1,721,007 research outputs found
Characterization of Besov spaces with dominating mixed smoothness by differences
Full text linkBesov spaces with dominating mixed smoothness, on the product of the real line and the torus as well as bounded domains, are studied. A characterization of these function spaces in terms of differences is provided. Applications to random fields, like Gaussian fields and the stochastic heat equation, are discussed, based on a Kolmogorov criterion for Besov regularity with dominating mixed smoothness
Uncertainties in Generative Deep Learning and Data Amplification for High Energy Physics
No full textThe upcoming high-luminosity upgrade of the LHC requires an increase in simulated data. Due to the high computational cost of detector simulation, this demand threatens to surpass the computational resources. As a consequence, it is important to develop faster, less compute intensive alternatives to classical detector simulation with Markov chain Monte Carlo (MCMC). Generative Deep Learning surrogates are one possible candidate for speeding up the simulation and are already applied in ATLAS fast simulation tools. However, the quality of the surrogate data is intrinsically limited by the training statistics.
We demonstrate that the amount of training data poses as an upper limit on the precision of global properties of observables constructed from the data. Such global properties include for example means or variances. Nevertheless, the inductive bias of the Neural Network fit allows to surpass the training statistics when analyzing smaller regions of the data space. We show that the relaxed limit, which still depends on the training data, can be estimated from uncertainties predicted by Bayesian Neural Networks. To achieve a truthful estimate, the uncertainty prediction needs to be well calibrated. We show one way to calibrate uncertainties for generative Bayesian Neural Networks and find that the common variational inference method is hard to calibrate.
We therefore develop a new method based on stochastic gradient MCMC. This method is called AdamMCMC. It is easy to apply and replaces the stochastic optimization commonly employed in Deep Learning. In contrast to variational inference, the variance of the uncertainty prediction can be adapted e!ectively through variation
of a single parameter. Diverse predictions indicate out-of-distribution application. Overall, we find that the stochastic gradient MCMC produces more reliable predictions than variational inference in multiple applications. Classifier Surrogates are one possible application of generative Machine Learning, where reliable uncertainties are crucial. This class of surrogates predicts the behavior of jet taggers working on detector data from more accessible data. Experimental analysis employing such taggers can be reinterpreted without the need for detector simulation. This cuts computational cost and enables sharing of the analysis outside the collaboration. However, the uncertainties introduced by the approximation need to be controlled and application to new data spaces needs to be prevented. We show that Continuous Normalizing Flows, in combination with AdamMCMC, can fulfill these requirements. Similar surrogates can be of high value for the community and could be implemented with every jet tagger employed at ATLAS or CMS
Sharp adaptive similarity testing with pathwise stability for ergodic diffusions
No full textWithin the nonparametric diffusion model, we develop a multiple test to infer about similarity of an unknown drift b to some reference drift b_0: At prescribed significance, we simultaneously identify those regions where violation from similarity occurs, without a priori knowledge of their number, size and location. This test is shown to be minimax-optimal and adaptive. At the same time, the procedure is robust under small deviation from Brownian motion as the driving noise process. A detailed investigation for fractional driving noise, which is neither a semimartingale nor a Markov process, is provided for Hurst indices close to the Brownian motion case
Information bounds for inverse problems with application to deconvolution and Lévy models
No full textStatistical methods for probabilistic forecasts of real-valued outcomes
Full text linkThe work considers three essential steps to properly perform a forecasting task where the target variable is real-valued and the corresponding forecasts are probabilistic. The three steps involve data analysis, fitting a model and forecast evaluation.
For binary target variables, there exist various methods to properly perform each of the forecasting steps while for continuous data the set of possible approaches is considerably smaller. To overcome this shortcoming, three new tools are presented which are either extensions of binary approaches or motivated by their binary counterpart.
First, a natural extension of classical ROC analysis is presented in the form of ROC movies, UROC curve and CPA. Then EasyUQ and smooth EasyUQ are introduced which produce calibrated statistical forecast distributions based on real-valued model output. Finally, a CRPS decomposition is proposed to obtain more informative components for forecast evaluation. In the last chapter of this thesis, the new tools are successively applied to a challenge weather forecasting problem, namely producing probabilistic forecasts for accumulated precipitation over the tropics
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe 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
From deconvolution to Lévy processes
Full text linkDie Schätzung von Quantilen und verwandten Funktionalen wird in zwei inversen Problemen behandelt: dem klassischen Dekonvolutionsmodell sowie dem Lévy-Modell in dem ein Lévy-Prozess beobachtet wird und Funktionale des Sprungmaßes geschätzt werden. Im einem abstrakteren Rahmen wird semiparametrische Effizienz im Sinne von Hájek-Le Cam für Funktionalschätzung in regulären, inversen Modellen untersucht. Ein allgemeiner Faltungssatz wird bewiesen, der auf eine große Klasse von statistischen inversen Problem anwendbar ist. Im Dekonvolutionsmodell beweisen wir, dass die Plugin-Schätzer der Verteilungsfunktion und der Quantile effizient sind. Auf der Grundlage von niederfrequenten diskreten Beobachtungen des Lévy-Prozesses wird im nichtlinearen Lévy-Modell eine Informationsschranke für die Schätzung von Funktionalen des Sprungmaßes hergeleitet. Die enge Verbindung zwischen dem Dekonvolutionsmodell und dem Lévy-Modell wird präzise beschrieben. Quantilschätzung für Dekonvolutionsprobleme wird umfassend untersucht. Insbesondere wird der realistischere Fall von unbekannten Fehlerverteilungen behandelt. Wir zeigen unter minimalen und natürlichen Bedingungen, dass die Plugin-Methode minimax optimal ist. Eine datengetriebene Bandweitenwahl erlaubt eine optimale adaptive Schätzung. Quantile werden auf den Fall von Lévy-Maßen, die nicht notwendiger Weise endlich sind, verallgemeinert. Mittels äquidistanten, diskreten Beobachtungen des Prozesses werden nichtparametrische Schätzer der verallgemeinerten Quantile konstruiert und minimax optimale Konvergenzraten hergeleitet. Als motivierendes Beispiel von inversen Problemen untersuchen wir ein Finanzmodell empirisch, in dem ein Anlagengegenstand durch einen exponentiellen Lévy-Prozess dargestellt wird. Die Quantilschätzer werden auf dieses Modell übertragen und eine optimale adaptive Bandweitenwahl wird konstruiert. Die Schätzmethode wird schließlich auf reale Daten von DAX-Optionen angewendet.The estimation of quantiles and realated functionals is studied in two inverse problems: the classical deconvolution model and the Lévy model, where a Lévy process is observed and where we aim for the estimation of functionals of the jump measure. From a more abstract perspective we study semiparametric efficiency in the sense of Hájek-Le Cam for functional estimation in regular indirect models. A general convolution theorem is proved which applies to a large class of statistical inverse problems. In particular, we consider the deconvolution model, where we prove that our plug-in estimators of the distribution function and of the quantiles are efficient. In the nonlinear Lévy model based on low-frequent discrete observations of the Lévy process, we deduce an information bound for the estimation of functionals of the jump measure. The strong relationship between the Lévy model and the deconvolution model is given a precise meaning. Quantile estimation in deconvolution problems is studied comprehensively. In particular, the more realistic setup of unknown error distributions is covered. Under minimal and natural conditions we show that the plug-in method is minimax optimal. A data-driven bandwidth choice yields optimal adaptive estimation. The concept of quantiles is generalized to the possibly infinite Lévy measures by considering left and right tail integrals. Based on equidistant discrete observations of the process, we construct a nonparametric estimator of the generalized quantiles and derive minimax convergence rates. As a motivating financial example for inverse problems, we empirically study the calibration of an exponential Lévy model for asset prices. The estimators of the generalized quantiles are adapted to this model. We construct an optimal adaptive quantile estimator and apply the procedure to real data of DAX-options
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