Modeling and Analysis of Some Heavy Tailed Time Series

The thesis has covered various aspects of modeling and analysis of finite mean time series with symmetric stable distributed innovations. Time series analysis based on Box and Jenkins methods are the most popular approaches where the models are linear and errors are Gaussian. We highlighted the limitations of classical time series analysis tools and explored some generalized tools and organized the approach parallel to the classical set up. In the present thesis we mainly studied the estimation and prediction of signal plus noise model. Here we assumed the signal and noise follow some models with symmetric stable innovations.We start the thesis with some motivating examples and application areas of alpha stable time series models. Classical time series analysis and corresponding theories based on finite variance models are extensively discussed in second chapter. We also surveyed the existing theories and methods correspond to infinite variance models in the same chapter. We present a linear filtering method for computing the filter weights assigned to the observation for estimating unobserved signal under general noisy environment in third chapter. Here we consider both the signal and the noise as stationary processes with infinite variance innovations. We derived semi infinite, double infinite and asymmetric signal extraction filters based on minimum dispersion criteria. Finite length filters based on Kalman-Levy filters are developed and identified the pattern of the filter weights. Simulation studies show that the proposed methods are competent enough in signal extraction for processes with infinite variance.Parameter estimation of autoregressive signals observed in a symmetric stable noise environment is discussed in fourth chapter. Here we used higher order Yule-Walker type estimation using auto-covariation function and exemplify the methods by simulation and application to Sea surface temperature data. We increased the number of Yule-Walker equations and proposed a ordinary least square estimate to the autoregressive parameters. Singularity problem of the auto-covariation matrix is addressed and derived a modified version of the Generalized Yule-Walker method using singular value decomposition.In fifth chapter of the thesis we introduced partial covariation function as a tool for stable time series analysis where covariance or partial covariance is ill defined. Asymptotic results of the partial auto-covariation is studied and its application in model identification of stable auto-regressive models are discussed. We generalize the Durbin-Levinson algorithm to include infinite variance models in terms of partial auto-covariation function and introduce a new information criteria for consistent order estimation of stable autoregressive model.In chapter six we explore the application of the techniques discussed in the previous chapter in signal processing. Frequency estimation of sinusoidal signal observed in symmetric stable noisy environment is discussed in this context. Here we introduced a parametric spectrum analysis and frequency estimate using power transfer function. Estimate of the power transfer function is obtained using the modified generalized Yule-Walker approach. Another important problem in statistical signal processing is to identify the number of sinusoidal components in an observed signal. We used a modified version of the proposed information criteria for this purpose.

Sedimentological and Geotechnical Studies of Coastal Sediments of Central Kerala.

All over the world, several Quaternary proxy data have been used to reconstruct past sea levels, mainly radiocarbon or OSL dating of exposures of marine facies or shore line indicators (e.g. Carr et al., 2010) as well as paleoenvironmental indicators in lagoon or estuary sediments (e.g. Baxter and Meadows, 1999). Estuaries and deltas develop at river mouths during transgressive and regressive phases, respectively (Boyd et al., 1992). In particular, the postglacial Holocene sea-level rise has contributed importantly to the estuary-to-delta transition (Hori et al. 2004). By analyzing radiocarbon ages of the basal or near-basal sediments of the world’s deltas, Stanley and Warne (1994) showed that delta initiation occurred on a worldwide scale after about 8500–6500 years BP and concluded that the initiation was controlled principally by the declining rate of the Holocene sea-level rise. Worldwide there were different regional sea-level changes since the last glacial maximum (LGM) (Irion et al., 2012). Along the northern Canadian coast, for example, sea level has been falling throughout the Holocene due to the glacial rebound of the crust after the last glaciation (Peltier, 1988). This is comparable to the development in Scandinavia (Steffen and Kaufmann, 2005) where sea level drops today. From about Virginia/USA to Mexico there is a constant sea-level rise similar to the Holocene sea-level development of the southern North Sea (e.g. Vink et al., 2007). From the border of Ceará/Rio Grande do Norte down to Patagonia, indicators of Holocene sea level point to a level that was up to 5 m higher than today's mean sea level (Angulo et al., 1999; Martin et al., 2003; Caldas et al., 2006a, b)