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.

Spectrum shaping and its application: spread-spectrum clock generator and circuits for ultra wide band

Electromagnetic spectrum can be identified as a resource for the designer, as well as for the manufacturer, from two complementary points of view: first, because it is a good in great demand by many different kind of applications; second, because despite its scarce availability, it may be advantageous to use more spectrum than necessary. This is the case of Spread-Spectrum Systems, those systems in which the transmitted signal is spread over a wide frequency band, much wider, in fact, than the minimum bandwidth required to transmit the information being sent. Part I of this dissertation deals with Spread-Spectrum Clock Generators (SSCG) aiming at reducing Electro Magnetic Interference (EMI) of clock signals in integrated circuits (IC) design. In particular, the modulation of the clock and the consequent spreading of its spectrum are obtained through a random modulating signal outputted by a chaotic map, i.e. a discrete-time dynamical system showing chaotic behavior. The advantages offered by this kind of modulation are highlighted. Three different prototypes of chaos-based SSCG are presented in all their aspects: design, simulation, and post-fabrication measurements. The third one, operating at a frequency equal to 3GHz, aims at being applied to Serial ATA, standard de facto for fast data transmission to and from Hard Disk Drives. The most extreme example of spread-spectrum signalling is the emerging ultra-wideband (UWB) technology, which proposes the use of large sections of the radio spectrum at low amplitudes to transmit high-bandwidth digital data. In part II of the dissertation, two UWB applications are presented, both dealing with the advantages as well as with the challenges of a wide-band system, namely: a chaos-based sequence generation method for reducing Multiple Access Interference (MAI) in Direct Sequence UWB Wireless-Sensor-Networks (WSNs), and design and simulations of a Low-Noise Amplifier (LNA) for impulse radio UWB. This latter topic was studied during a study-abroad period in collaboration with Delft University of Technology, Delft, Netherlands.

Enabling Blocks for Integrated CMOS UWB Transceivers

The last decades have seen an unrivaled growth and diffusion of mobile telecommunications. Several standards have been developed to this purposes, from GSM mobile phone communications to WLAN IEEE 802.11, providing different services for the the transmission of signals ranging from voice to high data rate digital communications and Digital Video Broadcasting (DVB). In this wide research and market field, this thesis focuses on Ultra Wideband (UWB) communications, an emerging technology for providing very high data rate transmissions over very short distances. In particular the presented research deals with the circuit design of enabling blocks for MB-OFDM UWB CMOS single-chip transceivers, namely the frequency synthesizer and the transmission mixer and power amplifier. First we discuss three different models for the simulation of chargepump phase-locked loops, namely the continuous time s-domain and discrete time z-domain approximations and the exact semi-analytical time-domain model. The limitations of the two approximated models are analyzed in terms of error in the computed settling time as a function of loop parameters, deriving practical conditions under which the different models are reliable for fast settling PLLs up to fourth order. Besides, a phase noise analysis method based upon the time-domain model is introduced and compared to the results obtained by means of the s-domain model. We compare the three models over the simulation of a fast switching PLL to be integrated in a frequency synthesizer for WiMedia MB-OFDM UWB systems. In the second part, the theoretical analysis is applied to the design of a 60mW 3.4 to 9.2GHz 12 Bands frequency synthesizer for MB-OFDM UWB based on two wide-band PLLs. The design is presented and discussed up to layout level. A test chip has been implemented in TSMC CMOS 90nm technology, measured data is provided. The functionality of the circuit is proved and specifications are met with state-of-the-art area occupation and power consumption. The last part of the thesis deals with the design of a transmission mixer and a power amplifier for MB-OFDM UWB band group 1. The design has been carried on up to layout level in ST Microlectronics 65nm CMOS technology. Main characteristics of the systems are the wideband behavior (1.6 GHz of bandwidth) and the constant behavior over process parameters, temperature and supply voltage thanks to the design of dedicated adaptive biasing circuits.

Advanced CMOS Interfaces for Bio-Nanosensors

The improvement of devices provided by Nanotechnology has put forward new classes of sensors, called bio-nanosensors, which are very promising for the detection of biochemical molecules in a large variety of applications. Their use in lab-on-a-chip could gives rise to new opportunities in many fields, from health-care and bio-warfare to environmental and high-throughput screening for pharmaceutical industry. Bio-nanosensors have great advantages in terms of cost, performance, and parallelization. Indeed, they require very low quantities of reagents and improve the overall signal-to-noise-ratio due to increase of binding signal variations vs. area and reduction of stray capacitances. Additionally, they give rise to new challenges, such as the need to design high-performance low-noise integrated electronic interfaces. This thesis is related to the design of high-performance advanced CMOS interfaces for electrochemical bio-nanosensors. The main focus of the thesis is: 1) critical analysis of noise in sensing interfaces, 2) devising new techniques for noise reduction in discrete-time approaches, 3) developing new architectures for low-noise, low-power sensing interfaces. The manuscript reports a multi-project activity focusing on low-noise design and presents two developed integrated circuits (ICs) as examples of advanced CMOS interfaces for bio-nanosensors. The first project concerns low-noise current-sensing interface for DC and transient measurements of electrophysiological signals. The focus of this research activity is on the noise optimization of the electronic interface. A new noise reduction technique has been developed so as to realize an integrated CMOS interfaces with performance comparable with state-of-the-art instrumentations. The second project intends to realize a stand-alone, high-accuracy electrochemical impedance spectroscopy interface. The system is tailored for conductivity-temperature-depth sensors in environmental applications, as well as for bio-nanosensors. It is based on a band-pass delta-sigma technique and combines low-noise performance with low-power requirements.

Essays in Financial Econometrics

The first paper sheds light on the informational content of high frequency data and daily data. I assess the economic value of the two family models comparing their performance in forecasting asset volatility through the Value at Risk metric. In running the comparison this paper introduces two key assumptions: jumps in prices and leverage effect in volatility dynamics. Findings suggest that high frequency data models do not exhibit a superior performance over daily data models. In the second paper, building on Majewski et al. (2015), I propose an affine-discrete time model, labeled VARG-J, which is characterized by a multifactor volatility specification. In the VARG-J model volatility experiences periods of extreme movements through a jump factor modeled as an Autoregressive Gamma Zero process. The estimation under historical measure is done by quasi-maximum likelihood and the Extended Kalman Filter. This strategy allows to filter out both volatility factors introducing a measurement equation that relates the Realized Volatility to latent volatility. The risk premia parameters are calibrated using call options written on S&P500 Index. The results clearly illustrate the important contribution of the jump factor in the pricing performance of options and the economic significance of the volatility jump risk premia. In the third paper, I analyze whether there is empirical evidence of contagion at the bank level, measuring the direction and the size of contagion transmission between European markets. In order to understand and quantify the contagion transmission on banking market, I estimate the econometric model by Aït-Sahalia et al. (2015) in which contagion is defined as the within and between countries transmission of shocks and asset returns are directly modeled as a Hawkes jump diffusion process. The empirical analysis indicates that there is a clear evidence of contagion from Greece to European countries as well as self-contagion in all countries.

Propagation modelling and resource allocation in mobile radio communications

Over the past years, ray tracing (RT) models popularity has been increasing. From the nineties, RT has been used for field prediction in environment such as indoor and urban environments. Nevertheless, with the advent of new technologies, the channel model has become decidedly more dynamic and to perform RT simulations at each discrete time instant become computationally expensive. In this thesis, a new dynamic ray tracing (DRT) approach is presented in which from a single ray tracing simulation at an initial time t0, through analytical formulas we are able to track the motion of the interaction points. The benefits that this approach bring are that Doppler frequencies and channel prediction can be derived at every time instant, without recurring to multiple RT runs and therefore shortening the computation time. DRT performance was studied on two case studies and the results shows the accuracy and the computational gain that derives from this approach. Another issue that has been addressed in this thesis is the licensed band exhaustion of some frequency bands. To deal with this problem, a novel unselfish spectrum leasing scheme in cognitive radio networks (CRNs) is proposed that offers an energy-efficient solution minimizing the environmental impact of the network. In addition, a network management architecture is introduced and resource allocation is proposed as a constrained sum energy efficiency maximization problem. System simulations demonstrate an increment in the energy efficiency of the primary users’ network compared with previously proposed algorithms.

Dynamic identification of structures: experimental assessment of modal parameteres through methods in frequency domain, in time-frequency domain and model updating

Among the experimental methods commonly used to define the behaviour of a full scale system, dynamic tests are the most complete and efficient procedures. A dynamic test is an experimental process, which would define a set of characteristic parameters of the dynamic behaviour of the system, such as natural frequencies of the structure, mode shapes and the corresponding modal damping values associated. An assessment of these modal characteristics can be used both to verify the theoretical assumptions of the project, to monitor the performance of the structural system during its operational use. The thesis is structured in the following chapters: The first introductive chapter recalls some basic notions of dynamics of structure, focusing the discussion on the problem of systems with multiply degrees of freedom (MDOF), which can represent a generic real system under study, when it is excited with harmonic force or in free vibration. The second chapter is entirely centred on to the problem of dynamic identification process of a structure, if it is subjected to an experimental test in forced vibrations. It first describes the construction of FRF through classical FFT of the recorded signal. A different method, also in the frequency domain, is subsequently introduced; it allows accurately to compute the FRF using the geometric characteristics of the ellipse that represents the direct input-output comparison. The two methods are compared and then the attention is focused on some advantages of the proposed methodology. The third chapter focuses on the study of real structures when they are subjected to experimental test, where the force is not known, like in an ambient or impact test. In this analysis we decided to use the CWT, which allows a simultaneous investigation in the time and frequency domain of a generic signal x(t). The CWT is first introduced to process free oscillations, with excellent results both in terms of frequencies, dampings and vibration modes. The application in the case of ambient vibrations defines accurate modal parameters of the system, although on the damping some important observations should be made. The fourth chapter is still on the problem of post processing data acquired after a vibration test, but this time through the application of discrete wavelet transform (DWT). In the first part the results obtained by the DWT are compared with those obtained by the application of CWT. Particular attention is given to the use of DWT as a tool for filtering the recorded signal, in fact in case of ambient vibrations the signals are often affected by the presence of a significant level of noise. The fifth chapter focuses on another important aspect of the identification process: the model updating. In this chapter, starting from the modal parameters obtained from some environmental vibration tests, performed by the University of Porto in 2008 and the University of Sheffild on the Humber Bridge in England, a FE model of the bridge is defined, in order to define what type of model is able to capture more accurately the real dynamic behaviour of the bridge. The sixth chapter outlines the necessary conclusions of the presented research. They concern the application of a method in the frequency domain in order to evaluate the modal parameters of a structure and its advantages, the advantages in applying a procedure based on the use of wavelet transforms in the process of identification in tests with unknown input and finally the problem of 3D modeling of systems with many degrees of freedom and with different types of uncertainty.

Estimating persistent Betti numbers for discrete shape analysis

Persistent Topology is an innovative way of matching topology and geometry, and it proves to be an effective mathematical tool in shape analysis. In order to express its full potential for applications, it has to interface with the typical environment of Computer Science: It must be possible to deal with a finite sampling of the object of interest, and with combinatorial representations of it. Following that idea, the main result claims that it is possible to construct a relation between the persistent Betti numbers (PBNs; also called rank invariant) of a compact, Riemannian submanifold X of R^m and the ones of an approximation U of X itself, where U is generated by a ball covering centered in the points of the sampling. Moreover we can state a further result in which, this time, we relate X with a finite simplicial complex S generated, thanks to a particular construction, by the sampling points. To be more precise, strict inequalities hold only in "blind strips'', i.e narrow areas around the discontinuity sets of the PBNs of U (or S). Out of the blind strips, the values of the PBNs of the original object, of the ball covering of it, and of the simplicial complex coincide, respectively.

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.

Frequency domain analysis of stationary time series

This thesis provides a necessary and sufficient condition for asymptotic efficiency of a nonparametric estimator of the generalised autocovariance function of a Gaussian stationary random process. The generalised autocovariance function is the inverse Fourier transform of a power transformation of the spectral density, and encompasses the traditional and inverse autocovariance functions. Its nonparametric estimator is based on the inverse discrete Fourier transform of the same power transformation of the pooled periodogram. The general result is then applied to the class of Gaussian stationary ARMA processes and its implications are discussed. We illustrate that for a class of contrast functionals and spectral densities, the minimum contrast estimator of the spectral density satisfies a Yule-Walker system of equations in the generalised autocovariance estimator. Selection of the pooling parameter, which characterizes the nonparametric estimator of the generalised autocovariance, controlling its resolution, is addressed by using a multiplicative periodogram bootstrap to estimate the finite-sample distribution of the estimator. A multivariate extension of recently introduced spectral models for univariate time series is considered, and an algorithm for the coefficients of a power transformation of matrix polynomials is derived, which allows to obtain the Wold coefficients from the matrix coefficients characterizing the generalised matrix cepstral models. This algorithm also allows the definition of the matrix variance profile, providing important quantities for vector time series analysis. A nonparametric estimator based on a transformation of the smoothed periodogram is proposed for estimation of the matrix variance profile.