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.

Methods for Clearance Influence Analysis in Planar and Spatial Mechanisms

This doctoral dissertation presents a new method to asses the influence of clearancein the kinematic pairs on the configuration of planar and spatial mechanisms. The subject has been widely investigated in both past and present scientific literature, and is approached in different ways: a static/kinetostatic way, which looks for the clearance take-up due to the external loads on the mechanism; a probabilistic way, which expresses clearance-due displacements using probability density functions; a dynamic way, which evaluates dynamic effects like the actual forces in the pairs caused by impacts, or the consequent vibrations. This dissertation presents a new method to approach the problem of clearance. The problem is studied from a purely kinematic perspective. With reference to a given mechanism configuration, the pose (position and orientation) error of the mechanism link of interest is expressed as a vector function of the degrees of freedom introduced in each pair by clearance: the presence of clearance in a kinematic pair, in facts, causes the actual pair to have more degrees of freedom than the theoretical clearance-free one. The clearance-due degrees of freedom are bounded by the pair geometry. A proper modelling of clearance-affected pairs allows expressing such bounding through analytical functions. It is then possible to study the problem as a maximization problem, where a continuous function (the pose error of the link of interest) subject to some constraints (the analytical functions bounding clearance- due degrees of freedom) has to be maximize. Revolute, prismatic, cylindrical, and spherical clearance-affected pairs have been analytically modelled; with reference to mechanisms involving such pairs, the solution to the maximization problem has been obtained in a closed form.

Improvement of photon transport model by including coupled photon-electron transport and kernel refinement

The first part of this work deals with the inverse problem solution in the X-ray spectroscopy field. An original strategy to solve the inverse problem by using the maximum entropy principle is illustrated. It is built the code UMESTRAT, to apply the described strategy in a semiautomatic way. The application of UMESTRAT is shown with a computational example. The second part of this work deals with the improvement of the X-ray Boltzmann model, by studying two radiative interactions neglected in the current photon models. Firstly it is studied the characteristic line emission due to Compton ionization. It is developed a strategy that allows the evaluation of this contribution for the shells K, L and M of all elements with Z from 11 to 92. It is evaluated the single shell Compton/photoelectric ratio as a function of the primary photon energy. It is derived the energy values at which the Compton interaction becomes the prevailing process to produce ionization for the considered shells. Finally it is introduced a new kernel for the XRF from Compton ionization. In a second place it is characterized the bremsstrahlung radiative contribution due the secondary electrons. The bremsstrahlung radiation is characterized in terms of space, angle and energy, for all elements whit Z=1-92 in the energy range 1–150 keV by using the Monte Carlo code PENELOPE. It is demonstrated that bremsstrahlung radiative contribution can be well approximated with an isotropic point photon source. It is created a data library comprising the energetic distributions of bremsstrahlung. It is developed a new bremsstrahlung kernel which allows the introduction of this contribution in the modified Boltzmann equation. An example of application to the simulation of a synchrotron experiment is shown.

Spatial and temporal characterisation of ground deformation recorded by geodetic techniques

A critical point in the analysis of ground displacements time series is the development of data driven methods that allow the different sources that generate the observed displacements to be discerned and characterised. A widely used multivariate statistical technique is the Principal Component Analysis (PCA), which allows reducing the dimensionality of the data space maintaining most of the variance of the dataset explained. Anyway, PCA does not perform well in finding the solution to the so-called Blind Source Separation (BSS) problem, i.e. in recovering and separating the original sources that generated the observed data. This is mainly due to the assumptions on which PCA relies: it looks for a new Euclidean space where the projected data are uncorrelated. The Independent Component Analysis (ICA) is a popular technique adopted to approach this problem. However, the independence condition is not easy to impose, and it is often necessary to introduce some approximations. To work around this problem, I use a variational bayesian ICA (vbICA) method, which models the probability density function (pdf) of each source signal using a mix of Gaussian distributions. This technique allows for more flexibility in the description of the pdf of the sources, giving a more reliable estimate of them. Here I present the application of the vbICA technique to GPS position time series. First, I use vbICA on synthetic data that simulate a seismic cycle (interseismic + coseismic + postseismic + seasonal + noise) and a volcanic source, and I study the ability of the algorithm to recover the original (known) sources of deformation. Secondly, I apply vbICA to different tectonically active scenarios, such as the 2009 L'Aquila (central Italy) earthquake, the 2012 Emilia (northern Italy) seismic sequence, and the 2006 Guerrero (Mexico) Slow Slip Event (SSE).

Cooperative communication and distributed detection in wireless sensor networks

Recent progress in microelectronic and wireless communications have enabled the development of low cost, low power, multifunctional sensors, which has allowed the birth of new type of networks named wireless sensor networks (WSNs). The main features of such networks are: the nodes can be positioned randomly over a given field with a high density; each node operates both like sensor (for collection of environmental data) as well as transceiver (for transmission of information to the data retrieval); the nodes have limited energy resources. The use of wireless communications and the small size of nodes, make this type of networks suitable for a large number of applications. For example, sensor nodes can be used to monitor a high risk region, as near a volcano; in a hospital they could be used to monitor physical conditions of patients. For each of these possible application scenarios, it is necessary to guarantee a trade-off between energy consumptions and communication reliability. The thesis investigates the use of WSNs in two possible scenarios and for each of them suggests a solution that permits to solve relating problems considering the trade-off introduced. The first scenario considers a network with a high number of nodes deployed in a given geographical area without detailed planning that have to transmit data toward a coordinator node, named sink, that we assume to be located onboard an unmanned aerial vehicle (UAV). This is a practical example of reachback communication, characterized by the high density of nodes that have to transmit data reliably and efficiently towards a far receiver. It is considered that each node transmits a common shared message directly to the receiver onboard the UAV whenever it receives a broadcast message (triggered for example by the vehicle). We assume that the communication channels between the local nodes and the receiver are subject to fading and noise. The receiver onboard the UAV must be able to fuse the weak and noisy signals in a coherent way to receive the data reliably. It is proposed a cooperative diversity concept as an effective solution to the reachback problem. In particular, it is considered a spread spectrum (SS) transmission scheme in conjunction with a fusion center that can exploit cooperative diversity, without requiring stringent synchronization between nodes. The idea consists of simultaneous transmission of the common message among the nodes and a Rake reception at the fusion center. The proposed solution is mainly motivated by two goals: the necessity to have simple nodes (to this aim we move the computational complexity to the receiver onboard the UAV), and the importance to guarantee high levels of energy efficiency of the network, thus increasing the network lifetime. The proposed scheme is analyzed in order to better understand the effectiveness of the approach presented. The performance metrics considered are both the theoretical limit on the maximum amount of data that can be collected by the receiver, as well as the error probability with a given modulation scheme. Since we deal with a WSN, both of these performance are evaluated taking into consideration the energy efficiency of the network. The second scenario considers the use of a chain network for the detection of fires by using nodes that have a double function of sensors and routers. The first one is relative to the monitoring of a temperature parameter that allows to take a local binary decision of target (fire) absent/present. The second one considers that each node receives a decision made by the previous node of the chain, compares this with that deriving by the observation of the phenomenon, and transmits the final result to the next node. The chain ends at the sink node that transmits the received decision to the user. In this network the goals are to limit throughput in each sensor-to-sensor link and minimize probability of error at the last stage of the chain. This is a typical scenario of distributed detection. To obtain good performance it is necessary to define some fusion rules for each node to summarize local observations and decisions of the previous nodes, to get a final decision that it is transmitted to the next node. WSNs have been studied also under a practical point of view, describing both the main characteristics of IEEE802:15:4 standard and two commercial WSN platforms. By using a commercial WSN platform it is realized an agricultural application that has been tested in a six months on-field experimentation.

3D probability tomography: theoretical developments and applications to high-resolution geophysical prospecting

The theory of the 3D multipole probability tomography method (3D GPT) to image source poles, dipoles, quadrupoles and octopoles, of a geophysical vector or scalar field dataset is developed. A geophysical dataset is assumed to be the response of an aggregation of poles, dipoles, quadrupoles and octopoles. These physical sources are used to reconstruct without a priori assumptions the most probable position and shape of the true geophysical buried sources, by determining the location of their centres and critical points of their boundaries, as corners, wedges and vertices. This theory, then, is adapted to the geoelectrical, gravity and self potential methods. A few synthetic examples using simple geometries and three field examples are discussed in order to demonstrate the notably enhanced resolution power of the new approach. At first, the application to a field example related to a dipole–dipole geoelectrical survey carried out in the archaeological park of Pompei is presented. The survey was finalised to recognize remains of the ancient Roman urban network including roads, squares and buildings, which were buried under the thick pyroclastic cover fallen during the 79 AD Vesuvius eruption. The revealed anomaly structures are ascribed to wellpreserved remnants of some aligned walls of Roman edifices, buried and partially destroyed by the 79 AD Vesuvius pyroclastic fall. Then, a field example related to a gravity survey carried out in the volcanic area of Mount Etna (Sicily, Italy) is presented, aimed at imaging as accurately as possible the differential mass density structure within the first few km of depth inside the volcanic apparatus. An assemblage of vertical prismatic blocks appears to be the most probable gravity model of the Etna apparatus within the first 5 km of depth below sea level. Finally, an experimental SP dataset collected in the Mt. Somma-Vesuvius volcanic district (Naples, Italy) is elaborated in order to define location and shape of the sources of two SP anomalies of opposite sign detected in the northwestern sector of the surveyed area. The modelled sources are interpreted as the polarization state induced by an intense hydrothermal convective flow mechanism within the volcanic apparatus, from the free surface down to about 3 km of depth b.s.l..