Efficient computation of updated lower expectations for imprecise continuous-time hidden Markov chains

We consider the problem of performing inference with imprecise continuous-time hidden Markov chains, that is, imprecise continuous-time Markov chains that are augmented with random output variables whose distribution depends on the hidden state of the chain. The prefix `imprecise' refers to the fact that we do not consider a classical continuous-time Markov chain, but replace it with a robust extension that allows us to represent various types of model uncertainty, using the theory of imprecise probabilities. The inference problem amounts to computing lower expectations of functions on the state-space of the chain, given observations of the output variables. We develop and investigate this problem with very few assumptions on the output variables; in particular, they can be chosen to be either discrete or continuous random variables. Our main result is a polynomial runtime algorithm to compute the lower expectation of functions on the state-space at any given time-point, given a collection of observations of the output variables

Hitting Times and Probabilities for Imprecise Markov Chains

We consider the problem of characterising expected hitting times and hitting probabilities for imprecise Markov chains. To this end, we consider three distinct ways in which imprecise Markov chains have been defined in the literature: as sets of homogeneous Markov chains, as sets of more general stochastic processes, and as game-theoretic probability models. Our first contribution is that all these different types of imprecise Markov chains have the same lower and upper expected hitting times, and similarly the hitting probabilities are the same for these three types. Moreover, we provide a characterisation of these quantities that directly generalises a similar characterisation for precise, homogeneous Markov chains

Hitting times and probabilities for imprecise Markov chains

We consider the problem of characterising expected hitting times and hitting probabilities for imprecise Markov chains. To this end, we consider three distinct ways in which imprecise Markov chains have been defined in the literature: as sets of homogeneous Markov chains, as sets of more general stochastic processes, and as game-theoretic probability models. Our first contribution is that all these different types of imprecise Markov chains have the same lower and upper expected hitting times, and similarly the hitting probabilities are the same for these three types. Moreover, we provide a characterisation of these quantities that directly generalises a similar characterisation for precise, homogeneous Markov chains

Imprecise Continuous-Time Markov Chains

Continuous-time Markov chains are mathematical models that are used to describe the state-evolution of dynamical systems under stochastic uncertainty, and have found widespread applications in various fields. In order to make these models computationally tractable, they rely on a number of assumptions that may not be realistic for the domain of application; in particular, the ability to provide exact numerical parameter assessments, and the applicability of time-homogeneity and the eponymous Markov property. In this work, we extend these models to imprecise continuous-time Markov chains (ICTMC's), which are a robust generalisation that relaxes these assumptions while remaining computationally tractable. More technically, an ICTMC is a set of "precise" continuous-time finite-state stochastic processes, and rather than computing expected values of functions, we seek to compute lower expectations, which are tight lower bounds on the expectations that correspond to such a set of "precise" models. Note that, in contrast to e.g. Bayesian methods, all the elements of such a set are treated on equal grounds; we do not consider a distribution over this set. The first part of this paper develops a formalism for describing continuous-time finite-state stochastic processes that does not require the aforementioned simplifying assumptions. Next, this formalism is used to characterise ICTMC's and to investigate their properties. The concept of lower expectation is then given an alternative operator-theoretic characterisation, by means of a lower transition operator, and the properties of this operator are investigated as well. Finally, we use this lower transition operator to derive tractable algorithms (with polynomial runtime complexity w.r.t. the maximum numerical error) for computing the lower expectation of functions that depend on the state at any finite number of time points

Bridge over Troubled Water: Linking Capacities of Sport and Non-Sport Organizations

Community Sport Development Programs (CSDPs) that use an intersectoral capacity building approach have shown potential in reaching individuals in disadvantaged situations. This study has investigated how the application of capacity building principles in disadvantaged communities results in higher sport participation rates in these communities. A multiple case design was used, including six similar disadvantaged communities in Antwerp, Belgium; four communities implemented the CSDP, two communities served as control communities without CSDP. In total, 52 face-to-face interviews were held with sport, social, health, cultural, and youth organizations in these communities. Four key findings were crucial to explain the success of the CSDP according to the principles of capacity building. First, the CSDP appeared to be the missing link between sport organizations on the one hand and health, social, youth, and cultural organizations on the other hand. Second, shifting from a sport-oriented staff to a mix of sport staff, social workers and representatives of people in disadvantaged situations helped increase trust through a participatory approach. Third, CSDPs assisted sport clubs to deal with financial, organizational, and cultural pressures that arose from the influx of new members in disadvantaged situations. Finally, the CSDPs developed well-planned and integrated strategies focusing on reinforcing the existing local organizations already using sport to reach their goals. These capacity building principles were key in attaining higher sport participation for people living in disadvantaged communities

A new search for distant radio galaxies in the Southern hemisphere -- III. Optical spectroscopy and analysis of the MRCR--SUMSS sample

We have compiled a sample of 234 ultra-steep-spectrum(USS)-selected radio sources in order to find high-redshift radio galaxies (HzRGs). The sample is in the southern sky at -40 deg < DEC < -30 deg which is the overlap region of the 408-MHz Revised Molonglo Reference Catalogue, 843-MHz Sydney University Molonglo Sky Survey (the MRCR--SUMSS sample) and the 1400-MHz NRAO VLA Sky Survey. This is the third in a series of papers on the MRCR--SUMSS sample. Here we present optical spectra from the ANU 2.3-m telescope, ESO New Technology Telescope and ESO Very Large Telescope for 52 of the identifications from Bryant et al. (2009, Paper II), yielding redshifts for 36 galaxies, 13 of which have z>2. We analyse the K-z distribution and compare 4-arcsec-aperture magnitudes with 64-kpc aperture magnitudes in several surveys from the literature; the MRCR--SUMSS sample is found to be consistent with models for 10^{11}-10^{12} solar mass galaxies. Dispersions about the fits in the K-z plot support passive evolution of radio galaxy hosts since z>3. By comparing USS-selected samples in the literature, we find that the resultant median redshift of the samples shown is not dependent on the flux density distribution or selection frequency of each sample. In addition, our finding that the majority of the radio spectral energy distributions remain straight over a wide frequency range suggests that a k-correction is not responsible for the success of USS-selection in identifying high redshift radio galaxies and therefore the steep radio spectra may be intrinsic to the source or a product of the environment. Two galaxies have been found to have both compact radio structures and strong self-absorption in the Ly-alpha line, suggesting they are surrounded by a dense medium...abridged.Comment: Accepted for MNRAS. 25 page

Two-state imprecise Markov chains for statistical modelling of two-state non-Markovian processes.

This paper proposes a method for fitting a two-state imprecise Markov chain to time series data from a twostate non-Markovian process. Such non-Markovian processes are common in practical applications. We focus on how to fit modelling parameters based on data from a process where time to transition is not exponentially distributed, thereby violating the Markov assumption. We do so by first fitting a many-state (i.e. having more than two states) Markov chain to the data, through its associated phase-type distribution. Then, we lump the process to a two-state imprecise Markov chain. In practical applications, a two-state imprecise Markov chain might be more convenient than a many-state Markov chain, as we have closed analytic expressions for typical quantities of interest (including the lower and upper expectation of any function of the state at any point in time). A numerical example demonstrates how the entire inference process (fitting and prediction) can be done using Markov chain Monte Carlo, for a given set of prior distributions on the parameters. In particular, we numerically identify the set of posterior densities and posterior lower and upper expectations on all model parameters and predictive quantities. We compare our inferences under a range of sample sizes and model assumptions. Keywords: imprecise Markov chain, estimation, reliability, Markov assumption, MCM

Pattern formation in directional solidification under shear flow. I: Linear stability analysis and basic patterns

An asymptotic interface equation for directional solidification near the absolute stabiliy limit is extended by a nonlocal term describing a shear flow parallel to the interface. In the long-wave limit considered, the flow acts destabilizing on a planar interface. Moreover, linear stability analysis suggests that the morphology diagram is modified by the flow near the onset of the Mullins-Sekerka instability. Via numerical analysis, the bifurcation structure of the system is shown to change. Besides the known hexagonal cells, structures consisting of stripes arise. Due to its symmetry-breaking properties, the flow term induces a lateral drift of the whole pattern, once the instability has become active. The drift velocity is measured numerically and described analytically in the framework of a linear analysis. At large flow strength, the linear description breaks down, which is accompanied by a transition to flow-dominated morphologies, described in a companion paper. Small and intermediate flows lead to increased order in the lattice structure of the pattern, facilitating the elimination of defects. Locally oscillating structures appear closer to the instability threshold with flow than without.Comment: 20 pages, Latex, accepted for Physical Review