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Register a new userThe Levy State Space Model
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Marina Riabiz Ms.
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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In this paper we introduce a new class of state space models based on shot-noise simulation representations of non-Gaussian Levy-driven linear systems, represented as stochastic differential equations. In particular a conditionally Gaussian version of the models is proposed that is able to capture heavy-tailed non-Gaussianity while retaining tractability for inference procedures. We focus on a canonical class of such processes, the $alpha$-stable Levy processes, which retain important properties such as self-similarity and heavy-tails, while emphasizing that broader classes of non-Gaussian Levy processes may be handled by similar methodology. An important feature is that we are able to marginalise both the skewness and the scale parameters of these challenging models from posterior probability distributions. The models are posed in continuous time and so are able to deal with irregular data arrival times. Example modelling and inference procedures are provided using Rao-Blackwellised sequential Monte Carlo applied to a two-dimensional Langevin model, and this is tested on real exchange rate data.
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Quantifying the impact of parametric and model-form uncertainty on the predictions of stochastic models is a key challenge in many applications. Previous work has shown that the relative entropy rate is an effective tool for deriving path-space uncertainty quantification (UQ) bounds on ergodic averages. In this work we identify appropriate information-theoretic objects for a wider range of quantities of interest on path-space, such as hitting times and exponentially discounted observables, and develop the corresponding UQ bounds. In addition, our method yields tighter UQ bounds, even in cases where previous relative-entropy-based methods also apply, e.g., for ergodic averages. We illustrate these results with examples from option pricing, non-reversible diffusion processes, stochastic control, semi-Markov queueing models, and expectations and distributions of hitting times.
An alternative to current mainstream preprocessing methods is proposed: Value Selection (VS). Unlike the existing methods such as feature selection that removes features and instance selection that eliminates instances, value selection eliminates the values (with respect to each feature) in the dataset with two purposes: reducing the model size and preserving its accuracy. Two probabilistic methods based on information theorys metric are proposed: PVS and P + VS. Extensive experiments on the benchmark datasets with various sizes are elaborated. Those results are compared with the existing preprocessing methods such as feature selection, feature transformation, and instance selection methods. Experiment results show that value selection can achieve the balance between accuracy and model size reduction.
Consider a spectrally positive Stable($1+alpha$) process whose jumps we interpret as lifetimes of individuals. We mark the jumps by continuous excursions assigning sizes varying during the lifetime. As for Crump-Mode-Jagers processes (with characteristics), we consider for each level the collection of individuals alive. We arrange their sizes at the crossing height from left to right to form an interval partition. We study the continuity and Markov properties of the interval-partition-valued process indexed by level. From the perspective of the Stable($1+alpha$) process, this yields new theorems of Ray-Knight-type. From the perspective of branching processes, this yields new, self-similar models with dense sets of birth and death times of (mostly short-lived) individuals. This paper feeds into projects resolving conjectures by Feng and Sun (2010) on the existence of certain measure-valued diffusions with Poisson--Dirichlet stationary laws, and by Aldous (1999) on the existence of a continuum-tree-valued diffusion.
Consider a stationary real-valued time series ${X_n}_{n=0}^{infty}$ with a priori unknown distribution. The goal is to estimate the conditional expectation $E(X_{n+1}|X_0,..., X_n)$ based on the observations $(X_0,..., X_n)$ in a pointwise consistent way. It is well known that this is not possible at all values of $n$. We will estimate it along stopping times.
This paper introduces a generalization of the so-called space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form. It turns out that this generalization can be put in relation to a specific subordination of a homogeneous Poisson process by means of a subordinator for which it is possible to express the characterizing Levy measure explicitly. Moreover, the law of this subordinator solves a one-sided first-order differential equation in which a particular convolution-type integral operator appears, called Prabhakar derivative. In the last section of the paper, a similar model is introduced in which the Prabhakar derivative also acts in time. In this case, too, the probability generating function of the corresponding process and the probability distribution are determined.
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