Tsallis Entropy and Conditional Tsallis Entropy of Fuzzy Partitions

Document Type: research paper


Department of Mathematics, Zahedan Branch, Islamic Azad University, Zahedan, Iran


The purpose of this study is to define the concepts of Tsallis entropy and conditional Tsallis entropy of fuzzy partitions and to obtain some results concerning this kind entropy. We show that the Tsallis entropy of fuzzy partitions has the subadditivity and concavity properties. We study this information measure under the refinement and zero mode subset relations. We check the chain rules for this information measure and prove some properties about the entropy of independent fuzzy partitions. Some results of the relationship between the Tsallis entropy and conditional Tsallis entropy of fuzzy partitions are obtained and, by using conditional Tsallis entropy of fuzzy partitions, we show that the subadditivity property for Tsallis entropy of  fuzzy partitions is not established in the case that the parameter of this entropy is smaller than one. In general, the Tsallis entropy of fuzzy partitions has similar properties to the shannon entropy, where the parameter of this entropy is larger than one, and therefore can be used in addition to the Shannon entropy, to measure the amount of information to be extracted from a fuzzy experiment.


Article Title [Persian]

آنتروپی تیسالیس و آنتروپی تیسالیس شرطی افرازهای فازی

Authors [Persian]

  • محمدحسین زارع‌نژاد
  • ابوالفضل ابراهیم‌زاده
گروه ریاضی، واحد زاهدان، دانشگاه آزاد اسلامی، زاهدان، ایران
Abstract [Persian]

هدف این پژوهش این است که مفاهیم آنتروپی تیسالیس و آنتروپی تیسالیس شرطی افرازهای فازی را تعریف کرده و نتایجی در مورد این نوع آنتروپی بدست ‌آوریم. نشان می‌دهیم آنتروپی تیسالیس افرازهای فازی دارای ویژگی‌های زیرجمعی و تقعر می‌باشد. این اندازه اطلاعات را تحت روابطه‌ تظریف و زیرمجموعه به مد صفر مورد مطالعه قرار می‌دهیم. قوانین زنجیره‌ای را برای این اندازه اطلاعات بررسی کرده و خواصی در مورد آنتروپی افرازهای فازی مستقل اثبات می‌نماییم. نتایجی درباره‌ی رابطه‌ی بین آنتروپی تیسالیس و آنتروپی تیسالیس شرطی افرازهای فازی بدست آورده و به کمک آنتروپی تیسالیس شرطی افرازهای فازی، نشان می‌دهیم که ویژگی زیرجمعی برای آنتروپی تیسالیس افرازهای فازی در حالتی که پارامتر این آنتروپی از یک کوچکتر است، برقرار نمی‌باشد. به طور کلی، آنتروپی تیسالیس افرازهای فازی در حالتی که پارامتر آنتروپی تیسالیس از یک بزرگتر است دارای خواصی شبیه به آنتروپی شانون افرازهای فازی می‌باشد و بنابراین می‌تواند علاوه بر آنتروپی شانون، برای اندازه‌گیری مقدار اطلاعات مستخرج از یک آزمایش فازی مورد استفاده قرار گیرد.

Keywords [Persian]

  • افراز فازی
  • آنتروپی تیسالیس
  • آنتروپی تیسالیس شرطی
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