Heavy-Tailed Prediction Error: A Difficulty in Predicting Biomedical Signals of 1/f Noise Type

A fractal signal x(t) in biomedical engineering may be characterized by 1/f noise, that is, the power spectrum density (PSD) divergences at f = 0. According the Taqqu's law, 1/f noise has the properties of long-range dependence and heavy-tailed probability density function (PDF). The contribution of this paper is to exhibit that the prediction error of a biomedical signal of 1/f noise type is long-range dependent (LRD). Thus, it is heavy-tailed and of 1/f noise. Consequently, the variance of the prediction error is usually large or may not exist, making predicting biomedical signals of 1/f noise type difficult.

[1]  U. Rajendra Acharya,et al.  Heart rate variability: a review , 2006, Medical and Biological Engineering and Computing.

[2]  Nitish V. Thakor,et al.  Long range correlations in the heart rate variability following the injury of cardiac arrest , 2007 .

[3]  P. Szendrő,et al.  Pink-noise behaviour of biosystems , 2001, European Biophysics Journal.

[4]  Alejandro Rodríguez,et al.  Bootstrap prediction mean squared errors of unobserved states based on the Kalman filter with estimated parameters , 2012, Comput. Stat. Data Anal..

[5]  Bernd Blasius,et al.  Extinction risk, coloured noise and the scaling of variance. , 2005, Theoretical population biology.

[6]  Zhang Hai,et al.  SCIENCE CHINA Information Sciences , 2010 .

[7]  N. Cressie,et al.  Mean squared prediction error in the spatial linear model with estimated covariance parameters , 1992 .

[8]  X. Xie,et al.  Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule. , 2004, Physical review letters.

[9]  Paolo Grigolini,et al.  A theory of 1/f noise in human cognition , 2009 .

[10]  Aidong Zhang,et al.  Predicting Protein Function by Frequent Functional Association Pattern Mining in Protein Interaction Networks , 2010, IEEE Transactions on Information Technology in Biomedicine.

[11]  D. T. Kaplan,et al.  Nonstationarity and 1/ f noise characteristics in heart rate. , 1999, American Journal of Physiology. Regulatory Integrative and Comparative Physiology.

[12]  M. Emiliano.  Mathematical Problems in Engineering , 2014 .

[13]  D. Navajas,et al.  Lung tissue rheology and 1/f noise , 1994, Annals of Biomedical Engineering.

[14]  Qiang Lv,et al.  A parallel ant colonies approach to de novo prediction of protein backbone in CASP8/9 , 2011, Science China Information Sciences.

[15]  J. Crocker,et al.  Mechanics of single cells: rheology, time dependence, and fluctuations. , 2007, Biophysical journal.

[16]  Z R Struzik,et al.  Autonomic Imbalance Induced Breakdown of Long-range Dependence in Healthy Heart Rate , 2007, Methods of Information in Medicine.

[17]  Richard T. Baillie,et al.  Asymptotic prediction mean squared error for vector autoregressive models , 1979 .

[18]  B.S. Atal.  The history of linear prediction , 2006, IEEE Signal Processing Magazine.

[19]  Adriana Isvoran,et al.  Long range correlation of hydrophilicity and flexibility along the calcium binding protein chains , 2009 .

[20]  D. Percival,et al.  Physiological time series , 2000 .

[21]  R. J. Bhansali,et al.  Asymptotic Mean-square Error of Predicting more than One-step ahead Using the Regression Method , 1974 .

[22]  Dawei Huang,et al.  LEVINSON-TYPE RECURSIVE ALGORITHMS FOR LEAST-SQUARES AUTOREGRESSION , 1990 .

[23]  Marc Joyeux,et al.  1/f fluctuations of DNA temperature at thermal denaturation. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Bruce J. West.  Fractal Physiology and the Fractional Calculus: A Perspective , 2010, Front. Physio..

[25]  Takahiro Harada,et al.  Generality of a power-law long-term correlation in beat timings of single cardiac cells. , 2009, Biochemical and biophysical research communications.

[26]  N. Wiener.  The Wiener RMS (Root Mean Square) Error Criterion in Filter Design and Prediction , 1949 .

[27]  Linh Hoang,et al.  1/f Neural Noise Reduction and Spike Feature Extraction Using a Subset of Informative Samples , 2010, Annals of Biomedical Engineering.

[28]  N. Levinson.  A Heuristic Exposition of Wiener's Mathematical Theory of Prediction and Filtering , 1947 .

[29]  Bruce J. West,et al.  Fractal physiology for physicists: Lévy statistics , 1994 .

[30]  D. Applebaum.  Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[31]  Y. Chen,et al.  Fractional Processes and Fractional-Order Signal Processing , 2012 .

[32]  Angel Rubio,et al.  The challenge of predicting optical properties of biomolecules: What can we learn from time-dependent density-functional theory? , 2009 .

[33]  R. Adler,et al.  A practical guide to heavy tails: statistical techniques and applications , 1998 .

[34]  Dario Farina,et al.  Compression of Multidimensional Biomedical Signals With Spatial and Temporal Codebook-Excited Linear Prediction , 2009, IEEE Transactions on Biomedical Engineering.

[35]  Silvano Bordignon,et al.  Mean square prediction error for long-memory processes , 2002 .

[36]  Edouard Laroche,et al.  Motion Prediction for Computer-Assisted Beating Heart Surgery , 2009, IEEE Transactions on Biomedical Engineering.

[37]  Qionghai Dai,et al.  An overview of computational photography , 2012, Science China Information Sciences.

[38]  Takahiro Harada,et al.  Singular behavior of slow dynamics of single excitable cells. , 2009, Biophysical journal.

[39]  Malte Meesmann,et al.  Further study on 1/f fluctuations observed in central single neurons during REM sleep , 2004, Biological Cybernetics.

[40]  B. Mandelbrot.  Multifractals And 1/F Noise , 1999 .

[41]  M. Riley,et al.  IN FRACTAL PHYSIOLOGY , 2022 .

[42]  J. L. Hock,et al.  An exact recursion for the composite nearest‐neighbor degeneracy for a 2×N lattice space , 1984 .

[43]  Vakhtang Putkaradze,et al.  Relaxation dynamics of nucleosomal DNA. , 2009, Physical chemistry chemical physics : PCCP.

[44]  L. Zadeh,et al.  An Extension of Wiener's Theory of Prediction , 1950 .

[45]  A. Papoulis.  A note on the predictability of band-limited processes , 1985, Proceedings of the IEEE.

[46]  Mark A. McComb.  A Practical Guide to Heavy Tails , 2000, Technometrics.

[47]  Nobuyasu Ito,et al.  Effects of demographic stochasticity on biological community assembly on evolutionary time scales. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[48]  Carolyn L. Beck,et al.  On the use of multivariable piecewise-linear models for predicting human response to anesthesia , 2004, IEEE Transactions on Biomedical Engineering.

[49]  William W. Edmonson,et al.  Linear prediction of bandlimited processes with flat spectral densities , 2001, IEEE Trans. Signal Process..

[50]  B Kaulakys,et al.  Tsallis distributions and 1/f noise from nonlinear stochastic differential equations. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[51]  M. S. Keshner.  1/f noise , 1982, Proceedings of the IEEE.

[52]  Bruce J. West,et al.  Physiology in fractal dimensions: Error tolerance , 2006, Annals of Biomedical Engineering.

[53]  Adam Gadomski,et al.  On the Protein Crystal Formation as an Interface-Controlled Process with Prototype Ion-Channeling Effect , 2007, Journal of biological physics.

[54]  Carlo Cattani,et al.  Simplicial Approach to Fractal Structures , 2012 .

[55]  Robert Kohn,et al.  Prediction mean squared error for state space models with estimated parameters , 1986 .

[56]  Piotr Kokoszka,et al.  Prediction of Long-Memory Time Series: A Tutorial Review , 2003 .

[57]  Peter Herman,et al.  Nonlinear Analysis of Blood Cell Flux Fluctuations in the Rat Brain Cortex during Stepwise Hypotension Challenge , 2006, Journal of cerebral blood flow and metabolism : official journal of the International Society of Cerebral Blood Flow and Metabolism.

[58]  Massimo Panella,et al.  Advances in biological time series prediction by neural networks , 2011, Biomed. Signal Process. Control..

[59]  A. N. Kolmogorov,et al.  Interpolation and extrapolation of stationary random sequences. , 1962 .

[60]  P. Neelamegam,et al.  Prediction of calcium concentration in human blood serum using an artificial neural network , 2011 .

[61]  Frank Beckers,et al.  Aging and nonlinear heart rate control in a healthy population. , 2006, American journal of physiology. Heart and circulatory physiology.

[62]  Norbert Wiener,et al.  Extrapolation, Interpolation, and Smoothing of Stationary Time Series , 1964 .

[63]  Markus Abt.  Estimating the Prediction Mean Squared Error in Gaussian Stochastic Processes with Exponential Correlation Structure , 1999 .

[64]  Carlo Cattani,et al.  On the Existence of Wavelet Symmetries in Archaea DNA , 2011, Comput. Math. Methods Medicine.

[65]  J. Makhoul,et al.  Linear prediction: A tutorial review , 1975, Proceedings of the IEEE.

[66]  Ming Li,et al.  On the Predictability of Long-Range Dependent Series , 2010 .

[67]  Mathias Baumert,et al.  Long-term correlations and fractal dimension of beat-to-beat blood pressure dynamics , 2005 .

[68]  Yoshiharu Yamamoto,et al.  Aging of complex heart rate dynamics , 2006, IEEE Transactions on Biomedical Engineering.

[69]  G. Xiao,et al.  Fault Detection and Prediction of Clocks and Timers Based on Computer Audition and Probabilistic Neural Networks , 2005, IWANN.

[70]  Rohit S. Deo.  Improved forecasting of autoregressive series by weighted least squares approximate REML estimation , 2012 .

[71]  Carlo Cattani,et al.  Fractals and Hidden Symmetries in DNA , 2010 .

[72]  Maurizio Carlini,et al.  Modelling the Vertical Heat Exchanger in Thermal Basin , 2007, ICCSA.

[73]  Tianshuang Qiu,et al.  Heavy-Tailed Distribution and Local Memory in Time Series of Molecular Motion on the Cell Membrane , 2011 .

[74]  D. Percival,et al.  Physiological time series: distinguishing fractal noises from motions , 2000, Pflügers Archiv.

[75]  Cleve Moler,et al.  Mathematical Handbook for Scientists and Engineers , 1961 .

[76]  Nikolai Dokuchaev.  The predictability of band-limited, high-frequency and mixed processes in the presence of ideal low-pass filters , 2008 .

[77]  Efthimios S. Skordas,et al.  Heart rate variability in natural time and 1/f “noise” , 2009 .

[78]  David F. Garway-Heath,et al.  The Pseudotemporal Bootstrap for Predicting Glaucoma From Cross-Sectional Visual Field Data , 2010, IEEE Transactions on Information Technology in Biomedicine.

[79]  E. Schwiebert,et al.  ABC transporter-facilitated ATP conductive transport. , 1999, American journal of physiology. Cell physiology.

[80]  Bingru Yang,et al.  A new intelligent prediction system model-the compound pyramid model , 2011, Science China Information Sciences.

[81]  W. Kunin,et al.  Extinction risk and the 1/f family of noise models. , 1999, Theoretical population biology.

[82]  Maurizio Carlini,et al.  Photovoltaic Greenhouses: Comparison of Optical and Thermal Behaviour for Energy Savings , 2012 .

[83]  William W. Edmonson,et al.  The predictability of continuous-time, bandlimited processes , 2000, IEEE Trans. Signal Process..

[84]  R. Bhansali.  Asymptotic Properties of the Wiener‐Kolmogorov Predictor. I , 1974 .

[85]  Maurizio Carlini,et al.  Optical Modelling of Square Solar Concentrator , 2011, ICCSA.