markov

# markov - Correlated And Time-Varying Events Correlated And...

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Correlated And Time Correlated And Time - - Varying Varying Events Events

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PR and ANN Correlated And Time-Varying Events ± So far, events and statistics are ² independent ² time (space) invariant ± Many applications violate the above ² speech signals ² images (textures) ± Need a more sophisticated model
PR and ANN Possibilities ± This problem (and its variants) has been studied in many fields over a long period of time ± We will briefly discuss ² Finite automata (computer science, syntactic pattern recognition) ² Markov chain (mathematics, statistics) ² Hidden Markov model (signal processing, speech processing)

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PR and ANN Finite Automata ± As a language (or more generally, a pattern ) recognition device ± A machine with a fixed number of states (memory) ± It parses a string (a pattern ) and based on the current state and current input to decide on the next state ± Eventually, when the string (the pattern ) is exhausted, the machine will halt and either accept (recognize) or reject (not recognize) the input
PR and ANN Finite Automata (cont.) 1 q 2 q 3 q 4 q 5 q n q 1 a 2 a 3 a m a input control

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PR and ANN ± A deterministic finite automata is a quintuple ² is a finite set of states ² is an alphabet ² is the initial state ² is the set of final states ² transition function ) , , , , ( F s K M δ Σ = K Σ K s K F K K Σ × ) ( Finite Automata (cont.)
PR and ANN Example ± A finite automata that accepts all strings in {a,b}* that have an even number of b’s ² alphabet: {a,b} ² state: {q o , q 1 } ² initial state: q o ² final states: {q o } ² transition rules 1 q 0 q a a b b 0 1 1 1 1 0 0 0 ) , ( q b q q a q q b q q a q q q σ δ

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PR and ANN More Example ± A finite automata that accepts strings over {a,b}* that does not contain three consecutive b’s 2 q 0 q a a b a 3 q 1 q bb a b
PR and ANN More Example ± A finite automata that accepts * ) ( aba ab L = 0 q 1 q 2 q 3 q 4 q a a a a a b b b b b

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PR and ANN An Opposite Definition ± Grammar: as a language (or pattern) generation device ± A grammar is a quadruple ² V an alphabet ² subset of V (terminals) ² rewriting rules ² S start symbol Σ * ) ( : V V R Σ