Bayesian Analysis of Linear Inverse Problems with Applications in Economics and Finance

In my PhD thesis I propose a Bayesian nonparametric estimation method for structural econometric models where the functional parameter of interest describes the economic agent's behavior. The structural parameter is characterized as the solution of a functional equation, or by using more technical words, as the solution of an inverse problem that can be either ill-posed or well-posed. From a Bayesian point of view, the parameter of interest is a random function and the solution to the inference problem is the posterior distribution of this parameter. A regular version of the posterior distribution in functional spaces is characterized. However, the infinite dimension of the considered spaces causes a problem of non continuity of the solution and then a problem of inconsistency, from a frequentist point of view, of the posterior distribution (i.e. problem of ill-posedness). The contribution of this essay is to propose new methods to deal with this problem of ill-posedness. The first one consists in adopting a Tikhonov regularization scheme in the construction of the posterior distribution so that I end up with a new object that I call regularized posterior distribution and that I guess it is solution of the inverse problem. The second approach consists in specifying a prior distribution on the parameter of interest of the g-prior type. Then, I detect a class of models for which the prior distribution is able to correct for the ill-posedness also in infinite dimensional problems. I study asymptotic properties of these proposed solutions and I prove that, under some regularity condition satisfied by the true value of the parameter of interest, they are consistent in a "frequentist" sense. Once I have set the general theory, I apply my bayesian nonparametric methodology to different estimation problems. First, I apply this estimator to deconvolution and to hazard rate, density and regression estimation. Then, I consider the estimation of an Instrumental Regression that is useful in micro-econometrics when we have to deal with problems of endogeneity. Finally, I develop an application in finance: I get the bayesian estimator for the equilibrium asset pricing functional by using the Euler equation defined in the Lucas'(1978) tree-type models.

A moment symbolic representation of Lévy processes with applications

By using a symbolic method, known in the literature as the classical umbral calculus, a symbolic representation of Lévy processes is given and a new family of time-space harmonic polynomials with respect to such processes, which includes and generalizes the exponential complete Bell polynomials, is introduced. The usefulness of time-space harmonic polynomials with respect to Lévy processes is that it is a martingale the stochastic process obtained by replacing the indeterminate x of the polynomials with a Lévy process, whereas the Lévy process does not necessarily have this property. Therefore to find such polynomials could be particularly meaningful for applications. This new family includes Hermite polynomials, time-space harmonic with respect to Brownian motion, Poisson-Charlier polynomials with respect to Poisson processes, Laguerre and actuarial polynomials with respect to Gamma processes , Meixner polynomials of the first kind with respect to Pascal processes, Euler, Bernoulli, Krawtchuk, and pseudo-Narumi polynomials with respect to suitable random walks. The role played by cumulants is stressed and brought to the light, either in the symbolic representation of Lévy processes and their infinite divisibility property, either in the generalization, via umbral Kailath-Segall formula, of the well-known formulae giving elementary symmetric polynomials in terms of power sum symmetric polynomials. The expression of the family of time-space harmonic polynomials here introduced has some connections with the so-called moment representation of various families of multivariate polynomials. Such moment representation has been studied here for the first time in connection with the time-space harmonic property with respect to suitable symbolic multivariate Lévy processes. In particular, multivariate Hermite polynomials and their properties have been studied in connection with a symbolic version of the multivariate Brownian motion, while multivariate Bernoulli and Euler polynomials are represented as powers of multivariate polynomials which are time-space harmonic with respect to suitable multivariate Lévy processes.

The spatial representation of time

Numerosi studi mostrano che gli intervalli temporali sono rappresentati attraverso un codice spaziale che si estende da sinistra verso destra, dove gli intervalli brevi sono rappresentati a sinistra rispetto a quelli lunghi. Inoltre tale disposizione spaziale del tempo può essere influenzata dalla manipolazione dell’attenzione-spaziale. La presente tesi si inserisce nel dibattito attuale sulla relazione tra rappresentazione spaziale del tempo e attenzione-spaziale attraverso l’uso di una tecnica che modula l’attenzione-spaziale, ovvero, l’Adattamento Prismatico (AP). La prima parte è dedicata ai meccanismi sottostanti tale relazione. Abbiamo mostrato che spostando l’attenzione-spaziale con AP, verso un lato dello spazio, si ottiene una distorsione della rappresentazione di intervalli temporali, in accordo con il lato dello spostamento attenzionale. Questo avviene sia con stimoli visivi, sia con stimoli uditivi, nonostante la modalità uditiva non sia direttamente coinvolta nella procedura visuo-motoria di AP. Questo risultato ci ha suggerito che il codice spaziale utilizzato per rappresentare il tempo, è un meccanismo centrale che viene influenzato ad alti livelli della cognizione spaziale. La tesi prosegue con l’indagine delle aree corticali che mediano l’interazione spazio-tempo, attraverso metodi neuropsicologici, neurofisiologici e di neuroimmagine. In particolare abbiamo evidenziato che, le aree localizzate nell’emisfero destro, sono cruciali per l’elaborazione del tempo, mentre le aree localizzate nell’emisfero sinistro sono cruciali ai fini della procedura di AP e affinché AP abbia effetto sugli intervalli temporali. Infine, la tesi, è dedicata allo studio dei disturbi della rappresentazione spaziale del tempo. I risultati ci indicano che un deficit di attenzione-spaziale, dopo danno emisferico destro, provoca un deficit di rappresentazione spaziale del tempo, che si riflette negativamente sulla vita quotidiana dei pazienti. Particolarmente interessanti sono i risultati ottenuti mediante AP. Un trattamento con AP, efficace nel ridurre il deficit di attenzione-spaziale, riduce anche il deficit di rappresentazione spaziale del tempo, migliorando la qualità di vita dei pazienti.

On the renormalization flow representation of field theory and some applications

In this thesis we discuss a representation of quantum mechanics and quantum and statistical field theory based on a functional renormalization flow equation for the one-particle-irreducible average effective action, and we employ it to get information on some specific systems.

Synthesis and aplication of linear and macrocyclic nitrogen ligand

Linear and macrocyclic nitrogen ligands have been found wide application during the years. Nitrogen has a much strong association with transition-metal ions because the electron pair is partucularly available for complexing purposes. We started our investigation with the synthesis of new chiral perazamacrocycles containing four pyrrole rings. This ligand was synthesized by the [2+2]condensation of (R,R)-diaminocyclohexane and dipirranedialdehydes and was tested, after a complexation with Cu(OAc)2, in Henry reactions. The best yields (96%) and higher ee’s (96%) were obtained when the meso-substituent on the dipyrrandialdehyde was a methyl group. The positive influence of the pyrrole-containing macrocyclic structure on the efficiency/enantioselectivity of the catalytic system was demonstrated by comparison with the Henry reactions performed using analogous ligands. Henry product was obtain in good yield but only 73% of ee, when the dialdehyde unit was replaced by a triheteroaromatic dialdehye (furan-pyrrol-furan). Another well known macrocyclic ligand is calix[4]pyrrole. We decided to investigate, in collaboration with Neier’s group, the metal-coordinating properties of calix[2]pyrrole[2]pyrrolidine compounds obtained by the reduction of calix[4]pyrrole. We focused our attention on the reduction conditions, and tested different Pd supported (charcoal, grafite) catalysts at different condition. Concerning the synthesis of linear polyamine ligands. We focused our attention to the synthesis of 2-heteroaryl- and 2,5-diheteroarylpyrrolidines. The reductive amination reaction of diarylketones and aryl-substitutedketo-aldehydes with different chiral amines was exploited to prepare a small library of diastereo-enriched substituted pyrrolidines. We have also described a new synthetic route to 1,2-disubstituted 1,2,3,4-tetrahydropyrrole[1,2-a]pyrazines, which involves the diastereoselective addition of Grignard reagents to chiral oxazolidines. The best diastereoselectivity (98:2) was dependent on the nature of both the chiral auxiliary, (S)-1-phenylglycinol, and the nature of the organometallic reagent (MeMgBr).

Sparse Signal Representation of Ultrasonic Signals for Structural Health Monitoring Applications

Assessment of the integrity of structural components is of great importance for aerospace systems, land and marine transportation, civil infrastructures and other biological and mechanical applications. Guided waves (GWs) based inspections are an attractive mean for structural health monitoring. In this thesis, the study and development of techniques for GW ultrasound signal analysis and compression in the context of non-destructive testing of structures will be presented. In guided wave inspections, it is necessary to address the problem of the dispersion compensation. A signal processing approach based on frequency warping was adopted. Such operator maps the frequencies axis through a function derived by the group velocity of the test material and it is used to remove the dependence on the travelled distance from the acquired signals. Such processing strategy was fruitfully applied for impact location and damage localization tasks in composite and aluminum panels. It has been shown that, basing on this processing tool, low power embedded system for GW structural monitoring can be implemented. Finally, a new procedure based on Compressive Sensing has been developed and applied for data reduction. Such procedure has also a beneficial effect in enhancing the accuracy of structural defects localization. This algorithm uses the convolutive model of the propagation of ultrasonic guided waves which takes advantage of a sparse signal representation in the warped frequency domain. The recovery from the compressed samples is based on an alternating minimization procedure which achieves both an accurate reconstruction of the ultrasonic signal and a precise estimation of waves time of flight. Such information is used to feed hyperbolic or elliptic localization procedures, for accurate impact or damage localization.

Advanced representation of the ocean/sea ice dynamics at high latitudes

Sea ice is a fundamental element of global climate system, with numerous impacts on the polar environment. The ongoing drastic changes in the Earth’s sea ice cover highlight the necessity of monitoring the polar regions and systematically evaluating the quality of different numerical products. The main objective of this thesis is to improve our knowledge of the representation of Arctic and Antarctic sea ice using comprehensive global ocean reanalyses and coupled climate models. The dissertation will explore (i) the Antarctic marginal ice zone (MIZ) and pack ice area in the ensemble mean of four global ocean reanalyses called GREP; (ii) historical representation of the Arctic and Antarctic sea ice state in HighResMIP models; (iii) the future evolution of Arctic sea ice in HighResMIP models. Global ocean reanalyses and GREP are found to adequately capture interannual and seasonal variability in both pack ice and MIZ areas at hemispheric and regional scales. The advantage of the ensemble-mean approach is proved as GREP smooths the strengths and weaknesses of single systems and provides the most consistent and reliable estimates. This work is intended to encourage the use of GREP in a wide range of applications. The analysis of sea ice representation in the coupled climate models shows no systematic impact of the increased horizontal resolution. We argue that a few minor improvements in sea ice representation with the enhanced horizontal resolution are presumably not worth the major effort of costly computations. The thesis highlights the critical importance to distinguish the MIZ from consolidated pack ice both for investigating changes in sea ice distribution and evaluating the product’s performance. Considering that the MIZ is predicted to dominate the Arctic sea ice cover, the model physics parameterizations and sea ice rheology might require modifications. The results of the work can be useful for modelling community.

Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

Emotional engagement and orienting: the effects of picture size on affective response

For their survival, humans and animals can rely on motivational systems which are specialized in assessing the valence and imminence of dangers and appetitive cues. The Orienting Response (OR) is a fundamental response pattern that an organism executes whenever a novel or significant stimulus is detected, and has been shown to be consistently modulated by the affective value of a stimulus. However, detecting threatening stimuli and appetitive affordances while they are far away compared to when they are within reach constitutes an obvious evolutionary advantage. Building on the linear relationship between stimulus distance and retinal size, the present research was aimed at investigating the extent to which emotional modulation of distinct processes (action preparation, attentional capture, and subjective emotional state) is affected when reducing the retinal size of a picture. Studies 1-3 examined the effects of picture size on emotional response. Subjective feeling of engagement, as well as sympathetic activation, were modulated by picture size, suggesting that action preparation and subjective experience reflect the combined effects of detecting an arousing stimulus and assessing its imminence. On the other hand, physiological responses which are thought to reflect the amount of attentional resources invested in stimulus processing did not vary with picture size. Studies 4-6 were conducted to substantiate and extend the results of studies 1-3. In particular, it was noted that a decrease in picture size is associated with a loss in the low spatial frequencies of a picture, which might confound the interpretation of the results of studies 1-3. Therefore, emotional and neutral images which were either low-pass filtered or reduced in size were presented, and affective responses were measured. Most effects which were observed when manipulating image size were replicated by blurring pictures. However, pictures depicting highly arousing unpleasant contents were associated with a more pronounced decrease in affective modulation when pictures were reduced in size compared to when they were blurred. The present results provide important information for the study of processes involved in picture perception and in the genesis and expression of an emotional response. In particular, the availability of high spatial frequencies might affect the degree of activation of an internal representation of an affectively charged scene, and might modulate subjective emotional state and preparation for action. Moreover, the manipulation of stimulus imminence revealed important effects of stimulus engagement on specific components of the emotional response, and the implications of the present data for some models of emotions have been discussed. In particular, within the framework of a staged model of emotional response, the tactic and strategic role of response preparation and attention allocation to stimuli varying in engaging power has been discussed, considering the adaptive advantages that each might represent in an evolutionary view. Finally, the identification of perceptual parameters that allow affective processing to be carried out has important methodological applications in future studies examining emotional response in basic research or clinical contexts.

The Dynamics of Passive Torsional Fatigue Test Rigs: Innovative Applications of Universal Joints

The dynamics of a passive back-to-back test rig have been characterised, leading to a multi-coordinate approach for the analysis of arbitrary test configurations. Universal joints have been introduced into a typical pre-loaded back-to-back system in order to produce an oscillating torsional moment in a test specimen. Two different arrangements have been investigated using a frequency-based sub-structuring approach: the receptance method. A numerical model has been developed in accordance with this theory, allowing interconnection of systems with two-coordinates and closed multi-loop schemes. The model calculates the receptance functions and modal and deflected shapes of a general system. Closed form expressions of the following individual elements have been developed: a servomotor, damped continuous shaft and a universal joint. Numerical results for specific cases have been compared with published data in literature and experimental measurements undertaken in the present work. Due to the complexity of the universal joint and its oscillating dynamic effects, a more detailed analysis of this component has been developed. Two models have been presented. The first represents the joint as two inertias connected by a massless cross-piece. The second, derived by the dynamic analysis of a spherical four-link mechanism, considers the contribution of the floating element and its gyroscopic effects. An investigation into non-linear behaviour has led to a time domain model that utilises the Runge-Kutta fourth order method for resolution of the dynamic equations. It has been demonstrated that the torsional receptances of a universal joint, derived using the simple model, result in representation of the joint as an equivalent variable inertia. In order to verify the model, a test rig has been built and experimental validation undertaken. The variable inertia of a universal joint has lead to a novel application of the component as a passive device for the balancing of inertia variations in slider-crank mechanisms.

Interpreting Judgements using Knowledge Representation Methods and Computational Models of Argument

The goal of the present research is to define a Semantic Web framework for precedent modelling, by using knowledge extracted from text, metadata, and rules, while maintaining a strong text-to-knowledge morphism between legal text and legal concepts, in order to fill the gap between legal document and its semantics. The framework is composed of four different models that make use of standard languages from the Semantic Web stack of technologies: a document metadata structure, modelling the main parts of a judgement, and creating a bridge between a text and its semantic annotations of legal concepts; a legal core ontology, modelling abstract legal concepts and institutions contained in a rule of law; a legal domain ontology, modelling the main legal concepts in a specific domain concerned by case-law; an argumentation system, modelling the structure of argumentation. The input to the framework includes metadata associated with judicial concepts, and an ontology library representing the structure of case-law. The research relies on the previous efforts of the community in the field of legal knowledge representation and rule interchange for applications in the legal domain, in order to apply the theory to a set of real legal documents, stressing the OWL axioms definitions as much as possible in order to enable them to provide a semantically powerful representation of the legal document and a solid ground for an argumentation system using a defeasible subset of predicate logics. It appears that some new features of OWL2 unlock useful reasoning features for legal knowledge, especially if combined with defeasible rules and argumentation schemes. The main task is thus to formalize legal concepts and argumentation patterns contained in a judgement, with the following requirement: to check, validate and reuse the discourse of a judge - and the argumentation he produces - as expressed by the judicial text.

Exact and heuristic algorithms for the integrated planning of multi-energy systems

The aim of this thesis is to present exact and heuristic algorithms for the integrated planning of multi-energy systems. The idea is to disaggregate the energy system, starting first with its core the Central Energy System, and then to proceed towards the Decentral part. Therefore, a mathematical model for the generation expansion operations to optimize the performance of a Central Energy System system is first proposed. To ensure that the proposed generation operations are compatible with the network, some extensions of the existing network are considered as well. All these decisions are evaluated both from an economic viewpoint and from an environmental perspective, as specific constraints related to greenhouse gases emissions are imposed in the formulation. Then, the thesis presents an optimization model for solar organic Rankine cycle in the context of transactive energy trading. In this study, the impact that this technology can have on the peer-to-peer trading application in renewable based community microgrids is inspected. Here the consumer becomes a prosumer and engages actively in virtual trading with other prosumers at the distribution system level. Moreover, there is an investigation of how different technological parameters of the solar Organic Rankine Cycle may affect the final solution. Finally, the thesis introduces a tactical optimization model for the maintenance operations’ scheduling phase of a Combined Heat and Power plant. Specifically, two types of cleaning operations are considered, i.e., online cleaning and offline cleaning. Furthermore, a piecewise linear representation of the electric efficiency variation curve is included. Given the challenge of solving the tactical management model, a heuristic algorithm is proposed. The heuristic works by solving the daily operational production scheduling problem, based on the final consumer’s demand and on the electricity prices. The aggregate information from the operational problem is used to derive maintenance decisions at a tactical level.