Lecture9

State j after n epochs after the nfh transition

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Unformatted text preview: ctor) - State of the system in epoch n, given as vector A@‘=[ a,(* 4j* 4,(* ] Initial State Vector - If n=0, then nlo) describes the initial conditions of the system l The absolute probability for state n can be computed according to the following: Matrix arithmetic can be used to find the 3rd step absolute probabilities: l A”’ = [~;‘)17;J)...l?:))] = ,#O)p(J) = /fl’p’Z’ = /fZ)fHl) n-step Matrices l l Example Chapman-Kolomogorov Eq’s n-step Probabilities The Chapman-Kolomorov Equations show that successive states can be described according to: l Find P(2), Glven the One Step Matrix v”) p = p(” =[I: ;] The equivalent matrix relationship is as follows: p(n) = pen-Mp(N r.52 p(z) = AS1 1.36 .6-I] EM-602 I QM-710 (NJ) Lecture 9 Page 9-4 Example (cont’d) Example (cont’d) n-step Probabilities n-step Probabilities Find PCS l PC’) # ::I pm It is interesting to observe that as the number of transitions gets larger, the rows become more similar l =[z :;I (*u) J4286 57141 P-1.4286 57141 .60f31 po, =r.392 1.456 5441 Example Example (cont’d) nth State Vectors nth State Vectors Flnd A(z), Given the Initial Condition Ato) l A'O' = [ .7 .3] l r52 A81 PC*) =1.X 6-IJ . Find A@) Ii@) =[.7 31 r392 P')=~e4s r52 .rsl #' = ‘4’“‘P’z’ =[ .7 .3][_x &J A"' = A'"'PJ' =[.7 A'*' = [A72 .528] .6061 WI r.392 .6081 .156 jerj 3 1 A"' =[A112 _5888] Markov Chains l l Markov Chains fenninology Classifications Markov Chain - A combination of OneStep Transition matrix and an Initial State Vector n-step Matrix - A stochastic matrix showing probability of transitions over n epochs l Recurrent state - A state in which revisiting is certain Pii = 1 l put”) < 1 l n<oc Absorbing state - A state from which there is no escape piiw = 1 EM-602 / QM-710 (NJ) Lecture 9 Page 9-5 nla, Transient state - A state from which the system can exit and Possible not return Markov Chains Markov Chains Classitkations l l l Classitkations Transient class - a group of transient states Absorbing class - a group of recurrent states from which there is no escape Irreducible - A markov chain in which ail states are in one class (all states are reachable from all other states) ViJ or, Piif* -I n 4.23 ViJ Pij” > 0 1 I fl I Absorbing Class Transient Class j(3, cc Transient S...
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