Abstract
Generalized statistical complexity measures provide a means to jointly quantify inner information and relative structural richness of a system described in terms of a probability model. As a natural divergence-based extension in this context, generalized relative complexity measures have been proposed for the local comparison of two given probability distributions. In this paper, the behavior of generalized relative complexity measures is studied for assessment of structural dependence in a random vector leading to a concept of ‘generalized mutual complexity’. A related optimality criterion for sampling network design, which provides an alternative to mutual information based methods in the complexity context, is formulated. Aspects related to practical implementation, and conceptual issues regarding the meaning and potential use of this approach, are discussed. Numerical examples are used for illustration.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aczél J, Daròczy Z (1975) On measures of information and their characterization. In: Mathematics in Science and Engineering, vol 115. Academic Press, London
Alonso F J, Bueso M C, Angulo J M (2012) Effect of data transformations on predictive risk indicators. Method Comput Appl Probab 14:705–716
Angulo J M, Bueso M C (2001) Random perturbation methods applied to multivariate spatial sampling design. Environmetrics 12:631–646
Angulo J M, Bueso M C, Alonso F J (2000) A study on sampling design for optimal prediction of space-time stochastic processes. Stoch Environ Res Risk Assess 14:412–427
Angulo J M, Bueso M C, Alonso F J (2013) Space-time adaptive sampling and data transformations. In: Mateu J, Müller W G (eds) Spatial-temporal designs. Advances in efficient data acquisition. Wiley, Chichester, pp 231–248
Angulo J M, Esquivel F J (2014) Structural complexity in space-time seismic event data. Stoch Environ Res Risk Assess 28:1187–1206
Anteneodo C, Plastino A R (1996) Some features of the López-Ruiz-Mancini-Calbet (LMC) statistical measure of complexity. Phys Lett A 223:348–354
Botev Z I, Kroese D P (2011) The generalized cross entropy method, with applications to probability density estimation. Method Comput Appl Probab 13:1–27
Bouvrie P A, Angulo J C (2012) A generalized relative complexity: application to atomic one-particle densities. Chem Phys Lett 539:191–196
Bueso M C, Angulo J M, Alonso F J (1998) A state-space-model approach to optimal spatial sampling design based on entropy. Environ Ecol Stat 5:29–44
Caselton W F, Zidek J V (1984) Optimal monitoring network designs. Stat Probab Lett 2:223–227
Catalán R G, Garay J, López-Ruiz R (2002) Features of the extension of a statistical measure of complexity to continuous systems. Phys Rev E 66:011102
Csiszár I (2008) Axiomatic characterizations of information measures. Entropy 10:261–273
Erdi P (2008) Complexity explained. Springer-Verlag, Berlin Heidelberg
Furuichi S, Yanagi K, Kuriyama K (2005) Fundamental properties of Tsallis relative entropy. J Math Phys 45:4868–4877
Gell-Mann M (1988) Simplicity and complexity in the description of nature. Eng Sci 51:2–9
Gell-Mann M (1995) What is complexity? Complexity 1:16–19
Khader M, Hamza A B (2011) Nonrigid image registratioin using an entropic similarity. IEEE Trans Inf Tech Biomed 15:681–690
Kullback S, Leibler R A (1951) On information and sufficiency. Ann Math Stat 22:79–86
López-Ruiz R, Mancini H L, Calbet X (1995) A statistical measure of complexity. Phys Lett A 209:321–326
López-Ruiz R, Mancini H L, Calbet X (2013) A statistical measure of complexity. In: Kowalski A M, Rossignoli R D, Curado E M F (eds) Concepts and recent advances in generalized information measures and statistics. Bentham eBooks, Sharjah, pp 147–168
López-Ruiz R, Nagy Á, Romera E, Sañudo J (2009) A generalized statistical complexity measure: applications to quantum systems. J Math Phys 50:123528 (10)
Lovallo M, Telesca L (2011) Complexity measures and information planes of X-ray astrophysical sources. J Stat Mech 03:P03029
Martín M T, Plastino A R, Plastino A (2000) Tsallis-like information measures and the analysis of complex signals. Phys A 275:262–271
Martín M T, Plastino O, Rosso A (2006) Generalized statistical complexity measures: geometrical and analytical properties. Phys A 369:439–462
Mendes R S, Evangelista L R, Thomaz S M, Agostinho A A, Gomes L G (2008) A unified index to measure ecological diversity and species rarity. Ecography 31:450–456
Romera E, Sen K D, Nagy Á (2011) A generalized relative complexity measure. J Stat Mech Theory Expert 09:P09016
Rosso O A, Martín M T, Figliola A, Keller K, Plastino A (2006) EEG analysis using wavelet-based informational tools. J Neurosci Methods 153:163–182
Rosso O A, Martín M T, Larrondo H A, Kowalski A M, Plastino A (2013) Generalized statistical complexity: a new tool for dynamical systems. In: Kowalski A M, Rossignoli R D, Curado E M F (eds) Concepts and recent advances in generalized information measures and statistics. Bentham e-books, Rio de Janeiro, pp 169–215
Sen KD (2011) Statistical complexity: applications in electronic structure. Springer, Netherlands
Su Z Y, Wu T (2006) Multifractal analyses of music sequences. Phys D 221:188–194
Tsallis C (1998) Generalized entropy-based criterion for consistent testing. Phys Rev E 58:1442–1445
Wootters K W (1981) Statistical distance and Hilbert space. Phys Rev D 23:357–362
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Alonso, F.J., Bueso, M.C. & Angulo, J.M. Dependence Assessment Based on Generalized Relative Complexity: Application to Sampling Network Design. Methodol Comput Appl Probab 18, 921–933 (2016). https://doi.org/10.1007/s11009-016-9495-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-016-9495-6