Distances of Time Series Components by Means of Symbolic Dynamics

In this note we describe a simple method for visualizing time-dependent similarities and dissimilarities between the components of a high-dimensional time series. On the base of symbolic dynamics, the time series is turned into a series of matrices whose rows quantify pattern types in the components of the original series. For different scales we introduce distances between the components via the obtained pattern type distributions and approximate them in a one-dimensional manner. The method is illustrated for 19-channel EEG data.

[1]  U. Parlitz,et al.  Mathematical methods for forecasting bank transaction data , 2003 .

[2]  M. Rumpf,et al.  Morphological image sequence processing , 2004 .

[3]  H. Voss,et al.  Phase synchronization from noisy univariate signals. , 2004, Physical review letters.

[4]  G. Steidl,et al.  Robust local approximation of scattered data , 2006 .

[5]  Hanno Scharr,et al.  Optimal Filters for Extended Optical Flow , 2004, IWCM.

[6]  Rudolf Mester The generalization, optimization, and information-theoretic justification of filter-based and autocovariance-based motion estimation , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[7]  Gabriele Steidl,et al.  Coorbit Spaces and Banach Frames on Homogeneous Spaces with Applications to the Sphere , 2004, Adv. Comput. Math..

[8]  C. Finney,et al.  Symbol Statistics: A New Tool for Understanding Multiphase Flow Phenomena , 1998, Heat Transfer: Volume 5 — Numerical and Experimental Methods in Heat Transfer.

[9]  Werner Ebeling,et al.  Word frequency and entropy of symbolic sequences: a dynamical perspective , 1992 .

[10]  Andreas Martin,et al.  Exact and fast numerical algorithms for the stochastic wave equation , 2003, Int. J. Comput. Math..

[11]  R. Dahlhaus,et al.  Semiparametric estimation by model selection for locally stationary processes , 2006 .

[12]  Jürgen Kurths,et al.  Nonlinear Dynamical System Identification from Uncertain and Indirect Measurements , 2004, Int. J. Bifurc. Chaos.

[13]  U. Clarenz,et al.  Towards fast non-rigid registration , 2003 .

[14]  J. Polzehl,et al.  Local likelihood modeling by adaptive weights smoothing , 2004 .

[15]  Rudolf Mester,et al.  A Fast Algorithm for Statistically Optimized Orientation Estimation , 2005, DAGM-Symposium.

[16]  G. Winkler,et al.  An Elementary Rigorous Introduction to Exact Sampling , 2005 .

[17]  Hagen Spies,et al.  Dense Parameter Fields from Total Least Squares , 2002, DAGM-Symposium.

[18]  P. Mrázek,et al.  ON ROBUST ESTIMATION AND SMOOTHING WITH SPATIAL AND TONAL KERNELS , 2006 .

[19]  Martin Rumpf,et al.  Axioms and variational problems in surface parameterization , 2004, Comput. Aided Geom. Des..

[20]  Martin Rumpf,et al.  Computational Methods for Nonlinear Image Registration , 2006 .

[21]  Frank Scherbaum,et al.  Instantaneous polarization attributes in the time–frequency domain and wavefield separation , 2005 .

[22]  Thomas Brox,et al.  On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs , 2004, SIAM J. Numer. Anal..

[23]  M. Muhlich,et al.  Subspace Estimation with Uncertain and Correlated Data , 2006 .

[24]  Til Aach,et al.  Estimation of multiple orientations in multidimensional signals , 2004, 2004 International Conference on Image Processing, 2004. ICIP '04..

[25]  Jürgen Kurths,et al.  Surrogate-Based Hypothesis Test without Surrogates , 2004, Int. J. Bifurc. Chaos.

[26]  Hagen Spies,et al.  Estimation of complex motion from thermographic image sequences , 2003, SPIE Defense + Commercial Sensing.

[27]  R. Clarke,et al.  Theory and Applications of Correspondence Analysis , 1985 .

[28]  A. Martin,et al.  Small Ball Asymptotics for the Stochastic Wave Equation , 2004 .

[29]  G. Winkler,et al.  Parsimonious segmentation of time series by Potts models , 2005 .

[30]  James Douglas Saddy,et al.  Language Processing by Dynamical Systems , 2004, Int. J. Bifurc. Chaos.

[31]  Til Aach,et al.  Estimation of Multiple Orientations at Corners and Junctions , 2004, DAGM-Symposium.

[32]  Hanno Scharr,et al.  Image statistics and anisotropic diffusion , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[33]  Michael E. Taylor,et al.  Propagation of singularities , 1991 .

[34]  H. Voss,et al.  Parameter estimation in nonlinear delayed feedback systems from noisy data , 2002 .

[35]  Bernd Jähne,et al.  Estimation of Surface Flow and Net Heat Flux from Infrared Image Sequences , 2003, Journal of Mathematical Imaging and Vision.

[36]  Erich Novak,et al.  Optimal approximation of elliptic problems by linear and nonlinear mappings I , 2006, J. Complex..

[37]  Til Aach,et al.  Estimation of multiple local orientations in image signals , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[38]  C. S. Daw,et al.  Symbolic Time-Series Analysis of Engine Combustion Measurements , 1998 .

[39]  A. Schulze-Bonhage,et al.  Comparison of three nonlinear seizure prediction methods by means of the seizure prediction characteristic , 2004 .

[40]  Karsten Keller,et al.  Symbolic Analysis of High-Dimensional Time Series , 2003, Int. J. Bifurc. Chaos.

[41]  C. Bandt Ordinal time series analysis , 2005 .

[42]  F. Scherbaum,et al.  Characterization of dispersive surface waves using continuous wavelet transforms , 2005 .

[43]  C. Garbe,et al.  Local models for dynamic processes in image sequences , 2002 .

[44]  Rudolf Mester,et al.  Wiener-Optimized Discrete Filters for Differential Motion Estimation , 2004, IWCM.

[45]  Til Aach,et al.  Spatial and spectral analysis of occluded motions , 2005, Signal Process. Image Commun..

[46]  J. Timmer,et al.  Asymptotic scaling laws for precision of parameter estimates in dynamical systems , 2003 .

[47]  Rudolf Mester,et al.  A statistical extension of normalized convolution and its usage for image interpolation and filtering , 2004, 2004 12th European Signal Processing Conference.

[48]  Til Aach,et al.  Linear and regularized solutions for multiple motions , 2003, 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP '03)..

[49]  M. Winterhalder,et al.  Mixing properties of the Rössler system and consequences for coherence and synchronization analysis. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[50]  Rudolf Mester,et al.  Signal and Noise Adapted Filters for Differential Motion Estimation , 2005, DAGM-Symposium.

[51]  G. Keller,et al.  Entropy of interval maps via permutations , 2002 .

[52]  Jens Timmer,et al.  Identification of Rate Constants and nonobservable Absorption Spectra in Nonlinear Biochemical Reaction Dynamics , 2004, Int. J. Bifurc. Chaos.

[53]  J. Kurths,et al.  Sensitivity and specificity of coherence and phase synchronization analysis , 2006 .

[54]  H. Thielemann Bounds for smoothness of refinable functions , 2004 .

[55]  Werner Ebeling,et al.  Entropy and Complexity Analysis of Intracranially recorded EEG , 2004, Int. J. Bifurc. Chaos.

[56]  J. Timmer,et al.  Data-driven optimal filtering for phase and frequency of noisy oscillations: Application to vortex flow metering. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[57]  Annette Witt,et al.  Analysis of solar spike events by means of symbolic dynamics methods , 1993 .

[58]  Rudolf Mester,et al.  ON THE MATHEMATICAL STRUCTURE OF DIRECTION AND MOTION ESTIMATION , 2002 .

[59]  Robert Strzodka,et al.  Real-time motion estimation and visualization on graphics cards , 2004, IEEE Visualization 2004.

[60]  Karsten Koch Interpolating Scaling Vectors , 2005, Int. J. Wavelets Multiresolution Inf. Process..

[61]  Hanno Scharr,et al.  Towards a Multi-camera Generalization of Brightness Constancy , 2004, IWCM.

[62]  G. Winkler,et al.  Numerical solution of boundary value problems for stochastic differential equations on the basis of the Gibbs sampler , 2003 .

[63]  J. Timmer,et al.  Fitting parameters in partial differential equations from partially observed noisy data , 2002 .

[64]  Michael Eichler,et al.  Abstract Journal of Neuroscience Methods xxx (2005) xxx–xxx Testing for directed influences among neural signals using partial directed coherence , 2005 .

[65]  Til Aach,et al.  Estimation of multiple motions: regularization and performance evaluation , 2003, IS&T/SPIE Electronic Imaging.

[66]  Michael Felsberg,et al.  Noise Adaptive Channel Smoothing of Low-Dose Images , 2003, 2003 Conference on Computer Vision and Pattern Recognition Workshop.

[67]  Erhardt Barth,et al.  Categorization of Transparent-Motion Patterns Using the Projective Plane , 2003, SNPD.

[68]  Matthias Mh,et al.  Derivation of optimal equilibration transformations for general covariance tensors of random matrices , 2005 .

[69]  Martin Rumpf,et al.  Flow field clustering via algebraic multigrid , 2004, IEEE Visualization 2004.

[70]  G. Teschke,et al.  Tikhonov replacement functionals for iteratively solving nonlinear operator equations , 2005 .

[71]  C. Eckart,et al.  The approximation of one matrix by another of lower rank , 1936 .

[72]  Marc Droske,et al.  A Mumford-Shah Level-Set Approach for Geometric Image Registration , 2006, SIAM J. Appl. Math..

[73]  G. Winkler,et al.  Exact numerical algorithms for linear stochastic wave equation and stochastic Klein-Gordon equation , 2003 .

[74]  Jens Timmer,et al.  Direct or indirect? Graphical models for neural oscillators , 2004, Journal of Physiology-Paris.

[75]  Bernd Jähne,et al.  A surface renewal model to analyze infrared image sequences of the ocean surface for the study of air‐sea heat and gas exchange , 2004 .

[76]  Henning Thielemann Audio Processing using Haskell , 2004 .

[77]  B. Pompe,et al.  Permutation entropy: a natural complexity measure for time series. , 2002, Physical review letters.

[78]  Martin Rumpf,et al.  Image Registration by a Regularized Gradient Flow. A Streaming Implementation in DX9 Graphics Hardware , 2004, Computing.

[79]  H. Scharr,et al.  A linear model for simultaneous estimation of 3D motion and depth , 2002, Workshop on Motion and Video Computing, 2002. Proceedings..

[80]  Klaus Lehnertz,et al.  POSSIBLE CLINICAL AND RESEARCH APPLICATIONS OF NONLINEAR EEG ANALYSIS IN HUMANS , 2000 .

[81]  Rudolf Mester,et al.  Unbiased Errors-In-Variables Estimation Using Generalized Eigensystem Analysis , 2004, ECCV Workshop SMVP.

[82]  Hanno Scharr,et al.  Estimation of transparent motions with physical models for additional brightness variation , 2005, 2005 13th European Signal Processing Conference.

[83]  K. Bredies,et al.  Mathematical concepts of multiscale smoothing , 2005 .

[84]  Dirk Kohler,et al.  A comparison of denoising methods for one dimensional time series , 2005 .

[85]  M. S. Diallo,et al.  Modeling of Wave Dispersion Using Continuous Wavelet Transforms , 2005 .

[86]  R. Küsters,et al.  Simultaneous estimation of local and global parameters in image sequences , 2002 .

[87]  D. Lorenz Variational Denoising in Besov Spaces and Interpolation of Hard and Soft Wavelet Shrinkage , 2003 .

[88]  H. Kantz,et al.  FREQUENCY DOMAIN CONDITIONS FOR THE EXISTENCE OF ALMOST PERIODIC SOLUTIONS IN EVOLUTIONARY VARIATIONAL INEQUALITIES , 2004 .

[89]  S. Rao On multiple regression models with nonstationary correlated errors , 2004 .

[90]  M. Droske,et al.  A level set formulation for Willmore flow , 2004 .

[91]  J. D. Saddy,et al.  Symbolic dynamics of event-related brain potentials. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[92]  Rudolf Mester,et al.  A New View at Differential and Tensor-Based Motion Estimation Schemes , 2003, DAGM-Symposium.

[93]  W. Panzer,et al.  An adaptive algorithm for the detection of microcalcifications in simulated low-dose mammography. , 2003, Physics in medicine and biology.

[94]  F. Takens Detecting strange attractors in turbulence , 1981 .

[95]  Bernd Jähne,et al.  Mixed OLS-TLS for the Estimation of Dynamic Processes with a Linear Source Term , 2002, DAGM-Symposium.

[96]  Bernd Jähne,et al.  Towards objective performance analysis for estimation of complex motion: analytic motion modeling, filter optimization, and test sequences , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).