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.

Molecular genetic characterization of endemic red -tailed barb,Gonoproktoptems curmuca (Hamilton - Buchanan, 1807)

The family Cyprinidae is the largest of freshwater fishes and, with the possible exception of Gobiidae, the largest family of vertebrates.Various members of this family are important as food fish, as aquarium fish, and in biological research. In this study, a fish species from this family exclusively found in the west flowing rivers originating from the Western Ghat region — Gonoproktopterus curmuca — was taken for population genetic analysis.There was an urgent need for restoration ecology by the development of apt management strategies to exploit resources judiciously. One of the strategies thus developed for the scientific management of these resources was to identify the natural units of the fishery resources under exploitation (Altukov, 1981). These natural units of a species can otherwise be called as stocks. A stock can be defined as a panmictic population of related individuals within a single species that is genetically distinct from other such populations.It is believed that a species may undergo micro evolutionary process and differentiate into genetically distinct sub-populations or stocks in course of time, if reproductively and geographically isolated.In recent times, there has been a wide spread degradation of natural aquatic environment due to anthropogenic activities and this has resulted in the decline and even extinction of some fish species. In such situations, evaluation of the genetic diversity of fish resources assumes important to conservation.The species selected for the study, was short-listed as one of the candidates for stock-specific, propagation assisted rehabilitation and management programme in rivers where it is naturally distributed. In connection with this, captive breeding and milt cryopreservation techniques of the species have been developed by the National Bureau of Fish Genetic Resources, Lucknow. However, for a scientific stock-specific rehabilitation programme, information on the stock structure and basic genetic profile of the species are essential and that is not available in case of G. curmuca. So the present work was taken up to identify molecular genetic markers like allozymes, microsatellites and RAPDs and, to use these markers to discriminate the distinct populations of the species, if any, in areas of its natural distribution. The genetic markers were found to be powerful tools to analyze the population genetic structure of the red-tailed barb and demonstrated clear cut genetic differentiation between pairs of populations examined. Geographic isolation by land distance is likely to be the factor that contributed to the restricted gene flow between the river systems. So the present study emphasizes the need for stock-wise, propagation assisted-rehabilitation of the natural populations of red-tailed barb, Gonoprokfopterus curmuca.