78,026 research outputs found
An Algorithm for Pattern Discovery in Time Series
We present a new algorithm for discovering patterns in time series and other
sequential data. We exhibit a reliable procedure for building the minimal set
of hidden, Markovian states that is statistically capable of producing the
behavior exhibited in the data -- the underlying process's causal states.
Unlike conventional methods for fitting hidden Markov models (HMMs) to data,
our algorithm makes no assumptions about the process's causal architecture (the
number of hidden states and their transition structure), but rather infers it
from the data. It starts with assumptions of minimal structure and introduces
complexity only when the data demand it. Moreover, the causal states it infers
have important predictive optimality properties that conventional HMM states
lack. We introduce the algorithm, review the theory behind it, prove its
asymptotic reliability, use large deviation theory to estimate its rate of
convergence, and compare it to other algorithms which also construct HMMs from
data. We also illustrate its behavior on an example process, and report
selected numerical results from an implementation.Comment: 26 pages, 5 figures; 5 tables;
http://www.santafe.edu/projects/CompMech Added discussion of algorithm
parameters; improved treatment of convergence and time complexity; added
comparison to older method
On Hidden States in Quantum Random Walks
It was recently pointed out that identifiability of quantum random walks and
hidden Markov processes underlie the same principles. This analogy immediately
raises questions on the existence of hidden states also in quantum random walks
and their relationship with earlier debates on hidden states in quantum
mechanics. The overarching insight was that not only hidden Markov processes,
but also quantum random walks are finitary processes. Since finitary processes
enjoy nice asymptotic properties, this also encourages to further investigate
the asymptotic properties of quantum random walks. Here, answers to all these
questions are given. Quantum random walks, hidden Markov processes and finitary
processes are put into a unifying model context. In this context, quantum
random walks are seen to not only enjoy nice ergodic properties in general, but
also intuitive quantum-style asymptotic properties. It is also pointed out how
hidden states arising from our framework relate to hidden states in earlier,
prominent treatments on topics such as the EPR paradoxon or Bell's
inequalities.Comment: 26 page
Asymptotic operating characteristics of an optimal change point detection in hidden Markov models
Let \xi_0,\xi_1,...,\xi_{\omega-1} be observations from the hidden Markov
model with probability distribution P^{\theta_0}, and let
\xi_{\omega},\xi_{\omega+1},... be observations from the hidden Markov model
with probability distribution P^{\theta_1}. The parameters \theta_0 and
\theta_1 are given, while the change point \omega is unknown. The problem is to
raise an alarm as soon as possible after the distribution changes from
P^{\theta_0} to P^{\theta_1}, but to avoid false alarms. Specifically, we seek
a stopping rule N which allows us to observe the \xi's sequentially, such that
E_{\infty}N is large, and subject to this constraint, sup_kE_k(N-k|N\geq k) is
as small as possible. Here E_k denotes expectation under the change point k,
and E_{\infty} denotes expectation under the hypothesis of no change whatever.
In this paper we investigate the performance of the Shiryayev-Roberts-Pollak
(SRP) rule for change point detection in the dynamic system of hidden Markov
models. By making use of Markov chain representation for the likelihood
function, the structure of asymptotically minimax policy and of the Bayes rule,
and sequential hypothesis testing theory for Markov random walks, we show that
the SRP procedure is asymptotically minimax in the sense of Pollak [Ann.
Statist. 13 (1985) 206-227]. Next, we present a second-order asymptotic
approximation for the expected stopping time of such a stopping scheme when
\omega=1. Motivated by the sequential analysis in hidden Markov models, a
nonlinear renewal theory for Markov random walks is also given.Comment: Published at http://dx.doi.org/10.1214/009053604000000580 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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