Approximate Entropy as an Irregularity Measure for Financial Data

The need to assess subtle, potentially exploitable changes in serial structure is paramount in the analysis of financial and econometric data. We demonstrate the utility of approximate entropy (ApEn), a model-independent measure of sequential irregularity, towards this goal, via several distinct applications, both empirical data and model-based. We also consider cross-ApEn, a related two-variable measure of asynchrony that provides a more robust and ubiquitous measure of bivariate correspondence than does correlation, and the resultant implications to diversification strategies. We provide analytic expressions for and statistical properties of ApEn, and compare ApEn to nonlinear (complexity) measures, correlation and spectral analyses, and other entropy measures.

[1]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[2]  A. Stuart,et al.  Portfolio Selection: Efficient Diversification of Investments , 1959 .

[3]  A. Kolmogorov,et al.  On Strong Mixing Conditions for Stationary Gaussian Processes , 1960 .

[4]  W. Sharpe.  CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER CONDITIONS OF RISK* , 1964 .

[5]  Gregory J. Chaitin,et al.  On the Length of Programs for Computing Finite Binary Sequences , 1966, JACM.

[6]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[7]  D. Ornstein.  Bernoulli shifts with the same entropy are isomorphic , 1970 .

[8]  B. Mandelbrot.  When Can Price Be Arbitraged Efficiently? A Limit to the Validity of the Random Walk and Martingale Models , 1971 .

[9]  F. Black.  Capital Market Equilibrium with Restricted Borrowing , 1972 .

[10]  P. Young,et al.  Time series analysis, forecasting and control , 1972, IEEE Transactions on Automatic Control.

[11]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[12]  G. Chaitin.  Randomness and Mathematical Proof , 1975 .

[13]  Kalman J. Cohen,et al.  The Returns Generation Process, Returns Variance, and the Effect of Thinness in Securities Markets , 1978 .

[14]  Robert Shaw.  Strange Attractors, Chaotic Behavior, and Information Flow , 1981 .

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

[16]  P. Grassberger,et al.  Estimation of the Kolmogorov entropy from a chaotic signal , 1983 .

[17]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[18]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[19]  Richard E. Blahut,et al.  Principles and practice of information theory , 1987 .

[20]  A. Lo,et al.  Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test , 1987 .

[21]  Benjamin Weiss,et al.  How Sampling Reveals a Process , 1990 .

[22]  Chris Chatfield,et al.  The Analysis of Time Series: An Introduction, 4th edn. , 1990 .

[23]  David J. Thomson,et al.  Coherence established between atmospheric carbon dioxide and global temperature , 1990, Nature.

[24]  David J. Thomson,et al.  Time series analysis of Holocene climate data , 1990, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[25]  D. A. Dickey.  The Analysis of Time Series: An Introduction (4th ed.) , 1991 .

[26]  S M Pincus,et al.  Approximate entropy as a measure of system complexity. , 1991, Proceedings of the National Academy of Sciences of the United States of America.

[27]  Stefun D. Leigh.  U-Statistics Theory and Practice , 1992 .

[28]  Steven M. Pincus,et al.  Approximate entropy: Statistical properties and applications , 1992 .

[29]  Steven M. Pincus.  Approximating Markov chains. , 1992, Proceedings of the National Academy of Sciences of the United States of America.

[30]  Steven M. Pincus,et al.  Quantification of hormone pulsatility via an approximate entropy algorithm. , 1992, The American journal of physiology.

[31]  G G Haddad,et al.  Heart rate control in normal and aborted-SIDS infants. , 1993, The American journal of physiology.

[32]  S M Pincus,et al.  Greater signal regularity may indicate increased system isolation. , 1994, Mathematical biosciences.

[33]  A L Goldberger,et al.  Physiological time-series analysis: what does regularity quantify? , 1994, The American journal of physiology.

[34]  S. Pincus.  Approximate entropy (ApEn) as a complexity measure. , 1995, Chaos.

[35]  Steven M. Pincus,et al.  Older males secrete luteinizing hormone and testosterone more irregularly, and jointly more asynchronously, than younger males. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[36]  S. Pincus,et al.  Randomness and degrees of irregularity. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[37]  B. LeBaron,et al.  A test for independence based on the correlation dimension , 1996 .

[38]  R E Kalman,et al.  Not all (possibly) "random" sequences are created equal. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[39]  Use of approximate entropy measurements to classify ventricular tachycardia and fibrillation , 1998 .

[40]  S. Pincus,et al.  A recipe for randomness. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[41]  J D Veldhuis,et al.  Patients with Cushing's disease secrete adrenocorticotropin and cortisol jointly more asynchronously than healthy subjects. , 1998, The Journal of clinical endocrinology and metabolism.

[42]  S. Schuckers,et al.  Use of approximate entropy measurements to classify ventricular tachycardia and fibrillation. , 1998, Journal of electrocardiology.

[43]  T. Andersen.  THE ECONOMETRICS OF FINANCIAL MARKETS , 1998, Econometric Theory.

[44]  H. Huikuri,et al.  Altered complexity and correlation properties of R-R interval dynamics before the spontaneous onset of paroxysmal atrial fibrillation. , 1999, Circulation.

[45]  J D Veldhuis,et al.  Jointly amplified basal and pulsatile growth hormone (GH) secretion and increased process irregularity in women with anorexia nervosa: indirect evidence for disruption of feedback regulation within the GH-insulin-like growth factor I axis. , 1999, The Journal of clinical endocrinology and metabolism.

[46]  J. Bruhn,et al.  Electroencephalogram Approximate Entropy Correctly Classifies the Occurrence of Burst Suppression Pattern as Increasing Anesthetic Drug Effect , 2000, Anesthesiology.

[47]  M. Yilmaz,et al.  An approximate entropy test for randomness , 2000 .

[48]  Karl M Newell,et al.  The dynamics of resting and postural tremor in Parkinson's disease , 2000, Clinical Neurophysiology.

[49]  Andrew L. Rukhin,et al.  Approximate entropy for testing randomness , 2000, Journal of Applied Probability.

[50]  M L Johnson,et al.  Impact of pulsatility on the ensemble orderliness (approximate entropy) of neurohormone secretion. , 2001, American journal of physiology. Regulatory, integrative and comparative physiology.

[51]  P. Embrechts,et al.  Risk Management: Correlation and Dependence in Risk Management: Properties and Pitfalls , 2002 .

[52]  M. Weinstein,et al.  Timing of menopause and patterns of menstrual bleeding. , 2003, American journal of epidemiology.

[53]  Karl M. Newell,et al.  Inter- and intra-limb coordination in arm tremor , 1996, Experimental Brain Research.

[54]  A. Schatzberg,et al.  Process irregularity of cortisol and adrenocorticotropin secretion in men with major depressive disorder , 2004, Psychoneuroendocrinology.

[55]  R. Kálmán,et al.  Irregularity, volatility, risk, and financial market time series. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[56]  Roberto Hornero,et al.  Analysis of regularity in the EEG background activity of Alzheimer's disease patients with Approximate Entropy , 2005, Clinical Neurophysiology.

[57]  T. Papaioannou.  Information, Measures of , 2006 .