IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v361y2006i1p124-138.html
   My bibliography  Save this article

Nonextensive triangle equality and other properties of Tsallis relative-entropy minimization

Author

Listed:
  • Dukkipati, Ambedkar
  • Murty, M. Narasimha
  • Bhatnagar, Shalabh

Abstract

Kullback–Leibler relative-entropy has unique properties in cases involving distributions resulting from relative-entropy minimization. Tsallis relative-entropy is a one-parameter generalization of Kullback–Leibler relative-entropy in the nonextensive thermostatistics. In this paper, we present the properties of Tsallis relative-entropy minimization and present some differences with the classical case. In the representation of such a minimum relative-entropy distribution, we highlight the use of the q-product, an operator that has been recently introduced to derive the mathematical structure behind the Tsallis statistics. One of our main results is the generalization of triangle equality of relative-entropy minimization to the nonextensive case.

Suggested Citation

  • Dukkipati, Ambedkar & Murty, M. Narasimha & Bhatnagar, Shalabh, 2006. "Nonextensive triangle equality and other properties of Tsallis relative-entropy minimization," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 361(1), pages 124-138.
    • Handle: RePEc:eee:phsmap:v:361:y:2006:i:1:p:124-138
    DOI: 10.1016/j.physa.2005.06.072
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437105007065
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2005.06.072?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Mendes, R.S., 1997. "Some general relations in arbitrary thermostatistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 242(1), pages 299-308.
    2. Yamano, Takuya, 2002. "Some properties of q-logarithm and q-exponential functions in Tsallis statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 305(3), pages 486-496.
    3. Anastasiadis, Aristoklis D. & Magoulas, George D., 2004. "Nonextensive statistical mechanics for hybrid learning of neural networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(3), pages 372-382.
    4. Tsallis, Constantino & Stariolo, Daniel A., 1996. "Generalized simulated annealing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 233(1), pages 395-406.
    5. Tsallis, Constantino & Mendes, RenioS. & Plastino, A.R., 1998. "The role of constraints within generalized nonextensive statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 261(3), pages 534-554.
    6. Borges, Ernesto P., 2004. "A possible deformed algebra and calculus inspired in nonextensive thermostatistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 340(1), pages 95-101.
    7. Martı́nez, S & Nicolás, F & Pennini, F & Plastino, A, 2000. "Tsallis’ entropy maximization procedure revisited," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 286(3), pages 489-502.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Dukkipati, Ambedkar & Bhatnagar, Shalabh & Murty, M. Narasimha, 2007. "On measure-theoretic aspects of nonextensive entropy functionals and corresponding maximum entropy prescriptions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 384(2), pages 758-774.
    2. Mao, Xuegeng & Shang, Pengjian & Wang, Jianing & Yin, Yi, 2020. "Fractional cumulative residual Kullback-Leibler information based on Tsallis entropy," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Dukkipati, Ambedkar & Bhatnagar, Shalabh & Murty, M. Narasimha, 2007. "On measure-theoretic aspects of nonextensive entropy functionals and corresponding maximum entropy prescriptions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 384(2), pages 758-774.
    2. da Silva, Sérgio Luiz Eduardo Ferreira, 2021. "Newton’s cooling law in generalised statistical mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 565(C).
    3. Nelson, Kenric P., 2015. "A definition of the coupled-product for multivariate coupled-exponentials," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 422(C), pages 187-192.
    4. Deeb, Omar El, 2023. "Entropic spatial auto-correlation of voter uncertainty and voter transitions in parliamentary elections," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 617(C).
    5. Suyari, Hiroki & Wada, Tatsuaki, 2008. "Multiplicative duality, q-triplet and (μ,ν,q)-relation derived from the one-to-one correspondence between the (μ,ν)-multinomial coefficient and Tsallis entropy Sq," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(1), pages 71-83.
    6. Oikonomou, Th., 2007. "Tsallis, Rényi and nonextensive Gaussian entropy derived from the respective multinomial coefficients," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 386(1), pages 119-134.
    7. Aristoklis D. Anastasiadis & Marcelo P. Albuquerque & Marcio P. Albuquerque & Diogo B. Mussi, 2010. "Tsallis q-exponential describes the distribution of scientific citations—a new characterization of the impact," Scientometrics, Springer;Akadémiai Kiadó, vol. 83(1), pages 205-218, April.
    8. Suyari, Hiroki, 2006. "Mathematical structures derived from the q-multinomial coefficient in Tsallis statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 368(1), pages 63-82.
    9. Oikonomou, Thomas & Tirnakli, Ugur, 2009. "Generalized entropic structures and non-generality of Jaynes’ Formalism," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3027-3034.
    10. da Silva, G.B. & Ramos, R.V., 2019. "The Lambert–Tsallis Wq function," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 164-170.
    11. Koponen, Ismo T. & Palmgren, Elina & Keski-Vakkuri, Esko, 2021. "Characterising heavy-tailed networks using q-generalised entropy and q-adjacency kernels," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 566(C).
    12. Bafghi, Seyed Mohammad Amin Tabatabaei & Kamalvand, Mohammad & Morsali, Ali & Bozorgmehr, Mohammad Reza, 2018. "Radial distribution function within the framework of the Tsallis statistical mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 506(C), pages 857-867.
    13. López-Rosa, S. & Angulo, J.C. & Dehesa, J.S. & Yáñez, R.J., 2008. "Existence conditions and spreading properties of extreme entropy D-dimensional distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(10), pages 2243-2255.
    14. Naif Alotaibi & A. S. Al-Moisheer & Ibrahim Elbatal & Mansour Shrahili & Mohammed Elgarhy & Ehab M. Almetwally, 2023. "Half Logistic Inverted Nadarajah–Haghighi Distribution under Ranked Set Sampling with Applications," Mathematics, MDPI, vol. 11(7), pages 1-32, April.
    15. Rubenthaler, Sylvain & Rydén, Tobias & Wiktorsson, Magnus, 2009. "Fast simulated annealing in with an application to maximum likelihood estimation in state-space models," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1912-1931, June.
    16. Kin Keung Lai & Shashi Kant Mishra & Ravina Sharma & Manjari Sharma & Bhagwat Ram, 2023. "A Modified q-BFGS Algorithm for Unconstrained Optimization," Mathematics, MDPI, vol. 11(6), pages 1-24, March.
    17. Moriguchi, Kai & Ueki, Tatsuhito & Saito, Masashi, 2020. "Establishing optimal forest harvesting regulation with continuous approximation," Operations Research Perspectives, Elsevier, vol. 7(C).
    18. Masi, Marco, 2007. "On the extended Kolmogorov–Nagumo information-entropy theory, the q→1/q duality and its possible implications for a non-extensive two-dimensional Ising model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 377(1), pages 67-78.
    19. Chang-Yong Lee & Dongju Lee, 2014. "Determination of initial temperature in fast simulated annealing," Computational Optimization and Applications, Springer, vol. 58(2), pages 503-522, June.
    20. Jinyu Zhang & Kang Gao & Yong Li & Qiaosen Zhang, 2022. "Maximum Likelihood Estimation Methods for Copula Models," Computational Economics, Springer;Society for Computational Economics, vol. 60(1), pages 99-124, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:361:y:2006:i:1:p:124-138. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.