ON THE SPECTRAL PROPERTIES OF MATRICES ASSOCIATED WITH TREND FILTERS

This note is concerned with the spectral properties of matrices associated with linear smoothers. We derive analytical results on the eigenvalues and eigenvectors of smoothing matrices by interpreting the latter as perturbations of matrices belonging to algebras with known spectral properties, such as the circulant and the generalized tau. These results are used to characterize the properties of a smoother in terms of an approximate eigen-decomposition of the associated smoothing matrix.

[1]  Tommaso Proietti,et al.  A Beveridge-Nelson smoother , 2000 .

[2]  Paola Favati,et al.  On a matrix algebra related to the discrete Hartley transform , 1993 .

[3]  Marianne Baxter,et al.  Measuring Business Cycles Approximate Band-Pass Filters for Economic Time Series , 1999 .

[4]  A. Cantoni,et al.  Eigenvalues and eigenvectors of symmetric centrosymmetric matrices , 1976 .

[5]  Dario Bini,et al.  SPECTRAL AND COMPUTATIONAL PROPERTIES OF BAND SYMMETRIC TOEPLITZ MATRICES , 1983 .

[6]  R. Tibshirani,et al.  Linear Smoothers and Additive Models , 1989 .

[7]  W. Cleveland,et al.  Smoothing by Local Regression: Principles and Methods , 1996 .

[8]  David F. Findley,et al.  New Capabilities and Methods of the X-12-ARIMA Seasonal-Adjustment Program , 1998 .

[9]  C. Leser.  A Simple Method of Trend Construction , 1961 .

[10]  D. G. Watts,et al.  Spectral analysis and its applications , 1968 .

[11]  B. Ripley,et al.  Semiparametric Regression: Preface , 2003 .

[12]  Guohua Pan,et al.  Local Regression and Likelihood , 1999, Technometrics.

[13]  A. Walden,et al.  Wavelet Methods for Time Series Analysis , 2000 .

[14]  Tommaso Proietti,et al.  Real Time Estimation in Local Polynomial Regression, with Application to Trend-Cycle Analysis , 2008, 0901.4219.

[15]  D. B. Preston.  Spectral Analysis and Time Series , 1983 .

[16]  B. Silverman,et al.  Nonparametric Regression and Generalized Linear Models: A roughness penalty approach , 1993 .

[17]  W. Härdle,et al.  Statistical Theory and Computational Aspects of Smoothing , 1996 .

[18]  P. Whittle,et al.  Prediction and Regulation by Linear Least‐Squares Methods , 1984 .

[19]  J. Makhoul.  On the eigenvectors of symmetric Toeplitz matrices , 1981 .

[20]  C. T. Fike,et al.  Norms and exclusion theorems , 1960 .

[21]  J. Weaver.  Centrosymmetric (Cross-Symmetric) Matrices, Their Basic Properties, Eigenvalues, and Eigenvectors , 1985 .

[22]  Robert J. Valenza,et al.  Eigenvalues and Eigenvectors , 1993 .

[23]  P. Whittle.  Prediction and Regulation by Linear Least-Square Methods , 1983 .

[24]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[25]  Jeff Dean,et al.  Time Series , 2009, Encyclopedia of Database Systems.

[26]  E. Hannan,et al.  The statistical theory of linear systems , 1989 .

[27]  M. Wand.  Local Regression and Likelihood , 2001 .

[28]  B. Silverman,et al.  Nonparametric Regression and Generalized Linear Models: A roughness penalty approach , 1993 .

[29]  Peter Bloomfield,et al.  Fourier analysis of time series , 1976 .

[30]  M. Kendall,et al.  The advanced theory of statistics , 1945 .

[31]  G. Wahba.  Spline models for observational data , 1990 .

[32]  Donggyu Sul,et al.  Transition Modeling and Econometric Convergence Tests , 2007 .

[33]  Enrico Bozzo,et al.  On the Use of Certain Matrix Algebras Associated with Discrete Trigonometric Transforms in Matrix Displacement Decomposition , 1995, SIAM J. Matrix Anal. Appl..

[34]  Michael R. Chernick,et al.  Wavelet Methods for Time Series Analysis , 2001, Technometrics.

[35]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

[36]  Albrecht Böttcher,et al.  Spectral properties of banded Toeplitz matrices , 1987 .

[37]  Ali H. Sayed,et al.  Displacement Structure: Theory and Applications , 1995, SIAM Rev..

[38]  S. R. Simanca,et al.  On Circulant Matrices , 2012 .

[39]  Robert M. Gray,et al.  Toeplitz and Circulant Matrices: A Review , 2005, Found. Trends Commun. Inf. Theory.

[40]  B. Silverman,et al.  Nonparametric regression and generalized linear models , 1994 .

[41]  C. Chatfield,et al.  Fourier Analysis of Time Series: An Introduction , 1977, IEEE Transactions on Systems, Man, and Cybernetics.

[42]  Debashis Paul,et al.  Tie-respecting bootstrap methods for estimating distributions of sets and functions of eigenvalues , 2009, 0906.2128.

[43]  C. Nelson,et al.  A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘business cycle’☆ , 1981 .

[44]  D. Pollock,et al.  Circulant matrices and time-series analysis , 2000 .

[45]  R. Fisher.  The Advanced Theory of Statistics , 1943, Nature.

[46]  Robert M. Gray,et al.  Toeplitz And Circulant Matrices: A Review (Foundations and Trends(R) in Communications and Information Theory) , 2006 .

[47]  B. Quenneville,et al.  Seasonal adjustment with the X-11 method , 2001 .

[48]  U. Grenander,et al.  Statistical analysis of stationary time series , 1957 .

[49]  E. J. Hannan,et al.  Multiple time series , 1970 .

[50]  Edmund Taylor Whittaker.  On a New Method of Graduation , 1922, Proceedings of the Edinburgh Mathematical Society.

[51]  E. Prescott,et al.  Postwar U.S. Business Cycles: An Empirical Investigation , 1997 .

[52]  J. Durbin,et al.  Local trend estimation and seasonal adjustment of economic and social time series (with discussion) , 1982 .

[53]  Tommaso Proietti,et al.  Real Time Estimation in Local Polynomial Regression, with Application to Trend-Cycle Analysis , 2008 .

[54]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[55]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .

[56]  Alessandra Luati,et al.  A linear transformation and its properties with special applications in time series filtering , 2004 .

[57]  T. Anderson.  Statistical analysis of time series , 1974 .