# Case Study: Maximization of Log-Lokelihood in Hidden Markov Model (hmm_discrete, hmm_normal, linear, linearmulti)

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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
——————————————————————–

Dataset 1 14 N/A -196.56695 0.01 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) 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
——————————————————————–

Dataset 1 12 N/A 204.8211262 0.02 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) 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

• 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