Fractional State Space Analysis of Temperature Time Series

Abstract Atmospheric temperatures characterize Earth as a slow dynamics spatiotemporal system, revealing long-memory and complex behavior. Temperature time series of 54 worldwide geographic locations are considered as representative of the Earth weather dynamics. These data are then interpreted as the time evolution of a set of state space variables describing a complex system. The data are analyzed by means of multidimensional scaling (MDS), and the fractional state space portrait (fSSP). A centennial perspective covering the period from 1910 to 2012 allows MDS to identify similarities among different Earth’s locations. The multivariate mutual information is proposed to determine the “optimal” order of the time derivative for the fSSP representation. The fSSP emerges as a valuable alternative for visualizing system dynamics.

[1]  António M. Lopes,et al.  Dynamical Analysis of the Global Warming , 2012 .

[2]  Gilney Figueira Zebende,et al.  DCCA cross-correlation coefficient apply in time series of air temperature and air relative humidity , 2012 .

[3]  Christos Giannakopoulos,et al.  Analysis of mean, maximum, and minimum temperature in Athens from 1897 to 2001 with emphasis on the last decade: trends, warm events, and cold events , 2004 .

[4]  José António Tenreiro Machado,et al.  Fractional dynamics and MDS visualization of earthquake phenomena , 2013, Comput. Math. Appl..

[5]  Marcelo A. Savi,et al.  Analysis of the global warming dynamics from temperature time series. , 2010 .

[6]  António M Lopes,et al.  Fractional order models of leaves , 2014 .

[7]  David B. Stephenson,et al.  Statistical methods for interpreting Monte Carlo ensemble forecasts , 2000 .

[8]  I. Petráš.  Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation , 2011 .

[9]  John A. Hartigan,et al.  Clustering Algorithms , 1975 .

[10]  Alan Trounson,et al.  The persistence of memory , 2010, Nature Medicine.

[11]  T. Subba Rao,et al.  Statistical analysis and time-series models for minimum/maximum temperatures in the Antarctic Peninsula , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[12]  Y. Chen,et al.  Fractional Order Motion Controls , 2012 .

[13]  P. Groenen,et al.  Modern Multidimensional Scaling: Theory and Applications , 1999 .

[14]  Y. Chen,et al.  Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications , 2011 .

[15]  Sunil Srinivasa.  A REVIEW ON MULTIVARIATE MUTUAL INFORMATION , 2005 .

[16]  Clara-Mihaela Ionescu,et al.  The Human Respiratory System: An Analysis of the Interplay between Anatomy, Structure, Breathing and Fractal Dynamics , 2013 .

[17]  C. Deser,et al.  Twentieth century tropical sea surface temperature trends revisited , 2010 .

[18]  J. T. Tenreiro Machado,et al.  Dynamic Analysis and Pattern Visualization of Forest Fires , 2014, PloS one.

[19]  J. Grieser,et al.  Statistical time series decomposition into significant components and application to European temperature , 2002 .

[20]  Jeffrey S. Rosenthal,et al.  Introduction to mathematical systems theory. a behavioral approach [Book Review] , 2002, IEEE Transactions on Automatic Control.

[21]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[22]  P. Jones,et al.  Uncertainty estimates in regional and global observed temperature changes: A new data set from 1850 , 2006 .

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

[24]  I. Podlubny.  Fractional differential equations , 1998 .

[25]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[26]  António M Lopes,et al.  State space analysis of forest fires , 2016 .

[27]  R. Rudra,et al.  Trend and Periodicity of Temperature Time Series in Ontario , 2014 .

[28]  J. J. Oñate,et al.  TEMPERATURE VARIATIONS IN SPAIN SINCE 1901: A PRELIMINARY ANALYSIS , 1996 .

[29]  António M. Lopes,et al.  Fractional Order Control of a Hexapod Robot , 2004 .

[30]  F. Mainardi,et al.  Recent history of fractional calculus , 2011 .

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

[32]  António M. Lopes,et al.  Analysis of temperature time-series: Embedding dynamics into the MDS method , 2014, Commun. Nonlinear Sci. Numer. Simul..

[33]  Trevor F. Cox,et al.  Metric multidimensional scaling , 2000 .

[34]  J. Machado,et al.  A Review of Definitions for Fractional Derivatives and Integral , 2014 .

[35]  J. Tenreiro Machado,et al.  Science metrics on fractional calculus development since 1966 , 2013 .

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

[37]  Angel R. Martinez,et al.  : Exploratory data analysis with MATLAB ® , 2007 .

[38]  Sunita Gakkhar,et al.  Complex dynamics in a prey predator system with multiple delays , 2012 .

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

[40]  J. Hansen,et al.  GLOBAL SURFACE TEMPERATURE CHANGE , 2010 .

[41]  Norden E. Huang,et al.  On the time-varying trend in global-mean surface temperature , 2011 .

[42]  Joydeep Ghosh,et al.  Cluster Ensembles --- A Knowledge Reuse Framework for Combining Multiple Partitions , 2002, J. Mach. Learn. Res..

[43]  V. Pelino,et al.  Complexity in rainfall phenomena , 2006 .

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

[45]  António M. Lopes,et al.  Rhapsody in fractional , 2014 .

[46]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[47]  F. Mainardi.  Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , 2010 .

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

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

[50]  Quansheng Ge,et al.  Temperature variation through 2000 years in China: An uncertainty analysis of reconstruction and regional difference , 2010 .

[51]  L. Dorcak.  Numerical Models for the Simulation of the Fractional-Order Control Systems , 2002 .

[52]  José António Tenreiro Machado,et al.  Some pioneers of the applications of fractional calculus , 2013 .

[53]  M. Rivero,et al.  Fractional calculus: A survey of useful formulas , 2013, The European Physical Journal Special Topics.