Innovations and Wold decompositions of stable sequences

Cambanis, Stamatis; Hardin, Clyde D.; Weron, Aleksander

doi:10.1007/bf00319099

Cited by 47 publications

(21 citation statements)

References 13 publications

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“…Not observing an ordinal pattern does not qualify it as “ forbidden ”, only as “ missing ”, and this could be due to the time series finite length. A similar observation also holds for the case of real data that always possess a stochastic component due to the omnipresence of dynamical noise [30]–[32]. Thus, “missing ordinal patterns” could be either related to stochastic processes (correlated or uncorrelated) or to deterministic noisy processes (always the case for observational time series).…”

Section: Introductionmentioning

confidence: 65%

Distinguishing Noise from Chaos: Objective versus Subjective Criteria Using Horizontal Visibility Graph

et al. 2014

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A recently proposed methodology called the Horizontal Visibility Graph (HVG) [Luque et al., Phys. Rev. E., 80, 046103 (2009)] that constitutes a geometrical simplification of the well known Visibility Graph algorithm [Lacasa et al., Proc. Natl. Sci. U.S.A. 105, 4972 (2008)], has been used to study the distinction between deterministic and stochastic components in time series [L. Lacasa and R. Toral, Phys. Rev. E., 82, 036120 (2010)]. Specifically, the authors propose that the node degree distribution of these processes follows an exponential functional of the form , in which is the node degree and is a positive parameter able to distinguish between deterministic (chaotic) and stochastic (uncorrelated and correlated) dynamics. In this work, we investigate the characteristics of the node degree distributions constructed by using HVG, for time series corresponding to chaotic maps, 2 chaotic flows and different stochastic processes. We thoroughly study the methodology proposed by Lacasa and Toral finding several cases for which their hypothesis is not valid. We propose a methodology that uses the HVG together with Information Theory quantifiers. An extensive and careful analysis of the node degree distributions obtained by applying HVG allow us to conclude that the Fisher-Shannon information plane is a remarkable tool able to graphically represent the different nature, deterministic or stochastic, of the systems under study.

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Section: Introductionmentioning

confidence: 65%

Distinguishing Noise from Chaos: Objective versus Subjective Criteria Using Horizontal Visibility Graph

et al. 2014

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“…Remark 2. Cambanis, Hardin & Weron (1988) point out that operator linearity implies result (i) for processes in L, (R, S,, p ) and for projection into arbitrary L,(sZ, S,, p)-spaces. This result, however, is trivial and does not anticipate the dual relationship between orthogonality and iterated projections (i) + (ii) without invoking operator linearity.…”

Section: Resultsmentioning

confidence: 87%

“…Similarly, Cambanis, Hardin & Weron (1988) establish an asymmetric decomposition theory for L, (R, St, p ) processes in Spt. The authors prove (i) projection operator linearity; (ii) iterated projections; and (iii) the existence of strong orthogonal innovations are equivalent when the innovations are restricted to the space Spt -Pt-lSp,.…”

Section: ( 2 )mentioning

confidence: 84%

“…In the literature, therefore, either independence is assumed, only weak orthogonality is proven, or explicit properties of closed linear spans are exploited to promote strong orthogonality. For projections into arbitrary (non-linear) L,-spaces, Cambanis, Hardin & Weron (1988) point out that operator linearity sufficiently renders strong orthogonal innovations (2). This result, however, is trivial: see Theorem 3, below.…”

Section: ( 2 )mentioning

confidence: 99%

“…The innovations in this literature are typically assumed to be independent and identically distributed , hence strongly orthogonal to far more than subspaces of Spt . Except for the special case of symmetric stable process (Cambanis, Hardin & Weron 1988), nowhere in this literature are necessary and sufficient conditions for the existence of such moving averages derived.…”

Section: ( 2 )mentioning

confidence: 99%

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Strong orthogonal decompositions and non‐linear impulse response functions for infinite‐variance processes

Hill

2006

Can J Statistics

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Abstract:The author proves that Wold-type decompositions with strong orthogonal prediction innovations exist in smooth, reflexive Banach spaces of discrete time processes if and only if the projection operator generating the innovations satisfies the property of iterations. His theory includes as special cases all previous Wold-type decompositions of discrete time processes, completely characterizes when non-linear heavy-tailed processes obtain a strong-orthogonal moving average representation, and easily promotes a theory of non-linear impulse response functions for infinite-variance processes. The author exemplifies his theory by developing a non-linear impulse response function for smooth transition threshold processes, and discusses how to test decomposition innovations for strong orthogonality and whether the proposed model represents the best predictor. A data set on currency exchange rates allows him to illustrate his methodology. Decompositions orthogonales fortes et fonctions reponses a impulsion non lineaire pour des processus a variance infinieR6ssumi : L'auteur ddmontre que pour qu'un espace de Banach dflexif lisse de processus ti temps discret admette une dkomposition de type Wold 21 innovations prkvisionnelles fortement oahogonales, il faut et il suffit que I'op6rateur projectif engendrant les innovations posshle la propriktk d'itdrations. Son rdsultat. qui englobe comme cas particuliers toutes les ddcompositions de type Wold obtenues antdrieurement pour des processus ti temps discret, caractkrise complktement les situations oil les processus non lindaires ti queues lourdes possMent une reprdsentation en moyenne mobile fortement orthogonale, en plus de conduire naturellement ti une thdorie des fonctions rdponses ti impulsion non linkaire pour les processus ti variance infinie. Il se sert de cette thdorie pour ddvelopper une fonction ti impulsion non lindaire pour des processus ti seuil de transition lisse et montre comment tester I'orthogonalitk forte des innovations de la dkcomposition et vkrifier si le modkle proposk est be1 et bien le meilleur pr6dicteur. Un jeu de donnks sur des taux de change de devises lui permet d'illustrer son propos.

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An Extremal Problem in H p of the Upper Half Plane with Application to Prediction of Stochastic Processes

Rajput

Ramamurthy

Sundberg

1991

Stable Processes and Related Topics

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Innovations and Wold decompositions of stable sequences

Cited by 47 publications

References 13 publications

Distinguishing Noise from Chaos: Objective versus Subjective Criteria Using Horizontal Visibility Graph

Distinguishing Noise from Chaos: Objective versus Subjective Criteria Using Horizontal Visibility Graph

Strong orthogonal decompositions and non‐linear impulse response functions for infinite‐variance processes

An Extremal Problem in H p of the Upper Half Plane with Application to Prediction of Stochastic Processes

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