Tiago M. Montanher Walter F. Mascarenhas · Tiago M. Montanher Walter F. Mascarenhas Global...
Transcript of Tiago M. Montanher Walter F. Mascarenhas · Tiago M. Montanher Walter F. Mascarenhas Global...
Tiago M. Montanher
Walter F. Mascarenhas
Global Estimation of
Hidden Markov Models
using Interval Arithmetic
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OutlineSC
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14• The Problem
• Statement
• Objectives
• Interval arithmetic algorithm
• Interval branch and bound
• Evaluating the objective function
• Discarding boxes
• Numerical experiments
• Plotting solution sets
• Basins of attraction
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What is a Hidden Markov Model?SC
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High Low InitialState
High 70% 30% 60%
Low 50% 50% 40%
Increase 30% 60%
Unchanged 50% 30%
Decrease 20% 10%
Dilma’s voting intention
Petrobras Stock value PETR4($)
We observe only the Market performance
(Increase, Decrease, Unchanged, Decrease....)3
Hidden Markov ModelsSC
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Problem StatementSC
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Computing ProbabilitiesSC
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ObjectivesSC
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14State of the Art
Our Aims
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Baum-Welch AlgorithmSC
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Interval Branch and BoundSC
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Computing probabilities – Interval ArithmeticSC
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Computing probabilities – Interval ArithmeticSC
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KKT ConditionsSC
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Discarding BoxesSC
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Discarding BoxesSC
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Implementation detailsSC
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Numerical ExperimentsSC
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Solution set for all instances with N = 2, M = 2 and
T = 8 (254 instances)16
Numerical ExperimentsSC
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Solution set for all instances with N = 2, M = 2 and
T = 9 (510 instances)17
Numerical ExperimentsSC
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Solution set for all instances with N = 2, M = 2 and
T = 10 (1022 instances)18
Basins of AttractionSC
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Basins of Attraction – N = 2, M = 2 and O = 0000011001
Global Solution Global Solution
Local Solution
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Basins of AttractionSC
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Global Solution Global Solution
Local Solution
Basins of Attraction – N = 2, M = 2 and O = 1001000100001001000101111
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Final RemarksSC
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• We provide an alternative to estimate parameters of
Hidden Markov Models.
• We derive a new interval extension and discarding
tests based on the structure of the problem.
• Only few models can be correctly predicted when we
have a small set of observations.
• The Baum Welch algorithm does not find a global
maximum likelihood estimator for more than 50% of
initial points.
Questions?SC
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