1,720,973 research outputs found
Topics in the probabilistic solution of ordinary differential equations
Get PDFThis thesis concerns several new developments in the probabilistic solution of ordinary differential equations. Probabilistic numerical methods are differentiated from their classical counterparts through the key property of returning a probability measure as output, rather than simply a point value. When properly calibrated, this measure can then be taken to probabilistically represent the output uncertainty arising from the application of the numerical procedure.
After giving some introductory context, we start with a concise survey of the still-developing field of probabilistic ODE solvers, highlighting how several different paradigms have developed somewhat in parallel. One of these, established by Conrad et al. (2016), defines randomised one-step solvers for initial value problems, where the outputs are empirical measures arising from Monte Carlo repetitions of the algorithm. We extend this to multistep solvers of Adams-Bashforth type using a novel Gaussian process construction. The properties of this method are explored and its convergence is rigorously proved.
We continue by defining a class of implicit probabilistic ODE solvers, the first in the literature. Unlike explicit methods, these modified Adams-Moulton algorithms incorporate information from the ODE dynamics beyond the current time-point, and as such are able to enhance the accuracy of the probabilistic model of numerical error. In their full form, they output a non-parametric description of the stepwise error, though we also propose a parametric approximation that aids computation. Once again, we explore the properties of the method and prove its convergence in the small step-size limit.
We follow with a discussion on the problem of calibration for these classes of algorithms, and generalise a proposal from Conrad et al. in order to implement it for our methods. We then apply the new integrators to two test differential equation models, first in the solution of the forward model, then later in the setting of a Bayesian inverse problem. We contrast the effect of using probabilistic integrators instead of classical ones on posterior inference over the model parameters, as well as derived functions of the forward solution.
We conclude with a brief discussion on the advantages and shortcomings of the proposed methods, and posit several suggestions for potential future research.Open Acces
Parallel Markov chain quasi-Monte Carlo methods
Get PDFQuasi-Monte Carlo (QMC) methods for estimating integrals are attractive since the resulting estimators typically converge at a faster rate than pseudo-random Monte Carlo. However, they can be difficult to set up on arbitrary posterior densities within the Bayesian framework, in particular for inverse problems. We propose a principled and efficient way of applying QMC to drive algorithms based on a general parallel Markov chain Monte Carlo (MCMC) framework, in which multiple proposals are generated per iteration.
We provide numerous methodological extensions of the original algorithm, including the use of non-reversible
transition kernels, adaptive proposal kernels and Rao-Blackwellisation. Further, we prove a law of large numbers
and a central limit theorem, ergodicity of the proposed adaptive methods and asymptotic unbiasedness for
estimates based on the Rao-Blackwellisation scheme.
We consider the use of completely uniformly distributed (CUD) numbers within the previously stated algorithms,
which leads to a general parallel Markov chain quasi-Monte Carlo (MCQMC) methodology. A scheme that
efficiently produces CUD seeds for arbitrary problem dimensions and proposal numbers based on an already
existing CUD sequence is developed. We prove ergodicity of the resulting adaptive methods and asymptotic
unbiasedness for the Rao-Blackwellised estimates. For the latter, we demonstrate numerically in a number of
statistical models that this approach scales close to n^{−2} as we increase parallelisation, instead of the usual
n^{−1} that is typical of pseudo-random MCMC algorithms. The improved rate is proven theoretically in a special case. Simulations are performed for Bayesian linear and logistic regression, simple non-linear ODE models and a
complex model for cardiac excitation. The CUD driven Rao-Blackwellisation algorithms yield multiple orders
of magnitude reduction in the variance and MSE of the resulting estimates compared to their pseudo-driven
counterparts and to reference algorithms such as Metropolis-Hastings.Open Acces
Discovering the hidden structure of financial markets through bayesian modelling
Get PDFUnderstanding what is driving the price of a financial asset is a question that is currently mostly unanswered. In this work we go beyond the classic one step ahead prediction and instead construct models that create new information on the behaviour of these time series. Our aim is to get a better understanding of the hidden structures that drive the moves of each financial time series and thus the market as a whole.
We propose a tool to decompose multiple time series into economically-meaningful variables to explain the endogenous and exogenous factors driving their underlying variability. The methodology we introduce goes beyond the direct model forecast. Indeed, since our model continuously adapts its variables and coefficients, we can study the time series of coefficients and selected variables. We also present a model to construct the causal graph of relations between these time series and include them in the exogenous factors.
Hence, we obtain a model able to explain what is driving the move of both each specific time series and the market as a whole. In addition, the obtained graph of the time series provides new information on the underlying risk structure of this environment. With this deeper understanding of the hidden structure we propose novel ways to detect and forecast risks in the market. We investigate our results with inferences up to one month into the future using stocks, FX futures and ETF futures, demonstrating its superior performance according to accuracy of large moves, longer-term prediction and consistency over time. We also go in more details on the economic interpretation of the new variables and discuss the created graph structure of the market.Open Acces
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Dispelling the Myths Behind First-author Citation Counts
Get PDFWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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