Computational Methods for Hidden Markov Tree Models—An Application to Wavelet Trees

Durand, Jean-Baptiste; Gonçalvès, Paulo; Guédon, Yann

doi:10.1109/tsp.2004.832006

“…We have established that for in the interior of MARG , the value of the conjugate dual is given by the negative entropy of the distribution , where the pair and are dually coupled via (12). Moreover, it can be shown [23] that when lies outside the closure of the marginal polytope, then the value of the dual function is .…”

Section: B Conjugate Dual Functionmentioning

confidence: 94%

“…Therefore, if there exists a solution to the equation , then the supremum (9) is attained at this point. Accordingly, we compute the gradient using (10) and set it equal to zero (12) It can be shown [23] that this equation has a unique solution whenever belongs to the interior of MARG . Substituting the relation into the definition (9) of yields, for any in the interior of MARG , the important relation .…”

Section: B Conjugate Dual Functionmentioning

confidence: 99%

See 1 more Smart Citation

Log-determinant relaxation for approximate inference in discrete Markov random fields

Wainwright

¹

,

Jordan

²

2006

IEEE Trans. Signal Process.

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Abstract-Graphical models are well suited to capture the complex and non-Gaussian statistical dependencies that arise in many real-world signals. A fundamental problem common to any signal processing application of a graphical model is that of computing approximate marginal probabilities over subsets of nodes. This paper proposes a novel method, applicable to discrete-valued Markov random fields (MRFs) on arbitrary graphs, for approximately solving this marginalization problem. The foundation of our method is a reformulation of the marginalization problem as the solution of a low-dimensional convex optimization problem over the marginal polytope. Exactly solving this problem for general graphs is intractable; for binary Markov random fields, we describe how to relax it by using a Gaussian bound on the discrete entropy and a semidefinite outer bound on the marginal polytope. This combination leads to a log-determinant maximization problem that can be solved efficiently by interior point methods, thereby providing approximations to the exact marginals. We show how a slightly weakened log-determinant relaxation can be solved even more efficiently by a dual reformulation. When applied to denoising problems in a coupled mixture-of-Gaussian model defined on a binary MRF with cycles, we find that the performance of this log-determinant relaxation is comparable or superior to the widely used sum-product algorithm over a range of experimental conditions.

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“…The wavelet HMT model is directional in that information from longer time horizons directly influences shorter time horizons, but not vice-versa. We impose the following five properties on the structure of our wavelet HMT model (Durand and Gonçalvès 2001): 1. The wavelet coefficient W is modeled by a mixture distribution with probability density function…”

Section: Introductionmentioning

confidence: 99%

Asymmetry of information flow between volatilities across time scales

Gençay

¹

,

Gradojević

²

,

Selçuk

³

et al. 2010

Quantitative Finance

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Conventional time series analysis, focusing exclusively on a time series at a given scale, lacks the ability to explain the nature of the data generating process. A process equation that successfully explains daily price changes, for example, is unable to characterize the nature of hourly price changes. On the other hand, statistical properties of monthly price changes are often not fully covered by a model based on daily price changes. In this paper, we simultaneously model regimes of volatilities at multiple time scales through wavelet-domain hidden Markov models.We establish an important stylized property of volatility across different time scales. We call this property asymmetric vertical dependence. It is asymmetric in the sense that a low volatility state (regime) at a long time horizon is most likely followed by low volatility states at shorter time horizons. On the other hand, a high volatility state at long time horizons does not necessarily imply a high volatility state at shorter time horizons.Our analysis provides evidence that volatility is a mixture of high and low volatility regimes, resulting in a distribution that is non-Gaussian. This result has important implications regarding the scaling behavior of volatility, and consequently, the calculation of risk at different time scales.

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“…Durand and Gonçalvès (2001) comment that the directions of the directed acyclic graph (DAG) for the model of Crouse et al (1998) is not necessary according to a paper by Smyth et al (1996). In this paper, one can drop all directions in the graph and the conditional independence statements would still be valid.…”

Section: Introductionmentioning

confidence: 99%

Asymmetry of Information Flow between Volatilities Across Time Scales

Gençay

¹

,

Selçuk

²

,

Gradojević

³

et al. 2003

View full text Add to dashboard Cite

Conventional time series analysis, focusing exclusively on a time series at a given scale, lacks the ability to explain the nature of the data generating process. A process equation that successfully explains daily price changes, for example, is unable to characterize the nature of hourly price changes. On the other hand, statistical properties of monthly price changes are often not fully covered by a model based on daily price changes. In this paper, we simultaneously model regimes of volatilities at multiple time scales through wavelet-domain hidden Markov models.We establish an important stylized property of volatility across different time scales. We call this property asymmetric vertical dependence. It is asymmetric in the sense that a low volatility state (regime) at a long time horizon is most likely followed by low volatility states at shorter time horizons. On the other hand, a high volatility state at long time horizons does not necessarily imply a high volatility state at shorter time horizons.Our analysis provides evidence that volatility is a mixture of high and low volatility regimes, resulting in a distribution that is non-Gaussian. This result has important implications regarding the scaling behavior of volatility, and consequently, the calculation of risk at different time scales.

show abstract

Computational Methods for Hidden Markov Tree Models—An Application to Wavelet Trees

Cited by 58 publications

References 18 publications

Log-determinant relaxation for approximate inference in discrete Markov random fields

Log-determinant relaxation for approximate inference in discrete Markov random fields

Asymmetry of information flow between volatilities across time scales

Asymmetry of Information Flow between Volatilities Across Time Scales

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