Abstract
Wavelet-based procedures are now indispensable in many areas of modern statistics, for example in regression, density and function estimation, factor analysis, modeling and forecasting of time series, functional data analysis, data mining and classification, with ranges of application areas in science and engineering. Wavelets owe their initial popularity in statistics to shrinkage, a simple and yet powerful procedure efficient for many nonparametric statistical models.
It is error only, and not truth, that shrinks from inquiry.
Thomas Paine (1737–1809)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Filters are indexed by the number of taps and rounded at seven decimal places.
- 2.
This image of Lenna (Sjooblom) Soderberg, a Playboy centerfold from 1972, has become one of the most widely used standard test images in signal processing.
Bibliography
I. Daubechies, Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61 (Society for Industrial and Applied Mathematics, Philadelphia, 1992)
I. Daubechies, J. Lagarias, Two-scale difference equations I. Existence and global regularity of solutions. SIAM J. Math. Anal. 22(5), 1388–1410 (1991)
I. Daubechies, J. Lagarias, Two-scale difference equations II. Local regularity, infinite products of matrices and fractals. SIAM J. Math. Anal. 23(4), 1031–1079 (1992)
C. Heneghan, S.B. Lown, M.C. Teich, Two-dimensional fractional Brownian motion: wavelet analysis and synthesis, in Image Analysis and Interpretation, Proceedings of the IEEE Southwest Symposium. IEEE (1996), pp. 213–217
S.G. Mallat, A theory of multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989a)
S.G. Mallat, Multiresolution approximations and wavelet orthonormal bases of \(l_{2}(\mathbb{R})\). Trans. Am. Math. Soc. 315, 69–87 (1989b)
T. Ogden, Essential Wavelets for Statistical Applications and Data Analysis (Birkhäuser, Boston, 1996)
A. Pinheiro, B. Vidakovic, Estimating the square root of a density via compactly supported wavelets. Comput. Stat. Data Anal. 25, 399–415 (1997)
D. Pollen, SU I (2, F[z, 1∕z]) for F a subfield of c. J. Am. Math. Soc. 3, 611–624 (1990)
H.L. Resnikoff, R.O. Wells Jr., Wavelet Analysis: The Scalable Structure of Information (Springer, New York, 1998)
D. Veitch, P. Abry, Wavelet-based joint estimate of the long-range dependence parameters. IEEE Trans. Inf. Theory 45(3), 878–897 (1999). Special Issue on Multiscale Statistical Signal Analysis and its Applications
B. Vidakovic, Statistical Modeling by Wavelets (Wiley, New York, 1999)
G.G. Walter, X. Shen, Wavelets and Others Orthogonal Systems, 2nd edn. (Chapman & Hall/CRC, Boca Raton, FL, 2000)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 The Author(s)
About this chapter
Cite this chapter
Morettin, P.A., Pinheiro, A., Vidakovic, B. (2017). Wavelets. In: Wavelets in Functional Data Analysis. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-59623-5_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-59623-5_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-59622-8
Online ISBN: 978-3-319-59623-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)