304,800 research outputs found
A Bayesian framework for functional time series analysis
The paper introduces a general framework for statistical analysis of
functional time series from a Bayesian perspective. The proposed approach,
based on an extension of the popular dynamic linear model to Banach-space
valued observations and states, is very flexible but also easy to implement in
many cases. For many kinds of data, such as continuous functions, we show how
the general theory of stochastic processes provides a convenient tool to
specify priors and transition probabilities of the model. Finally, we show how
standard Markov chain Monte Carlo methods for posterior simulation can be
employed under consistent discretizations of the data
Computable de Finetti measures
We prove a computable version of de Finetti's theorem on exchangeable
sequences of real random variables. As a consequence, exchangeable stochastic
processes expressed in probabilistic functional programming languages can be
automatically rewritten as procedures that do not modify non-local state. Along
the way, we prove that a distribution on the unit interval is computable if and
only if its moments are uniformly computable.Comment: 32 pages. Final journal version; expanded somewhat, with minor
corrections. To appear in Annals of Pure and Applied Logic. Extended abstract
appeared in Proceedings of CiE '09, LNCS 5635, pp. 218-23
Assessing extrema of empirical principal component functions
The difficulties of estimating and representing the distributions of
functional data mean that principal component methods play a substantially
greater role in functional data analysis than in more conventional
finite-dimensional settings. Local maxima and minima in principal component
functions are of direct importance; they indicate places in the domain of a
random function where influence on the function value tends to be relatively
strong but of opposite sign. We explore statistical properties of the
relationship between extrema of empirical principal component functions, and
their counterparts for the true principal component functions. It is shown that
empirical principal component funcions have relatively little trouble capturing
conventional extrema, but can experience difficulty distinguishing a
``shoulder'' in a curve from a small bump. For example, when the true principal
component function has a shoulder, the probability that the empirical principal
component function has instead a bump is approximately equal to 1/2. We suggest
and describe the performance of bootstrap methods for assessing the strength of
extrema. It is shown that the subsample bootstrap is more effective than the
standard bootstrap in this regard. A ``bootstrap likelihood'' is proposed for
measuring extremum strength. Exploratory numerical methods are suggested.Comment: Published at http://dx.doi.org/10.1214/009053606000000371 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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