Fast State Discovery for HMM Model Selection and Learning Sajid M. Siddiqi Geoffrey J. Gordon Andrew W. Moore CMU

2 t OtOt Consider a sequence of real-valued observations (speech, sensor readings, stock prices …)

3 t We can model it purely based on contextual properties OtOt

4 t Consider a sequence of real-valued observations (speech, sensor readings, stock prices …) We can model it purely based on contextual properties OtOt

5 t Consider a sequence of real-valued observations (speech, sensor readings, stock prices …) We can model it purely based on contextual properties However, we would miss important temporal structure OtOt

6 t Consider a sequence of real-valued observations (speech, sensor readings, stock prices …) We can model it purely based on contextual properties However, we would miss important temporal structure OtOt

7 t Current efficient approaches learn the wrong model OtOt

8 t OtOt Our method successfully discovers the overlapping states

9 t Our goal: Efficiently discover states in sequential data while learning a Hidden Markov Model OtOt

10 Motion Capture

11 Definitions and Notation An HMM is ={A,B,  } where A : N  N transition matrix B : observation model {  s,  s } for each of N states  : N  1 prior probability vector T : size of observation sequence O 1,…,O T q t : the state the HMM is in at time t. q t  {s 1,…,s N }

12 ProblemAlgorithm Complexity Likelihood evaluation: L(  O) = P(O| ) Forward- Backward O(TN 2 ) Path inference : Q * = argmax Q P(O,Q| ) Viterbi O(TN 2 ) Parameter learning: *  argmax,Q P(O,Q|  *  argmax P(O|  (for fixed N) Viterbi Training Baum-Welch (EM) O(TN 2 ) Operations on HMMs

13 ProblemAlgorithm Complexity Likelihood evaluation: L(  O) = P(O| ) Forward- Backward O(TN 2 ) Path inference : Q * = argmax Q P(O,Q| ) Viterbi O(TN 2 ) Parameter learning: *  argmax,Q P(O,Q|  *  argmax P(O|  (for fixed N) Viterbi Training Baum-Welch (EM) O(TN 2 ) Model selection: *  argmax,Q,N P(O,Q|  *  argmax,N P(O|  ?? want O(TN 2 ) Operations on HMMs

14 Previous Approaches Multi-restart Baum-Welch N is inefficient, highly prone to local minima

15 Previous Approaches Multi-restart Baum-Welch N is inefficient, highly prone to local minima Bottom-up State merging [Stolcke & Omohundro 1994] Entropic state pruning [Brand 1999] Advantage: –More robust to local minima Problems: –Require a loose upper bound on N, which adds complexity –Difficult to decide which states to prune/merge

16 Previous Approaches Multi-restart Baum-Welch N is inefficient, highly prone to local minima Bottom-upTop-down State merging [Stolcke & Omohundro 1994] Entropic state pruning [Brand 1999] Advantage: –More robust to local minima Problems: –Require a loose upper bound on N, which adds complexity –Difficult to decide which states to prune/merge ML Successive State Splitting [Ostendorf & Singer 1997] Heuristic split-merge [ Li & Biswas 1999] Advantage: –More robust to local minima, and more scalable Problems: – Previous methods not effective at state discovery, and still slow for large N

17 Previous Approaches Multi-restart Baum-Welch N is inefficient, highly prone to local minima Bottom-upTop-down State merging [Stolcke & Omohundro 1994] Entropic state pruning [Brand 1999] Advantage: –More robust to local minima Problems: –Require a loose upper bound on N, which adds complexity –Difficult to decide which states to prune/merge ML Successive State Splitting [Ostendorf & Singer 1997] Heuristic split-merge [ Li & Biswas 1999] Advantage: –More robust to local minima, and more scalable Problems: – Previous methods not effective at state discovery, and still slow for large N We propose Simultaneous Temporal and Contextual Splitting (STACS) A top-down approach that is much better at state- discovery while being at least as efficient, and a variant V-STACS that is much faster.

18 Bayesian Information Criterion (BIC) for Model Selection - Would like to compute the posterior probability for model selection P(model size|data) / P(data|model size) P(model size) log P(model size|data) / log P(data|model size) + log P(model size)

19 Bayesian Information Criterion (BIC) for Model Selection - Would like to compute the posterior probability for model selection P(model size|data) / P(data|model size) P(model size) log P(model size|data) / log P(data|model size) + log P(model size) - BIC assumes a prior that penalizes complexity (favors smaller models): log P(model size|data) ¼ log P(data|model size, MLE ) – (#FP/2) log T where #FP = number of free parameters, T = length of data sequence, MLE is the ML parameter estimate

20 Bayesian Information Criterion (BIC) for Model Selection - Would like to compute the posterior probability for model selection P(model size|data) / P(data|model size) P(model size) log P(model size|data) / log P(data|model size) + log P(model size) - BIC assumes a prior that penalizes complexity (favors smaller models): log P(model size|data) ¼ log P(data|model size, MLE ) – (#FP/2) log T where #FP = number of free parameters, T = length of data sequence, MLE is the ML parameter estimate - BIC is an asymptotic approximation to the true posterior

21 Algorithm Summary (STACS/VSTACS) Initialize n 0 -state HMM randomly for n = n 0 … Nmax –Learn model parameters –for i = 1 … n Split state i, optimize by constrained EM (STACS) or constrained Viterbi Training (VSTACS) Calculate approximate BIC score of split model –Choose best split based on approximate BIC –Compare to original model with exact BIC (STACS) or approximate BIC (VSTACS) –if larger model not chosen, stop

22 STACS input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat

23 STACS Learn parameters using EM, calculate the Viterbi path Q * input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat S1S1 S2S2

24 STACS Learn parameters using EM, calculate the Viterbi path Q * Consider splits on all states e.g. for state s 2 input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat S1S1 S2S2

25 Learn parameters using EM, calculate the Viterbi path Q * Consider splits on all states e.g. for state s 2 –Choose a subset D = {O t : Q * (t) = s 2 } STACS input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat S1S1 S2S2

26 Learn parameters using EM, calculate the Viterbi path Q * Consider splits on all states e.g. for state s 2 –Choose a subset D = {O t : Q * (t) = s 2 } –Note that | D | = O(T/N) STACS input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat S1S1 S2S2

27 STACS Split the state input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat S1S1 S2S2 S3S3

28 Split the state Constrain  s to except for offspring states’ observation densities and all their transition probabilities, both in and out S1S1 S2S2 S3S3 STACS input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat

29 Split the state Constrain  s to except for offspring states’ observation densities and all their transition probabilities, both in and out Learn the free parameters using two-state EM over D. This optimizes the partially observed likelihood P(O,Q * \ D |  s ) STACS input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat S1S1 S2S2 S3S3

30 Split the state Constrain  s to except for offspring states’ observation densities and all their transition probabilities, both in and out Learn the free parameters using two-state EM over D. This optimizes the partially observed likelihood P(O,Q * \ D |  s ) Update Q * over D to get R * STACS input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat S1S1 S2S2 S3S3

31 Scoring is of two types: STACS input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat

32 Scoring is of two types: The candidates are compared to each other according to their Viterbi path likelihoods STACS input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat S1S1 S2S2 S3S3 vs. S1S1 S2S2 S3S3

33 Scoring is of two types: The candidates are compared to each other according to their Viterbi path likelihoods The best candidate in this ranking is compared to the un-split model  using BIC, i.e. log P(model | data )  log  P(data | model) – complexity penalty STACS input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat S1S1 S2S2 S3S3 vs. S1S1 S2S2 S1S1 S2S2 S3S3 S1S1 S2S2 S3S3

34 Viterbi STACS (V-STACS) input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat

35 Recall that STACS learns the free parameters using two-state EM over D. However, EM also has “winner-take-all” variants Viterbi STACS (V-STACS) input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat S1S1 S2S2 S3S3

36 Recall that STACS learns the free parameters using two-state EM over D. However, EM also has “winner-take-all” variants V-STACS uses two-state Viterbi training over D to learn the free parameters, which uses hard updates vs STACS’ soft updates Viterbi STACS (V-STACS) input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat S1S1 S2S2 S3S3

37 Recall that STACS learns the free parameters using two-state EM over D. However, EM also has “winner-take-all” variants V-STACS uses two-state Viterbi training over D to learn the free parameters, which uses hard updates vs STACS’ soft updates The Viterbi path likelihood is used to approximate the BIC vs. the un-split model in V-STACS Viterbi STACS (V-STACS) input: n 0, data sequence O = {O 1,…,O T } output: HMM of appropriate size  n 0 -state initial HMM repeat optimize  over sequence O choose a subset of states  for each s   design a candidate model  s : choose a relevant subset of sequence O split state s, optimize  s over subset score  s end for if max s (score(  s )) > score( )  best-scoring candidate from {  s } else terminate, return current end if end repeat S1S1 S2S2 S3S3

38 Time Complexity Optimizing N candidates takes – N  O(T) time for STACS – N  O(T/N) time for V-STACS Scoring N candidates takes N  O(T) time  Candidate search and scoring is O(TN) Best-candidate evaluation is –O(TN 2 ) for BIC in STACS –O(TN) for approximate BIC in V-STACS

39 Other Methods Li-Biswas –Generates two candidates splits state with highest-variance merges pair of closest states (rarely chosen)

40 Other Methods Li-Biswas –Generates two candidates splits state with highest-variance merges pair of closest states (rarely chosen) –Optimizes all candidate parameters over entire sequence

41 Other Methods Li-Biswas –Generates two candidates splits state with highest-variance merges pair of closest states (rarely chosen) –Optimizes all candidate parameters over entire sequence ML-SSS –Generates 2 N candidates, splitting each state in two ways

42 Other Methods Li-Biswas –Generates two candidates splits state with highest-variance merges pair of closest states (rarely chosen) –Optimizes all candidate parameters over entire sequence ML-SSS –Generates 2 N candidates, splitting each state in two ways Contextual split: optimizes offspring states’ observation densities with 2-Gaussian mixture EM, assumes offspring connected ``in parallel”

43 Other Methods Li-Biswas –Generates two candidates splits state with highest-variance merges pair of closest states (rarely chosen) –Optimizes all candidate parameters over entire sequence ML-SSS –Generates 2 N candidates, splitting each state in two ways Contextual split: optimizes offspring states’ observation densities with 2-Gaussian mixture EM, assumes offspring connected ``in parallel” Temporal split: optimizes offspring states’ observation densities, self-transitions and mutual transitions with EM, assumes offspring ``in series”

44 Other Methods Li-Biswas –Generates two candidates splits state with highest-variance merges pair of closest states (rarely chosen) –Optimizes all candidate parameters over entire sequence ML-SSS –Generates 2 N candidates, splitting each state in two ways Contextual split: optimizes offspring states’ observation densities with 2-Gaussian mixture EM, assumes offspring connected ``in parallel” Temporal split: optimizes offspring states’ observation densities, self-transitions and mutual transitions with EM, assumes offspring ``in series” –Optimizes split of state s over all timesteps with nonzero posterior probability of being in state s [ i.e. O(T) data points]

45 Results

46 Data sets Australian Sign-Language data collected from 2 Flock 5DT instrumented gloves and Ascension flock-of-birds tracker [Kadous 2002 (available in UCI KDD Archive)] Other data sets obtained from the literature –Robot, MoCap, MLog, Vowel

47 Learning HMMs of Predetermined Size: Scalability Robot data (others similar)

48 Learning HMMs of Predetermined Size: Log-Likelihood Learning a 40-state HMM on Robot data (others similar)

49 Learning HMMs of Predetermined Size Learning 40-state HMMs

50 Model Selection: Synthetic Data Generalize (4 states, T = 1000) to (10 states, T = 10,000)

51 Model Selection: Synthetic Data Generalize (4 states, T = 1000) to (10 states, T = 10,000) Both STACS, VSTACS discovered 10 states and correct underlying transition structure

52 Model Selection: Synthetic Data Generalize (4 states, T = 1000) to (10 states, T = 10,000) Both STACS, VSTACS discovered 10 states and correct underlying transition structure Li-Biswas, ML-SSS failed to find 10-state model 10-state Baum-Welch also failed to find correct observation and transition models, even with 50 restarts!

53 Model Selection: BIC score MoCap data (others similar)

54 Model Selection

55 Sign-language recognition Initial results on sign-language word recognition 95 distinct words, 27 instances each, divided 8:1 Average classification accuracies and HMM sizes: Accuracy final N

56 Modeling motion capture data 35-dimensional data (thanks to Adrien Treuille)

57 Modeling motion capture data Original data:

58 Modeling motion capture data Original data: STACS simulation: (found 235 states)

59 Modeling motion capture data Original data: STACS simulation: Baum-Welch: (found 235 states) (on 235 states)

60 Modeling motion capture data Original data: STACS simulation: Baum-Welch: (found 235 states) (on 235 states) [Video]

61 Discovering Underlying Structure Sparse dynamics - difficult to learn using regular EM STACS smoothly tiles the low-dimensional manifold of observations along with correct dynamic structure

62 Conclusion –A better method for HMM model selection and learning discovers hidden states avoids local minima faster than Baum-Welch – Even when learning HMMs with known size, better to discover states using STACS up to the desired N –Widespread applicability classification, recognition and prediction for real-valued sequential data problems

63

64 t δ  t (1) δ  t (2) δ  t (3)… δ  t (N)

65 t δ  t (1) δ  t (2) δ  t (3)… δ  t (N) 1 2… 3…

66 t δ  t (1) δ  t (2) δ  t (3)… δ  t (N) The Viterbi path is denoted by Suppose we split state N into s 1,s 2

67 t δ  t (1) δ  t (2) δ  t (3)… δ  t (s 1 ) δ  t (s 2 ) ?? ?? ?? ?? The Viterbi path is denoted by Suppose we split state N into s 1,s 2

68 t δ  t (1) δ  t (2) δ  t (3)… δ  t (s 1 ) δ  t (s 2 ) The Viterbi path is denoted by Suppose we split state N into s 1,s 2

69 t δ  t (1) δ  t (2) δ  t (3)… δ  t (s 1 ) δ  t (s 2 ) The Viterbi path is denoted by Suppose we split state N into s 1,s 2