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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