Chapter 1
1 Introduction
2 The Type-I and Type-II Approaches
2.1 Implementation of the Type-I Approach
2.2 Implementation of the Type-II Approach
3 Multistep Prediction of the Long-Memory Time Series by the Type-II Approaches
4 Linear Long Memory Processes with Infinite Variance
5 Long Memory Chaotic Time Series
Acknowledgement
References
Chapter 2
1 Introduction
2 Self-similarity, Fractional Gaussian Noise, and Fractional Brownian Motion
2.1 Self-similarity and Fractional Gaussian Noise
2.3 The Power Spectra of fGn and fBm
3 The Continuous-Time Fractional Gaussian Noise and Fractional Brownian Motion
4 Estimations of Power Spectra and Their Statistical Accuracy
5 The Shape of Two-Dimensional fBm
6 The Fractal Geometry of fBm and fGn
7 Nonlinear Block Transformation and Stability of Its Fixed Point
8 Discussion
Acknowledgements
References
Chapter 3
1 Introduction and Motivation
1.1 Scaling Phenomena
1.2 Random Walks and Self-similarity
1.3 Wavelets
2 Self-similarity and Wavelets
2.1 Self-similarity
2.2 Wavelet Analysis
2.3 Self-similarity and Wavelets: Theory
2.4 Self-similarity and Wavelets: Application
3 Beyond Self-similarity
3.1 Practical Limitations
3.2 Beyond Finite Variance
3.3 Beyond Scaling over All Scales: Long Range Dependence, 1/f-Processes and Local Regularity
3.4 Beyond Second-Order Statistics – Multiplicative and Multifractal Processes
3.5 Beyond Power Laws – Infinitely Divisible Cascade
4 Conclusion
References
Chapter 4
1 Introduction
2 The Time Series
2.1 Linear Fractional Stable Motion
2.3 Asymptotic Self-similarity. Connection Between LFSM and FARIMA
3 Wavelet Estimators for the Self-similarity Parameter H of LFSM
3.1 The “power” and “log” Estimators
3.2 Asymptotic Properties of the “power” and “log” Estimators
4 Wavelet Estimators for the Hurst Exponent of a FARIMA Time Series
4.1 The Estimators ilde {H_{beta}} and ilde {Hlog}
4.2 Asymptotic Properties of the Estimators ilde{H_{beta}} and ilde{Hlog}
5 On Computing the Estimators in Practice
6 Computer Simulations
6.1 The Case of LFSM
6.2 The Choice of the Regression Weight Matrix G
6.4 The Coice of Scales n_1, . . . , n_m
6.6 The Case of FARIMA Time Series
Appendix
References
Chapter 5
1 Introduction
2 The Lamperti Transformation
2.1 Stationarity and Self-similarity
2.2 The Transform
2.3 From Stationarity to Self-similarity, and Back
2.4 Consequences
3 Examples and Applications
3.1 Tones and Chirps
3.2 Fractional Brownian Motion
3.3 Ornstein–Uhlenbeck Processes
3.4 Euler–Cauchy Processes
4 Variations
4.1 Nonstationary Tools
4.2 From Global to Local
4.3 Discrete Scale Invariance
5 Conclusion
References
Chapter 6
1 Introduction
1.1 “Normal” and “Anomalous” Fluctuations, the Noah and Joseph Effects and the Distinction Between the Local and the Global Form of Statistical Dependence
1.2 Inspiration for the Partly Random Fractal Sums of Pulses (PFSP)
1.3 Sketch of the Construction of the PFSP
1.4 The Roles of delta as Tail Exponent and H = 1/ delta as Self-affinity Exponent
1.5 When delta < 2, the Link Between the Tail and Self-affinity Exponents Complicates the Testing for Global Dependence
1.6 Lateral Rescaling and Lateral Attractors
1.7 Summary
1.8 Related Papers
2 Deffnitions
2.1 Stationarity and Self-affinity
2.2 Pulse Templates, Pulses, Affine Convolutions, and Fractal Sums of Pulses
2.3 The Levy Measure for the Probability Distribution of Pulse Height and Position
2.4 Semi-random Pulses and the Probability Distribution of Pulse Widths and Position
3 Self-affinity and the Exponent H = 1/delta; Existence of Global Dependence
3.1 The Self-affinity Property of all PFSP
3.2 First Corollary of Self-affinity: Each PFSP Defines a Special Domain of Attraction, Hence the Standard Limit Problem Concerning Random Processes Becomes Degenerate
3.3 Second Corollary of Self-affinity: All PFSP Are Globally Dependent
3.4 Thoughts on the Role of Limit Theorems, Given That, in the Case of PFSP, the Standard Limit Problem Is Degenerate
4 The Concept of “Lateral Limit Problem” and the Exponent alpha; for PFSP, the Lateral Attractor Can Be Either Uniscaling (alpha = delta = 1/H) or Pluriscaling (alpha= min[2, delta]
eq 1/H
4.1 Background of the New “Lateral” Limit Problem
4.2 The Lateral Limit Problem as Applied to PFSP; Reason for the Term “Lateral”
4.3 An Important Corollary of the Results in Table 1: Global Dependence Can Be “Pluriscaling” (H
eq 1/alpha), Like in the Limit Case FBM; but It Can Also Be “Uniscaling” (H = 1/alpha), Like in the Limit Case LSM
5 Address Diagrams and the Mechanism of Non-linear Global Dependence in PFSP
5.1 Address Function and Diagram; Characteristic Function of F
5.2 Joint Address Diagrams and Pictorial Illustration of the Reason for Interdependence Between the Delta F Corresponding to Two Non-overlapping Intervals; Its Strongly Non-linear Character
6 Discussion of Table 1: Effects of Pulse Shape on the Admissible delta, and on the Lateral Attractor
7 Proofs of the Claims in Table 1 for the Cylindrical Pulses
7.1 The Logarithms of the Characteristic Function (l.c.f.) of F(T ; C, delta).
7.2 The Attractor in the Case delta < 2. Lateral Attraction to Symmetric Levi Stable Increments with alpha=delta
7.4 The Attractor in the Case delta > 2. Lateral Attraction to Gaussian Increments
7.6 Some Semi-random PFSP Belonging to the Domain of Standard Attraction of a Semi-random Self-affine PFSP
Acknowledgments
References
Chapter 7
1 Introduction
2 Master Equations for Random Walks
3 Decoupled Memory: A Diffusive Case
4 Coupled Memory: A Diffusion Front
5 Long Random Flights with Constant Velocity
6 Random Flights in a Turbulent Fluid
7 Relativistic Turbulent Diffusion[9]
8 Accelerating Random Flights in a Gravitational Field
9 Conclusions
References
Chapter 8
1 Introduction
2 The Space-Time Fractional Diffusion Equation
3 The Green Function for Space-Time Fractional Diffusion
4 From CTRW to Fractional Diffusion
5 Simulations
6 Conclusions
Acknowledgements
References
Chapter 9
1 Introduction
2 Fractional Fokker-Planck Equation for Levy Type Anomalous Diffusion with Drift
3 FPT Density Function for Levy Type Anomalous Diffusion with Zero Drift
4 Laplace Transform of FPT Density Function for Levy Type Anomalous Diffusion with Drift
5 Summary
Acknowledgments
Appendix: Properties of H-functions
References
Chapter 10
1 Introduction
2 Deviation from Gaussian
3 Diffusion in Connected Porous Media
4 Time Dependent Diffusion Coeffcient in a Disordered Medium
5 Effect of Relaxation
6 Dispersion
7 Conclusion
Acknowledgement
References
Chapter 11
1 Introduction
2 AC-Driven Hamiltonian: The Model
3 The Role of Regular Islands
4 Generalized Asymmetric CTRW-Model
5 Temporary Symmetry Breaking Action: Manipulation of Systems
6 Conclusion
Acknowledgment
Appendix
References
Chapter 12
1 Introduction to Patterns in Economics
2 Classic Approaches to Finance Patterns
3 Patterns in Finance Fluctuations
4 Patterns Resembling “Diffusion in a Tsunami Wave”
5 Patterns Resembling Critical Point Phenomena
6 Cross-Correlations in Price Fluctuations of Different Stocks
7 Patterns in Firm Growth
8 Universality of the Firm Growth Problem
9 “Take-Home Message”
Acknowledgements
References
Chapter 13
1 Introduction
2 The Model
2.1 Deffnition
2.2 Intuitive Explanation of the Deffnition
2.3 Some Economic Motivation
3 Estimation of SEMIFAR Models – A Review
3.1 Kernel Estimation of the Trend Function
3.2 Maximum Likelihood Estimation
3.3 Estimation of the Whole Model
4 SEMIFAR Forecasting
4.1 Extrapolation of the Trend Function
4.2 Prediction of the Stochastic Component
4.3 Prediction Intervals
5 Examples
5.1 Commodities and Exchange Rates
5.2 Volatility of Stock Market Indices
5.3 Simulated Examples
5.4 Comparison Between SEMIFAR and AR
6 Final Remarks
Acknowledgements
References
Chapter 14
1 Long-Range Dependence in Finance
2 Long-Range Dependent vs. Change-Point Processes
2.1 Statistical Inference
2.2 Long-Memory Volatility Models
2.3 Multivariate Analysis
2.4 Change-Point Processes
3 Interaction Models
4 Simulation Study
Acknowledgments
References
Chapter 15
1 Introduction
2 Long Memory: Definition and Statistical Tests
3 Studies of Long Memory in Real and Financial Economic Variables
4 Theoretical Basis for Long Memory in Macroeconomic Variables
5 Long Memory and Economic Growth Revisited
6 Discussion and Conclusions
References
Chapter 16
1 Introduction
2 Correlated Firing in Sensory Neurons
2.1 The Firing Model
2.2 Interspike Interval Correlations
2.3 Detection of Weak Signals
2.4 Discussion of Correlated Firing
3 Delayed Neurodynamical Systems
3.1 Delay-Differential Equations
3.2 Correlations
3.3 Delayed Neural Control
4 Noise Induced Stabilization of Bumps
4.1 Background
4.2 Stochastic Working Memory
4.3 Discussion of Noisy Bumps
5 Conclusion
References
Chapter 17
1 Introduction
2 Long Range Dependence of Synchronization Timing Errors
2.1 Methods
2.2 Data Analyses and Results
3 Dependence of Scaling Exponent on Task Conditions and Coordination Strategies
3.1 Methods
3.2 Results and Discussion
4 Discussion
Acknowledgments
References
Chapter 18
1 Introduction
2 Self-organization in Neuronal Systems
2.1 Brain Ontogeny
2.2 Oscillations in Neuronal Networks
2.3 Low-Dimensional Chaos in 10-Hz Oscillations?
3 Self-organization and Complexity
3.1 Self-organized Criticality
3.2 SOC in Large-Scale Neuronal Activity?
4 Evidence for SOC in Neuronal Systems
4.1 Materials and Methods
4.2 Long-Range Temporal Correlations and Scaling Behavior
5 General Discussion
Acknowledgements
References
Chapter 19
1 Introduction
2 1/f Fluctuations in Heartbeat Dynamics
3 Monofractal Analysis:Long-Range Anticorrelations in the Heartbeat Fluctuations
4 Long-Range Correlations in the Magnitudes and Signs of Heartbeat Fluctuations
5 Self-similar Cascades in the Heartbeat Fluctuations
6 Multifractality: Nonstationarity in Local Scaling
7 Multifractality in Heartbeat Dynamics
8 Multifractality and Nonlinearity
9 Summary
Acknowledgments
References
Chapter 20
1 Introduction
2 Modeling Broadband Tele Traffic: Shifting Paradigms
2.1 Multiple Scaling: From Self Similarity to Multifractals
3 Multiplicative Multifractal Cascades
4 Broadband Traffic Inter Arrival Time Modeling Using V.V.G.M Model
4.1 Development of V.V.G.M Multifractal Model
4.2 Estimation of Multiplier Distributions
4.3 Synthesis Algorithm
5 Comparison of Queuing Performance Analysis
6 Analysis of Multiplexing and Aggregation of Multifractal Traffic
6.1 Analysis Using VVGM Model
6.2 Analysis Using the f(alpha) Curve
6.3 Analysis of Multiplexing Bursty Traces Using Entropy
7 Information Theoretic Analysis of Multifractal Systems
7.1 Variation of Entropy with Scale
7.2 Relation with Generalized Dimensions
8 Control of Broadband Traffic: Packet Count or Interarrival Times
9 Conclusion
References