Statistical modeling by wavelets [[electronic resource] /] / Brani Vidakovic

• Record Nr.: UNINA9910144684203321
• Autore: Vidakovic Brani <1955->
• Pubbl/distr/stampa: New York, : Wiley, 1999
• ISBN: 1-282-30775-4; 9786612307751; 0-470-31702-7; 0-470-31786-8
• Descrizione fisica: 1 online resource (410 p.)
• Collana: Wiley series in probability and mathematical statistics. Applied probability and statistics section
• Disciplina: 515.2433; 519.5
• Soggetti: Mathematical statistics; Wavelets (Mathematics)
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: "A Wiley-Interscience publication."
• Nota di bibliografia: Includes bibliographical references (p. 345-370) and indexes.
• Nota di contenuto: Statistical Modeling by Wavelets; Contents; Preface; Acknowledgments; 1. Introduction; 1.1. Wavelet Evolution; 1.2. Wavelet Revolution; 1.3. Wavelets and Statistics; 1.4. An Appetizer: California Earthquakes; 2. Prerequisites; 2.1. General; 2.2. Hilben Spaces; 2.2.1. Projection Theorem; 2.2.2. 0rthonomal Sets; 2.2.3. Reproducing Kernel Hilberf Spaces; 2.3. Fourier Transformation; 2.3.1. Basic Properties; 2.3.2. Poisson Summation Formula and Sampling Theorem; 2.3.3. Fourier Series; 2.3.4. Discrete Fourier Transform; 2.4. Heisenberg's Uncertainty Principle; 2.5. Some Important Function Spaces; 2.6. Fundanzentals of Signal Processing2.7. Exercises; 3. Wavelets; 3.1. Continuous Wavelet Transformation; 3.1.1. Basic Properties; 3.1.2. Wavelets for Continuous Transfonnations; 3.2. Discretization of the Continuous Wavelet Transform; 3.3. Multiresolution Analysis; 3.3.1. Derivation of a Wavelet Function; 3.4. Same Important Wavelet Bases; 3.4.1. Haar's Wavelets; 3.4.2. Shannon's Wavelets; 3.4.3. Meyer's Wavelets; 3.4.4. Franklin s Wavelets; 3.4.5. Daubechies ' Conzpactly Supporled Wavelets; 3.5. Some Extensions; 3.5.1. Regularity of Wavelets; 3.5.2. The Least Asytnmetric Daubechies ' Wavelets: Symrnlets3.5.3. Approxintations and Characterizations of Functional Spaces; 3.5.4. Daubechies-Lagarias Algorithm; 3.5.5. Moment Conditions; 3.5.6. Interpolating (Cardinal) Wavelets; 3.5.7. Pollen-Type Parameterization of Wavelets; 3.6. Exercises; 4. Discrete Wavelet Transformations; 4.1. Introduction; 4.2. The Cascade Algorithnt; 4.3. The Operator Notation of DWT; 4.3.1. Discrete Wavelet Transfomiations as Linear Transfonnations; 4.4. Exercises; 5. Some Generalizations; 5.1. Coiflets; 5.1.1. Construction of Coifrets; 5.2. Biorthogonal Wavelets5.2.1. Construction of Biorthogonal Wavelets; 5.2.2. B-Spline Wavelets; 5.3. Wavelet Packets; 5.3.1. Basic Properties of Wavelet Packets; 5.3.2. Wavelet Packet Tables; 5.4. Best Basis Selection; 5.4.1. Some Cost Measures and the Best Basis Algorithm; 5.5. ε-Decimated and Stationary Wavelet Transformations; 5.5.1. ε-Decimated Wavelet Transformation; 5.5.2. Stationary (Non-Decimated) Wavelet Transformation; 5.6. Periodic Wavelet Transformations; 5.7. Multivariate Wavelet Transfornations; 5.8. Discussion; 5.9. Exercises; 6. Wavelet Shrinkage; 6.1. Shrinkage Method; 6.2. Lineur Wavelet Regression Estimators6.2.1. Wavelet Kernels; 6.2.2. Local Constant Fit Estimators; 6.3. The Simplest Non-Linear Wavelet Shrinkage: Tliresholding; 6.3.1. Variable Selection and Thresholding; 6.3.2. Oracular Risk for Thresholding Rules; 6.3.3. Why the Wavelet Shrinkage Works; 6.3.4. Almost Sure Convergence of Wavelet Sh rinkuge Est imaf ors; 6.4. General Minimax Paradigm; 6.4.1. Translation of Minimaxity Results to the Wavelet Domain; 6.5. Thresholding Policies and Thresholdkg Rides; 6.5.1. Exact Risk Analysis of Thresholding Rules; 6.5.2. Large Sample Properties; 6.5.3. Some Orher Shrinkage Rules
• Sommario/riassunto: A comprehensive, step-by-step introduction to wavelets in statistics.What are wavelets? What makes them increasingly indispensable in statistical nonparametrics? Why are they suitable for ""time-scale"" applications? How are they used to solve such problems as denoising, regression, or density estimation? Where can one find up-to-date information on these newly ""discovered"" mathematical objects? These are some of the questions Brani Vidakovic answers in Statistical Modeling by Wavelets. Providing a much-needed introduction to the latest tools afforded statisticians by wavelet theory,