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 xed 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.
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369
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Statistics for Long-Memory Processes
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- 1994
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169
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Ergodic theory of chaos and strange attractors
– Eckman, Ruelle
- 1985
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163
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Non-linear Time Series - A Dynamic System Approach
– Tong
- 1990
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114
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Prediction chaotic time series
– Farmer, Sidorowich
- 1987
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107
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Consistent nonparametric regression
– Stone
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102
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Nonlinear prediction of chaotic time series
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- 1989
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91
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Ergodic theory
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49
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Nonparametric Statistics for Stochastic Processes: Estimation and Prediction
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- 1998
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44
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Ergodic properties of invariant measures for piecewise monotonic transformations
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- 1982
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39
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Chaos and deterministic versus stochastic nonlinear modeling
– Casdagli
- 1991
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27
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Nonparametric Curve Estimation from Time Series
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- 1989
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22
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Finding chaos in noisy systems
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22
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Curve estimates
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17
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Asymptotically minimax adaptive estimation I: Upper bounds. Optimally adaptive estimates. Theory Probab
– Lepskii
- 1991
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17
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Multivariate local polynomial regression for time series: Uniform strong consistency and rates
– Masry
- 1996
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17
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Memory-universal prediction of stationary random processes
– Modha, Masry
- 1998
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15
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Predicting chaotic time series, Phys
– Farmer, Sidorowich
- 1987
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13
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Density Estimates and Markov Sequences
– Rosenblatt
- 1970
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13
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Nonparametric density and regression estimation from Markov sequences without mixing assumptions
– Yakowitz
- 1989
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9
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Noise reduction: finding the simplest dynamical system consistent with the data
– Kostelich, Yorke
- 1990
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8
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Statistical aspects of chaos: a review
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7
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Nonparametric estimation in Markov processes
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7
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Nonparametric estimation of the transition distribution function of a Markov process
– Roussas
- 1969
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6
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Nonparametric estimation of the chaotic function and the invariant measure of a dynamical system
– Bosq, Guegan
- 1995
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6
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Rates of convergence of nearest neighbor estimation under arbitrary sampling
– Kulkarni, Posner
- 1995
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6
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Limits to classification and regression estimation from ergodic processes
– Nobel
- 1999
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5
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Rigorous statistical procedures for data from dynamical systems
– Denker, Keller
- 1986
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5
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Chaotic dynamical systems with a view towards statistics: a review
– Jensen
- 1993
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5
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Approach to equilibrium for locally expanding maps in R k
– Mayer
- 1984
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5
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Regression estimation from an individual stable sequence
– Morvai, Kulkarni, et al.
- 1999
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4
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Density estimation from an individual numerical sequence
– Nobel, Morvai, et al.
- 1998
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3
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Strongly consistent density estimate from ergodic sample
– Györfi
- 1981
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3
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Estimating local Lyapunov exponents
– Lu, Smith
- 1997
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3
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Noise reduction: the simplest dynamical system consistent with the data
– Kostelich, Yorke
- 1990
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3
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Limits to classi and regression estimation from ergodic processes
– Nobel
- 1999
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2
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Finitary reconstruction of a measure preserving transformation
– Adams, Nobel
- 2001
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2
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Families of ergodic processes without consistent density or regression estimates
– Adams
- 1999
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2
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Sur l'estimation et la pr'evision non-param'etrique des processus ergodiques. Thesis at Univ. of Lille Flandres Artois
– Delecroix
- 1987
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2
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Nonparametric estimation of a regression function and its derivatives under an ergodic hypothesis
– Delecroix, Rosa
- 1996
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2
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The L1 and L2 strong consistency of recursive kernel density estimation from dependent samples
– Györfi, Masry
- 1990
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2
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Nearest neighbor regression estimation for null-recurrent Markov time series
– Yakowitz
- 1993
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2
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CLSI-Solutions: " VFORMAL User's Manual ", Version 1.0
– unknown authors
- 1993
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2
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The L 1 and L 2 Strong Consistency of Recursive Kernel Density Estimation from Dependent Samples
– Gyorfi, Masry
- 1990
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2
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Long range dependency eects with implications for forecasting and queuing inference. Preprint, submitted for publication
– Yakowitz, Heyde
- 1997
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1
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Strongly-consistent nonparametric estimation of smooth regression functions for stationary ergodic sequences. Under revision
– Yakowitz, Gyor, et al.
- 1997
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1
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Strongly consistent density estimate from ergodic sample
– Gyor
- 1981
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