Multiscale interpretation of taut string estimation and its connection to Unbalanced Haar wavelets
Tóm tắt
We compare two state-of-the-art non-linear techniques for nonparametric function estimation via piecewise constant approximation: the taut string and the Unbalanced Haar methods. While it is well-known that the latter is multiscale, it is not obvious that the former can also be interpreted as multiscale. We provide a unified multiscale representation for both methods, which offers an insight into the relationship between them as well as suggesting lessons both methods can learn from each other.
Tài liệu tham khảo
Barlow, R., Bartholomew, D., Bremner, J., Brunk, H.: Statistical Inference under Order Restrictions. Wiley, New York (1972)
Breiman, L., Friedman, J.H., Olshen, RA, Stone, C.J., Steinberg, D., Colla, P.: CART: Classification and Regression Trees. Wadsworth, Belmont (1983)
Brodsky, B.E., Darkhovsky, B.S.: Nonparametric Methods in Change-point Problems. Springer, Berlin (1993)
Comte, F., Rozenholc, Y.: A new algorithm for fixed design regression and denoising. Ann. Inst. Stat. Math. 56, 449–473 (2004)
Davies, P.L., Kovac, A.: Local extremes, runs, strings, and multiresolution. Ann. Stat. 29, 1–65 (2001)
DeVore, R.: Nonlinear approximation. Acta Numer. 7, 51–150 (1998)
Donoho, D.: CART and best-ortho-basis: A connection. Ann. Stat. 25, 1870–1911 (1997)
Donoho, D.L., Johnstone, J.M.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81, 425–455 (1994)
Dümbgen, L., Kovac, A.: Extensions of smoothing via taut strings. Electron. J. Stat. 3, 41–75 (2009)
Engel, J.: The multiresolution histogram. Metrika 46, 41–57 (1997)
Fryzlewicz, P.: Unbalanced Haar technique for nonparametric function estimation. J. Am. Stat. Assoc. 102, 1318–1327 (2007)
Fryzlewicz, P.: Data-driven wavelet-Fisz methodology for nonparametric function estimation. Electron. J. Stat. 2, 863–896 (2008)
Girardi, M., Sweldens, W.: A new class of unbalanced Haar wavelets that form an unconditional basis for Lp on general measure spaces. J. Fourier Anal. Appl. 3, 457–474 (1997)
Kolaczyk, E., Nowak, R.: Multiscale generalised linear models for nonparametric function estimation. Biometrika 92, 119–133 (2005)
Mammen, E., van de Geer, S.: Penalized quasi-likelihood estimation in partial linear models. Ann. Stat. 25, 1014–1035 (1997)
Polzehl, J., Spokoiny, V.: Adaptive weights smoothing with applications to image restoration. J. R. Stat. Soc., Ser. B 62, 335–354 (2000)
Venkatraman, E.S.: Consistency results in multiple change-point problems. Ph.D. Thesis, Stanford University (1993)
Vidakovic, B.: Statistical Modeling by Wavelets. Wiley, New York (1999)