Stochastic discrete scale invariance

Borgnat, Pierre; Flandrin, Patrick; Amblard, Pierre‐Olivier

doi:10.1109/lsp.2002.800504

Cited by 34 publications

(24 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The basis functions are the Mellin chirps {t H +i2πf , t > 0}, with f ∈ R. Using this equivalence, note that one can obtain a harmonic-like representation of a self-similar process X(t) as an inverse Mellin transform, namely an integral of uncorrelated spectral increments dξ X (f ) on the Mellin basis [10,26]:…”

Section: Covariance and Spectrum Of H-ss Processesmentioning

confidence: 99%

“…We will not detail other aspects, except for discrete scale invariance hereafter, and the reader will find elsewhere details on higher order distributions that can be introduced on this grounding [3], on multiplicative harmonizability for non-H-ss processes [10], or on the analysis of locally asymptotically self-similar processes by the use of L H [26,11].…”

Section: (T F ) = M X (T)tmentioning

confidence: 99%

“…This was advocated as a central concept in the study of many critical systems [47,51,52]. More attention has been given to the deterministic framework and we rely here on the stochastic extension of the property [10].…”

Section: Discrete Scale Invariancementioning

confidence: 99%

See 2 more Smart Citations

Scale invariances and Lamperti transformations for stochastic processes

Borgnat

Amblard

Flandrin

2005

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

Scale-invariant processes, and hereafter processes with broken versions of this symmetry, are studied by means of the Lamperti transformation, a one-toone transformation linking stationary and self-similar processes. A general overview of the use of the transformation, and of the stationary generators it builds, is given for modelling and analysis of scale invariance. We put an emphasis on generalizations to non-strictly scale-invariant situations. The examples of discrete scale invariance and finite-size scale invariance are developed by means of the Lamperti transformation framework, and some specific examples of processes with these generalized symmetries are given. Scale invariance and beyondScale invariance, once acknowledged as an important feature [39], has often been used as a fundamental property to handle physical phenomena. The idea that some quantity behaves the same at each scale, irrelevant to the scale at which it is observed, has made its way into the study of geometrical fractal sets [21], 1/f spectra, long memory [6], simple dimensional analysis [5] or involved analysis of critical systems in statistical physics [22], textures in geophysics [52] or image processing [38], turbulence of fluids [27], data of network traffic [45] and so on.Common as this invariance may be in physics, it often eludes general and convenient methods or models. Even though there is no single definition of scale invariance [19], it is often described as a symmetry of the system relatively to a transformation of scale, that is

show abstract

Section: Covariance and Spectrum Of H-ss Processesmentioning

confidence: 99%

Section: (T F ) = M X (T)tmentioning

confidence: 99%

See 1 more Smart Citation

Scale invariances and Lamperti transformations for stochastic processes

Borgnat

Amblard

Flandrin

2005

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

show abstract

“…We will only consider here the DSI property but some results about different classes of processes and their description are proposed in [6]. A useful property is that one can give the (potentially nonstationary) correlation function of the Lamperti transform X of a process Y:…”

Section: Lamperti Transformationmentioning

confidence: 99%

Stochastic discrete scale invariance and Lamperti transformation

Borgnat

Flandrin²,

Amblard³

Proceedings of the 11th IEEE Signal Processing Workshop on Statistical Signal Processing (Cat. No.01TH8563)

Self Cite

View full text Add to dashboard Cite

We define and study stochastic discrete scale invariance (DSI), a property which requires invariance by dilation for certain preferred scaling factors only. We prove that the Lamperti transformation, known to map self-similar processes to stationary processes, is an important tool to study these processes and gives a more general connection: in particular between DSI and cyclostationarity. Some general properties of DSI processes are given. Examples of random sequences with DSI are then constructed and illustrated. We address finally the problem of analysis of DSI processes, first using the inverse Lamperti transformation to analyse DSI processes by means of cyclostationary methods. Second we propose to re-write these tools directly in a Mellin formalism.

show abstract

“…Another developement is to break the self-similarity by supposing discrete scale invariance (DSI), as in [13]. This is achieved by taking coefficients a(t) and σ(t) (variance of the input noise) as periodic functions in log t. The same procedure leads to DSI with a log-periodic function multiplying the continous Green function that was used before.…”

Section: Direct Sampling Of Ec Systemsmentioning

confidence: 99%

On Sampling Methods for Linear Scale-Invariant Systems

Borgnat

2006 IEEE International Conference on Acoustics Speed and Signal Processing Proceedings

Self Cite

View full text Add to dashboard Cite

Stochastic discrete scale invariance

Cited by 34 publications

References 18 publications

Scale invariances and Lamperti transformations for stochastic processes

Scale invariances and Lamperti transformations for stochastic processes

Stochastic discrete scale invariance and Lamperti transformation

On Sampling Methods for Linear Scale-Invariant Systems

Contact Info

Product

Resources

About