Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Predicting dynamical systems with too few time-delay measurements: error estimates

Krzysztof Barański, Yonatan Gutman, Adam Śpiewak

Abstract

We study the problem of reconstructing and predicting the future of a dynamical system by the use of time-delay measurements of typical observables. Considering the case of too few measurements, we prove that for Lipschitz systems on compact sets in Euclidean spaces, equipped with an invariant Borel probability measure $μ$ of Hausdorff dimension $d$, one needs at least $d$ measurements of a typical (prevalent) Lipschitz observable for $μ$-almost sure reconstruction and prediction. Consequently, the Hausdorff dimension of $μ$ is the precise threshold for the minimal delay (embedding) dimension for such systems in a probabilistic setting. Furthermore, we establish a lower bound postulated in the Schroer--Sauer--Ott--Yorke prediction error conjecture from 1998, after necessary modifications (whereas the upper estimates were obtained in our previous work). To this aim, we prove a general theorem on the dimensions of conditional measures of $μ$ with respect to time-delay coordinate maps.

Predicting dynamical systems with too few time-delay measurements: error estimates

Abstract

We study the problem of reconstructing and predicting the future of a dynamical system by the use of time-delay measurements of typical observables. Considering the case of too few measurements, we prove that for Lipschitz systems on compact sets in Euclidean spaces, equipped with an invariant Borel probability measure of Hausdorff dimension , one needs at least measurements of a typical (prevalent) Lipschitz observable for -almost sure reconstruction and prediction. Consequently, the Hausdorff dimension of is the precise threshold for the minimal delay (embedding) dimension for such systems in a probabilistic setting. Furthermore, we establish a lower bound postulated in the Schroer--Sauer--Ott--Yorke prediction error conjecture from 1998, after necessary modifications (whereas the upper estimates were obtained in our previous work). To this aim, we prove a general theorem on the dimensions of conditional measures of with respect to time-delay coordinate maps.
Paper Structure (29 sections, 33 theorems, 170 equations)

This paper contains 29 sections, 33 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Time-delayed measurements
  3. Time-delay measurements from a probabilistic point of view
  4. New results
  5. Dimension of conditional measures
  6. Note on the proofs
  7. Structure of the paper
  8. Preliminaries
  9. Singular values
  10. Measures
  11. Prevalence
  12. Dimensions
  13. Relations between conditional measures and injectivity/predictability properties
  14. Technical tools
  15. Energy integral estimates
  16. ...and 14 more sections

Key Result

Theorem 1.2

Let $X \subset \mathbb R^N$, $N \in {\mathbb N}$, be a compact set and let $T\colon X \to X$ be a Lipschitz map. Then a prevalent Lipschitz observable $h \colon X \to \mathbb R$ is $k$-predictable, with a continuous prediction map, for every $k > 2\overline{\dim}_B\, X$. If, additionally, $T$ is inj

Theorems & Definitions (80)

  • Definition 1.1
  • Theorem 1.2: Time-delay prediction and embedding theorem
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6: Probabilistic time-delay prediction and embedding theorem
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 70 more