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pdf markov2 - 馬科夫鏈 Markov Chain...

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馬科夫鏈 Markov Chain
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± 馬可夫過程 (Markov Process) :即為機率過程的一 種,由俄國數學家 A.A. Markov 1907 年提出, 對於一項稱為布朗寧運動 (Brownian Motion) 的物 理現象之數學說明,其基本意義係經由鏈環 (Chain) 的程序,從一項已知的情況,推估未來的 情況。 ± 在多次研究試驗中發現,某些事象的機率轉換過 程中,其第 n 次驗的結果,常決定於前一次 ( 即第 n-1 ) 試驗的結果,而與前二次 ( 即第 n-2 ) 前的結果無關,而以後在學術研究上對於由一種 狀態轉換至另一種狀態的過程中具有移轉機率 (Transition Probability) 者,而且此種轉換機率可 以依據其緊接的前次狀態而推算出來,即稱為馬 可夫過程,一連串的此種轉換過程之整體則稱為 馬可夫鏈 (Markov Chain )
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Definition ± 狀態(States):一個試驗或觀察具有各種可能的結果,其中每一種結 果即稱為狀態,而所有狀態的集合,稱為狀態空間(States Space)。 ± 若狀態空間中元素的個數為有限個,則稱該馬可夫鏈為有限馬可夫鏈 (Finite Markov Chain)。否則稱為無限馬可夫鏈(Infinite Markov Chain)。 ± State space: The set of possible values (or states) that X(t) may take on is called its state space ± Discrete-state process often say chain ² The state space for a chain is usually the set of integers { 0, 1, 2, …} ± Markov Process with a discrete state space is referred to as a Markov chain ² If the probability that the next value (state) is x n+1 depends only upon the current value (state) x n and not upon any previous values (memoryless) P[X(t n+1 )=x n+1 |X(t n )=x n , X(t n-1 )=x n-1 , . .,X(t 1 )=x 1 ] = P[X(t n+1 )=x n+1 |x(t n )=x n ] ± Discrete-time Markov chain : the instants when state changes may occur are preordained to be at the integers 0, 1, 2, … ± Continuous-time Markov chain : the transitions between states may take place at any instant in time
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pdf markov2 - 馬科夫鏈 Markov Chain...

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