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Heterogeneous Beliefs and Routes to Chaos in a Simple Asset Pricing Model
, 1998
"... This paper investigates the dynamics in a simple present discounted value asset pricing model with heterogeneous beliefs. Agents choose from a finite set of predictors of future prices of a risky asset and revise their `beliefs' in each period in a boundedly rational way, according to a `fitnes ..."
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Cited by 370 (23 self)
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This paper investigates the dynamics in a simple present discounted value asset pricing model with heterogeneous beliefs. Agents choose from a finite set of predictors of future prices of a risky asset and revise their `beliefs' in each period in a boundedly rational way, according to a `fitness measure' such as past realized profits. Price fluctuations are thus driven by an evolutionary dynamics between different expectation schemes (`rational animal spirits'). Using a mixture of local bifurcation theory and numerical methods, we investigate possible bifurcation routes to complicated asset price dynamics. In particular, we present numerical evidence of strange, chaotic attractors when the intensity of choice to switch prediction strategies is high.
A multifractal wavelet model with application to TCP network traffic
 IEEE TRANS. INFORM. THEORY
, 1999
"... In this paper, we develop a new multiscale modeling framework for characterizing positivevalued data with longrangedependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the mo ..."
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Cited by 213 (34 self)
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In this paper, we develop a new multiscale modeling framework for characterizing positivevalued data with longrangedependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the model provides a rapid O(N) cascade algorithm for synthesizing Npoint data sets. We study both the secondorder and multifractal properties of the model, the latter after a tutorial overview of multifractal analysis. We derive a scheme for matching the model to real data observations and, to demonstrate its effectiveness, apply the model to network traffic synthesis. The flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics (variancetime plots and moment scaling) and queuing behavior. Although for illustrative purposes we focus on applications in network traffic modeling, the multifractal wavelet model could be useful in a number of other areas involving positive data, including image processing, finance, and geophysics.
Nearestneighbor searching and metric space dimensions
 In NearestNeighbor Methods for Learning and Vision: Theory and Practice
, 2006
"... Given a set S of n sites (points), and a distance measure d, the nearest neighbor searching problem is to build a data structure so that given a query point q, the site nearest to q can be found quickly. This paper gives a data structure for this problem; the data structure is built using the distan ..."
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Cited by 106 (0 self)
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Given a set S of n sites (points), and a distance measure d, the nearest neighbor searching problem is to build a data structure so that given a query point q, the site nearest to q can be found quickly. This paper gives a data structure for this problem; the data structure is built using the distance function as a “black box”. The structure is able to speed up nearest neighbor searching in a variety of settings, for example: points in lowdimensional or structured Euclidean space, strings under Hamming and edit distance, and bit vector data from an OCR application. The data structures are observed to need linear space, with a modest constant factor. The preprocessing time needed per site is observed to match the query time. The data structure can be viewed as an application of a “kdtree ” approach in the metric space setting, using Voronoi regions of a subset in place of axisaligned boxes. 1
The Dimensions of Individual Strings and Sequences
 INFORMATION AND COMPUTATION
, 2003
"... A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary ..."
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Cited by 99 (11 self)
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A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that
Probabilistic and fractal aspects of Lévy trees
 Probab. Th. Rel. Fields
, 2005
"... We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete GaltonWatson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted Rtrees, which i ..."
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Cited by 92 (21 self)
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We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete GaltonWatson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted Rtrees, which is equipped with the GromovHausdorff distance. To construct Lévy trees, we make use of the coding by the height process which was studied in detail in previous work. We then investigate various probabilistic properties of Lévy trees. In particular we establish a branching property analogous to the wellknown property for GaltonWatson trees: Conditionally given the tree below level a, the subtrees originating from that level are distributed as the atoms of a Poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root. We study regularity properties of local times in the space variable, and prove that the support of local time is the full level set, except for certain exceptional values of a corresponding to local extinctions. We also compute several fractal dimensions of Lévy trees, including Hausdorff and packing dimensions, in terms of lower and upper indices for the branching
Conformally invariant processes in the plane
 Mathematical Surveys and Monographs, 114. American Mathematical Society
, 2005
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Effective strong dimension in algorithmic information and computational complexity
 SIAM Journal on Computing
, 2004
"... The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded exten ..."
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Cited by 75 (25 self)
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The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. Lutz (2000) has recently proven a simple characterization of Hausdorff dimension in terms of gales, which are betting strategies that generalize martingales. Imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of Hausdorff dimension, including constructive, computable, polynomialspace, polynomialtime, and finitestate dimensions. Work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science. In this paper we show that packing dimension can also be characterized in terms of gales. Moreover, even though the usual definition of packing dimension is considerably more complex than that of Hausdorff dimension, our gale characterization of packing dimension is an exact dual
Measure Theoretic Laws for lim sup Sets
, 2004
"... Given a compact metric space (Ω, d) equipped with a nonatomic, probability measure m and a positive decreasing function ψ, we consider a natural class of limsup subsets Λ(ψ) of Ω. The classical limsup set W(ψ) of ‘ψ–approximable ’ numbers in the theory of metric Diophantine approximation fall withi ..."
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Cited by 63 (11 self)
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Given a compact metric space (Ω, d) equipped with a nonatomic, probability measure m and a positive decreasing function ψ, we consider a natural class of limsup subsets Λ(ψ) of Ω. The classical limsup set W(ψ) of ‘ψ–approximable ’ numbers in the theory of metric Diophantine approximation fall within this class. We establish sufficient conditions (which are also necessary under some natural assumptions) for the m–measure of Λ(ψ) to be either positive or full in Ω and for the Hausdorff fmeasure to be infinite. The classical theorems of KhintchineGroshev and Jarník concerning W(ψ) fall into our general framework. The main results provide a unifying treatment of numerous problems in metric Diophantine approximation including those for real, complex and padic fields associated with both independent and dependent quantities. Applications also include those to Kleinian groups and rational maps. Compared to earlier works our framework allows us to successfully remove many unnecessary conditions and strengthen results such as Jarník’s theorem and the BakerSchmidt theorem. In particular, the strengthening of Jarník’s theorem opens up the DuffinSchaeffer conjecture for Hausdorff measures.
A Mass Transference Principle and the Duffin–Schaeffer conjecture for Hausdorff measures
, 2005
"... A Hausdorff measure version of the DuffinSchaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements ..."
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Cited by 50 (10 self)
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A Hausdorff measure version of the DuffinSchaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for limsup subsets of R k to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of limsup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.
Faster rates in regression via active learning
 in Proceedings of NIPS
, 2005
"... In this paper we address the theoretical capabilities of active sampling for estimating functions in noise. Specifically, the problem we consider is that of estimating a function from noisy pointwise samples, that is, the measurements which are collected at various points over the domain of the fun ..."
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Cited by 49 (11 self)
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In this paper we address the theoretical capabilities of active sampling for estimating functions in noise. Specifically, the problem we consider is that of estimating a function from noisy pointwise samples, that is, the measurements which are collected at various points over the domain of the function. In the classical (passive) setting the sampling locations are chosen a priori, meaning that the choice of the sample locations precedes the gathering of the function observations. In the active sampling setting, on the other hand, the sample locations are chosen in an online fashion: the decision of where to sample next depends on all the observations made up to that point, in the spirit of the twenty questions game (as opposed to passive sampling where all the questions need to be asked before any answers are given). This extra degree of flexibility leads to improved signal reconstruction in comparison to the performance of classical (passive) methods. We present results characterizing the fundamental limits of active learning for various nonparametric function classes, as well as practical algorithms capable of exploiting the extra flexibility of the active setting and provably improving on classical techniques. In particular, significantly faster rates of convergence are achievable in cases involving functions whose complexity (in a the Kolmogorov sense) is highly concentrated in small regions of space (e.g., piecewise constant functions). Our active learning theory and methods show promise in a number of applications, including field estimation using wireless sensor networks and fault line detection. 1