IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v407y2014icp252-275.html
   My bibliography  Save this article

On the frequencies of patterns of rises and falls

Author

Listed:
  • Luck, J.M.

Abstract

We investigate the probability of observing a given pattern of n rises and falls in a random stationary data series. The data are modeled as a sequence of n+1 independent and identically distributed random numbers. This probabilistic approach has a combinatorial equivalent, where the data are modeled by a random permutation on n+1 objects. The probability of observing a long pattern of rises and falls decays exponentially with its length n in general. The associated decay rate α is interpreted as the embedding entropy of the pattern. This rate is evaluated exactly for all periodic patterns. In the most general case, it is expressed in terms of a determinant of generalized hyperbolic or trigonometric functions. Alternating patterns have the smallest rate αmin=ln(π/2)=0.451582…, while other examples lead to arbitrarily large rates. The probabilities of observing uniformly chosen random patterns are demonstrated to obey multifractal statistics. The typical value α0=0.806361… of the rate plays the role of a Lyapunov exponent. A wide range of examples of patterns, either deterministic or random, is also investigated.

Suggested Citation

  • Luck, J.M., 2014. "On the frequencies of patterns of rises and falls," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 407(C), pages 252-275.
    • Handle: RePEc:eee:phsmap:v:407:y:2014:i:c:p:252-275
    DOI: 10.1016/j.physa.2014.04.010
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437114003148
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2014.04.010?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Le Doussal, Pierre & Monthus, Cécile, 2003. "Exact solutions for the statistics of extrema of some random 1D landscapes, application to the equilibrium and the dynamics of the toy model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 317(1), pages 140-198.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Gregory Schehr & Satya N. Majumdar, 2013. "Exact record and order statistics of random walks via first-passage ideas," Papers 1305.0639, arXiv.org.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:407:y:2014:i:c:p:252-275. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.