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

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

The Role of Biofluid Mechanics in the Assessment of Clinical and Pathological Observations: Sixth International Bio-Fluid Mechanics Symposium and Workshop, March 28–30, 2008 Pasadena, California

Biofluid mechanics is increasingly applied in support of diagnosis and decision-making for treatment of clinical pathologies. Exploring the relationship between blood flow phenomena and pathophysiological observations is enhanced by continuing advances in the imaging modalities, measurement techniques, and capabilities of computational models. When combined with underlying physiological models, a powerful set of tools becomes available to address unmet clinical needs, predominantly in the direction of enhanced diagnosis, as well as assessment and prediction of treatment outcomes. This position paper presents an overview of current approaches and future developments along this theme that were discussed at the 5th International Biofluid Symposium and Workshop held at the California Institute of Technology in 2008. The introduction of novel mechanical biomarkers in device design and optimization, and applications in the characterization of more specific and focal conditions such as aneurysms, are at the center of attention. Further advances in integrative modeling, incorporating multiscale and multiphysics techniques are also discussed

Priors on network structures. Biasing the search for Bayesian networks

AbstractIn this paper we show how a user can influence recovery of Bayesian networks from a database by specifying prior knowledge. The main novelty of our approach is that the user only has to provide partial prior knowledge, which is then completed to a full prior over all possible network structures. This partial prior knowledge is expressed among variables in an intuitive pairwise way, which embodies the uncertainty of the user about his/her own prior knowledge. Thus, the uncertainty of the model is updated in the normal Bayesian way

Complex Data: Mining using Patterns

There is a growing need to analyse sets of complex data, i.e., data in which the individual data items are (semi-) structured collections of data themselves, such as sets of time-series. To perform such analysis, one has to redefine familiar notions such as similarity on such complex data types. One can do that either on the data items directly, or indi- rectly, based on features or patterns computed from the individual data items. In this paper, we argue that wavelet decomposition is a general tool for the latter approac

Wavelet transform in similarity paradigm

[INS-R9802] Searching for similarity in time series finds still broader applications in data mining. However, due to the very broad spectrum of data involved, there is no possibility of defining one single notion of similarity suitable to serve all applications. We present a powerful framework based on wavelet decomposition, which allows designing and implementing a variety of criteria for the evaluation of similarity between time series. As an example, two main classes of similarity measures are considered. One is the global, statistical similarity which uses the wavelet transform derived Hurst exponent to classify time series according to their global scaling properties. The second measure estimates similarity locally using the scale-position bifurcation representation derived from the wavelet transform modulus maxima representation of the time series. A variety of generic or custom designed matching criteria can be incorporated into the detail similarity measure. We demonstrate the ability of the technique to deal with the presence of scaling, translation and polynomial bias and we also test sensitivity to the addition of random noise. Other criteria can be designed and this flexibility can be built into the data mining system to allow for specific user requirements.#[INS-R9815] For the majority of data mining applications, there are no models of data which would facilitate the tasks of comparing records of time series, thus leaving one with `noise' as the only description. We propose a generic approach to comparing noise time series using the largest deviations from consistent statistical behaviour. For this purpose we use a powerful framework based on wavelet decomposition, which allows filtering polynomial bias, while capturing the essential singular behaviour. In particular we are able to reveal scale-wise ranking of singular events including their scale-free characteristic: the Hölder exponent. We use such characteristics to design a compact representation of the time series suitable for direct comparison, e.g. evaluation of the correlation product. We demonstrate that the distance between such representations closely corresponds to the subjective feeling of similarity between the time series. In order to test the validity of subjective criteria, we test the records of currency exchanges, finding convincing levels of (local) correlation

The Haar Wavelet Transform in the Time Series Similarity Paradigm

Similarity measures play an important role in many data mining algorithms. To allow the use of such algorithms on non-standard databases, such as databases of financial time series, their similarity measure has to be defined. We present a simple and powerful technique which allows for the rapid evaluation of similarity between time series in large data bases. It is based on the orthonormal decomposition of the time series into the Haar basis. We demonstrate that this approach is capable of providing estimates of the local slope of the time series in the sequence of multi-resolution steps. The Haar representation and a number of related represenations derived from it are suitable for direct comparison, e.g. evaluation of the correlation product. We demonstrate that the distance between such representations closely corresponds to the subjective feeling of similarity between the time series. In order to test the validity of subjective criteria, we test the records of currency exchanges, finding convincing levels of correlation