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Predicting the maximum or the minimum of a random signal

par Webmaster - 20 octobre 2006

One is often interested in predicting the maximum or minimum in a given time interval of a random but correlated temporal signal X(t).

Defining the probability P_<(M,t) that the process X(t') (and hence its maximum) remains below the value M (t'\in[0,t]), its knowledge provides valuable information on the underlying physical system associated to X. For large time, one observes P_<(M,t)\sim t^{-\theta} in many physical system, or P_<(M,t)\sim \exp{-\theta t} for stationary systems, thus defining the persistence exponent \theta. This general problem has applications in

  • Physics : X(t) can be the height of an atomic interface, the magnetization of a spin system or an individual spin, the position of a random walker... Persistence exponents have been measured in as different systems as breath figures, liquid crystals, laser-polarized Xe gas, fluctuating steps on a Si surface, or soap bubbles.
  • Finance : X(t) can be the prize of a stock, an option, or any financial signal.
  • Image processing : statistical bounds of noisy signals are extremely useful (for instance in medical imaging or astrophysics) in order to obtain cleaner images by correcting spurious bright or dark pixels.
  • Biology : in the context of genetic cartography, statistical methods to evaluate the maximum of a complex signal has been exploited to identify putative quantitative trait loci.
  • Experimental sciences : when X(t) is the experimental noise, it is certainly important to be able to bound it efficiently.

Previous analytical results for \theta and P_<(M,t) have addressed the case M=0 (when X is of zero average) and have focused on the simpler case where X is a Gaussian process. In a recent article, a member of the laboratory has obtained an explicit form for P_<(M,t) from a minimal knowledge of the properties of a general process X (for instance, its correlation function \langle
X(t_1)X(t_2)\rangle). In addition, the distributions of time intervals when the signal remains below or above the level M and the persistence exponent have been computed. The calculation relies on the approximation that the lengths of the intervals between successive crossings of the level M are uncorrelated, an assumption becoming exact for large |M|.

The corresponding paper Probability distribution of the maximum of a smooth temporal signal has been published in Physical Review Letters (2007), and is available on cond-mat.

The corresponding author in the laboratory is Clément Sire.