Long Memory Modelling of Inflation with Stochastic Variance and Structural Breaks

We investigate changes in the time series characteristics of postwar U.S. inflation. In a model-based analysis the conditional mean of inflation is specified by a long memory autoregressive fractionally integrated moving average process and the conditional variance is modelled by a stochastic volatility process. We develop a Monte Carlo maximum likelihood method to obtain efficient estimates of the parameters using a monthly data-set of core inflation for which we consider different subsamples of varying size. Based on the new modelling framework and the associated estimation technique, we find remarkable changes in the variance, in the order of integration, in the short memory characteristics and in the volatility of volatility

Modeling the ongoing dynamics of short and long-range temporal correlations in broadband EEG during movement

Electroencephalogram (EEG) undergoes complex temporal and spectral changes during voluntary movement intention. Characterization of such changes has focused mostly on narrowband spectral processes such as Event-Related Desynchronization (ERD) in the sensorimotor rhythms because EEG is mostly considered as emerging from oscillations of the neuronal populations. However, the changes in the temporal dynamics, especially in the broadband arrhythmic EEG have not been investigated for movement intention detection. The Long-Range Temporal Correlations (LRTC) are ubiquitously present in several neuronal processes, typically requiring longer timescales to detect. In this paper, we study the ongoing changes in the dynamics of long- as well as short-range temporal dependencies in the single trial broadband EEG during movement intention. We obtained LRTC in 2 s windows of broadband EEG and modeled it using the Autoregressive Fractionally Integrated Moving Average (ARFIMA) model which allowed simultaneous modeling of short- and long-range temporal correlations. There were significant (p < 0.05) changes in both broadband long- and short-range temporal correlations during movement intention and execution. We discovered that the broadband LRTC and narrowband ERD are complementary processes providing distinct information about movement because eliminating LRTC from the signal did not affect the ERD and conversely, eliminating ERD from the signal did not affect LRTC. Exploring the possibility of applications in Brain Computer Interfaces (BCI), we used hybrid features with combinations of LRTC, ARFIMA, and ERD to detect movement intention. A significantly higher (p < 0.05) classification accuracy of 88.3 ± 4.2% was obtained using the combination of ARFIMA and ERD features together, which also predicted the earliest movement at 1 s before its onset. The ongoing changes in the long- and short-range temporal correlations in broadband EEG contribute to effectively capturing the motor command generation and can be used to detect movement successfully. These temporal dependencies provide different and additional information about the movement

Computational aspects of Bayesian spectral density estimation

Gaussian time-series models are often specified through their spectral density. Such models present several computational challenges, in particular because of the non-sparse nature of the covariance matrix. We derive a fast approximation of the likelihood for such models. We propose to sample from the approximate posterior (that is, the prior times the approximate likelihood), and then to recover the exact posterior through importance sampling. We show that the variance of the importance sampling weights vanishes as the sample size goes to infinity. We explain why the approximate posterior may typically multi-modal, and we derive a Sequential Monte Carlo sampler based on an annealing sequence in order to sample from that target distribution. Performance of the overall approach is evaluated on simulated and real datasets. In addition, for one real world dataset, we provide some numerical evidence that a Bayesian approach to semi-parametric estimation of spectral density may provide more reasonable results than its Frequentist counter-parts

Special Functions: Fractional Calculus and the Pathway for Entropy

Historically, the notion of entropy emerged in conceptually very distinct contexts. This book deals with the connection between entropy, probability, and fractional dynamics as they appeared, for example, in solar neutrino astrophysics since the 1970's (Mathai and Rathie 1975, Mathai and Pederzoli 1977, Mathai and Saxena 1978, Mathai, Saxena, and Haubold 2010). The original solar neutrino problem, experimentally and theoretically, was resolved through the discovery of neutrino oscillations and was recently enriched by neutrino entanglement entropy. To reconsider possible new physics of solar neutrinos, diffusion entropy analysis, utilizing Boltzmann entropy, and standard deviation analysis was undertaken with Super-Kamiokande solar neutrino data. This analysis revealed a non-Gaussian signal with harmonic content. The Hurst exponent is different from the scaling exponent of the probability density function and both Hurst exponent and scaling exponent of the Super-Kamiokande data deviate considerably from the value of ½, which indicates that the statistics of the underlying phenomenon is anomalous. Here experiment may provide guidance about the generalization of theory of Boltzmann statistical mechanics. Arguments in the so-called Boltzmann-Planck-Einstein discussion related to Planck's discovery of the black-body radiation law are recapitulated mathematically and statistically and emphasize from this discussion is pursued that a meaningful implementation of the complex ‘entropy-probability-dynamics’ may offer two ways for explaining the results of diffusion entropy analysis and standard deviation analysis. One way is to consider an anomalous diffusion process that needs to use the fractional space-time diffusion equation (Gorenflo and Mainardi) and the other way is to consider a generalized Boltzmann entropy by assuming a power law probability density function. Here new mathematical framework, invented by sheer thought, may provide guidance for the generalization of Boltzmann statistical mechanics. In this book Boltzmann entropy, generalized by Tsallis and Mathai, is considered. The second one contains a varying parameter that is used to construct an entropic pathway covering generalized type-1 beta, type-2 beta, and gamma families of densities. Similarly, pathways for respective distributions and differential equations can be developed. Mathai's entropy is optimized under various conditions reproducing the well-known Boltzmann distribution, Raleigh distribution, and other distributions used in physics. Properties of the entropy measure for the generalized entropy are examined. In this process the role of special functions of mathematical physics, particularly the H-function, is highlighted

Long range dependence in South African Platinum prices under heavy tailed error distributions

South Africa is rich in platinum group metals (PGMs) and these metals are important in providing jobs as well as investments some of which have been seen in the Johannesburg Securities Exchange (JSE). In this country this sector has experienced some setbacks in recent times. The most notable ones are the 2008/2009 global nancial crisis and the 2012 major nationwide labour unrest. Worrisomely, these setbacks keep simmering. These events usually introduce jumps and breaks in data which changes the structure of the underlying information thereby inducing spurious long memory (long range dependence). Thus it is recommended that these two phenomena must be addressed together. Further, it is well-known that nancial returns are dominated by stylized facts. In this thesis we carried out an investigation on distributional properties of platinum returns, structural changes, long memory and stylized facts in platinum returns and volatility series. To understand the distributional properties of the returns, we used two classes of heavy tailed distributions namely the alpha-Stable distributions and generalized hyperbolic distributions. We then investigated structural changes in the platinum return series and changes in long range dependence and volatility. Using Akaike information criterion, the ARFIMA-FIAPARCH under the Student distribution was selected as the best model for platinum although the ARCH e ects were slightly signi cant, while using the Schwarz information criteria the ARFIMA-FIAPARCH under the Normal distribution. Further, ARFIMA-FIEGARCH under the skewed Student distribution and ARFIMA-HYGARCH under the Normal distribution models were able to capture the ARCH effects. The best models with respect to prediction excluded the ARFIMA-FIGARCH model and were dominated by ARFIMA-FIAPARCH model with non-Normal error distributions which indicates the importance of asymmetry and heavy tailed error distributions.StatisticsM. Sc. (Statistics

Time Varying Parameter Models for Inflation and Exchange Rates

Een van de onderwerpen binnen de Econometrie is het zoeken naar de structuur die schuil gaat achter macroeconomische reeksen. Dit proefschrift gaat in op twee van der gelijke reeksen, namelijk inflatie en wisselkoers reeksen. Uiteraard zijn dergelijke reeksen al vele malen geanalyseerd, met wisselend resultaat. In dit proefschrift wordt gepoogd op nieuwe wijze de data te analyseren, waarbij speciaal aandacht is voor modellen met tijdsvariÄerende parameters. Uit de gereedschapskist van de econometrie worden diverse technieken tevoorschijn gehaald die de beste (lees: meest gedetailleerde, waarheidsgetrou we) beschrijving beloven te leveren van de structuren die aan d

Ongoing temporal dynamics of broadband EEG during movement intention for BCI

Brain Computer Interface (BCI) empowers individuals with severe movement impairing conditions to interact with the computers directly by their thoughts, without the involvement of any motor pathways. Motor-based BCIs can offer intuitive control by merely intending to move. Hence, to develop effective motor-based non-invasive BCIs, it is essential to understand the mechanisms of neural processes involved in motor command generation in electroencephalography (EEG). The EEG consists of complex narrowband oscillatory and broadband arrhythmic processes. However, there is more focus on the oscillations in different frequency bands for studying motor command generation in the literature. The narrowband processes such as event-related (de)synchronisation (ERD/S) and movement-related cortical potential (MRCP) are commonly used for movement detection. Analysis of these narrowband EEG components disregards the information existing in the rest of the frequencies and their dynamics. Hence, this thesis investigates various facets of previously unexplored temporal dynamics of neuronal processes in the broadband arrhythmic EEG to fill the gap in the knowledge of motor command generation on a single trial basis in the BCI framework. The temporal dynamics of the broadband EEG were characterised by the decay of its autocorrelation. The autocorrelation decayed according to the power-law resulting in the longrange temporal correlations (LRTC). The instantaneous ongoing changes in the broadband LRTC were uniquely quantified by the Hurst exponent on very short EEG sliding windows. There was an increase in the temporal dependencies in the EEG leading to slower decay of autocorrelation during the movement and significant increase in the LRTC (p<0.05). Different types of temporal dependencies in the broadband EEG were comprehensively examined further by modelling the long and short-range correlations together using autoregressive fractionally integrated moving average model (ARFIMA). The short-range correlations also changed significantly (p<0.05) during the movement. These ongoing changes in the dynamics of the broadband EEG were able to predict the movement 1 s before its onset with accuracy higher than ERD and MRCP. The LRTCs were robust across participants and did not require determination of participant specific parameters such as most responsive spectral or spatial components