495 research outputs found

    Splitting of liftings in products of probability spaces

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    We prove that if (X,\mathfrakA,P) is an arbitrary probability space with countably generated \sigma-algebra \mathfrakA, (Y,\mathfrakB,Q) is an arbitrary complete probability space with a lifting \rho and \hat R is a complete probability measure on \mathfrakA \hat \otimes_R \mathfrakB determined by a regular conditional probability {S_y:y\in Y} on \mathfrakA with respect to \mathfrakB, then there exist a lifting \pi on (X\times Y,\mathfrakA \hat \otimes_R \mathfrakB,\hat R) and liftings \sigma_y on (X,\hat \mathfrakA_y,\hat S_y), y\in Y, such that, for every E\in\mathfrakA \hat \otimes_R \mathfrakB and every y\in Y, [\pi(E)]^y=\sigma_y\bigl([\pi(E)]^y\bigr). Assuming the absolute continuity of R with respect to P\otimes Q, we prove the existence of a regular conditional probability {T_y:y\in Y} and liftings \varpi on (X\times Y,\mathfrakA \hat \otimes_R \mathfrakB,\hat R), \rho' on (Y,\mathfrakB,\hat Q) and \sigma_y on (X,\hat \mathfrakA_y,\hat S_y), y\in Y, such that, for every E\in\mathfrakA \hat \otimes_R \mathfrakB and every y\in Y, [\varpi(E)]^y=\sigma_y\bigl([\varpi(E)]^y\bigr) and \varpi(A\times B)=\bigcup_{y\in\rho'(B)}\sigma_y(A)\times{y}\qquadif A\times B\in\mathfrakA\times\mathfrakB. Both results are generalizations of Musia\l, Strauss and Macheras [Fund. Math. 166 (2000) 281-303] to the case of measures which are not necessarily products of marginal measures. We prove also that liftings obtained in this paper always convert \hat R-measurable stochastic processes into their \hat R-measurable modifications.Comment: Published at http://dx.doi.org/10.1214/009117904000000018 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    The weak Radon-Nikodym property

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    DICE: Deep intelligent contextual embedding for twitter sentiment analysis

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    © 2019 IEEE. The sentiment analysis of the social media-based short text (e.g., Twitter messages) is very valuable for many good reasons, explored increasingly in different communities such as text analysis, social media analysis, and recommendation. However, it is challenging as tweet-like social media text is often short, informal and noisy, and involves language ambiguity such as polysemy. The existing sentiment analysis approaches are mainly for document and clean textual data. Accordingly, we propose a Deep Intelligent Contextual Embedding (DICE), which enhances the tweet quality by handling noises within contexts, and then integrates four embeddings to involve polysemy in context, semantics, syntax, and sentiment knowledge of words in a tweet. DICE is then fed to a Bi-directional Long Short Term Memory (BiLSTM) network with attention to determine the sentiment of a tweet. The experimental results show that our model outperforms several baselines of both classic classifiers and combinations of various word embedding models in the sentiment analysis of airline-related tweets

    Multifunctions determined by integrable functions

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    Integral properties of multifunctions determined by vector valued functions are presented. Such multifunctions quite often serve as examples and counterexamples. In particular it can be observed that the properties of being integrable in the sense of Bochner, McShane or Birkhoff can be transferred to the generated multifunction while Henstock integrability does not guarantee i

    Set valued Kurzweil-Henstock-Pettis integral

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    It is shown that the obvious generalization of the Pettis integral of a multifunction obtained by replacing the Lebesgue integrability of the support functions by the Kurzweil--Henstock integrability, produces an integral which can be described -- in case of multifunctions with (weakly) compact convex values -- in terms of the Pettis set-valued integral

    Decompositions of Weakly Compact Valued Integrable Multifunctions

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    We give a short overview on the decomposition property for integrable multifunctions, i.e., when an "integrable in a certain sense" multifunction can be represented as a sum of one of its integrable selections and a multifunction integrable in a narrower sense. The decomposition theorems are important tools of the theory of multivalued integration since they allow us to see an integrable multifunction as a translation of a multifunction with better properties. Consequently, they provide better characterization of integrable multifunctions under consideration. There is a large literature on it starting from the seminal paper of the authors in 2006, where the property was proved for Henstock integrable multifunctions taking compact convex values in a separable Banach space X. In this paper, we summarize the earlier results, we prove further results and present tables which show the state of art in this topi

    Interacting Spreading Processes in Multilayer Networks: A Systematic Review

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    © 2013 IEEE. The world of network science is fascinating and filled with complex phenomena that we aspire to understand. One of them is the dynamics of spreading processes over complex networked structures. Building the knowledge-base in the field where we can face more than one spreading process propagating over a network that has more than one layer is a challenging task, as the complexity comes both from the environment in which the spread happens and from characteristics and interplay of spreads' propagation. As this cross-disciplinary field bringing together computer science, network science, biology and physics has rapidly grown over the last decade, there is a need to comprehensively review the current state-of-the-art and offer to the research community a roadmap that helps to organise the future research in this area. Thus, this survey is a first attempt to present the current landscape of the multi-processes spread over multilayer networks and to suggest the potential ways forward

    Next challenges for adaptive learning systems

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    Learning from evolving streaming data has become a 'hot' research topic in the last decade and many adaptive learning algorithms have been developed. This research was stimulated by rapidly growing amounts of industrial, transactional, sensor and other business data that arrives in real time and needs to be mined in real time. Under such circumstances, constant manual adjustment of models is in-efficient and with increasing amounts of data is becoming infeasible. Nevertheless, adaptive learning models are still rarely employed in business applications in practice. In the light of rapidly growing structurally rich 'big data', new generation of parallel computing solutions and cloud computing services as well as recent advances in portable computing devices, this article aims to identify the current key research directions to be taken to bring the adaptive learning closer to application needs. We identify six forthcoming challenges in designing and building adaptive learning (pre-diction) systems: making adaptive systems scalable, dealing with realistic data, improving usability and trust, integrat-ing expert knowledge, taking into account various application needs, and moving from adaptive algorithms towards adaptive tools. Those challenges are critical for the evolving stream settings, as the process of model building needs to be fully automated and continuous.</jats:p

    Change point detection in social networksCritical review with experiments

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    © 2018 Elsevier Inc. Change point detection in social networks is an important element in developing the understanding of dynamic systems. This complex and growing area of research has no clear guidelines on what methods to use or in which circumstances. This paper critically discusses several possible network metrics to be used for a change point detection problem and conducts an experimental, comparative analysis using the Enron and MIT networks. Bayesian change point detection analysis is conducted on different global graph metrics (Size, Density, Average Clustering Coefficient, Average Shortest Path) as well as metrics derived from the Hierarchical and Block models (Entropy, Edge Probability, No. of Communities, Hierarchy Level Membership). The results produced the posterior probability of a change point at weekly time intervals that were analysed against ground truth change points using precision and recall measures. Results suggest that computationally heavy generative models offer only slightly better results compared to some of the global graph metrics. The simplest metrics used in the experiments, i.e. nodes and links numbers, are the recommended choice for detecting overall structural changes
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