Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

A Multilevel Model with Time Series Components for the Analysis of Tribal Art Prices

In the present work we perform an econometric analysis of the Tribal art market. To this aim, we use a unique and original database that includes information on Tribal art market auctions worldwide from 1998 to 2011. In Literature, art prices are modelled through the hedonic regression model, a classic fixed-effect model. The main drawback of the hedonic approach is the large number of parameters, since, in general, art data include many categorical variables. In this work, we propose a multilevel model for the analysis of Tribal art prices that takes into account the influence of time on artwork prices. In fact, it is natural to assume that time exerts an influence over the price dynamics in various ways. Nevertheless, since the set of objects change at every auction date, we do not have repeated measurements of the same items over time. Hence, the dataset does not constitute a proper panel; rather, it has a two-level structure in that items, level-1 units, are grouped in time points, level-2 units. The main theoretical contribution is the extension of classical multilevel models to cope with the case described above. In particular, we introduce a model with time dependent random effects at the second level. We propose a novel specification of the model, derive the maximum likelihood estimators and implement them through the E-M algorithm. We test the finite sample properties of the estimators and the validity of the own-written R-code by means of a simulation study. Finally, we show that the new model improves considerably the fit of the Tribal art data with respect to both the hedonic regression model and the classic multilevel model.

Variability analysis of discrete event series and wavefront patterns in surface electrocardiogram: contributions to psychophysiological and clinical research

The surface electrocardiogram (ECG) is an established diagnostic tool for the detection of abnormalities in the electrical activity of the heart. The interest of the ECG, however, extends beyond the diagnostic purpose. In recent years, studies in cognitive psychophysiology have related heart rate variability (HRV) to memory performance and mental workload. The aim of this thesis was to analyze the variability of surface ECG derived rhythms, at two different time scales: the discrete-event time scale, typical of beat-related features (Objective I), and the “continuous” time scale of separated sources in the ECG (Objective II), in selected scenarios relevant to psychophysiological and clinical research, respectively. Objective I) Joint time-frequency and non-linear analysis of HRV was carried out, with the goal of assessing psychophysiological workload (PPW) in response to working memory engaging tasks. Results from fourteen healthy young subjects suggest the potential use of the proposed indices in discriminating PPW levels in response to varying memory-search task difficulty. Objective II) A novel source-cancellation method based on morphology clustering was proposed for the estimation of the atrial wavefront in atrial fibrillation (AF) from body surface potential maps. Strong direct correlation between spectral concentration (SC) of atrial wavefront and temporal variability of the spectral distribution was shown in persistent AF patients, suggesting that with higher SC, shorter observation time is required to collect spectral distribution, from which the fibrillatory rate is estimated. This could be time and cost effective in clinical decision-making. The results held for reduced leads sets, suggesting that a simplified setup could also be considered, further reducing the costs. In designing the methods of this thesis, an online signal processing approach was kept, with the goal of contributing to real-world applicability. An algorithm for automatic assessment of ambulatory ECG quality, and an automatic ECG delineation algorithm were designed and validated.

Financial Development and Economic Growth: Time Series Evidence from Albania

The main objective of this thesis is to explore the short and long run causality patterns in the finance – growth nexus and finance-growth-trade nexus before and after the global financial crisis, in the case of Albania. To this end we use quarterly data on real GDP, 13 proxy measures for financial development and the trade openness indicator for the period 1998Q1 – 2013Q2 and 1998Q1-2008Q3. Causality patterns will be explored in a VAR-VECM framework. For this purpose we will proceed as follows: (i) testing for the integration order of the variables; (ii) cointegration analysis and (iii) performing Granger causality tests in a VAR-VECM framework. In the finance-growth nexus, empirical evidence suggests for a positive long run relationship between finance and economic growth, with causality running from financial development to economic growth. The global financial crisis seems to have not affected the causality direction in the finance and growth nexus, thus supporting the finance led growth hypothesis in the long run in the case of Albania. In the finance-growth-trade openness nexus, we found evidence for a positive long run relationship the variables, with causality direction depending on the proxy used for financial development. When the pre-crisis sample is considered, we find evidence for causality running from financial development and trade openness to economic growth. The global financial crisis seems to have affected somewhat the causality direction in the finance-growth-trade nexus, which has become sensible to the proxy used for financial development. On the short run, empirical evidence suggests for a clear unidirectional relationship between finance and growth, with causality mostly running from economic growth to financial development. When we consider the per-crisis sub sample results are mixed, depending on the proxy used for financial development. The same results are confirmed when trade openness is taken into account.

Local Trigonometric Methods for Time Series Smoothing.

The thesis is concerned with local trigonometric regression methods. The aim was to develop a method for extraction of cyclical components in time series. The main results of the thesis are the following. First, a generalization of the filter proposed by Christiano and Fitzgerald is furnished for the smoothing of ARIMA(p,d,q) process. Second, a local trigonometric filter is built, with its statistical properties. Third, they are discussed the convergence properties of trigonometric estimators, and the problem of choosing the order of the model. A large scale simulation experiment has been designed in order to assess the performance of the proposed models and methods. The results show that local trigonometric regression may be a useful tool for periodic time series analysis.

The Perception of American Adolescent Culture Through the Dubbing and Fansubbing of a Selection of US Teen Series from 1990 to 2013

The present dissertation aims at analyzing the construction of American adolescent culture through teen-targeted television series and the shift in perception that occurs as a consequence of the translation process. In light of the recent changes in television production and consumption modes, largely caused by new technologies, this project explores the evolution of Italian audiences, focusing on fansubbing (freely distributed amateur subtitles made by fans for fan consumption) and social viewing (the re-aggregation of television consumption based on social networks and dedicated platforms, rather than on physical presence). These phenomena are symptoms of a sort of ‘viewership 2.0’ and of a new type of active viewing, which calls for a revision of traditional AVT strategies. Using a framework that combines television studies, new media studies, and fandom studies with an approach to AVT based on Descriptive Translation Studies (Toury 1995), this dissertation analyzes the non-Anglophone audience’s growing need to participation in the global dialogue and appropriation process based on US scheduling and informed by the new paradigm of convergence culture, transmedia storytelling, and affective economics (Jenkins 2006 and 2007), as well as the constraints intrinsic to multimodal translation and the different types of linguistic and cultural adaptation performed through dubbing (which tends to be more domesticating; Venuti 1995) and fansubbing (typically more foreignizing). The study analyzes a selection of episodes from six of the most popular teen television series between 1990 and 2013, which has been divided into three ages based on the different modes of television consumption: top-down, pre-Internet consumption (Beverly Hills, 90210, 1990 – 2000), emergence of audience participation (Buffy the Vampire Slayer, 1997 – 2003; Dawson’s Creek, 1998 – 2003), age of convergence and Viewership 2.0 (Gossip Girl, 2007 – 2012; Glee, 2009 – present; The Big Bang Theory, 2007 - present).