7,548 research outputs found
Approximate Bayesian Computation in State Space Models
A new approach to inference in state space models is proposed, based on
approximate Bayesian computation (ABC). ABC avoids evaluation of the likelihood
function by matching observed summary statistics with statistics computed from
data simulated from the true process; exact inference being feasible only if
the statistics are sufficient. With finite sample sufficiency unattainable in
the state space setting, we seek asymptotic sufficiency via the maximum
likelihood estimator (MLE) of the parameters of an auxiliary model. We prove
that this auxiliary model-based approach achieves Bayesian consistency, and
that - in a precise limiting sense - the proximity to (asymptotic) sufficiency
yielded by the MLE is replicated by the score. In multiple parameter settings a
separate treatment of scalar parameters, based on integrated likelihood
techniques, is advocated as a way of avoiding the curse of dimensionality. Some
attention is given to a structure in which the state variable is driven by a
continuous time process, with exact inference typically infeasible in this case
as a result of intractable transitions. The ABC method is demonstrated using
the unscented Kalman filter as a fast and simple way of producing an
approximation in this setting, with a stochastic volatility model for financial
returns used for illustration
The Extended Parameter Filter
The parameters of temporal models, such as dynamic Bayesian networks, may be
modelled in a Bayesian context as static or atemporal variables that influence
transition probabilities at every time step. Particle filters fail for models
that include such variables, while methods that use Gibbs sampling of parameter
variables may incur a per-sample cost that grows linearly with the length of
the observation sequence. Storvik devised a method for incremental computation
of exact sufficient statistics that, for some cases, reduces the per-sample
cost to a constant. In this paper, we demonstrate a connection between
Storvik's filter and a Kalman filter in parameter space and establish more
general conditions under which Storvik's filter works. Drawing on an analogy to
the extended Kalman filter, we develop and analyze, both theoretically and
experimentally, a Taylor approximation to the parameter posterior that allows
Storvik's method to be applied to a broader class of models. Our experiments on
both synthetic examples and real applications show improvement over existing
methods
Auxiliary Likelihood-Based Approximate Bayesian Computation in State Space Models
A computationally simple approach to inference in state space models is
proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation
of an intractable likelihood by matching summary statistics for the observed
data with statistics computed from data simulated from the true process, based
on parameter draws from the prior. Draws that produce a 'match' between
observed and simulated summaries are retained, and used to estimate the
inaccessible posterior. With no reduction to a low-dimensional set of
sufficient statistics being possible in the state space setting, we define the
summaries as the maximum of an auxiliary likelihood function, and thereby
exploit the asymptotic sufficiency of this estimator for the auxiliary
parameter vector. We derive conditions under which this approach - including a
computationally efficient version based on the auxiliary score - achieves
Bayesian consistency. To reduce the well-documented inaccuracy of ABC in
multi-parameter settings, we propose the separate treatment of each parameter
dimension using an integrated likelihood technique. Three stochastic volatility
models for which exact Bayesian inference is either computationally
challenging, or infeasible, are used for illustration. We demonstrate that our
approach compares favorably against an extensive set of approximate and exact
comparators. An empirical illustration completes the paper.Comment: This paper is forthcoming at the Journal of Computational and
Graphical Statistics. It also supersedes the earlier arXiv paper "Approximate
Bayesian Computation in State Space Models" (arXiv:1409.8363
Measures of Analysis of Time Series (MATS): A MATLAB Toolkit for Computation of Multiple Measures on Time Series Data Bases
In many applications, such as physiology and finance, large time series data
bases are to be analyzed requiring the computation of linear, nonlinear and
other measures. Such measures have been developed and implemented in commercial
and freeware softwares rather selectively and independently. The Measures of
Analysis of Time Series ({\tt MATS}) {\tt MATLAB} toolkit is designed to handle
an arbitrary large set of scalar time series and compute a large variety of
measures on them, allowing for the specification of varying measure parameters
as well. The variety of options with added facilities for visualization of the
results support different settings of time series analysis, such as the
detection of dynamics changes in long data records, resampling (surrogate or
bootstrap) tests for independence and linearity with various test statistics,
and discrimination power of different measures and for different combinations
of their parameters. The basic features of {\tt MATS} are presented and the
implemented measures are briefly described. The usefulness of {\tt MATS} is
illustrated on some empirical examples along with screenshots.Comment: 25 pages, 9 figures, two tables, the software can be downloaded at
http://eeganalysis.web.auth.gr/indexen.ht
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