A Theory of State Space Reconstruction in the Presence of Noise

Casdagli, Martin; Eubank, Stephen; Farmer, J. Doyne; Gibson, John

doi:10.1007/978-1-4899-2305-9_5

Cited by 8 publications

(4 citation statements)

References 16 publications

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“…Markovian dynamics is rigorously applicable in fundamental physical systems and we capture this ideal with an approach to state space reconstruction of unknown dynamics that maximizes the predictive information of the state. Building upon the results of Takens and others [34][35][36][37][38], who have shown that past information can be leveraged to reconstruct a Markovian state [39], we include previous time points and examine how the predictability improves.…”

Section: Predictive Information For State Space Reconstructionmentioning

confidence: 99%

Maximally predictive ensemble dynamics from data

Costa

Ahamed

Jordan³

et al. 2021

Preprint

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We leverage the interplay between microscopic variability and macroscopic order to connect physical descriptions across scales directly from data, without underlying equations. We reconstruct a state space by concatenating measurements in time, building a maximum entropy partition of the resulting sequences, and choosing the sequence length to maximize predictive information. Trading non-linear trajectories for linear, ensemble evolution, we analyze reconstructed dynamics through transfer operators. The evolution is parameterized by a transition time τ: capturing the source entropy rate at small τ and revealing timescale separation with collective, coherent states through the operator spectrum at larger τ. Applicable to both deterministic and stochastic systems, we illustrate our approach through the Langevin dynamics of a particle in a double-well potential and the Lorenz system. Applied to the behavior of the nematode worm C. elegans, we derive a "run-and-pirouette" navigation strategy directly from posture dynamics. We demonstrate how sequences simulated from the ensemble evolution recover effective diffusion in the worm's centroid trajectories and introduce a top-down, operator-based clustering which reveals subtle subdivisions of the "run" behavior.

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Section: Predictive Information For State Space Reconstructionmentioning

confidence: 99%

Maximally predictive ensemble dynamics from data

Costa

Ahamed

Jordan³

et al. 2021

Preprint

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“…Some of them consider the chaotic functioning of transport systems. Table 1 contains works on chaos [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], assessing the characteristics of the influence of chaos on the functioning of transport systems [17,[24][25][26][27][28], and the influence of chaos on human behavioral processes and sociodynamics [29][30][31], studying the properties of dynamic systems [32][33][34][35][36][37][38]. Skeptics of applying chaos theory to the study of traffic processes formulate the following common question: "in what way are the chaos and complexity theories specifically applied to urban transport systems?"…”

Section: The Place and Role Of Chaos In Operation Of The Sustainable ...mentioning

confidence: 99%

Ideology of Urban Road Transport Chaos and Accident Risk Management for Sustainable Transport Systems

Morozov,

Petrov,

Shepelev

et al. 2024

Sustainability

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Transport systems are complex systems present in modern cities. The sustainability of all other urban systems depends on the sustainable functioning of urban transport. Various processes occur within transport systems. Road traffic is one of them. At the same time, road traffic is a rather complex process to manage, which is explained by the influence of many different internal and external environmental factors. The unpredictable and chaotic behavior of each vehicle in a traffic flow complicates predicting the transport situation and traffic management. This problem gave rise to several unsolved problems, including traffic congestion and road accident rates. The solution to these problems is connected with sustainably managing transport systems in terms of road traffic. However, numerous regularities between elements within the system should be understood in order to implement the management process. Unfortunately, the results of many previous studies often reflect only partial regularities and have limited functionality. Therefore, a new approach to urban traffic management is needed. As opposed to the existing solutions, the authors of this paper propose to implement management based on the regularities of changes in the chaos of the transport system. In this regard, the purpose of this research is to establish the relationship between road traffic chaos and road accident rates. The general methodological basis of this research is the system approach and its methods: analysis and synthesis. The theoretical studies were mostly based on the theories of chaos, dynamic systems, and traffic flows. The experimental studies were based on the theories of experimental design, probability, and mathematical statistics. To achieve this goal, a profound analysis covered studies on the sustainability of transport and dynamic systems, sociodynamics, and traffic. The authors proposed considering the relative entropy of lane occupancy at signal-controlled intersections as a measure for assessing traffic flow chaos and sustainability. Notably, as the main conclusions, the authors established regularities for the influence of entropy on the kinetic energy of traffic flows and injury risk. It also makes sense to emphasize that the initial data for the experiment were collected via real-time processing of video images using neural network technologies. These technologies will further allow for the implementation of traffic management and real-time forecasting of various events. Ultimately, the authors identified changes in injury risk depending on the level of road chaos. According to the authors, the obtained results can be used to improve the sustainability of urban transport systems. The research identified changes in injury risk depending on the level of road chaos, which could have significant implications for urban traffic management strategies.

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“…In our situation, the data is permitted to be noisy, and the embedding procedure amplifies noise (Casdagli et al 1991). Therefore, the optimal control approach of approximating the Takens field allows a global, continuous, and nonlinear representation of g to be estimated.…”

Section: Partially Observed Predictionmentioning

confidence: 99%

Optimal Reconstruction of Vector Fields from Data for Prediction and Uncertainty Quantification

McGowan,

Robertson,

Blachut

et al. 2024

J Nonlinear Sci

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Predicting the evolution of dynamics from a given trajectory history of an unknown system is an important and challenging problem. This paper presents a model-free method of forecasting unknown chaotic systems through reconstructing vector fields from noisy measured data via an adaptation of optimal control methods. This technique is also applicable to partially observed systems using a Takens delay embedding approach. The algorithms are validated on the Lorenz system and the four-dimensional hyperchaotic Rössler system, and demonstrate successful predictions well beyond the Lyapunov timescale. It is found that for small datasets or datasets with large levels of noise, the prediction accuracy of partially observed systems approaches that of fully observed systems. The presented approach also allows the model-free assessment of local predictability on the attractor by evolving initial condition density through the reconstructed vector fields via estimation of the transfer operator. The method is compared to predictions made by an imperfect model which highlights the utility of model-free approaches when the only available models have significant model error. The capability of this method for reconstruction of continuous and global vector fields may be applied to model validation, forecasting of initial conditions not in the training set, and model-free filtering.

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A Theory of State Space Reconstruction in the Presence of Noise

Cited by 8 publications

References 16 publications

Maximally predictive ensemble dynamics from data

Maximally predictive ensemble dynamics from data

Ideology of Urban Road Transport Chaos and Accident Risk Management for Sustainable Transport Systems

Optimal Reconstruction of Vector Fields from Data for Prediction and Uncertainty Quantification

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