Low-Pass Filter Design using Locally Weighted Polynomial Regression and Discrete Prolate Spheroidal Sequences

Proietti, Tommaso and Luati, Alessandra (2009): Low-Pass Filter Design using Locally Weighted Polynomial Regression and Discrete Prolate Spheroidal Sequences.

Abstract

The paper concerns the design of nonparametric low-pass filters that have the property of reproducing a polynomial of a given degree. Two approaches are considered. The first is locally weighted polynomial regression (LWPR), which leads to linear filters depending on three parameters: the bandwidth, the order of the fitting polynomial, and the kernel. We find a remarkable linear (hyperbolic) relationship between the cutoff period (frequency) and the bandwidth, conditional on the choices of the order and the kernel, upon which we build the design of a low-pass filter. The second hinges on a generalization of the maximum concentration approach, leading to filters related to discrete prolate spheroidal sequences (DPSS). In particular, we propose a new class of lowpass filters that maximize the concentration over a specified frequency range, subject to polynomial reproducing constraints. The design of generalized DPSS filters depends on three parameters: the bandwidth, the polynomial order, and the concentration frequency. We discuss the properties of the corresponding filters in relation to the LWPR filters, and illustrate their use for the design of low-pass filters by investigating how the three parameters are related to the cutoff frequency.

• Item Type: MPRA Paper
• Language: English
• Keywords: Trend filters; Kernels; Concentration; Filter Design.
• Subjects: E - Macroeconomics and Monetary Economics > E3 - Prices, Business Fluctuations, and Cycles > E32 - Business Fluctuations ; Cycles; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes

References

Anderson T.W. (1971), The Statistical Analysis of Time Series, Wiley.

Baxter, M., King, R.G. (1999). Measuring Business Cycles: Approximate Band-Pass Filters for Economic Time Series. The Review of Economics and Statistics, 81, 575-593.

Behrens R.T. (1994), Signal Processing Applications of Oblique Projection Operators, IEEE Transactions on Signal Processing, 42, 6, 1413-1424.

Christiano L.J., Fitzgerald T.J., (2003), The band pass filter, International Economic Review, 44, 435-465. Cleveland W.S. (1979). Robust Locally Weighted Regression and Smoothing Scatterplots, Journal of the American Statistical Society, 74, 368, 829-836.

Cleveland W.S., Cleveland R.B, McRae J.E., Terpenning I., (1990). STL: A Seasonal-Trend Decomposition Procedure Based on Loess (with Discussion), Journal of Official Statistics, 6, 3-73.

Cleveland W.S. and Loader C.L. (1996). Smoothing by Local Regression: Principles and Methods. In W. Haerdle and M. G. Schimek, editors, Statistical Theory and Computational Aspects of Smoothing, 1049. Springer, New York.

Dhillon I.S., Parlett B.N. (2004), Multiple Representations to Compute Orthogonal Eigenvectors of Symmetric Tridiagonal Matrices, Linear Algebra and its Applications, 387, 1-28.

Eberhard A. (1973), An Optimal DiscreteWindow for the Calculation of Power Spectra, IEEE Transactions on Audio and Electroacoustics, 21, 1, 37-43.

Epanechnikov V.A. (1969), Nonparametric Estimation of a Multivariate Probability Density, Theory of Probability and Applications, 14, 153-158.

Fan J. and Gjibels I. (1996). Local Polynomial Modelling and its Applications, Springer.

Fan J. and Yao Q. (2005). Nonlinear Time Series. Nonparametric and Parametric Methods, Chapman and Hall, New York.

Findley, D.F., Monsell, B.C., Bell, W.R., Otto, M.C., Chen B. (1998). New Capabilities and Methods of the X12-ARIMA Seasonal Adjustment Program, Journal of Business and Economic Statistics, 16, 2.

Hannan E.J. (1970), Multiple Time Series, John Wiley and Sons, New York.

Hastie T.J and Tibshirani R.J. (1990). Generalized Additive Models, Chapman and Hall, London.

Henderson R. (1916), Note on Graduation by Adjusted Average, Transaction of the Actuarial Society of America, 17, 43-48.

Kendall M. G. (1973). Time Series, Oxford University Press, Oxford.

Kendall M., Stuart, A., and Ord, J.K. (1983). The Advanced Theory of Statistics, Vol 3. C. Griffin.

Kenny P.B., and Durbin J. (1982). Local Trend Estimation and Seasonal Adjustment of Economic and Social Time Series, Journal of the Royal Statistical Society A, 145, I, 1-41.

Ladiray, D. and Quenneville, B. (2001). Seasonal Adjustment with the X-11 Method (Lecture Notes in Statistics), Springer-Verlag, New York.

Landau H.J., Pollak H.O. (1961), Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty, II, The Bell System Technical Journal, 40, 65-84.

Landau H.J., Pollak H.O. (1962), Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty, III, The Bell System Technical Journal, 41, 1295-1336.

Loader, C. (1999). Local Regression and Likelihood. Springer-Verlag, New York.

Lii K.S., Rosenblatt M. (2008), Prolate spheroidal spectral estimates, Statistics and Probability Letters, 78, 11, 1339-1348.

Macaulay F.R. (1931), The Smoothing of Time Series, NBER. Mathews J.D., Breakall J.K., Karawas G.D. (1994), The Discrete Prolate Spheroidal Filter as a Digital Signal Processing Tool, IEEE Transactions on Acoustics, Speech and Signal Processing, 33, 6, 1471-1478.

Meyer C.D., (2000), Matrix Analysis and Applied Linear Algebra, SIAM.

Musgrave J. (1964). A Set of End Weights to End All End Weights. Working paper. Census Bureau, Washington.

Nadaraya, E.A. (1964). On estimating regression, Theory of Probability and its Applications, 10, 186-190.

Papoulis A., Bertran M.S. (1970), Digital Filtering and Prolate Functions, IEEE Transactions on Circuit Theory, 19, 6, 674-681.

Percival D., Walden A. (1993). Spectral Analysis for Physical Applications. Cambridge University Press.

Proietti T., Luati A. (2008), Real Time Estimation in Local Polynomial Regression, Annals of Applied Statistics, 2, 4, 1523-1553.

Ruppert D., Wand, M.J. and Carroll R.J. (1989). Semiparametric regression, Cambridge University Press.

Slepian D., Pollak H.O. (1961), Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty, I, The Bell System Technical Journal, 40, 43-64.

Slepian D. (1978), Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - V: The Discrete Case, The Bell System Technical Journal, 57, 1371-1430.

Slepian D. (1983), Some Comments on Fourier Analysis, Uncertainty and Modeling, SIAM Review, 25, 3, 379-393.

Thomson D.J. (1982), Spectrum Estimation and Harmonic Analysis, Proceedings of the IEEE, 70, 9, 1055-1096.

Tufts D.W., Francis J.T. (1970), Designing Digital Low-Pass Filters- Comparison of Some Methods and Criteria, IEEE Transactions on Audio and Electroacoustics, 18, 4, 487-494.

Tufts D.W., Rorabacher D.W., MosierW.E. (1970), Designing Simple, Effective Digital Filters, IEEE Transactions on Audio and Electroacoustics, 18, 142-158.

Wand M.P. and Jones M.C. (1995). Kernel Smoothing, Monographs on Statistics and Applied Probability, 60, Chapman&Hall.

Watson, G.S. (1964). Smooth regression analysis, Sankhya Series A, 26, 359-372.

Xu W.Y., Chamaz C. (1984), On the Periodic Discrete Prolate Spheroidal Sequences, SIAM Journal on Applied Mathematics, 44, 6, 1210-1217.