A maximum entropy perspective of Pena’s synthetic indicators

Mishra, SK (2012): A maximum entropy perspective of Pena’s synthetic indicators.

Abstract

This paper uses mixed combinatorial-cum-real particle swarm method to obtain a heuristically optimal order in which the constituent variables can be arranged so as to yield some generalized maximum entropy synthetic indicators that represent the constituent variables in the best information-theoretic sense. It may help resolve the arbitrariness and indeterminacy of Pena’s method of construction of a synthetic indicator which at present is very sensitive to the order in which the constituent variables (whose linear aggregation yields the synthetic indicator) are arranged.

• Item Type: MPRA Paper
• Language: English
• Keywords: Synthetic indicators, Composite indices, Pena’s distance, Mixed Combinatorial Particle swarm, Sharma-Mittal entropy, Rényi entropy, Tsallis entropy. Kaniadakis entropy, Abe entropy
• Subjects: C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C43 - Index Numbers and Aggregation; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C44 - Operations Research ; Statistical Decision Theory; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis

References

1. Abe, S. (1997): "A Note on the q-Deformation-theoretic Aspect of the Generalized Entropies in Nonextensive Physics", Physics Letters - A 224: 326–330.

2. Abe, S. (2000): "Axioms and Uniqueness Theorem for Tsallis Entropy", Physics Letters - A 271(1-2): 74–79.

3. Abe, S. (2002): “Stability of Tsallis entropy and instabilities of Rényi and normalized Tsallis entropies: A basis for q-exponential distributions”, Physical Review - E 66 (4 pt 2), 046134. http://arxiv.org/abs/cond-mat/0206078

4. Aktürk, E., Bağci, G.B. & Sever, R. (2008): "Is Sharma-Mittal entropy really a step beyond Tsallis and Rényi entropies?", http://arxiv.org/pdf/cond-mat/0703277.pdf (visited 28.3.2012).

5. Beck, C. (2008): “Generalised Information and Entropy Measures in Physics”, Contemporary Physics, 50(4): 495–510.

6. Esteban, M.D. & Morales, D. (1995): "A Summary of Entropy Statistics", Kybernetika, 31(4): 337-346. http://www.cse.msu.edu/~cse902/S03/entropy_measures.pdf (visited: 29.03.2012).

7. García, E.C., Rodríguez Martín, J.A. & Pabsdorf, M.N. (2010): “The Features of Development in the Pacific Countries of the African, Caribbean and Pacific Group”, Social Indicators Research 99(3): 469-485.

8. Kaniadakis, G. (2002): "Statistical Mechanics in the Context of Special Relativity", Physical Review - E 66: 056125:1–17

9. Khinchin, A.I. (1957) Mathematical Foundations of Information Theory, Dover, New York.

10. Lesche, B. (1982): “Instabilities of Rényi entropies”, J. Stat. Phys. 27(2): 419-423.

11. Lutsko, J.F., Boon, J.P. & Grosfils, P. (2009): “Is the Tsallis entropy stable?”, Europhysics Letters 86(4): doi:10.1209/0295-5075/86/40005

12. Martína, J.A.R. & Fernández, J.A.S. (2011): “An Index of Maternal and Child Health in the Least Developed Countries of Asia”, Gac Sanit. 2011 [in press]; doi:10.1016/j.gaceta.2011.05.021.

13. Masi, M. (2005): “A Step beyond Tsallis and Renyi Entropies”, Physics Letters - A, 338: 217-224. http://144.206.159.178/ft/847/571592/12348937.pdf

14. Mishra, S.K. (2007): “A Note on Human Development Indices with Income Equalities”, http://ssrn.com/abstract=992854 or http://dx.doi.org/10.2139/ssrn.992854.

15. Mishra, S.K. (2010a): "Performance of Differential Evolution and Particle Swarm Methods on Some Relatively Harder Multi-modal Benchmark Functions", The IUP Journal of Computational Mathematics, III(1): 7-18.

16. Mishra, S.K. (2010b): "Construction of an Index: A New Method", in Growth and Human Development in North-East India, Nayak, P. (ed.), Oxford University Press: 24-35.

17. Mishra, S.K. (2011): "A Comparative Study of Various Inclusive Indices and the Index Constructed by the Principal Component Analysis", IUP Journal of Computational Mathematics, 4(2): 7-26.

18. Mishra, S.K. (2012a): “A Note on the Indeterminacy and Arbitrariness of Pena’s Method of Construction of Synthetic Indicators”, SSRN: http://ssrn.com/abstract=2026293 or http://dx.doi.org/10.2139/ssrn.2026293

19. Mishra, S.K. (2012b): “A Note on Construction of Heuristically Optimal Pena’s Synthetic Indicators by the Particle Swarm Method of Global Optimization”, SSRN: http://ssrn.com/abstract=2028395 or http://dx.doi.org/10.2139/ssrn.2028395

20. Montero, J.M., Chasco, C. & Lanaz, B. (2010): “Building an environmental quality index for a big city: a spatial interpolation approach combined with a distance indicator”, J. Geogr. Syst. 12: 435-459.

21. Munda, G. & Nardo, M (2005): “Constructing Consistent Composite Indicators: The Issue of Weights”, EUR 21834 EN, Institute for the Protection and Security of the citizen, European Commission, Luxembourg.

22. Naudts, J. (2011): Generalised Thermostatistics, New York, Springer.

23. Pena, J. B. (1977): Problemas de la medición del bienestar y conceptos afines. Una aplicación al Caso Espanól. (I.N.E.: Madrid).

24. Rényi, A. (1970): Probability Theory, North Holland,Amsterdam.

25. Rodrigues, P.S. & Giraldi, G.A. (2009): "Computing the q-index for Tsallis Nonextensive Image Segmentation", Computer Graphics and Image Processing (SIBGRAPI), 2009 XXII Brazilian Symposium, 11-15 Oct. 2009: 232 – 237. http://fei.edu.br/~psergio/Visao-PEL205_arquivos/q-automatic.pdf visited 24.03.2012.

26. Sarker, S., Biswas, B. & Soundrs, P.J. (2006): “Distribution-Augmented Human Development Index: A Principal Component Analysis”, GSP, College of Business, Utah State Univ., USA. www.usu.edu/cob/econ/graduatestudents/documents/papers/developmentpaper.pdf. (visited on April1 24, 2007).

27. Sharma, B.D. & Mittal, D.P. (1975): “New Non-additive Measures of Entropy for a Discrete Probability Distribution”, J. Math. Sci. 10: 28-40.

28. Somarriba, N. & Pena, B. (2009): “Synthetic Indicators of Quality of Life in Europe”, Soc. Indic. Res. 94(1): 115–133.

29. Tasgetiren, F., Sevkli, M., Lian, Y.C., & Gencyilmaz, G. (2004) “Particle Swarm Optimization Algorithm for Single Machine Weighted Tardiness Problem”, Proceedings of IEEE Congress on Evolutionary Computation: 1412–1419.

30. Tsallis, C. (1988): “Possible Generalization of Boltzmann-Gibbs Statistics”, J. Stat. Phys. 52(1-2) : 479–487.

31. Zarzosa, P. (1996): Aproximación a la medición del bienestar social. Valladolid: Secretario de Publicaciones.