynamics of thebiped when one foot is on the ground while the other leg isswinging. The impact of the swinging feet with the ground,however, requires some additional considerations. Here, tomodel such impact we make the following simplifying as-sumptions: a the impact is perfectly plastic no-bouncecondition; b the legs do not slip no-slip condition; csupport is instantaneously transferred from the supportingleg to the swing nonsupporting leg; and d the angularmomentum is conserved. For further elaborations on this is-sue, refer to 24.

The controller applies torque to the hip at continuous timetkR0

+, with tk+1 tk, t0=0; in all our simulations, tk=kt.The control events can be described by the following equa-tions:

n = 2cn,

cn = 1 − 11 − cn−12 + 1f ,

f = s − ns, 24

where the control constant 2 determines the couplingstrength between the dynamics of the controller and the con-trolled system via torque, the parameter 1 sets the influ-ence of the feedback term onto the evolution equation of the

logistic map xn= 1−xn−12 , and is the bifurcation pa-

rameter. For all our simulations, =1.95, for which the lo-gistic map exhibits chaotic dynamics.

We studied two control strategies: 1 open-loop control1=0 and =20; the actuation dynamics and thus thetorque n applied to the hip joint are independent of the sen-sory feedback f; and 2 closed-loop control =1=2

0; the state cn of the chaotic map is affected by the con-trolled system via feedback, and conversely the controlleraffects the controlled system. In both control scenarios, wevaried the coupling parameter in the interval 0.03, 0.33with increments of 0.003 and measured the wavelet-basedtransfer entropy between the torque controlled variableand the interleg angular velocity s− ns the measured vari-able. All runs had a maximum length of 15 000 time stepsequivalent to 75 s and a minimum length of 1200 timesteps equivalent to 6 s. If the biped fell before the 6 s limit,the trial was labeled as unsuccessful, and the transfer entropywas set to zero. For every value of , we performed 30

runs with t=0 and t=0 uniformly sampled from−0.4,0.4 −0.4,0.4. An excerpt from a typical robot dataset for =1=2=0.11 is reproduced in Fig. 8. We note thatother control scenarios would have been possible, e.g., withasymmetric coupling between controller and controlled sys-tems, that is, 12 or 12. For the sake of simplicity,however, we studied only open-loop control and “symmet-ric” closed-loop control.

Figures 9–12 show the results of our analysis 0=5, s0=1, V=10, and n=256 scales. We can make a few observa-tions. First off, in the closed-loop case the information trans-fer from the controlled system body to the controller cha-otic “neural” system is clearly localized in frequency Fig.12. In other words, feedback of “meaningful” sensory infor-mation occurs at specific time scales. In the figure two peaksare discernible, one located in the scale interval 45, 60, theother in the interval 100, 140 roughly equivalent to thefrequency intervals 2.7–3.6 Hz and 1.1–1.8 Hz, respec-tively. Such localization is clearly a result of the feedbackloop and is absent in the open-loop case. The time scale ofthe natural dynamics of the biped lies in the same range. Forexample, in Fig. 8 the stride frequency is 1.47 Hz. A secondobservation is that in the closed-loop case there is an evidentasymmetry between information flowing into the controlledsystem Tc,b→n peak: 0.445 bit and information flow out of itTc,n→b peak: 0.021 bit—despite symmetry in the coupling

TABLE I. Parameters used for the simulated compasswalker.

mH

kgm

kga

mb

ml=a+b

mg

m /s2

10.0 5.0 0.5 0.5 1.0 9.81

23 24 25 26−0.1

−0.05

0

0.05

0.1

time (s)

torq

ue(N

m)

23 24 25 26

−0.2

0

0.2

time (s)

inte

r−le

gan

gle

(rad

)

23 24 25 26−0.5

0

0.5

1

1.5

time (s)

inte

r−le

gan

g.sp

eed

(rad

/s)

FIG. 8. Time series excerpt for closed-loop control scenario=0.11. Torque applied to hip joint in Nm; interleg angle in rad;interleg angular speed in rad/s. The sampling frequency is 200 Hz.

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1=2. One possible explanation is that for “good” valuesof , the self-stabilizing properties of the biped robot allow itto operate at a stable pseudoperiodic limit cycle. In a sense,the natural dynamics of the coupled system and the physicalinteraction induce essential information structure in the sen-sory feedback. This hypothesis is validated by Figs. 10 and12 which show that the information flow is concentrated atlower frequencies. In both control scenarios, the controlleraffects the controlled system over a larger range of frequen-cies than vice versa see Figs. 9 and 10—as it ought to be,given that the controller is a chaotic system.

We further observe that a small amount of control infor-mation can have a large effect. In the closed-loop case, forinstance, the flow of information from controller to con-trolled system Tc,n→b is one order of magnitude smaller thanthe flow from controlled system to controller Tc,b→n. Thisresult might be relevant from a design point of view. Byfocusing more on the natural dynamics of the controlled sys-tem i.e., its body dynamics and less on control that is,strong coupling between controlled system and controlled, itmight be possible to design more energy-efficient roboticsystems e.g., 25,26. For instance, the positive mechanicalwork per cycle in the case of the symmetric closed-loop con-troller for =0.11 is approximately E+=29 mJ /cycle due tothe instantaneous inelastic collisions of the feet with the

ground energy is lost; whereas for the open-loop controllerE+=47 mJ /cycle. These are average values obtained overten successful runs. A last observation concerns the open-loop case. For 0.4, the causal influence of the controlleron the controlled system is small, and the self-stabilizingproperties of the latter are sufficient for walking to be ratherstable the robot walks stably for more than 10 s see Fig.13. For 0.4, however, such influence is too strong andwalking does not occur. The robot falls often and on theaverage walks stably only for less than 7.5 s data notshown.

IV. SUMMARY AND DISCUSSION

Systems in the real world are typically regulated by acomplex web of dependencies and couplings exchanging in-formation at multiple spatial and temporal scales. In the fieldof experimental signal analysis, a fundamental problem is totell if a given set of scalar observations originates from in-teracting or noninteracting systems. In this paper we haveproposed a wavelet-based extension of transfer entropy toanalyze the causal dependencies between bivariate time se-ries across multiple time scales. Using five different data setscollected from different dynamical systems, we have shown

1 2 4 8 16 32 64 128 256

0.030.09

0.150.21

0.270.33

0

0.03

0.06

0.09

0.12

scalecoupling

Tc,

b→

n

FIG. 10. Wavelet-based transfer entropy for open-loop system.Displayed is Tc,b→n, where “b” indicates the controlled systembody and “n” indicates the controller neural system.

1 2 4 8 16 32 64 128 256

0.030.09

0.150.21

0.270.33

0

0.01

0.02

0.03

0.04

scalecoupling

Tc,

n→

b

FIG. 11. Wavelet-based transfer entropy for closed-loop system.Displayed is Tc,n→b, where “n” indicates the controller neural sys-tem and “b” indicates the controlled system body.

1 2 4 8 16 32 64 128 256

0.030.09

0.150.21

0.270.33

0

0.1

0.2

0.3

0.4

0.5

scalecoupling

Tc,

b→

n

FIG. 12. Wavelet-based transfer entropy for closed-loop system.Displayed is Tc,b→n, where “b” indicates the controlled systembody and “n” indicates the controller neural system.

1 2 4 8 16 32 64 128 256

0.030.09

0.150.21

0.270.33

0

0.01

0.02

0.03

0.04

scalecoupling

Tc,

n→

b

FIG. 9. Wavelet-based transfer entropy for open-loop system.Displayed is Tc,n→b, where “n” indicates the controller neural sys-tem and “b” indicates the controlled system body.

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that our approach yields consistent findings when applied toassess multiscale information flow between artificial, physi-ological, and robotic time series. Our experimental resultsare consistent with previous work confirming the effective-ness and validity of our approach. Our method is furthermorecapable of detecting scale-dependent causal dependencies ina physiological data set undetectable with previous methodsand provides an innovative view on information flow in asimulated chaos-based biped walker. Although at smallscales high frequencies our approach delivers qualitativelysimilar results to other techniques 2,3,20,27, it yields in-sights about potential interactions at higher scales lower fre-quencies. Our wavelet-based method appears to yield ameaningful approach to capture the coarse-to-fine “causal”structure of the interaction between coupled dynamical sys-tems.

One particular problem affecting the current implementa-tion is the finite sample size effect which grows fast as thenumber of data points is reduced. The method proposed hererequires an adequate amount of data in order to provide suf-ficient statistical accuracy to estimate the transfer entropy.Throughout the paper, it was made sure that NNmin3 /r3 11 e.g., for the physiological time series r=0.16and N=4000 samples, whereas Nmin=733. The finite sampleeffect is a known problem, and could be addressed either byreplacing transfer entropy with another measure of causality21, or by choosing a better kernel density estimator 28.Alternatively, one could replace the kernel density estimatorby some adaptive binning strategy, such as the one proposedin 21. A related issue, particularly relevant in the case of alow number of data, is how to assess the significance of thedetected multiscale information transfer. Here we built sur-rogate time series by shuffling randomly reordering the se-quence of data points. In this way, we destroyed the correla-tions among the data points while preserving the statisticalproperties of the distribution, particularly, the first and sec-ond order moments. A similar technique was used in 18.

Another limitation of the proposed method is its bivariatenature. Some recent studies have emphasized the importanceof a multivariate approach 21,29. Moreover, many multi

scale approaches to assess the complexity or the informationflow between interconnected parts of a complex system havebeen suggested e.g., 30–32. Typically, multivariate non-linear interdependencies between subsystems at multiplescales are analyzed by resampling the original time series atvarious scales yielding a collection of data possessing differ-ent coarse-graining from which various measures can be cal-culated e.g., entropy, mutual information. For instance, in33 it was shown that the heart-beat time series of healthypeople asymptotically approaches a constant value of en-tropy as the measurement scale is increased. On the otherhand, the analysis demonstrates that in patients sufferingfrom “atrial fibrillation” or “congestive heart failure” the en-tropy as a function of time scale shows a consistently differ-ent pattern. The usage of such tools for diagnostic purposesis invaluable. Especially because the relevant time scales ofsuch measured time series, such as the cardiac cycle, are notknown a priori. Furthermore, they can be modified by auto-nomic blockades and under pathophysiological conditions.The approach introduced in this paper can be readily ex-tended to incorporate multivariate techniques. One could, forinstance, replace the original time series of any multivariatetechnique by the wavelet coefficients and then perform theanalysis in scale space. Or, one could achieve the same resultby appropriately conditioning the transition probabilities inEq. 4 7,21.

In this paper, we have considered the information transfercausal structure on the same scale. It might be worthwhileconsidering also information flow between different scales34. This could lead to meaningful insights into the dynam-ics of many biological systems in which processes workingat dissimilar time scales coexist, e.g., the time scale of neuralactivity and the time scale of actions in the real world 30.Such processes describe, in a sense, the informational em-bedding of organisms within their ecological niches at mul-tiple time scales defining continuous and dynamic couplingbetween sensory, neural, and motor variables 35. The com-parison of the relative influence such variables exert on eachother helps extract patterns of interaction between the net-work’s elements that may support biological informationprocessing. The measure introduced in this paper, which byusing transfer entropy allows capturing directed exchangesof information information flow between sensory and mo-tor variables in a physically embedded system, might proveuseful for mapping the intrinsic dynamics of such sensorimo-tor networks.

To conclude, we suggest that the combination of wavelettransform and transfer entropy or any other time seriesbased measure to detect information transfer or couplingscan help shed light on the hidden causal dependencies in thecoarse structure of the data from a wide range of complexsystems, which a sample-based approach working at onesingle time scale is not able to disclose.

ACKNOWLEDGMENTS

We thank two anonymous reviewers for their valuablecomments on the draft of the manuscript.

0.03 0.09 0.15 0.21 0.27 0.33

5

15

25

35

45

55

65

75

coupling γ

aver

age

wal

king

time

[sec

]open,avgopen,minclosed,avgclosed,min

FIG. 13. Average walking time before falling of the compass-gait biped. On the average open-loop control leads to shorter walk-ing times due to its inherent instability.

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