Optimal sampling strategies for multiscale stochastic processes

In this paper, we determine which non-random sampling of fixed size gives the best linear predictor of the sum of a finite spatial population. We employ different multiscale superpopulation models and use the minimum mean-squared error as our optimality criterion. In multiscale superpopulation tree models, the leaves represent the units of the population, interior nodes represent partial sums of the population, and the root node represents the total sum of the population. We prove that the optimal sampling pattern varies dramatically with the correlation structure of the tree nodes. While uniform sampling is optimal for trees with ``positive correlation progression'', it provides the worst possible sampling with ``negative correlation progression.'' As an analysis tool, we introduce and study a class of independent innovations trees that are of interest in their own right. We derive a fast water-filling algorithm to determine the optimal sampling of the leaves to estimate the root of an independent innovations tree.Comment: Published at http://dx.doi.org/10.1214/074921706000000509 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Wavelet analysis of fractional Brownian motion in multifractal time

- Nous étudions le mouvement Brownien fractionnaire en temps multifractal, un modèle de processus multifractal proposé récemment dans le cadre de l'étude de séries financières. Notre intérêt porte sur les propriétés statistiques des coefficients d'ondelette issus de la décomposition de ces processus. Parmi ces propriétés nous nous intéressons particulièrement aux corrélations résiduelles (longue dépendance), à la stationnarité, qui sont les composantes essentielles permettant de caractériser les performances statistiques des estimateurs de spectre multifractal, construits à partir de transformées en ondelettes

Fractional Brownian motion and data traffic modeling: The other end of the spectrum

International audienceWe analyze the fractal behavior of the high frequency part of the Fourier spectrum of fBm using multifractal analysis and show that it is not consistent with what is measured on real traffic traces. We propose two extensions of fBm which come closer to actual traffic traces multifractal properties

Wavelet and Multiscale Analysis of Network Traffic

The complexity and richness of telecommunications traffic is such that one may despair to find any regularity or explanatory principles. Nonetheless, the discovery of scaling behaviour in tele-traffic has provided hope that parsimonious models can be found. The statistics of scaling behavior present many challenges, especially in non-stationary environments. In this paper we describe the state of the art in this area, focusing on the capabilities of the wavelet transform as a key tool for unravelling the mysteries of traffic statistics and dynamics

On non scale invariant Infinitely Divisible Cascades

- Nous présentons les définitions et synthèses de processus stochastiques respectant des lois d'échelles qui s'écartent de façon contrôlée d'un comportement en loi de puissance. Nous définissons des bruit, mouvement et marche aléatoire issus de cascades infiniment divisibles (IDC). Nous étudions analytiquement le comportement des moments des accroissements de ces processus à travers les échelles. Ces résultats théoriques sont illustrés sur l'exemple d'une cascade log-Normale non invariante d'échelle. Les algorithmes de synthèse et les fonctions MATLAB utilisés sont disponibles sur nos pages web

Multifractal products of stochastic processes: Construction and some basic properties

Abstract In various fields, such as teletraffic and economics, measured time series have been reported to adhere to multifractal scaling. Classical cascading measures possess multifractal scaling, but their increments form a non-stationary process. To overcome this problem we introduce a construction of random multifractal measures based on iterative multiplication of stationary stochastic processes, a special form of T-martingales. We study L 2 -convergence, non-degeneracy and continuity of the limit process. Establishing a power law for its moments we obtain a formula for the multifractal spectrum and hint at how to prove the full formalism

Explicit Lower Bounds of the Hausdorff Dimension of Certain Self Affine Sets

Journal PaperA lower bound of the Hausdorff dimension of certain self-affine sets is given. Moreover, this and other known bounds such as the box dimension are expressed in terms of solutions of simple equations involving the singular values of the affinities

Numerical Estimates of Generalized Dimensions D_q for Negative q

Journal PaperUsual fixed-size box-counting algorithms are inefficient for computing generalized fractal dimensions D(<i>q</i>) in the range of <i>q</i><0. In this Letter we describe a new numerical algorithm specifically devised to estimate generalized dimensions for large negative <i>q</i>, providing evidence of its better performance. We compute the complete spectrum of the Hénon attractor, and interpret our results in terms of a "phase transition" between different multiplicative laws

An introduction to multifractals

Conference PaperThis is an easy read introduction to multifractals. We start with a thorough study of the Binomial measure from a multifractal point of view, introducing the main multifractal tools. We then continue by showing how to generate more general multiplicative measures and close by presenting an extensive set of examples on which we elaborate how to 'read' a multifractal spectrum