State sequence prediction in imprecise hidden Markov models

Transkript

1 State sequence prediction in imprecise hidden Markov models Jasper De Bock & Gert de Cooman 27 July 2011

2 Jasper De Bock & Gert de Cooman

3 Jasper De Bock & Gert de Cooman

4 Research group SYSTeMS Jasper De Bock Gert de Cooman

5 Research group SYSTeMS Jasper De Bock Gert de Cooman Filip Hermans Erik Quaeghebeur Keivan Shariatmadar Arthur Van Camp

6 State sequence prediction in imprecise hidden Markov models

7 State sequence prediction in imprecise hidden Markov models The imprecise hidden Markov model

8 Jasper De Bock 8 Imprecise hidden Markov model A sequence of hidden state variables X 1 X 2 X 3 O O O S 1 (O 1 X 1 ) S 2 (O 2 X 2 ) S 3 (O 3 X 3 ) A sequence of observable variables

9 Jasper De Bock 9 Imprecise hidden Markov model Q 1 (X 1 ) A sequence Qof hidden 2 (X 2 Xstate 1 ) variables Q 2 (X 3 X 2 ) X 1 X 2 X 3 O O O S 1 (O 1 X 1 ) S 2 (O 2 X 2 ) S 3 (O 3 X 3 ) A sequence of observable variables

10 Jasper De Bock 10 Imprecise hidden Markov model Q 1 (X 1 ) Q 2 (X 2 X 1 ) Q 2 (X 3 X 2 ) X 1 X 2 X 3 All local models are coherent lower previsions O O O S 1 (O 1 X 1 ) S 2 (O 2 X 2 ) S 3 (O 3 X 3 )

11 Jasper De Bock 11 Imprecise hidden Markov model Q 1 (X 1 ) Q 2 (X 2 X 1 ) Q 2 (X 3 X 2 ) X 1 X 2 X 3 O O O S 1 (O 1 X 1 ) S 2 (O 2 X 2 ) S 3 (O 3 X 3 )

12 State sequence prediction in imprecise hidden Markov models Epistemic Irrelevance

13 Epistemic irrelevance X 1 X 2 X 3 O O O Conditional on its mother variable, the non-parent nondescendants of any variable in the tree are epistemically irrelevant to this variable and its descendants Jasper De Bock 13

14 State sequence prediction in imprecise hidden Markov models Recursive construction of a joint model for the imprecise hidden Markov model

15 Recursive construction of a joint model

16 Recursive construction of a joint model Marginal extension Independent natural extension

17 Recursive construction of a joint model Marginal extension Independent natural extension

18 State sequence prediction in imprecise hidden Markov models Conditioning the model on the observations

19 State sequence prediction in imprecise hidden Markov models Conditioning the model on the observations Generalised Bayes rule: An extension of the Bayes rule to imprecise probabilities

20 State sequence prediction in imprecise hidden Markov models Maximal state sequences

21 We predict the state sequence by calculating a set of optimal sequences Notion of optimality: maximality Strict partial ordening: Maximal state sequences: State sequence prediction in imprecise hidden Markov models Maximal state sequences

22 We predict the state sequence by calculating a set of optimal sequences Notion of optimality: maximality Strict partial ordening: Maximal state sequences: State sequence prediction in imprecise hidden Markov models Maximal state sequences

23 State sequence prediction in imprecise hidden Markov models EstiHMM: an efficient algorithm to determine the maximal state sequences in an imprecise hidden Markov model

24 State sequence prediction in imprecise hidden Markov models EstiHMM: an efficient algorithm to determine the maximal state sequences in an imprecise hidden Markov model

25 EstiHMM: an efficient algorithm to determine the maximal sequences

26 EstiHMM: an efficient algorithm to determine the maximal sequences Principle of optimality

27 EstiHMM: an efficient algorithm to determine the maximal sequences Principle of optimality X 2 X 3 O 2 O 3

28 EstiHMM: an efficient algorithm to determine the maximal sequences Principle of optimality X 1 X 2 X 3 O 1 O 2 O 3

29 EstiHMM: an efficient algorithm to determine the maximal sequences Principle of optimality Deriving an alternative Roptimality criterion

30 EstiHMM: an efficient algorithm to determine the maximal sequences Principle of optimality Deriving an alternative Roptimality criterion

31 EstiHMM: an efficient algorithm to determine the maximal sequences Principle of optimality Deriving an alternative Roptimality criterion A recursive approach

32 EstiHMM: an efficient algorithm to determine the maximal sequences Principle of optimality Deriving an alternative Roptimality criterion A recursive approach Complexity

33 EstiHMM: an efficient algorithm to determine the maximal sequences Principle of optimality Deriving an alternative Roptimality criterion A recursive approach Complexity Theoretical analysis

34 EstiHMM: an efficient algorithm to determine the maximal sequences Principle of optimality Deriving an alternative Roptimality criterion A recursive approach Complexity Theoretical analysis Linear in the number of Rmaximal sequences

35 EstiHMM: an efficient algorithm to determine the maximal sequences Principle of optimality Deriving an alternative Roptimality criterion A recursive approach Complexity Theoretical analysis Linear in the number of Rmaximal sequences Empirical confirmation

36 State sequence prediction in imprecise hidden Markov models A first experiment

37 A first experiment

38 A first experiment

39 A first experiment See you at the poster session!