A unified approach to nonparametric spectrum estimation algorithms

Journal ArticleAbstract-Different approaches to spectrum estimation can be broadly classified as parametric and nonparametric methods. In the parametric techniques, an underlying model is assumed in the formulation of the spectrum estimation problem and one estimates the parameters of the model. For nonparametric methods, no such model is assumed. In this paper, several nonparametric spectrum estimation algorithms are brought under a unified framework by the introduction of a generalized nonparametric spectrum estimation algorithm A four-stage approach is employed. It contains as special cases the Blackman-Tukey algorithm, the weighted, overlapped segment averagin(Wg OSA) method, the lag-reshape approach, Rader's algorithm, and the short-time unbiased spectrum estimation (STUSE) algorithm. The framework proposed in the paper is more general than the one recently proposed by Nuttall and Carter. Theoretical expressions for the spectrum estimation variance of the generalized algorithm are derived, and then verified via simulation example. Also, several nonparametric approaches for obtaining unbiased spectrum estimates are discussed and compared. Finally we conclude the paper with a brief discussion of the applicability and usefulness of several methods in specific situations

Nonlinear Time Series Modeling: A Unified Perspective, Algorithm, and Application

A new comprehensive approach to nonlinear time series analysis and modeling is developed in the present paper. We introduce novel data-specific mid-distribution based Legendre Polynomial (LP) like nonlinear transformations of the original time series Y(t) that enables us to adapt all the existing stationary linear Gaussian time series modeling strategy and made it applicable for non-Gaussian and nonlinear processes in a robust fashion. The emphasis of the present paper is on empirical time series modeling via the algorithm LPTime. We demonstrate the effectiveness of our theoretical framework using daily S&P 500 return data between Jan/2/1963 - Dec/31/2009. Our proposed LPTime algorithm systematically discovers all the `stylized facts' of the financial time series automatically all at once, which were previously noted by many researchers one at a time.Comment: Major restructuring has been don

Dynamic Experiment Design Regularization Approach to Adaptive Imaging with Array Radar/SAR Sensor Systems

We consider a problem of high-resolution array radar/SAR imaging formalized in terms of a nonlinear ill-posed inverse problem of nonparametric estimation of the power spatial spectrum pattern (SSP) of the random wavefield scattered from a remotely sensed scene observed through a kernel signal formation operator and contaminated with random Gaussian noise. First, the Sobolev-type solution space is constructed to specify the class of consistent kernel SSP estimators with the reproducing kernel structures adapted to the metrics in such the solution space. Next, the “model-free” variational analysis (VA)-based image enhancement approach and the “model-based” descriptive experiment design (DEED) regularization paradigm are unified into a new dynamic experiment design (DYED) regularization framework. Application of the proposed DYED framework to the adaptive array radar/SAR imaging problem leads to a class of two-level (DEED-VA) regularized SSP reconstruction techniques that aggregate the kernel adaptive anisotropic windowing with the projections onto convex sets to enforce the consistency and robustness of the overall iterative SSP estimators. We also show how the proposed DYED regularization method may be considered as a generalization of the MVDR, APES and other high-resolution nonparametric adaptive radar sensing techniques. A family of the DYED-related algorithms is constructed and their effectiveness is finally illustrated via numerical simulations