The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.

/pdf/toeplitz-and-circulant-matrices-3e6lj1a9jc.pdf

Toeplitz and circulant matrices

Summary. Variable selection plays an important role in high dimensional statistical modelling which nowadays appears in many areas and is key to various scientific discoveries. For problems of large scale or dimensionality p, accuracy of estimation and computational cost are two top concerns. Recently, Candes and Tao have proposed the Dantzig selector using L1-regularization and showed that it achieves the ideal risk up to a logarithmic factor  log (p). Their innovative procedure and remarkable result are challenged when the dimensionality is ultrahigh as the factor  log (p) can be large and their uniform uncertainty principle can fail. Motivated by these concerns, we introduce the concept of sure screening and propose a sure screening method that is based on correlation learning, called sure independence screening, to reduce dimensionality from high to a moderate scale that is below the sample size. In a fairly general asymptotic framework, correlation learning is shown to have the sure screening property for even exponentially growing dimensionality. As a methodological extension, iterative sure independence screening is also proposed to enhance its finite sample performance. With dimension reduced accurately from high to below sample size, variable selection can be improved on both speed and accuracy, and can then be accomplished by a well-developed method such as smoothly clipped absolute deviation, the Dantzig selector, lasso or adaptive lasso. The connections between these penalized least squares methods are also elucidated.

/pdf/sure-independence-screening-for-ultrahigh-dimensional-3pasjczpn0.pdf

Sure independence screening for ultrahigh dimensional feature space

Variable selection plays an important role in high dimensional statistical modeling which nowadays appears in many areas and is key to various scientific discoveries. For problems of large scale or dimensionality $p$, estimation accuracy and computational cost are two top concerns. In a recent paper, Candes and Tao (2007) propose the Dantzig selector using $L_1$ regularization and show that it achieves the ideal risk up to a logarithmic factor $\log p$. Their innovative procedure and remarkable result are challenged when the dimensionality is ultra high as the factor $\log p$ can be large and their uniform uncertainty principle can fail. 
Motivated by these concerns, we introduce the concept of sure screening and propose a sure screening method based on a correlation learning, called the Sure Independence Screening (SIS), to reduce dimensionality from high to a moderate scale that is below sample size. In a fairly general asymptotic framework, the correlation learning is shown to have the sure screening property for even exponentially growing dimensionality. As a methodological extension, an iterative SIS (ISIS) is also proposed to enhance its finite sample performance. With dimension reduced accurately from high to below sample size, variable selection can be improved on both speed and accuracy, and can then be accomplished by a well-developed method such as the SCAD, Dantzig selector, Lasso, or adaptive Lasso. The connections of these penalized least-squares methods are also elucidated.

Sure Independence Screening for Ultra-High Dimensional Feature Space

Measures of linear dependence and feedback for multiple time series are defined. The measure of linear dependence is the sum of the measure of linear feedback from the first series to the second, linear feedback from the second to the first, and instantaneous linear feedback. The measures are nonnegative, and zero only when feedback (causality) of the relevant type is absent. The measures of linear feedback from one series to another can be additively decomposed by frequency. A readily usable theory of inference for all of these measures and their decompositions is described; the computations involved are modest.

https://www.jstor.org/stable/pdfplus/2287238.pdf

Measurement of Linear Dependence and Feedback between Multiple Time Series

http://automatica.dei.unipd.it/public/Schenato/PSC/2010_2011/gruppo4-Building_termo_identification/IdentificazioneTermodinamica20072008/Biblio/Biblio-PLS/Performance%20of%20some%20variable%20selection%20methods%20when%20multicollinearity.pdf

Performance of some variable selection methods when multicollinearity is present

Part I: Toeplitz Forms: Preliminaries Orthogonal polynomials. Algebraic properties Orthogonal polynomials. Limit properties The trigonometric moment problem Eigenvalues of Toeplitz forms Generalizations and analogs of Toeplitz forms Further generalizations Certain matrices and integral equations of the Toeplitz type Part II: Applications of Toeplitz Forms: Applications to analytic functions Applications to probability theory Applications to statistics Appendix: Notes and references Bibliography Index.

Gabor Szego

Papers

Toeplitz Forms and Their Applications.

Toeplitz forms and their applications