679 research outputs found
The Bayesian Context Trees State Space Model for time series modelling and forecasting
A hierarchical Bayesian framework is introduced for developing rich mixture
models for real-valued time series, along with a collection of effective tools
for learning and inference. At the top level, meaningful discrete states are
identified as appropriately quantised values of some of the most recent
samples. This collection of observable states is described as a discrete
context-tree model. Then, at the bottom level, a different, arbitrary model for
real-valued time series - a base model - is associated with each state. This
defines a very general framework that can be used in conjunction with any
existing model class to build flexible and interpretable mixture models. We
call this the Bayesian Context Trees State Space Model, or the BCT-X framework.
Efficient algorithms are introduced that allow for effective, exact Bayesian
inference; in particular, the maximum a posteriori probability (MAP)
context-tree model can be identified. These algorithms can be updated
sequentially, facilitating efficient online forecasting. The utility of the
general framework is illustrated in two particular instances: When
autoregressive (AR) models are used as base models, resulting in a nonlinear AR
mixture model, and when conditional heteroscedastic (ARCH) models are used,
resulting in a mixture model that offers a powerful and systematic way of
modelling the well-known volatility asymmetries in financial data. In
forecasting, the BCT-X methods are found to outperform state-of-the-art
techniques on simulated and real-world data, both in terms of accuracy and
computational requirements. In modelling, the BCT-X structure finds natural
structure present in the data. In particular, the BCT-ARCH model reveals a
novel, important feature of stock market index data, in the form of an enhanced
leverage effect.Comment: arXiv admin note: text overlap with arXiv:2106.0302
Infinite systems of interacting chains with memory of variable length beyond the Dobrushin condition
We study conditions that allow infinite systems of interacting chains with
memory of variable length to go beyond the usual Dobrushin condition. Then, we
derive an analytical characterization of the invariant state based on Replica
Mean Field limits. As a result we extend the Galves-L\"ocherbach model beyond
the restrictive Dobrushin condition of the model
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