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Robustness of a system has been defined in various ways and a lot of work has been done to model the system robustness , but quantifying or measuring robustness has always been very difficult. In this research we consider a simple system of a linear estimator and then attempt to model the system performance and robustness in a geometrical manner which admits an analysis using the differential geometric concepts of slope and curvature. We try to compare two different types of curvatures, namely the curvature along the maximum slope of a surface and the square-root of the absolute value of sectional curvature of a surface, and observe the values to see if both of them can alternately be used in the process of understanding or measuring system robustness. In this process we have worked on two different examples and taken readings for many points to find if there is any consistency in the two curvatures.

Progressive Type-II censoring was introduced by Cohen (1963) and has since been the topic of much research. The question stands whether it is sensible to use this sampling plan by design, instead of regular Type-II right censoring. We introduce an asymptotic progressive censoring model, and find optimal censoring schemes for location-scale families. Our optimality criterion is the determinant of the 2x2 covariance matrix of the asymptotic best linear unbiased estimators. We present an explicit expression for this criterion, and conditions for its boundedness. By means of numerical optimization, we determine optimal censoring schemes for the extreme value, the Weibull and the normal distributions. In many situations, it is shown that these progressive schemes significantly improve upon regular Type-II right censoring.

In this paper an overview is given on the application of directional basis functions, known under the name Curvelets/Contourlets, to various aspects of seismic processing and imaging. Key concepts in the approach are the use of (i) that localize in both domains (e.g. space and angle); (ii) non-linear estimation, which corresponds to localized muting on the coefficients, possibly supplemented by constrained optimization (iii) invariance of the basis functions under the imaging operators. We will discuss applications that include multiple and ground roll removal; sparseness-constrained least-squares migration and the computation of 4-D difference cubes.

In a large number of applications arising in various fields of study, time series are approximated using linear MMSE estimates. Such approximations include finite order moving average and autoregressive approximations as well as the causal Wiener filter. In this dissertation, we study two topics related to the estimation of wide sense stationary (WSS) time series using linear MMSE estimates. In the first part of this dissertation, we study the asymptotic behaviour of autoregressive (AR) and moving average (MA) approximations. Our objective is to investigate how faithfully such approximations replicate the original sequence, as the model order as well as the number of samples approach infinity. We consider two aspects: convergence of spectral density of MA and AR approximations when the covariances are known and when they are estimated. Under certain mild conditions on the spectral density and the covariance sequence, it is shown that the spectral densities of both approximations converge in L2 as the order of approximation increases. It is also shown that the spectral density of AR approximations converges at the origin under the same conditions. Under additional regularity assumptions, we show that similar results hold for approximations from empirical covariance estimates. In the second part of this dissertation, we address the problem of detecting interdependence relations within a group of time series. Ideally, in order to infer the complete interdependence structure of a complex system, dynamic behaviour of all the processes involved should be considered simultaneously. However, for large systems, use of such a method may be infeasible and computationally intensive, and pairwise estimation techniques may be used to obtain sub-optimal results. Here, we investigate the problem of determining Granger-causality in an interdependent group of jointly WSS time series by using pairwise causal Wiener filters. Analytical results are presented, along with simulations that compare the performance of a method based on finite impulse response Wiener filters to another using directed information, a tool widely used in literature. The problem is studied in the context of cyclostationary processes as well. Finally, a new technique is proposed that allows the determination of causal connections under certain sparsity conditions.

Progressive Type-II censoring was introduced by Cohen (1963) and has since been the topic of much research. The question stands whether it is sensible to use this sampling plan by design, instead of regular Type-II right censoring. We introduce an asymptotic progressive censoring model, and find optimal censoring schemes for location-scale families. Our optimality criterion is the determinant of the 2x2 covariance matrix of the asymptotic best linear unbiased estimators. We present an explicit expression for this criterion, and conditions for its boundedness. By means of numerical optimization, we determine optimal censoring schemes for the extreme value, the Weibull and the normal distributions. In many situations, it is shown that these progressive schemes significantly improve upon regular Type-II right censoring.