Assessing observability of chaotic systems using Delay Differential Analysis
Christopher E. Gonzalez, Claudia Lainscsek, Terrence J. Sejnowski, Christophe Letellier
AAssessing observability of chaotic systems using Delay DifferentialAnalysis
Christopher E. Gonzalez ∗ , , , Claudia Lainscsek ∗ , , , Terrence J. Sejnowski , , , and ChristopheLetellier ∗ These authors contributed equally to this work. . Corresponding author: Christopher Gonzalez [email protected] Computational Neurobiology Laboratory, The Salk Institute for Biological Studies,10010 North Torrey Pines Road, La Jolla, CA 92037, USA Department of Neurosciences, University of California San Diego, La Jolla, California 92093,USA Institute for Neural Computation, University of California San Diego, La Jolla, CA 92093,USA Division of Biological Sciences, University of California San Diego, La Jolla, CA 92093,USA Rouen Normandie Universit´e, CORIA, Campus Universitaire du Madrillet,F-76800 Saint-Etienne du Rouvray, France (Dated: 25 September 2020)
Observability can determine which recorded variables of a given system are optimal for dis-criminating its different states. Quantifying observability requires knowledge of the equationsgoverning the dynamics. These equations are often unknown when experimental data are consid-ered. Consequently, we propose an approach for numerically assessing observability using DelayDifferential Analysis (DDA). Given a time series, DDA uses a delay differential equation forapproximating the measured data. The lower the least squares error between the predicted andrecorded data, the higher the observability. We thus rank the variables of several chaotic systemsaccording to their corresponding least square error to assess observability. The performance ofour approach is evaluated by comparison with the ranking provided by the symbolic observabil-ity coefficients as well as with two other data-based approaches using reservoir computing andsingular value decomposition of the reconstructed space. We investigate the robustness of ourapproach against noise contamination.
A popular approach for studying nonlineardynamical systems from a recorded time se-ries is to reconstruct the original systemusing delay or derivative coordinates. Itis known that the choice of the measuredvariable can affect the quality of attrac-tor reconstruction. Unlike in linear sys-tems for which the state space is observ-able or not from the measurements, non-linear systems are more or less observablefrom measurements depending on the statespace location. Moreover the observabil-ity strongly depends on the measured vari-ables. It is therefore useful to assess theobservability provided by a variable usinga real number within the unit interval be-tween two extreme values: 0 for nonobserv-able, and 1 for full observability. Analyti-cal techniques for determining observabilityrequire knowledge of the underlying equa-tions which are typically unknown when anexperimental system is investigated. This isoften the case for social and biological net-works. It is thus of primary importance toassess observability directly from recordedtime series. In this paper, we show how De-lay Differential Analysis (DDA) can assess observability from time series. The perfor-mance of this approach is evaluated by com-paring our results obtained for simulatedchaotic systems with the symbolic observ-ability coefficients obtained from the gov-erning equations.
I. INTRODUCTION
Studying dynamical systems from real world datacan be difficult as they are often high-dimensionaland nonlinear; moreover, it is typically not pos-sible to measure all the variables spanning theassociated state space.
In theory, it is pos-sible to reconstruct the non-measured variablesby using delay or differential embeddings froma single measurement. However, when perform-ing state-space reconstruction, the dimension re-quired to obtain a diffeomorphical equivalence —required for correctly distinguishing the differentstates of the system — with the original statespace may depend on the measured variable(s). Indeed, a d -dimensional system can be optimallyreconstructed from a given variable with a d -dimensional embedding but a higher-dimensional a r X i v : . [ n li n . AO ] S e p space may be required when another variable ismeasured. For instance, the R¨ossler attractor iseasily reproduced with a three-dimensional globalmodel from variable y but a four-dimensionalmodel or a quite sophisticated procedure isneeded when variable z is measured. It was shownthat data analysis often (if not always) dependson the observability provided by the measuredvariable. In the 1960s, the concept of observability was in-troduced by Rudolf K´alm´an in control theory. Observability assesses whether different states ofthe original system can be distinguished from themeasured variable. A system is said to be fullyobservable from some measurement if the rank ofthe observability matrix is equal to the dimensionof the system.
With such an approach, the an-swer is either fully observable or non observable.This approach is sufficient for linear systems be-cause the observability matrix does not depend onthe location in the state space.This is not true for nonlinear systems and ob-servability coefficients were introduced to over-come this inaccurate answer.
Observability co-efficients are real numbers within the unit intervalbetween two extreme values, 0 for nonobservable,1 for fully observable. These coefficients are esti-mated at every point of the trajectory producedby the governing equations in the state space,and then averaged along that trajectory.
It isalso possible to construct symbolic observabilitycoefficients from the Jacobian matrix of the sys-tem studied.
In this way, observability takesa graded value according to the probability withwhich the attractor intersects the singular observ-ability manifold, that is, the subset of the orig-inal space for which the determinant of the ob-servability matrix is zero. The great advantageof these coefficients is that they allow comparingthe observability provided by variables from differ-ent systems and they can be computed for high-dimensional systems. It is then possible to rankthe variables according to the observability of theoriginal state space they provide. The dependencyof the observability on the measured variable isdue to the way variables are coupled in the origi-nal system. Symmetries are often sources of dif-ficulty for assessing observability, particularly be-cause reconstructing the original symmetry is notpossible from a single variable if the symmetry dif-fers from an inversion.
The weakness of these analytical approaches isthat the governing equations must be known and itis not possible to assess observability from experi-mental data. A first attempt to overcome this wasbased on a singular value decomposition of somematrices built from local data. Results were en-couraging but some slight discrepancies with an-alytical results were noticed. Another approach,based on a model built directly from the data us- ing reservoir computing was also proposed. Inboth cases, some discpreancies with the symbolicobservability coefficients were observed. It there-fore remains challenging to develop a reliable tech-nique which always matches with theoretical re-sults. In this work we propose a measure for as-sessing observability from recorded data by usingdelay differential analysis (DDA) and compare ourresults and those obtained — when available inthe literature — with the two techniques discussedabove with the symbolic observability coefficientscomputed for several well-studied chaotic systems.Here, DDA is based on a delay differential equationwhich approximates the dynamics underlying themeasured time series. Contrary to what is donewith global modeling or reservoir computing, there is no need for an accurate model. Previouswork showed a rough model with a very limitednumber of terms (typically three) is sufficient todetect dynamical changes or classify various dy-namical regime. The subsequent part of this paper is organized asfollows. Section II A is a brief introduction tothe computation of symbolic observability coeffi-cients. Section II B provides an introduction toDDA and explains how it can be used for rank-ing variables according to the observability of thestate space they provide. Section III introducesthe investigated chaotic systems and provides thecorresponding symbolic observability coefficients.Section IV is the main section of this paper: itdiscusses the performance of DDA for assessing ob-servability of the chaotic systems and compares itwith those of the two other data-based techniques.Section V provide some conclusions.
II. THEORETICAL BACKGROUNDA. Symbolic observability coefficients
Let us consider a d -dimensional dynamical systemrepresented by the state vector x ∈ R d whose com-ponents are given by˙ x i = f i ( x , x , x , ..., x d ) , i = 1 , , , ..., d (1)where f i is the i th component of the vector field f . Let us introduce the measurement function h ( x ) : R d (cid:55)→ R m of m variables chosen amongthe d ones spanning the original state space. It isthen required to reconstruct a space R d r ( d r ≥ d )from the m measured variables. One has to choose d r − m derivatives of these m measured variablesto get a d r -dimensional vector X spanning the re-constructed space. Commonly, observability is as-sessed by using d r = d . In the present work, weare only working with scalar time series ( m = 1).The change of coordinate between the originalstate space and the reconstructed one is thus themap Φ : R d ( x ) (cid:55)→ R d ( X ) . (2)When the derivative coordinates are used for span-ning the reconstructed space, the map can be an-alytically computed. The observability of a sys-tem from a variable is defined as follows.
Forthe sake of simplicity, let us limit ourselves to thecase m = 1 (a generalization to the others cases isstraightforward). Definition 1
The dynamical system (1) is said tobe state observable at time t f if every initial state x (0) can be uniquely determined from the knowl-edge of the vector s ( τ ) , ≤ τ ≤ t f . To test whether a system is observable or not is toconstruct the observability matrix which is de-fined as the Jacobian matrix of the Lie derivativesof h ( x ). Differentiating h ( x ) yieldsdd t h ( x ) = ∂h∂ x ˙ x = ∂h∂ x f ( x ) = L f h ( x ) , where L f h ( x ) is the Lie derivative of h ( x ) alongthe vector field f . The k th order Lie derivative isgiven by L k f h ( x ) = ∂ L k − f h ( x ) ∂ x f ( x ) , being the zero order Lie derivative the measuredvariable itself, L f h ( x ) = h ( x ). Therefore, the ob-servability matrix O ∈ R d × d is written as O ( x ) = d h ( x )d L f h ( x )...d L d − f h ( x ) (3)where d ≡ ∂∂ x . The observability Definition 2
The dynamical system (1) is said tobe state observable if and only if the observabilitymatrix has full rank, that is, rank ( O ) = d . The observability matrix O is equal to the Jaco-bian matrix of the change of coordinates Φ : x → X when derivative coordinates are used. In thisapproach, the observability is either full or zero.The term structural was introduced when the re-sults do not depend on parameter values . Com-puting the rank of the observability matrix is in-dependent of parameter values and, consequently,is an example of structural observability. Com-puting observability with graphs is also a structural approach. We term observability as-sessed from recorded data — necessarily depen-dent on the parameter values used for simulat-ing the trajectory of the system —as dynamical observability. This type of approach returns areal number within the unit interval: variablescan be ranked between the two extreme cases, 1.0(0.0) for a full (null) observability There is a thirdtype of observability, symbolic observability, whichdoes not depend on parameter values but allowsranking the variables. All types of observabilityare not sensitive to symmetry-related problems.This is due to the fact that observability is a localproperty while symmetry is a global one. Conse-quently, symmetry may degrade the assessment ofobservability. The procedure to compute symbolic observabil-ity coefficients is implemented in three steps asfollows.
First, the Jacobian matrix J of the sys-tem (1), composed of elements J ij , is transformedinto the symbolic Jacobian matrix ˜ J by replacingeach constant element J ij by 1, each polynomialelement J ij by ¯1, and each rational element J ij by ¯¯1 when the j th variable is present in the de-nominator, or by ¯1 otherwise. Rational terms inthe governing equations (1) are distinguished frompolynomial terms since the formers reduce morestrongly the observability than the latters. Then the symbolic observability matrix ˜ O is con-structed. The first row of ˜ O is defined by thederivative of the measurement function d h ( x ), thatis, ˜ O j = 1 if j = i and 0 otherwise when the i thvariable is measured. The second row is the i throw of ˜ J . The k th row is obtained as follows.First, each element of the i th row of ˜ J is multi-plied by the corresponding i th component of thevector v = ( ˜ O (cid:96) , · · · , ˜ O (cid:96)d ) T where (cid:96) = k − k − O . The rules to perform the symbolicproduct ˜ J ij ⊗ v i are such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⊗ a = 0 , ⊗ a = a, ¯1 ⊗ a = a for a = ¯1 , ¯¯1 , ¯¯1 ⊗ a = ¯¯1 for a (cid:54) = 0 . (4)Second, the matrix ˜ J (cid:48) is reduced into a row whereeach element ˜ O kj = (cid:80) i ˜ J (cid:48) ij according to the addi-tion law (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⊕ a = a, ⊕ a = a for a (cid:54) = 0 , ¯1 ⊕ a = a for a (cid:54) = 0 , , ¯¯1 ⊕ a = ¯¯1 . (5)The last step is associated with the computationof the symbolic observability coefficients. The de-terminant of ˜ O is computed according to the sym-bolic product rule defined in (4) and expressed asproducts and addends of the symbolic terms 1, ¯1and ¯¯1, whose number of occurrences are N , N ¯1 and N ¯¯1 , respectively. It is convenient to imposethat, if N ¯1 = 0 and N ¯¯1 (cid:54) = 0 then N ¯1 = N ¯¯1 . Thesymbolic observability coefficient is thus defined as η = 1 D N + 1 D N ¯1 + 1 D N ¯¯1 (6)with D = N + N ¯1 + N ¯¯1 . This coefficient is inthe unit interval, η = 1 for a variable providingfull observability of the original state space. Anobservability is said to be good when η ≥ . B. Delay Differential Analysis
Let us assume that a time series { X } is recordedin a d -dimensional system. From this time series,it is possible to obtain a global model reproducingthe underlying dynamics. There are typically twomain approaches working with either derivative ordelay coordinates. When derivatives are used,it is possible to construct a d -dimensional differ-ential model ˙ X = X ˙ X = X ... ˙ X d = F ( X , X , ..., X d ) (7)where X i is the ( i − X . The function F can be numericallyestimated by using a least-square technique with astructure selection. F can be polynomial or rational. This model requires d -ordinarydifferential equations whose variables are the d successive derivatives of X : this model works ina differentiable embedding.Second, it is possible to construct a model whoseequations have the form of a difference equation X ( k + 1) = F (cid:0) X τ j ( k ) (cid:1) = N (cid:88) i =0 a i ϕ i (8)where ϕ i is a monomial of delay coordinates X τ j ( k ) = X ( k − τ j ) with τ j = nδ t ( n ∈ N + )is a time delay expressed in terms of the sam-pling time δ t with which the scalar time series { X ( k ) } is recorded: k is the discrete time. Sucha model has an auto-regressive form, and typi-cally the number N of terms is between 10 and20. The space in which this model is workingis thus spanned by delay coordinates: its dimen-sion is very often significantly larger than the di-mension d , the embedding dimension or eventhan the Takens criterion. An optimal form ofthe difference equation (8) is developed under theform of a nonlinear autoregressive-moving average(NARMA) model. Recently a third type of model was investigatedunder the name of reservoir computing . This ap-proach considers an oversized model with a func-tional structure based on a network whose nodesare characterized by some simple function. Forinstance, the Lorenz attractor was accurately re-produced with a Erd¨os-R´enyi network of 300 nodeswith a mean degree δ = 6, each node being madeof a difference equation. The model so-obtainedcorresponds to an accurate global model of the dy-namics. Notably, this model was constructed fromthe measurements of all the variables of the Lorenzsystem. The main advantage of such a large modelis its flexibility, that is, its ability to capture vari-ous dynamical regimes, but it has the disadvantagethat the space in which it is working is not clearlydefined and has a very large dimension ( d r > X = F X = N (cid:88) i =1 a i ϕ i ( X τ j ) (9)where X = X designates the measured vari-able and X τ j some delay coordinates. The pur-pose is not to construct a global model repro-ducing accurately the dynamics but only an ap-proximated model for detecting dynamical changes(nonstationarity) or classifying different dynami-cal regimes. We therefore use a rough modelwith very few terms ( N ≤ Indeed, delay differential equations areknown to already produce complex dynamics withonly two terms.
Many characteristics of themeasured dynamics can be captured with two orthree terms and appropriate time delays. Basedon previous works, it is assumed that thesecharacteristics are sufficient to distinguish differ-ent dynamical regimes. This DDA model (9) is adifferential equation whose state space is spannedby delay coordinates X τ j .Model (9) has two sets of parameters, the fixed pa-rameters τ j (set during the structure selection) andthe free parameters a i (estimated independentlyfrom each data window). The structure of model(9) as well as the delays are determined for eachtime series. Then, the free coefficients a i are de-termined for each window of the recorded time se-ries. The data in each window { X } is normalizedto have zero mean and unit variance to removeamplitude information before estimating the freeparameters a i by using a singular value decompo-sition (SVD). The least-square error ρ X = (cid:118)(cid:117)(cid:117)(cid:116) K K (cid:88) k =1 (cid:16) ˙ X ( k ) − F X ( k ) (cid:17) (10)between the derivatives returned by the DDAmodel and the derivatives computed from themeasured time series quantifies the ability of themodel to capture the underlying dynamics. It isknown that there is a relationship between themodel quality and observability. The signalderivative ˙ X is computed using a five-point centerderivative. In this work, structure selection (i.e.choosing the model form of Eq. (9) and the fixedparameters τ j ) was performed via an exhaustivesearch over all possible three-term models (threemonomials: N = 3) with two delays such that τ j ∈ [ m + 1; 60] δ t , where m = 5 is equal to thenumber of points for estimating the derivative and δ t is the sampling time. Function F is made ofthree monomials selected from the possible candi-dates ϕ i ∈ (cid:8) X τ , X τ , X τ , X τ X τ , X τ ,X τ , X τ X τ , X τ X τ , X τ (cid:9) . (11)Monomials and delays are selected in an exhaus-tive search over all possible model forms, i.e.44, and delay combinations under the restrictionsspecified above. Each model is thus characterizedby the set of “fixed” parameters ( τ , τ ), the corre-sponding monomials ϕ i , and the free parameters a i which are estimated for each time window ofthe measuredndata. The structure providing themodel with the lowest ρ X is retained to assess ob-servability according to the model error ρ X .As used with reservoir computing, the error ρ X between the model and the measured data pro-vides a measure of how the system dynamics maybe reconstructed from these data. Indeed, to ob-tain a reliable deterministic model, it is necessaryto distinguish every different state of the systemfor retrieving the underlying causality. Since theerror is used as a relative measure, it is only neededto have a sufficiently flexible functional form forthe model as observed with reservoir computingor with a delay differential equation. Consequentlythe smaller the error ρ X , the higher the observabil-ity provided by the variable X . This results fromprevious works where it was shown that the com-plexity of the model to approximate was correlatedto the observability: the better the observabilityprovided by the measured variable, the simpler themodel to approximate. The error ρ X from thebest DDA model is computed with an increasingnoise amplitude. For each three-dimensional sys-tem and each signal-to-noise ratio (no noise, 20,10 and 0 dB: where 0dB indicates the variance ofthe noise matches the variance of the signal), theerror ρ X was computed over several hundred pseu-doperiods for each time series. III. DYNAMICAL SYSTEMS ANDOBSERVABILITY COEFFICIENTSA. Low-dimensional systems
The governing equations of the systems here inves-tigated are reported in Table I. The symbolic ob-servability coefficients (SOC) and the model error ρ X are reported for each variable of every systemin Table I. Parameter values are reported in TableII.The R¨ossler 76 , Lorenz 84 , Cord ,Hindmarsh-Rose (HR) and Fisher sys-tems have no symmetry. The Hindmarsh-Rosesystem is known to be problematic when variable x or z is measured, for two different reasons. When variable z is measured, the observabilitymatrix O z = rs − rrs ( xχ − r ) rs r ( r − s ) (12)where χ = 2 b − ax becomes singular when r istoo small (Det O z = r s ): the observability canbe null for r = 0 and full for r (cid:54) = 0 (this is alsotrue for s , but s is commonly significantly differentfrom 0). When variable x is measured, althoughthe observability matrix O x is never singular (Det O x = r − x does notreveal the chaotic nature of the underlying dynam-ics, contrary to what is clearly provided by vari-able z (Fig. 1). As discussed by Aguirre et al , the observability matrix O x = χx O x χx − − χx + r (13)where O x = χ x − rs − bx + 2( b − a ) × (cid:2)(cid:0) I + x ( b − ax ) + y − z (cid:1)(cid:3) (14)has a determinant Det O x whose polynomial na-ture is cancelled by the contributions of O and O but this is not structurally stable. Any pertur-bation in one of these two elements would lead toa determinant vanishing for a subset of the statespace. This is not detected by the symbolic ob-servability coefficients. If we keep the polynomialnature of elements O and O , the symbolic ob-servability matrix would be O x = . (15) TABLE I. Governing equations of each system forwhich the symbolic observability coefficients (SOC) η s and ρ X between the DDA model and the mea-sured data with no noise contamination are reported.The SOC for variable x of the Hindmarsh-Rose (HR)system is corrected as discussed in the main text.Forthe Chua system, f ( x ) = bx + ( a − b )( | x + 1 | − | x − | )System Equations SOC ErrorR¨ossler 76 ˙ x = − y − z .
84 0.037 ˙ y = x + ay z = b + z ( x − c ) 0 .
56 0.106R¨ossler 77 ˙ x = − ax − y (1 − x ) 0.56 0 . ˙ y = µ ( bx + y − cz ) 0.84 0 . z = µ ( x + cy − dz ) 0.68 0 . x = σ ( y − x ) 0.78 0.02 ˙ y = Rx − y − xz z = − bz + xy x = − y − z − ax + aF ˙ y = xy − bxz − y + G z = bxy + xz − z x = − y − z − ax + aF ˙ y = xy − bxz − y + G z = bxy + xz − z x = y − ax + bx + I − z ˙ y = c − dx − y z = r [ s ( x − x R ) − z ] 1.00 0.002Fisher ˙ x = y ˙ y = − ax − by − z z = b + x − | x | x = α ( − x + y − f ( x )) 1.00 0.05 ˙ y = x − y + z z = − βy x = y ˙ y = − µy + x − x + u u = v v = − ω u x = − y − z ˙ y = x + ay + w z = b + xz w = − cz + dw x = u ˙ y = v u = − x − xy v = − y − y − x The corresponding corrected symbolic observabil-ity coefficient is thus η (cid:48) x = 0 .
68. The correctedranking of variables is therefore z (cid:66) x (cid:66) y . Thisranking will be used in the subsequent analysis.The other systems have symmetry properties as TABLE II. Parameter values of the investigated sys-tems.R¨ossler 76 a = 0 . b = 2 c = 4R¨ossler 77 a = 0 . b = 0 . c = 2 d = 0 . µ = 0 . σ = 10 b = 8 / R = 28Lorenz 84 a = 0 . b = 4 F = 8 G = 1Cord a = 0 . b = 4 F = 8 G = 1HR a = 1 b = 3 c = 1 d = 5 I = 3 . x R = Fisher a = 0 . b = 0 . α = 9 β = a = − b = − Duffing µ = 0 . ω = 1 . x = 1 y = 0 u = 0 . v = 0R¨ossler 79 a = 0 . b = 3 c = 0 . d = 0 . -1 -0,5 0 0,5 1 1,5 2 x -5-4-3-2-1012 x . -6 -4 -2 0 y -10-505 y . z z . FIG. 1. Differential embedding induced by each of thethree variables of the Hindmarsh-Rose system. follows. The Lorenz 63 system is equivariantunder a R z rotation symmetry around the z -axis. Variables x and y are mapped into theiropposite ( − x and − y , respectively) while vari-able z is invariant under the rotation symmetry.At least two variables must be measured to cor-rectly reconstruct the rotation symmetry. TheR¨ossler 77 , Chua circuit and the driven Duff-ing systems are equivariant under an inversionsymmetry. Such a symmetry can be recoveredfrom a single variable and, consequently, shouldnot blur the observability analysis. The drivenDuffing system is in fact a four-dimensional sys-tem, a conservative harmonic oscillator driving thedissipative Duffing oscillator: it is thus a semi-dissipative (or semi-conservative) system. Whenvariable u (or v ) is recorded , a periodic orbit isobtained while variable x (or y ) provides a chaoticstate portrait. Since a chaotic driving signal nec-essarily implies a chaotic response, it is obviousthat u drives x and not the opposite. It can there-fore be concluded, without further analysis, thatthe system is not observable from u (or v ). Thus,we only have to determine the observability fromvariable x and y , respectively. The Fisher systemand the Chua circuit have a piecewise nonlinearity.They will be useful to test whether DDA is robustagainst discontinuous nonlinearity.All these systems but three — the Lorenz 84, theCord, and the H´enon-Heiles systems — have atleast one variable providing a good observability( η > .
75) of the original state space. The H´enon-Heiles system is conservative and one may guessthat the observability problem will be more sensi-tive since the invariant domain of the state spacehas a dimension close to 3, and not 2 as for all theother systems which are strongly dissipative.
B. A higher-dimensional system
The Lorenz 63 system results from a Galerkinexpansion of the Navier-Stokes equations forRayleigh-B´enard convection. It is also possibleto have a higher-dimensional expansion in retain-ing more Fourier components. One of them leadto the 9D Lorenz system ˙ x = − σ ( b x + b x ) + x ( b x − x ) + b x x ˙ x = − σx + x x − x x + x x − σx x = σ ( b x − b x ) + x x − b x − b x x ˙ x = − σx − x x − x x + x x + σx x = − σb x + x − x x = − b x + x x − x x ˙ x = − b x − Rx + 2 x x − x x ˙ x = − b x + Rx − x x + x x ˙ x = − x + ( R + 2 x )( x − x ) + x x − x x (16)where b = 4 1 + a a b = 1 + 2 a a ) b = 2 1 − a a b = a a b = 8 a a b = 41 + 2 a (17)This 9D Lorenz system is equivariant. Depend-ing on the R -values, the attractor produced maybe asymmetric [Fig. 2(a)] or symmetric [Fig. 2(b)].The symbolic observability coefficients are η x = η x = η x = η x = 0 . η x = η x = 0 . η x = η x = η x = 0 (18)leading to x = x = x = x (cid:66) x = x (cid:66) x = x = x Notice that every variable offers an extremely poorobservability of the original state space. It wasshown that, at least five variables need to be mea-sured for having a good observability ( η > . Moreover, for a suf-ficiently large R -value ( R = 45), the behavior is -1,0 -0,5 0,0 0,5 1,0 x -1,0-0,50,00,51,0 x -4.5-3.5-2.5-1.5 x x x x x x x x x NoNoise 20dB 10dB 0dB l n () (a) -1,0 -0,5 0,0 0,5 1,0 x -1,0-0,50,00,51,0 x -4.5-3.5-2.5-1.5 No Noise 20dB 10dB 0dB l n () (b) -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 x -1,5-1,0-0,50,00,51,01,5 x -3.5-2.5-1.5 No Noise 20dB 10dB 0dB l n () (c) -200 -100 0 100 200 x -200-1000100200 x -2.5-2-1.5-1 No Noise 20dB 10dB 0dB l n () (d)FIG. 2. Chaotic attractor produced by the 9D Lorenzsystem (18). (a) R = 14 .
22, (b) R = 14 .
30, (c) R = 15 .
10, (d) R = 45 .
00. Other parameter values: a = 0 .
5, and σ = 0 .
5. When there are co-existingattractors, they are plotted in different colors in theplane projections of the state space. hyperchaotic. One of the characteristics of thishighly developed behavior is that there are twodifferent time scales. We will therefore investigatewhether the observability assessed with DDA isdependent on parameter values, that is, on bifur-cation affecting the symmetry properties (order-4or order-2 asymmetric chaos, symmetric chaos andhyperchaos).
IV. DDA RANKING
The structure of the best DDA models F X arereported in Appendix A, Table V along with thecorresponding time delays retained for identifyingthe free parameters. As examples, ρ X for somesystems with increasing noise are shown in Figs. 3.For no noise, ρ X is reported in Table I.The rankings for variables according to increasingsymbolic observability coefficients (SOC), decreas-ing ρ X for DDA, and when available in the lit-erature, for decreasing reservoir computing (RC)and singular value decomposition observability(SVDO) are summarized in Table III for all lowdimensional systems ( d ≤ systems are in a perfect agreementwith the SOC. The discontinuity of the Fisher sys-tem does not perturb the analysis. The hyper-chaotic nature of the R¨ossler 79 system was notproblematic for correctly assessing observability.The Lorenz 63, Lorenz 84, Cord and Hindmarsh-Rose systems show close agreement between DDAand SOC. For the Lorenz 63 system, variable x was correctly detected as providing the best ob-servability but variable z was found to offer worseobservability than variable y , a feature which isnot predicted by the SOC due to a problem in-herent to the symmetry involved. For the Cordsystem, while no single variable provides good ob-servability for the original state space, DDA cor-rectly ranks x as providing the best observability.However, DDA ranks z as providing worse observ-ability than variable y , while SOC ranks them withequivalent observability. For the Hindmarsh-Rosesystem, variable z provides full observability andis associated with the lowest ρ X . However, thereis some discrepancy between DDA and SOC since,as assessed with DDA, y provides a slightly higherobservability than x . Results for the H´enon-Heilessystem are quite equivalent to the SOC.Variables x and y are more observable than u and v , how-ever, x ( u ) is more observable than y ( v ) instead ofshowing equivalent observability.For the Chua circuit, the variable x contains apiecewise nonlinearity and has full observability,and DDA correctly ranks x as the most observ-able. DDA also ranks variable y with the worstobservability, which is in agreement with SOC.However, variable z has only slightly better ob-servability than y , whereas it should be equivalentto x .When compared to the two other data-based tech- niques, DDA performs better than RC for theR¨ossler 76, R¨ossler 79 and the Lorenz 63 systemsbut not for the Chua circuit. Compared to theSVDO, the DDA approach provides equivalent re-sults for all the systems investigated by these twotechniques, but does perform better for the hyper-chaotic R¨ossler 79 system in correctly identifyingthe variable y as providing the best observability,a feature which missed by the SVDO.For most of the systems, these results are ro-bust against noise contamination, at least up toa signal-to-noise ratio greater than 10 dB: belowthis ratio, results can be blurred and observabil-ity can no longer be reliably assessed using DDA.A similar robustness was observed with SVDO. Itwas not investigated with RC.Note that another interesting data-based tech-nique for assessing observability was proposed byParlitz and co-workers. It was only tested withthe R¨ossler 76 system (and the H´enon map, notinvestigated here). It would be interesting to fur-ther investigate its performance but this is out ofthe scope of this paper.The results for the 9D Lorenz system are not soclear. The first reason is that this system is nearlyunobservable from a single variable. The SOC arenearly saturated (close to 0) with nonlinear ele-ments as revealed by the symbolic Jacobian matrixof the 9D Lorenz system (16), namely J sym = , (19)which illustrates most of the couplings betweenvariables are nonlinear. Considering only the ob-servability provided by a single variable is here in-vestigated, and that the SOC are all close to 0,one may conclude that the 9D Lorenz system isnot observable from a single variable.Results provided by DDA are shown in Fig. 2where it is seen that variables cannot be easilyranked, particularly when R is increased. Resultsare summarized in Table IV as follows. For each R -values, the rankings of the variables are reported— from 1 for the variable offering the best ob-servability to 9 for the one providing the poorestobservability — and compared to the ranking pro-vided by the SOC. The results are strongly depen-dent on R -value in a way which does not allow toextract a clear tendency. Variable x with a nullobservability as assessed by the SOC (and analyt-ically) is found to provide the best observability TABLE III. Ranking variables according to the observability as assessed by the symbolic observability coefficients(SOC), DDA analysis, reservoir computing (RC) and singular value decomposition observability (SVDO). Aperfect agreement with the SOC is indicated by a • . When the variable providing the best observability iscorrectly detected or when = is replaced with ≈ or (cid:66) , a ◦ is reported.System SOC DDA RC SVDOR¨ossler 76 y (cid:66) x (cid:66) z • x (cid:66) y (cid:66) z • R¨ossler 77 y (cid:66) z (cid:66) x • — —Lorenz 63 x (cid:66) y = z ◦ y (cid:66) x (cid:66) z ◦ Lorenz 84 x = y = z ◦ — • Cord x (cid:66) y = z ◦ — ◦ Hindmarsh-Rose z (cid:66) x (cid:66) y ◦ — • Fisher x (cid:66) y (cid:66) z • — —Chua x = z (cid:66) y ◦ • ◦ Duffing x (cid:66) y (cid:66) u = v • — —R¨ossler 79 y (cid:66) x (cid:66) w (cid:66) z • x (cid:66) y (cid:66) z (cid:66) w x (cid:66) y (cid:66) w (cid:66) z H´enon-Heiles x = y (cid:66) u = v ◦ ◦ — xyz No Noise 20dB 10dB 0dB xyz
No Noise 20dB 10dB 0dB xyz
No Noise 20dB 10dB 0dB (a) (b) (c) xyz
No Noise 20dB 10dB 0dB -4 xyuv xyz NoNoise 20dB 10dB 0dB (d) (e) (f)
FIG. 3. Error ρ X versus a decreasing signal-to-noise ratio for some of the different systems investigated inthis paper. (a) R¨ossler 76 system, (b) Lorenz 63 system, (c) Cord system, (d) Hindmarsh-Rose system, (e)H´enon-Heiles system, (f) Chua system. as assessed by DDA. Nevertheless, this is in agree-ment with the successful three-dimensional globalmodel obtained from this variable for R = 14 . that is, at least for this R -value, the dynamics canbe correctly reconstructed for recovering the un-derlying determinism.It should be pointed out that looking for full ob-servability (i.e. being able to “reconstruct” each of the non-measured variables) is not the samething as looking for an embedding. Especially forlarge d -dimensional systems producing an attrac-tor which can be embedded within a space whosedimension d R is lower than the dimension d of theoriginal state space. Full observability ensures theexistence of an embedding, the opposite is not nec-essarily true. Here DDA selects the variable which0 TABLE IV. Observability of the 9D Lorenz systemas assessed with the symbolic observability coefficients(SOC) and DDA.
R x x x x x x x x x SOC — 1 2 1 2 3 3 1 1 3DDA 14.22 7 4 1 5 2 8 3 6 914.30 3 7 2 6 1 8 5 4 915.10 5 2 6 3 1 9 8 7 445.00 8 5 7 6 1 4 3 2 9 provides the best reconstructed space. If comparedwith the results provided by the SOC with multi-variate measurements, variables x , x , x , and x are always among the six variables selected forproviding a full observability. DDA returns threeof them as providing the best observability, x , x , and x (Table IV). Variable x , the single onewhich is invariant under the symmetry of this sys-tem, is identified as a variable providing a poorobservability. Once again, symmetry induces dif-ficulties for assessing observability. V. CONCLUSION
The ability to infer the state of a system from ascalar output depends on which system variableis measured. We have introduced a numerical ap-proach using the error between a DDA model andmeasured data to assess the observability providedby the measured variables in several chaotic sys-tems. We compared these measures with sym-bolic observability coefficients, which are deter-mined directly from the system’s equations. Ourmeasure overall reliably ranks variables accordingto the observability they provide about the origi-nal state space. The largest discrepancy was ob-tained for a large-dimensional (9D Lorenz) system.The smaller the model error, the better the observ-ability provided. The assessment of observabilityis quite robust against noise contamination in themajority of the systems here considered.There are two situations in which our approachesmay face some complications. The first one isa common one. Inconsistencies in assessing ob-servability are known for systems with symmetryproperties, particularly with variables left invari-ant. The second one is also a typical one: when thedimension of the system increases, the observabil-ity of the state space provided by a single variablebecomes very poor and assessing observability isdelicate. Our approach is thus very reliable forlow-dimensional systems without symmetry prop-erties, even with a signal-to-noise ratio as com-monly encountered in experiments.As in most of the other techniques, variables of dif- ferent systems cannot be compared to each other.This is a common limitation in assessing observ-ability that is only overcome by using an analyticalapproach, such as by computing explicitly the ob-servability matrix or by using the symbolic observ-ability coefficients. A kind of normalization shouldbe considered to have, for instance, the error ρ y ofvariable y of the R¨ossler 76 system (which has fullobservability) smaller than for variable y of theR¨ossler 77 system. This problem is more challeng-ing than it may appear. It was, for instance, neversolved for the observability coefficients computedalong a trajectory using a relationship extractedfrom the system’s equations or using SVD appliedto a reconstructed space. ACKNOWLEDGMENTS
C. Letellier wishes to thank Irene Sendi˜na-Nadalfor her assistance in computing the symbolic ob-servability coefficients for the 9D Lorenz sys-tem. This work was supported by the Na-tional Institute of Health (NIH)/NIBIB (GrantNo. R01EB026899-01) and by the National Sci-ence Foundation Graduate Research Fellowship(Grant No. DGE-1650112).
Data Availability
The data that support the findings of this studyare available from the corresponding author uponreasonable request.
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9D Lorenz F , , X τ X τ X τ δ t δ t R = 14 . F , , X τ X τ X τ δ t δ t F , X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t
9D Lorenz F − , , X τ X τ X τ δ t δ t R = 14 . F , X τ X τ X τ δ t δ t
9D Lorenz F , , , X τ X τ X τ δ t δ t R = 15 . F , , , X τ X τ X τ δ t δ t F X τ X τ X τ δ t δ t
9D Lorenz F X τ X τ X τ X τ δ t δ t R = 45 F X τ X τ X τ X τ δ t δ t F X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t TABLE V. Cont a a a τ τ R¨ossler 79 F x X τ X τ X τ δ t δ t F y X τ X τ X τ δ t δ t F z X τ X τ X τ δ t δ t F w X τ X τ X τ δ t δ t H´enon-Heiles F x X τ X τ X τ δδ
9D Lorenz F X τ X τ X τ X τ δ t δ t R = 45 F X τ X τ X τ X τ δ t δ t F X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t TABLE V. Cont a a a τ τ R¨ossler 79 F x X τ X τ X τ δ t δ t F y X τ X τ X τ δ t δ t F z X τ X τ X τ δ t δ t F w X τ X τ X τ δ t δ t H´enon-Heiles F x X τ X τ X τ δδ t δδ
9D Lorenz F X τ X τ X τ X τ δ t δ t R = 45 F X τ X τ X τ X τ δ t δ t F X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t TABLE V. Cont a a a τ τ R¨ossler 79 F x X τ X τ X τ δ t δ t F y X τ X τ X τ δ t δ t F z X τ X τ X τ δ t δ t F w X τ X τ X τ δ t δ t H´enon-Heiles F x X τ X τ X τ δδ t δδ t F y X τ X τ X τ δδ
9D Lorenz F X τ X τ X τ X τ δ t δ t R = 45 F X τ X τ X τ X τ δ t δ t F X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t TABLE V. Cont a a a τ τ R¨ossler 79 F x X τ X τ X τ δ t δ t F y X τ X τ X τ δ t δ t F z X τ X τ X τ δ t δ t F w X τ X τ X τ δ t δ t H´enon-Heiles F x X τ X τ X τ δδ t δδ t F y X τ X τ X τ δδ t δδ
9D Lorenz F X τ X τ X τ X τ δ t δ t R = 45 F X τ X τ X τ X τ δ t δ t F X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t TABLE V. Cont a a a τ τ R¨ossler 79 F x X τ X τ X τ δ t δ t F y X τ X τ X τ δ t δ t F z X τ X τ X τ δ t δ t F w X τ X τ X τ δ t δ t H´enon-Heiles F x X τ X τ X τ δδ t δδ t F y X τ X τ X τ δδ t δδ t F u X τ X τ X τ δδ
9D Lorenz F X τ X τ X τ X τ δ t δ t R = 45 F X τ X τ X τ X τ δ t δ t F X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t TABLE V. Cont a a a τ τ R¨ossler 79 F x X τ X τ X τ δ t δ t F y X τ X τ X τ δ t δ t F z X τ X τ X τ δ t δ t F w X τ X τ X τ δ t δ t H´enon-Heiles F x X τ X τ X τ δδ t δδ t F y X τ X τ X τ δδ t δδ t F u X τ X τ X τ δδ t δδ
9D Lorenz F X τ X τ X τ X τ δ t δ t R = 45 F X τ X τ X τ X τ δ t δ t F X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t TABLE V. Cont a a a τ τ R¨ossler 79 F x X τ X τ X τ δ t δ t F y X τ X τ X τ δ t δ t F z X τ X τ X τ δ t δ t F w X τ X τ X τ δ t δ t H´enon-Heiles F x X τ X τ X τ δδ t δδ t F y X τ X τ X τ δδ t δδ t F u X τ X τ X τ δδ t δδ t F v X τ X τ X τ δδ
9D Lorenz F X τ X τ X τ X τ δ t δ t R = 45 F X τ X τ X τ X τ δ t δ t F X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t F X τ X τ X τ X τ δ t δ t TABLE V. Cont a a a τ τ R¨ossler 79 F x X τ X τ X τ δ t δ t F y X τ X τ X τ δ t δ t F z X τ X τ X τ δ t δ t F w X τ X τ X τ δ t δ t H´enon-Heiles F x X τ X τ X τ δδ t δδ t F y X τ X τ X τ δδ t δδ t F u X τ X τ X τ δδ t δδ t F v X τ X τ X τ δδ t δδ