本文選題：廣義熵　切入點(diǎn)：Bayes估計(jì)　出處：《華中科技大學(xué)》2015年碩士論文

【摘要】：1948年,C.E.Shannon將熱力學(xué)系統(tǒng)中熵的概念引入到信息論中,標(biāo)志著現(xiàn)代信息論的誕生。隨后,Shannon熵的估計(jì)被應(yīng)用在工程學(xué)、生物醫(yī)藥學(xué)和生物化學(xué)等方面。學(xué)者們?yōu)榱私鉀Q生活中的實(shí)際問題,又提出了其他不同形式的廣義熵,如Tsallis熵和Renyi熵。在統(tǒng)計(jì)分析中,廣義熵的應(yīng)用研究越來越受到學(xué)者們的關(guān)注。本文主要討論離散型隨機(jī)變量的廣義熵,將熵的估計(jì)由Shannon熵推廣到Tsallis熵和Renyi熵。在Shannon熵極大似然估計(jì)方法的基礎(chǔ)上,得到Shannon熵的Bayes估計(jì)方法,并將其與極大似然估計(jì)方法進(jìn)行比較。進(jìn)一步,本文推導(dǎo)出廣義熵中具有代表性的Renyi熵和Tsallis熵的Bayes估計(jì)方法。運(yùn)用Matlab軟件,以概率(0.1,0.4,0.5)產(chǎn)生樣本容量為N的隨機(jī)數(shù)。用每個(gè)數(shù)的頻率代替概率,得到傳統(tǒng)熵估計(jì)的極大似然估計(jì)方法。將每個(gè)數(shù)出現(xiàn)的相應(yīng)次數(shù),代入到Bayes估計(jì)式中。通過Matlab運(yùn)算出樣本容量為N的Bayes估計(jì)值。改變N的大小,經(jīng)過多次試驗(yàn),得到一組極大似然估計(jì)值與Bayes估計(jì)值,并將兩種方法進(jìn)行比較。由誤差平方和可以看出Shannon熵的Bayes估計(jì)效果更好一些,然而Tsallis熵的Bayes估計(jì)并沒有顯現(xiàn)出更好的效果�？梢钥紤]改變q的值以及進(jìn)行大量的隨機(jī)試驗(yàn),再將Tsallis熵的Bayes估計(jì)方法與極大似然估計(jì)方法進(jìn)行對比。雖然廣義熵的Bayes估計(jì)方法并不一定是最優(yōu)的,但從廣義熵的應(yīng)用角度來看,仍具有一定的實(shí)際意義。最后,本文結(jié)合次序統(tǒng)計(jì)量,進(jìn)一步討論了連續(xù)型隨機(jī)變量的Bayes估計(jì)方法,推導(dǎo)出連續(xù)型隨機(jī)變量Shannon熵和Renyi熵的Bayes估計(jì)量。
[Abstract]:In 1948, C.E. Shannon introduced the concept of entropy into information theory, which marked the birth of modern information theory.Then Shannon entropy estimation was applied in engineering, biomedical and biochemistry.In order to solve the practical problems in life, scholars put forward other forms of generalized entropy, such as Tsallis entropy and Renyi entropy.In statistical analysis, more and more scholars pay attention to the application of generalized entropy.In this paper, the generalized entropy of discrete random variables is discussed. The estimation of entropy is extended from Shannon entropy to Tsallis entropy and Renyi entropy.Based on the Shannon entropy maximum likelihood estimation method, the Bayes estimation method of Shannon entropy is obtained and compared with the maximum likelihood estimation method.Furthermore, the Bayes estimation method for the representative Renyi entropy and Tsallis entropy in generalized entropy is derived.By using Matlab software, the random number with sample size of N is generated by probability 0. 1 / 0. 4 ~ 0. 5).The maximum likelihood estimation method of traditional entropy estimation is obtained by replacing the probability with the frequency of each number.The corresponding number of occurrences of each number is substituted into the Bayes estimator.The Bayes estimate of sample size N is calculated by Matlab.By changing the size of N, a set of maximum likelihood estimators and Bayes estimators are obtained through many experiments, and the two methods are compared.It can be seen from the sum of squared errors that the Bayes estimation of Shannon entropy is better, but the Bayes estimation of Tsallis entropy does not show better effect.We can consider changing the value of Q and carrying out a lot of random experiments, and then compare the Bayes estimation method of Tsallis entropy with the maximum likelihood estimation method.Although the Bayes estimation method of generalized entropy is not necessarily optimal, it still has some practical significance from the point of view of the application of generalized entropy.Finally, the Bayes estimation method for continuous random variables is discussed with order statistics, and the Bayes estimators of Shannon entropy and Renyi entropy of continuous random variables are derived.
【學(xué)位授予單位】：華中科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O212.8

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