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Applied Nonlinear Time Series Analysis Applications in Physics Physiology and Finance 1st Edition by Michael Small ISBN 981256117X 9789812561176

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Author(s): Michael Small
ISBN(s): 9789812561176, 981256117X
Edition: 1
File Details: PDF, 13.09 MB
Year: 2005
Language: english
SKU: EB-1103188 Category: Tag:

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ISBN 10: 981256117X 
ISBN 13: 9789812561176
Author: Michael Small

Nonlinear time series methods have developed rapidly over a quarter of a century and have reached an advanced state of maturity during the last decade. Implementations of these methods for experimental data are now widely accepted and fairly routine; however, genuinely useful applications remain rare. This book focuses on the practice of applying these methods to solve real problems.To illustrate the usefulness of these methods, a wide variety of physical and physiological systems are considered. The technical tools utilized in this book fall into three distinct, but interconnected areas: quantitative measures of nonlinear dynamics, Monte-Carlo statistical hypothesis testing, and nonlinear modeling. Ten highly detailed applications serve as case studies of fruitful applications and illustrate the mathematical techniques described in the text.
 

Applied Nonlinear Time Series Analysis Applications in Physics Physiology and Finance 1st Table of contents:

Chapter 1: Time Series Embedding and Reconstruction

1.1 Stochasticity and Determinism: Why Should We Bother?

  • Introduction to the concepts of stochastic (random) and deterministic (predictable) systems in time series analysis.

1.2 Embedding Dimension

  • The embedding dimension is crucial for reconstructing a phase space from a time series.

  • 1.2.1 False Nearest Neighbours: A method for determining the optimal embedding dimension.

  • 1.2.2 False Strands and So On: Likely discusses other phenomena or artifacts that arise in the process of embedding.

  • 1.2.3 Embed, Embed and Then Embed: A focus on iterative embedding processes to better reconstruct the state space.

  • 1.2.4 Embed and Model, and Then Embed Again: Combining embedding with modeling techniques.

1.3 Embedding Lag

  • Lag refers to the time delay used in embedding the series.

  • 1.3.1 Autocorrelation: Used to understand time-dependent relationships.

  • 1.3.2 Mutual Information: A measure of the dependency between the current and lagged values.

  • 1.3.3 Approximate Period: Estimation of periodicity in the series.

  • 1.3.4 Generalised Embedding Lags: General techniques for choosing embedding lags.

1.4 Which Comes First?

  • Likely a discussion on causality or the sequence of events in a time series.

1.5 An Embedding Zoo

  • Possibly an exploration of different types of embedding methods available in time series analysis.

1.6 Irregular Embeddings

  • 1.6.1 Finding Irregular Embeddings: Techniques for dealing with irregular or noisy data.

1.7 Embedding Window

  • Time windows in embedding are crucial for modeling.

  • 1.7.1 A Modelling Paradigm: Discusses how to frame embedding for modeling.

  • 1.7.2 Examples: Case studies and examples of embedding windows.

1.8 Application: Sunspots and Chaotic Laser Dynamics

  • Applying embedding and reconstruction techniques to real-world chaotic systems.

1.9 Summary

  • Recap of the techniques and concepts introduced in this chapter.

Chapter 2: Dynamic Measures and Topological Invariants

2.1 Correlation Dimension

  • A measure of the complexity of a dynamic system.

2.2 Entropy, Complexity, and Information

  • 2.2.1 Entropy: A measure of disorder or unpredictability.

  • 2.2.2 Complexity: How complex or structured the system is.

  • 2.2.3 Alternative Encoding Schemes: Techniques to represent the data differently for better analysis.

2.3 Application: Detecting Ventricular Arrhythmia

  • Applying dynamic measures to medical data, specifically heart arrhythmias.

2.4 Lyapunov Exponents and Nonlinear Prediction Error

  • Lyapunov exponents measure the sensitivity to initial conditions in chaotic systems.

2.5 Application: Potential Predictability in Financial Time Series

  • How these dynamic measures can be used to predict financial markets.

2.6 Summary

  • Summary of dynamic measures and topological invariants.

Chapter 3: Estimation of Correlation Dimension

3.1 Preamble

  • Introduction to methods for estimating the correlation dimension, a key dynamic measure.

3.2 Box-Counting and the Grassberger-Procaccia Algorithm

  • Techniques for calculating the correlation dimension.

3.3 Judd’s Algorithm

  • Another method for estimating the correlation dimension.

3.4 Application: Distinguishing Sleep States by Monitoring Respiration

  • Using the correlation dimension to classify sleep states.

3.5 The Gaussian Kernel Algorithm

  • A more advanced method for estimating dimensions.

3.6 Application: Categorising Cardiac Dynamics from Measured ECG

  • Using correlation dimension to analyze ECG data.

3.7 Even More Algorithms

  • Discusses other methods for estimating the correlation dimension.

Chapter 4: The Method of Surrogate Data

4.1 The Rationale and Language of Surrogate Data

  • Surrogate data is often used to test whether a time series is deterministic or stochastic.

4.2 Linear Surrogates

  • 4.2.1 Algorithm 0 and its Analogues: Basic surrogate data generation techniques.

  • 4.2.2 Algorithm 1 and its Applications: More advanced surrogate methods.

  • 4.2.3 Algorithm 2 and its Problems: Discusses limitations of certain surrogate methods.

4.3 Cycle Shuffled Surrogates

  • A type of surrogate data generated by shuffling cycles in the time series.

4.4 Test Statistics

  • Methods for validating the surrogate data against the original data.

4.5 Correlation Dimension: A Pivotal Test Statistic — Linear Hypotheses

  • Discusses using the correlation dimension as a key test statistic for surrogate analysis.

4.6 Application: Are Financial Time Series Deterministic?

  • Investigating financial data to see if they exhibit deterministic dynamics.

4.7 Summary

  • Recap of surrogate data methods and applications.

Chapter 5: Non-Standard and Non-Linear Surrogates

5.1 Generalized Nonlinear Null Hypotheses

  • Advanced null hypotheses for surrogate testing in nonlinear systems.

5.1.1 The “Pivotalness” of Dynamic Measures

  • The importance of dynamic measures in testing nonlinear hypotheses.

5.2 Application: Infant Sleep Apnea

  • Applying nonlinear surrogate data techniques to medical monitoring.

5.3 Pseudo-Periodic Surrogates

  • Creating surrogates that mimic periodic patterns.

5.4 Application: Mimicking Human Vocalization Patterns

  • Using surrogate data to study vocalizations.

5.5 Application: Are Financial Time Series Really Deterministic?

  • Further examination of financial time series, now considering nonlinear surrogates.

5.6 Simulated Annealing and Other Computational Methods

  • Techniques like simulated annealing used for finding surrogate data.

5.7 Summary

  • Summary of non-standard surrogate methods.

Chapter 6: Identifying the Dynamics

6.1 Phenomenological and Ontological Models

  • Discusses different types of models for identifying dynamics in time series.

6.2 Application: Severe Acute Respiratory Syndrome (SARS)

  • A case study on modeling governmental strategies during the SARS outbreak.

6.3 Local Models

  • Techniques for local modeling in time series.

6.4 The Importance of Embedding for Modelling

  • Emphasizes the importance of embedding techniques in dynamic modeling.

6.5 Semi-local Models

  • Semi-local models and their applications:

  • 6.5.1 Radial Basis Functions

  • 6.5.2 Minimum Description Length Principle

  • 6.5.3 Pseudo Linear Models

  • 6.5.4 Cylindrical Basis Models

6.6 Application: Predicting Onset of Ventricular Fibrillation

  • Predictive modeling for cardiac events.

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