Spectral filtering for trend estimation. (English) Zbl 1312.62117
Summary: This paper deals with trend estimation at the boundaries of a time series by means of smoothing methods. After deriving the asymptotic properties of sequences of matrices associated with linear smoothers, two classes of asymmetric filters that approximate a given symmetric estimator are introduced: the reflective filters and antireflective filters. The associated smoothing matrices, though non-symmetric, have analytically known spectral decomposition. The paper analyses the properties of the new filters and considers reflective and antireflective algebras for approximating the eigensystems of time series smoothing matrices. Based on this, a thresholding strategy for a spectral filter design is discussed.
MSC:
| 62M20 | Inference from stochastic processes and prediction |
| 62M15 | Inference from stochastic processes and spectral analysis |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
References:
| [1] | Aricò, A.; Donatelli, M.; Nagy, J.; Serra Capizzano, S., The anti-reflective transform and regularization by filtering, (Numerical Linear Algebra in Signals, Systems, and Control. Numerical Linear Algebra in Signals, Systems, and Control, Lect. Notes Electr. Eng., vol. 80 (2011), Springer Verlag: Springer Verlag Berlin), 1-21 · Zbl 1258.94014 |
| [2] | Aricò, A.; Donatelli, M.; Serra-Capizzano, S., Spectral analysis of the anti-reflective algebra, Linear Algebra Appl., 428, 2-3, 657-675 (2008) · Zbl 1140.15005 |
| [3] | Bhatia, R., Matrix Analysis, Grad. Texts in Math., vol. 169 (1997), Springer-Verlag: Springer-Verlag New York |
| [4] | Dagum, E. B.; Luati, A., A cascade linear filter to reduce revisions and turning points for real time trend-cycle estimation, Econometric Rev., 28, 40-59 (2009) · Zbl 1161.62062 |
| [5] | Dagum, E. B.; Luati, A., Asymmetric filters for trend-cycle estimation, (Bell, W. R.; Holan, S. H.; McElroy, T. S., Economic Time Series: Modeling and Seasonality (2012), Chapman & Hall, CRC: Chapman & Hall, CRC Boca Raton, FL), 213-230 · Zbl 1260.91190 |
| [6] | Findley, D. F.; Monsell, B. C.; Bell, W. R.; Otto, M. C.; Chen, B., New capabilities and methods of the x12-arima seasonal adjustment program, J. Bus. Econom. Statist., 16, 127-176 (1998) |
| [7] | Golinskii, L.; Serra-Capizzano, S., The asymptotic properties of the spectrum of nonsymmetrically perturbed Jacobi matrix sequences, J. Approx. Theory, 144, 1, 84-102 (2007) · Zbl 1111.15013 |
| [8] | Gray, A. G.; Thomson, P. J., Surrogate Henderson filters in x-11, Aust. N. Z. J. Stat., 43, 385-392 (2001) · Zbl 1009.62577 |
| [9] | Gray, A. G.; Thomson, P. J., On a family of finite moving-average trend filters for the ends of series, J. Forecast., 21, 125-149 (2002) |
| [10] | Hansen, P. C.; Nagy, J. G.; O’Leary, D. P., Deblurring Images. Matrices, Spectra, and Filtering, Fundam. Algorithms, vol. 3 (2006), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA · Zbl 1112.68127 |
| [11] | Henderson, R., Note on graduation by adjusted average, Trans. Actuar. Soc. Amer., 17, 43-48 (1916) |
| [12] | Fan, J.; Gjibels, I., Local Polynomial Modelling and Its Applications (1996), Chapman and Hall: Chapman and Hall New York · Zbl 0873.62037 |
| [13] | Ladiray, D.; Quenneville, B., Seasonal Adjustment with the X-11 Method, Lecture Notes in Statist., 127-176 (2001), Springer-Verlag: Springer-Verlag New York · Zbl 0961.62073 |
| [14] | Loader, C., Local Regression and Likelihood, Statist. Comput. (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0929.62046 |
| [15] | Luati, A.; Proietti, T., On the spectral properties of matrices associated with trend filters, Econometric Theory, 26, 321-354 (2010) |
| [16] | Pollock, D. S.G., Handbook of Time Series Analysis, Signal Processing and Dynamics, Signal Processing and Its Applications (1999), Academic Press: Academic Press London · Zbl 0953.62090 |
| [17] | Proietti, T.; Luati, A., Real time estimation in local polynomial regression, with applications to trend-cycle analysis, Ann. Appl. Stat., 2, 1523-1553 (2008) · Zbl 1454.62279 |
| [18] | Ruppert, David; Wand, M. P.; Carroll, R. J., Semiparametric Regression, Camb. Ser. Stat. Probab. Math., vol. 12 (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1038.62042 |
| [19] | Serra-Capizzano, S., A note on antireflective boundary conditions and fast deblurring models, SIAM J. Sci. Comput., 25, 4, 1307-1325 (2003-2004), (electronic) · Zbl 1062.65152 |
| [20] | Wand, M. P.; Jones, M. C., Kernel Smoothing, Monogr. Statist. Appl. Probab. (1995), Chapman and Hall: Chapman and Hall New York · Zbl 0854.62043 |
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