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Denoising Deterministic Time Series
, 2002
"... This paper addresses a problem of statistical inference from dependent processes, namely how to recover a deterministic time series from observations that are corrupted by additive, independent noise. We will refer to this as the denoising problem. A distinctive feature of the denoising problem i ..."
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This paper addresses a problem of statistical inference from dependent processes, namely how to recover a deterministic time series from observations that are corrupted by additive, independent noise. We will refer to this as the denoising problem. A distinctive feature of the denoising problem is that the available observations exhibit dependence across long time scales and, as a consequence, existing statistical theory and methods are not readily applicable. This paper describes one analysis of the denoising problem, beginning from rst principles. We establish both positive and negative results.
What Is Information?
, 1995
"... this paper knows that Shannon did no such thing. It must not be forgotten that Shannon called his theory "a general theory of communication ", not a theory of information. The distinction is crucial. As Shannon put it in [15]: The fundamental problem of communication is that of reproducing ..."
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this paper knows that Shannon did no such thing. It must not be forgotten that Shannon called his theory "a general theory of communication ", not a theory of information. The distinction is crucial. As Shannon put it in [15]: The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is, they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages. It would be impossible to overstress the fact that all aspects of "information" other than statistical phenomena are completely irrelevant to communication theory.
All Entropies Agree For An Sft
"... this paper I discuss a number of "entropies" which have definitions which are respectively probabilistic, topological, algebraic, and algorithmic. I shall explain how these entropies are all defined in the setting of shift dynamical systems. The main result of the paper, which should be re ..."
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this paper I discuss a number of "entropies" which have definitions which are respectively probabilistic, topological, algebraic, and algorithmic. I shall explain how these entropies are all defined in the setting of shift dynamical systems. The main result of the paper, which should be regarded as part of the folklore, is the fact that for topologically transitive shifts of finite type (the definitions of these terms will be found below) all these entropies agree numerically.
Estimating a Function from Ergodic Samples with Additive Noise
, 2001
"... We study the problem of estimating an unknown function from ergodic samples corrupted by additive noise. It is shown that one can consistently recover an unknown measurable function in this setting if the one dimensional distribution of the samples is comparable to a known reference distribution, an ..."
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We study the problem of estimating an unknown function from ergodic samples corrupted by additive noise. It is shown that one can consistently recover an unknown measurable function in this setting if the one dimensional distribution of the samples is comparable to a known reference distribution, and the noise is independent of the samples and has known mixing rates. The estimates are applied to deterministic sampling schemes, in which successive samples are obtained by repeatedly applying a fixed map to a given initial vector, and it is then shown how the estimates can be used to reconstruct an ergodic transformation from one of its trajectories.
Statistical Classification Of Chaotic Signals
"... The classification of chaotic signals generated by a lowdimensional deterministic models given a dictionary of possible model is considered. The proposed classification methods rely on the concept of "best predictor" of signal. A statistical interpretation of this concept based on the erg ..."
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The classification of chaotic signals generated by a lowdimensional deterministic models given a dictionary of possible model is considered. The proposed classification methods rely on the concept of "best predictor" of signal. A statistical interpretation of this concept based on the ergodic theory of chaotic system is presented. A sort of "bootstrapping" estimator of the statistical properties is introduced. The method is validated by numerical simulations. Directions for future research are suggested. 1. INTRODUCTION Also known under the popular name of "chaos theory," the theory of nonlinear dynamical system has been the subject of considerable advances over the past twenty years. Chaotic systems are deterministic dynamical systems with a small number of degrees of freedom whose behavior appears, in a sense, random and unpredictable. Most of the early research efforts on chaos theory have been the fact of mathematicians and physicists. Mathematical and computational tools for the...