Analysis of financial stock markets through multidimensional scaling based on information measures

In this paper, a novel multidimensional scaling (MDS) based on information measures method is proposed to analyze financial stock markets. In order to examine the effectiveness of this method, we applied it to the classification of two types of artificial series, the logistic map model and the cubic map model, as well as stock time series. Moreover, the traditional MDS using Euclidean dissimilarity is also provided as a reference for comparisons. The results show that the MDS based on information measures can give us more detailed, exact and clearer information on the classification of simulation series and stock time series than the MDS using Euclidean dissimilarity. In addition, the proposed graphical method may also assist in the construction of multivariate econometric models.

[1]  Ary L Goldberger,et al.  Statistical physics approach to categorize biologic signals: from heart rate dynamics to DNA sequences. , 2007, Chaos.

[2]  H. Eugene Stanley.  The Fragility of Interdependency: Coupled Networks Switching Phenomena , 2013 .

[3]  James O. Ramsay,et al.  Some small sample results for maximum likelihood estimation in multidimensional scaling , 1980 .

[4]  Yudong Wang,et al.  Detrended fluctuation analysis on spot and futures markets of West Texas Intermediate crude oil , 2011 .

[5]  Pengjian Shang,et al.  MULTISCALE ENTROPY ANALYSIS OF FINANCIAL TIME SERIES , 2012 .

[6]  H. Stanley,et al.  Detrended cross-correlation analysis: a new method for analyzing two nonstationary time series. , 2007, Physical review letters.

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

[8]  J. Sprott.  Chaos and time-series analysis , 2001 .

[9]  J. Kruskal.  Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis , 1964 .

[10]  Sheila Nirenberg,et al.  Decoding neuronal spike trains: How important are correlations? , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[11]  Bodo Ahrens,et al.  Distance in spatial interpolation of daily rain gauge data , 2005 .

[12]  Harvard Medical School,et al.  Effect of nonstationarities on detrended fluctuation analysis. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  G. F. Zebende.  DCCA cross-correlation coefficient: Quantifying level of cross-correlation , 2011 .

[14]  José António Tenreiro Machado,et al.  Analysis and Visualization of Seismic Data Using Mutual Information , 2013, Entropy.

[15]  J. A. Tenreiro Machado,et al.  Identifying economic periods and crisis with the multidimensional scaling , 2011 .

[16]  Raoul R. Nigmatullin,et al.  Universal distribution function for the strongly-correlated fluctuations: General way for description of different random sequences , 2010 .

[17]  H. Stanley,et al.  Quantifying cross-correlations using local and global detrending approaches , 2009 .

[18]  Huey-Wen Yien,et al.  Linguistic analysis of the human heartbeat using frequency and rank order statistics. , 2003, Physical review letters.

[19]  J. A. Tenreiro Machado,et al.  A review of power laws in real life phenomena , 2012 .

[20]  Boris Podobnik,et al.  Detrended cross-correlation analysis for non-stationary time series with periodic trends , 2011 .

[21]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[22]  Christopher K. I. Williams.  On a Connection between Kernel PCA and Metric Multidimensional Scaling , 2004, Machine Learning.

[23]  G. F. Zebende,et al.  Cross-correlation between time series of vehicles and passengers , 2009 .

[24]  Fernando B. M. Duarte,et al.  Dynamics of the Dow Jones and the NASDAQ stock indexes , 2010 .

[25]  W. Torgerson.  Multidimensional scaling: I. Theory and method , 1952 .

[26]  Bernhard Schölkopf,et al.  Kernel Principal Component Analysis , 1997, International Conference on Artificial Neural Networks.

[27]  Joseph Woelfel,et al.  Multidimensional scaling in Riemann space , 1982 .

[28]  Pengjian Shang,et al.  Classifying of financial time series based on multiscale entropy and multiscale time irreversibility , 2014 .

[29]  Pengjian Shang,et al.  The cross-correlations of stock markets based on DCCA and time-delay DCCA , 2012 .

[30]  S. Du,et al.  Yanget al. reply , 2004 .

[31]  A. Householder,et al.  Discussion of a set of points in terms of their mutual distances , 1938 .

[32]  Patrick J. F. Groenen,et al.  Modern Multidimensional Scaling: Theory and Applications , 2003 .

[33]  Dimitris K. Agrafiotis,et al.  Multidimensional scaling and visualization of large molecular similarity tables , 2001 .

[34]  Yi Yin,et al.  Modified DFA and DCCA approach for quantifying the multiscale correlation structure of financial markets , 2013 .

[35]  Philip Hans Franses,et al.  Visualizing time-varying correlations across stock markets , 2000 .

[36]  José António Tenreiro Machado,et al.  Fractional State Space Analysis of Economic Systems , 2015, Entropy.

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

[38]  Deborah F. Swayne,et al.  Data Visualization With Multidimensional Scaling , 2008 .

[39]  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.

[40]  J. A. Tenreiro Machado,et al.  Relativistic time effects in financial dynamics , 2014 .

[41]  J. Tenreiro Machado,et al.  Analysis of financial data series using fractional Fourier transform and multidimensional scaling , 2011 .

[42]  J. A. Tenreiro Machado,et al.  Multidimensional Scaling Analysis of Stock Market Indexes , 2011 .

[43]  António M. Lopes,et al.  Analysis of Natural and Artificial Phenomena Using Signal Processing and Fractional Calculus , 2015 .

[44]  Forrest W. Young,et al.  Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features , 1977 .

[45]  Joseph L. Zinnes,et al.  Theory and Methods of Scaling. , 1958 .

[46]  Borko Stosic,et al.  Correlations and cross-correlations in the Brazilian agrarian commodities and stocks , 2010 .

[47]  Boris Podobnik,et al.  1/f behavior in cross-correlations between absolute returns in a US market , 2012 .

[48]  J. A. Tenreiro Machado,et al.  Analysis of stock market indices through multidimensional scaling , 2011 .

[49]  Rosario N. Mantegna,et al.  Book Review: An Introduction to Econophysics, Correlations, and Complexity in Finance, N. Rosario, H. Mantegna, and H. E. Stanley, Cambridge University Press, Cambridge, 2000. , 2000 .

[50]  V. Plerou,et al.  Econophysics: financial time series from a statistical physics point of view , 2000 .

[51]  H. Stanley,et al.  Magnitude and sign correlations in heartbeat fluctuations. , 2000, Physical review letters.

[52]  R. Shepard.  The analysis of proximities: Multidimensional scaling with an unknown distance function. II , 1962 .

[53]  R. Mantegna,et al.  Scaling behaviour in the dynamics of an economic index , 1995, Nature.

[54]  Huey-Wen Yien,et al.  Information categorization approach to literary authorship disputes , 2003 .

[55]  H. Stanley,et al.  Cross-correlations between volume change and price change , 2009, Proceedings of the National Academy of Sciences.