** Case study background and problem formulations**

Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, and PSG R.

**PROBLEM1: Problem_HMM_Discrete**

Maximize Hmm_discrete

subject to

Linear= 1 (sum of probabilitiesof initial states constraint)

Linearmulti = 1 (sum of probabilities of transitions constraints)

Linearmulti = 1 (sum of probabilities of observations constraints)

Box constraints (lower bounds on probabilities)

——————————————————————–

Hmm_discrete = Log-likelihood function for Hidden Markov Model with discrete distributions of observations

Linear = linear function

Linearmulti = system of linear functions

Box constraints = constraints on individual decision variables

——————————————————————–

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset 1 | 14 | N/A | -196.56695 | 0.01 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab | Matlab Code | Data | |||||

R | R Code | Data |

**PROBLEM2: problem_HMM_normal**

Maximize Hmm_normal

subject to

Linear = 1 (sum of probabilities of initial states constraint)

Linearmulti = 1 (sum of probabilities of transitions constraints)

Box constraints (lower bounds on probabilities)

——————————————————————–

Hmm_normal = Log-likelihood function for Hidden Markov Model with normal distributions of observations

Linear = linear function

Linearmulti = system of linear functions

Box constraints = constraints on individual decision variables

——————————————————————–

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset 1 | 12 | N/A | 204.8211262 | 0.02 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab | Matlab Code | Data | |||||

R | R Code | Data |

**CASE STUDY SUMMARY**

This case study considers two variants of Hidden Markov Model. One with discrete distributions of observations and other with normal distributions of observations. Correspondently two Problem statements for maximization of Log-Lokelihood function in Hidden Markov Model are shown.

For maximization type of problem PSG uses an expectation modificaion (EM) procedure in form of Baum–Welch algorithm to find good initial point.

hmm_discrete and hmm_normal functions report probabilities of initial states, transition probabilities and probabilities of observations or parameters of normal distributions. Additionally they report Viterbi states vector.

References

References

• Lawrence R. Rabiner (1989).A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of IEEE, 77-2, p. 267-295.

http://www.ece.ucsb.edu/Faculty/Rabiner/ece259/Reprints/tutorial%20on%20hmm%20and%20applications.pdf