On the characterization of the deterministic/stochastic and linear/nonlinear nature of time series

The need for the characterization of real-world signals in terms of their linear, nonlinear, deterministic and stochastic nature is highlighted and a novel framework for signal modality characterization is presented. A comprehensive analysis of signal nonlinearity characterization methods is provided, and based upon local predictability in phase space, a new criterion for qualitative performance assessment in machine learning is introduced. This is achieved based on a simultaneous assessment of nonlinearity and uncertainty within a real-world signal. Next, for a given embedding dimension, based on the target variance of delay vectors, a novel framework for heterogeneous data fusion is introduced. The proposed signal modality characterization framework is verified by comprehensive simulations and comparison against other established methods. Case studies covering a range of machine learning applications support the analysis.

[1]  Karl J. Friston,et al.  Nonlinear Responses in fMRI: The Balloon Model, Volterra Kernels, and Other Hemodynamics , 2000, NeuroImage.

[2]  Mo Chen,et al.  Feature Fusion for the Detection of Microsleep Events , 2007, J. VLSI Signal Process..

[3]  Kumpati S. Narendra,et al.  Identification and control of dynamical systems using neural networks , 1990, IEEE Trans. Neural Networks.

[4]  T. Schreiber.  Interdisciplinary application of nonlinear time series methods , 1998, chao-dyn/9807001.

[5]  Antígona Martínez,et al.  Nonlinear temporal dynamics of the cerebral blood flow response , 2001, Human brain mapping.

[6]  G. Orban,et al.  Visual Motion Processing Investigated Using Contrast Agent-Enhanced fMRI in Awake Behaving Monkeys , 2001, Neuron.

[7]  A. Porta,et al.  Non-linear dynamics and chaotic indices in heart rate variability of normal subjects and heart-transplanted patients. , 1996, Cardiovascular Research.

[8]  T. Schreiber,et al.  Surrogate time series , 1999, chao-dyn/9909037.

[9]  Danilo P. Mandic,et al.  A non-parametric test for detecting the complex-valued nature of time series , 2003, Int. J. Knowl. Based Intell. Eng. Syst..

[10]  James Theiler,et al.  Statistical error in a chord estimator of correlation dimension: The rule of five'' , 1993 .

[11]  Danilo P. Mandic,et al.  Recurrent Neural Networks for Prediction: Learning Algorithms, Architectures and Stability , 2001 .

[12]  David Sommer,et al.  Automatic Knowledge Extraction: Fusion of Human Expert Ratings and Biosignal Features for Fatigue Monitoring Applications , 2008 .

[13]  D. T. Kaplan,et al.  Exceptional events as evidence for determinism , 1994 .

[14]  James Theiler,et al.  Testing for nonlinearity in time series: the method of surrogate data , 1992 .

[15]  D. Levy,et al.  Predicting survival in heart failure case and control subjects by use of fully automated methods for deriving nonlinear and conventional indices of heart rate dynamics. , 1997, Circulation.

[16]  M. G. Kendall,et al.  A Study in the Analysis of Stationary Time-Series. , 1955 .

[17]  James Theiler,et al.  Detecting Nonlinearity in Data with Long Coherence Times , 1993, comp-gas/9302003.

[18]  Danilo P. Mandic,et al.  A non-parametric test for detecting the complex-valued nature of time series , 2004 .

[19]  D. Heeger,et al.  Linear Systems Analysis of Functional Magnetic Resonance Imaging in Human V1 , 1996, The Journal of Neuroscience.

[20]  Christopher K. Merrill,et al.  Decrease of cardiac choas in congestive heart failure , 1997 .

[21]  M. Hulle,et al.  The Delay Vector Variance Method for Detecting Determinism and Nonlinearity in Time Series , 2004 .

[22]  D. T. Kaplan,et al.  Nonlinearity and nonstationarity: the use of surrogate data in interpreting fluctuations , 1997 .

[23]  T. Schreiber,et al.  Discrimination power of measures for nonlinearity in a time series , 1997, chao-dyn/9909043.

[24]  M. Casdagli.  Chaos and Deterministic Versus Stochastic Non‐Linear Modelling , 1992 .

[25]  Andreas S. Weigend,et al.  Time Series Prediction: Forecasting the Future and Understanding the Past , 1994 .

[26]  L. Cao.  Practical method for determining the minimum embedding dimension of a scalar time series , 1997 .

[27]  Schreiber,et al.  Improved Surrogate Data for Nonlinearity Tests. , 1996, Physical review letters.

[28]  K. Lutchen,et al.  Application of linear and nonlinear time series modeling to heart rate dynamics analysis , 1995, IEEE Transactions on Biomedical Engineering.

[29]  Clare Anderson,et al.  Electroencephalographic activities during wakefulness and sleep in the frontal cortex of healthy older people: links with "thinking". , 2003, Sleep.

[30]  H. Abarbanel,et al.  Determining embedding dimension for phase-space reconstruction using a geometrical construction. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[31]  P. Grassberger,et al.  Measuring the Strangeness of Strange Attractors , 1983 .

[32]  James Theiler,et al.  Constrained-realization Monte-carlo Method for Hypothesis Testing , 1996 .

[33]  Richard D. Braatz,et al.  On the "Identification and control of dynamical systems using neural networks" , 1997, IEEE Trans. Neural Networks.

[34]  Chi-Sang Poon,et al.  Decrease of cardiac chaos in congestive heart failure , 1997, Nature.

[35]  Konstantinos Kostikas,et al.  The effects of adaptive servo ventilation on cerebral vascular reactivity in patients with congestive heart failure and sleep-disordered breathing. , 2007, Sleep.

[36]  Kazuyuki Aihara,et al.  Complex-valued forecasting of wind profile , 2006 .

[37]  Karl J. Friston,et al.  Nonlinear Coupling between Evoked rCBF and BOLD Signals: A Simulation Study of Hemodynamic Responses , 2001, NeuroImage.

[38]  Cees Diks,et al.  Reversibility as a criterion for discriminating time series , 1995 .

[39]  Tryphon T. Georgiou.  Aspects of Modeling, Identification and Control of Dynamical Systems. , 1995 .

[40]  Danilo P. Mandic,et al.  Signal nonlinearity in fMRI: a comparison between BOLD and MION , 2003, IEEE Transactions on Medical Imaging.

[41]  Danilo P Mandic,et al.  Indications of nonlinear structures in brain electrical activity. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[42]  Timmer,et al.  What can Be inferred from surrogate data testing? , 2000, Physical review letters.

[43]  Farmer,et al.  Predicting chaotic time series. , 1987, Physical review letters.

[44]  D. Kugiumtzis,et al.  Test your surrogate data before you test for nonlinearity. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[45]  J. Rissanen,et al.  Modeling By Shortest Data Description* , 1978, Autom..

[46]  S. L. Goh,et al.  Sequential Data Fusion via Vector Spaces: Complex Modular Neural Network Approach , 2005, 2005 IEEE Workshop on Machine Learning for Signal Processing.

[47]  P. Grassberger,et al.  Measuring the Strangeness of Strange Attractors , 1983 .

[48]  Holger Kantz,et al.  Practical implementation of nonlinear time series methods: The TISEAN package. , 1998, Chaos.

[49]  Danilo P. Mandic,et al.  Recurrent Neural Networks for Prediction: Learning Algorithms, Architectures and Stability , 2001 .

[50]  Anthony G. Constantinides,et al.  Data Fusion for Modern Engineering Applications: An Overview , 2005, ICANN.

[51]  G. Weiss.  TIME-REVERSIBILITY OF LINEAR STOCHASTIC PROCESSES , 1975 .

[52]  William A. Barnett,et al.  A single-blind controlled competition among tests for nonlinearity and chaos , 1997 .