7,512 research outputs found
Statistical Physics Analysis of Maximum a Posteriori Estimation for Multi-channel Hidden Markov Models
The performance of Maximum a posteriori (MAP) estimation is studied
analytically for binary symmetric multi-channel Hidden Markov processes. We
reduce the estimation problem to a 1D Ising spin model and define order
parameters that correspond to different characteristics of the MAP-estimated
sequence. The solution to the MAP estimation problem has different operational
regimes separated by first order phase transitions. The transition points for
-channel system with identical noise levels, are uniquely determined by
being odd or even, irrespective of the actual number of channels. We
demonstrate that for lower noise intensities, the number of solutions is
uniquely determined for odd , whereas for even there are exponentially
many solutions. We also develop a semi analytical approach to calculate the
estimation error without resorting to brute force simulations. Finally, we
examine the tradeoff between a system with single low-noise channel and one
with multiple noisy channels.Comment: The paper has been submitted to Journal of Statistical Physics with
submission number JOSS-S-12-0039
Bayesian Structural Inference for Hidden Processes
We introduce a Bayesian approach to discovering patterns in structurally
complex processes. The proposed method of Bayesian Structural Inference (BSI)
relies on a set of candidate unifilar HMM (uHMM) topologies for inference of
process structure from a data series. We employ a recently developed exact
enumeration of topological epsilon-machines. (A sequel then removes the
topological restriction.) This subset of the uHMM topologies has the added
benefit that inferred models are guaranteed to be epsilon-machines,
irrespective of estimated transition probabilities. Properties of
epsilon-machines and uHMMs allow for the derivation of analytic expressions for
estimating transition probabilities, inferring start states, and comparing the
posterior probability of candidate model topologies, despite process internal
structure being only indirectly present in data. We demonstrate BSI's
effectiveness in estimating a process's randomness, as reflected by the Shannon
entropy rate, and its structure, as quantified by the statistical complexity.
We also compare using the posterior distribution over candidate models and the
single, maximum a posteriori model for point estimation and show that the
former more accurately reflects uncertainty in estimated values. We apply BSI
to in-class examples of finite- and infinite-order Markov processes, as well to
an out-of-class, infinite-state hidden process.Comment: 20 pages, 11 figures, 1 table; supplementary materials, 15 pages, 11
figures, 6 tables; http://csc.ucdavis.edu/~cmg/compmech/pubs/bsihp.ht
Computing the likelihood of sequence segmentation under Markov modelling
I tackle the problem of partitioning a sequence into homogeneous segments,
where homogeneity is defined by a set of Markov models. The problem is to study
the likelihood that a sequence is divided into a given number of segments.
Here, the moments of this likelihood are computed through an efficient
algorithm. Unlike methods involving Hidden Markov Models, this algorithm does
not require probability transitions between the models. Among many possible
usages of the likelihood, I present a maximum \textit{a posteriori} probability
criterion to predict the number of homogeneous segments into which a sequence
can be divided, and an application of this method to find CpG islands
Forecasting trends with asset prices
In this paper, we consider a stochastic asset price model where the trend is
an unobservable Ornstein Uhlenbeck process. We first review some classical
results from Kalman filtering. Expectedly, the choice of the parameters is
crucial to put it into practice. For this purpose, we obtain the likelihood in
closed form, and provide two on-line computations of this function. Then, we
investigate the asymptotic behaviour of statistical estimators. Finally, we
quantify the effect of a bad calibration with the continuous time mis-specified
Kalman filter. Numerical examples illustrate the difficulty of trend
forecasting in financial time series.Comment: 26 pages, 11 figure
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