×

zbMATH — the first resource for mathematics

Zipf-Mandelbrot law, \(f\)-divergences and the Jensen-type interpolating inequalities. (English) Zbl 1412.94137
Summary: Motivated by the method of interpolating inequalities that makes use of the improved Jensen-type inequalities, in this paper we integrate this approach with the well known Zipf-Mandelbrot law applied to various types of \(f\)-divergences and distances, such are Kullback-Leibler divergence, Hellinger distance, Bhattacharyya distance (via coefficient), \(\chi^{2}\)-divergence, total variation distance and triangular discrimination. Addressing these applications, we firstly deduce general results of the type for the Csiszár divergence functional from which the listed divergences originate. When presenting the analyzed inequalities for the Zipf-Mandelbrot law, we accentuate its special form, the Zipf law with its specific role in linguistics. We introduce this aspect through the Zipfian word distribution associated to the English and Russian languages, using the obtained bounds for the Kullback-Leibler divergence.

MSC:
94A17 Measures of information, entropy
94A15 Information theory (general)
26D15 Inequalities for sums, series and integrals
26A51 Convexity of real functions in one variable, generalizations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Csiszár, I., Information-type measures of difference of probability functions and indirect observations, Studia Sci. Math. Hung., 2, 299-318, (1967) · Zbl 0157.25802
[2] Csiszár, I., Information measures: a critical survey, 73-86, (1978) · Zbl 0401.94010
[3] Kullback, S.: Information Theory and Statistics. Wiley, New York (1959) · Zbl 0088.10406
[4] Kullback, S.; Leibler, R.A., On information and sufficiency, Ann. Math. Stat., 22, 79-86, (1951) · Zbl 0042.38403
[5] Dragomir, S.S.: Some inequalities for the Csiszár Φ-divergence, pp. 1-13. RGMIA (2001)
[6] Taneja, I.J.: Bounds on triangular discrimination, harmonic mean and symmetric Chi-square divergences. arXiv:math/0505238 · Zbl 1098.62007
[7] Mitrinović, D.S., Pečarić, J., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic, Dordrecht (1993) · Zbl 0771.26009
[8] Krnić, M.; Lovričević, N.; Pečarić, J., On mcshane’s functional’s properties and its applications, Period. Math. Hung., 66, 159-180, (2013) · Zbl 1299.26045
[9] Krnić, M.; Lovričević, N.; Pečarić, J., Superadditivity of the Levinson functional and applications, Period. Math. Hung., 71, 166-178, (2015) · Zbl 1363.26036
[10] Krnić, M.; Lovričević, N.; Pečarić, J., Jessen’s functional, its properties and applications, An. Ştiinţ. Univ. ‘Ovidius’ Constanţa, Ser. Mat., 20, 225-248, (2012) · Zbl 1274.26058
[11] Dragomir, S.S.; Pečarić, J.; Persson, L.E., Properties of some functionals related to jensen’s inequality, Acta Math. Hung., 70, 129-143, (1996) · Zbl 0847.26013
[12] Krnić, M., Lovričević, N., Pečarić, J., Perić, J.: Superadditivity and Monotonicity of the Jensen-Type Functionals. Element (2015) · Zbl 1358.26001
[13] Manin, D.Y., Mandelbrot’s model for zipf’s law: can mandelbrot’s model explain zipf’s law for language?, J. Quant. Linguist., 16, 274-285, (2009)
[14] Mandelbrot, B.; Jackson, W. (ed.), An informational theory of the statistical structure of language, 486-502, (1953), New York
[15] Mandelbrot, B.: Information Theory and Psycholinguistics. Scientific Psychology: Principles and Approaches. Basic Books, New York (1965)
[16] Montemurro, M.A.: Beyond the Zipf-Mandelbrot law in quantitative linguistics. arXiv:cond-mat/0104066 · Zbl 0978.68126
[17] Egghe, L., Rousseau, R.: Introduction to Informetrics. Quantitative Methods in Library, Documentation and Information Science. Elsevier, New York (1990)
[18] Silagadze, Z.K., Citations and the Zipf-Mandelbrot law, Complex Syst., 11, 487-499, (1997) · Zbl 0956.68541
[19] Mouillot, D.; Lepretre, A., Introduction of relative abundance distribution (RAD) indices, estimated from the rank-frequency diagrams (RFD), to assess changes in community diversity, Environ. Monit. Assess., 63, 279-295, (2000)
[20] Manaris, B.; Vaughan, D.; Wagner, C.S.; Romero, J.; Davis, R.B., Evolutionary music and the Zipf-Mandelbrot law: developing fitness functions for pleasant music, Essex · Zbl 1033.68858
[21] Horváth, L., Pečarić, Ð., Pečarić, J.: Estimations of \(f\)- and Rènyi divergences by using a cyclic refinement of the Jensen’s inequality. https://doi.org/10.1007/s40840-017-0526-4 · Zbl 1428.26040
[22] Renyi, A., On measures of entropy and information, San Diego
[23] Shannon, C.E., A mathematical theory of communication, Bell Syst. Tech. J., 27, 379-423, (1948) · Zbl 1154.94303
[24] Erven, T.; Harremoës, P., Rényi divergence and Kullback-Leibler divergence, J. Latex Class Files, 6, 1-24, (2007) · Zbl 1360.94180
[25] Gelbukh, A.; Sidorov, G., Zipf and heaps laws’ coefficients depend on language, Mexico City · Zbl 0976.68583
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.