pdf markov - CH10 10.1 Markov Chain Probability Vector Def...

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CH10 Markov Chain 10.1 Probability Vector Def 10.1 <Definition 10.1> Row vector U = (u 1 , u 2 , u 3 , …. , u n ) If U statisfies (1)u i 0 , for all i (2) i u i = 1 ,then U is said a probability vector eg. ( 4 1 , 4 1 , 0, 2 1 ) is a probability vector Def 10.2 <Definition 10.2> p is a square matrix , if every row in P is a probability vector then P is said to be a stochastic matrix eg. 2 1 3 2 4 3 4 3 3 1 4 1 0 3 2 3 1 3 1 6 1 2 1 0 1 0 X 因為非方陣以及元素和不為 1 <Theorem 10.1> If P and Q are two stochastic matrix then PQ is stochastic matrix (That is P* also a stochastic matrix )
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10.2 Fixed point of square matrix Def 10.2 Let P be a square matrix. If a non-zero row vector U = (u 1 , u 2 , u 3 , … , u n ) statisfies UP = U then U is said to be the fixed point of P on steady-state vector eg. P = 3 2 1 2 U = 1 2 1 2 3 2 1 2 = 1 2 Check 2 U = 2 4 3 2 1 2 = 2 4 Theorem 10.2 If U is a fixed point of square matrix P then λU is also a fixed point of P for every non-zero λ . ie. (λU)P = λ(UP) = λU < 例一 > 某轎車出租公司在 (1) 台北 .(2) 台中 .(3) 高雄三地設有出租據點,顧客可 以在任何租車點租車,在任何點還車。經過長期數據分析,顧客租還 車機率如 : P = 3 2 1 2 . 0 6 . 0 2 . 0 5 . 0 2 . 0 3 . 0 1 . 0 1 . 0 8 . 0 3 2 1 必為 probability vector. 某轎車由台中租出,則 initial state vector 0 1 0 P X n = X n-1 P → X 1 = X 0 P = 0 1 0 P = 5 . 0 2 . 0 3 . 0 X 2 = 5 . 0 2 . 0 3 . 0 P …… X 11 = 213 . 0 230 . 0 557 . 0 …… 都一樣 Transition matrix 租車點 還車
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< 觀察 > n值上升後 X n U( fixed UP = U → U(I - P) =0 ( 聯立方程式 ) 3 2 1 u u u 8 . 0 6 . 0 2 . 0 5 . 0 8 . 0 3 . 0 1 . 0 1 . 0 2 . 0 0 0 0 reduced row echlon matrix 3 2 1 u u u 0 13 14 13 34 0 1 0 0 0 1 = 0 0 0 → u1 = 13 34 (u3) u1 = 13 34 u3 u2 = 13 14 u3 u3 = λ U = λ 1 13 14 13 34 Letλ= 61 13 1 13 14 13 34 1 s.t U is a probability vector →U = 61 13 61 14 61 34 = 213 . 0 229 . 0 557 . 0 = X 11 ASK !假設公司有 1000 輛車時,在北 . . 高各有多少停車位最適當? 557 / 229 / 213 < 例二 > 興大 OK 50% 7-11 50% ← 95 / 5 市佔率 經驗顯示 7 . 0 3 . 0 2 . 0 8 . 0 每月顧客移動情形, OK 店長?六月市佔? 5 . 0 5 . 0 7 . 0 3 . 0 2 . 0 8 . 0 45 . 0 55 . 0 Find the fixed pt of P Long-term 市佔? U = U P, U is a probability vector (state vector) OK 7-11 OK 7-11
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Y X = Y X 7 . 0 3 . 0 2 . 0 8 . 0 → 0.8x + 0.3y = x 0.2x + 0.7y = y x + y = 1 去掉一個多餘的式子 → 0.2x 0.3y = 0 x + y = 1 OK 60% , 7-11 40% < 例二續 > 某天加入實習商店後 (nchu) 一年後,市佔率 9 . 0 0 1 . 0
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