Non-equispaced B-spline wavelets

This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new construction is based on the factorisation of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics.Comment: 42 pages, 2 figure

Nonseparable multivariate wavelets

We review the one-dimensional setting of wavelet theory and generalize it to nonseparable multivariate wavelets. This process presents significant technical difficulties. Some techniques of the one-dimensional setting carry over in a more or less straightforward way; some do not generalize at all.;The main results include the following: an algorithm for computing the moments for multivariate multiwavelets; a necessary and sufficient condition for the approximation order; the lifting scheme for multivariate wavelets; and a generalization of the method of Lai [12] for the biorthogonal completion of a polyphase matrix under suitable conditions.;One-dimensional techniques which cannot be generalized include the factorization of the polyphase matrix, and a general solution to the completion problem

Use of Anisotropic Radial Basis Functions

학위논문(박사) -- 서울대학교대학원 : 자연과학대학 통계학과, 2021.8. 오희석.Spatial inhomogeneity along the one-dimensional curve makes two-dimensional data non-stationary. Curvelet transform, first proposed by Candes and Donoho (1999), is one of the most well-known multiscale methods to represent the directional singularity, but it has a limitation in that the data needs to be observed on equally-spaced sites. On the other hand, radial basis function interpolation is widely used to approximate the underlying function from the scattered data. However, the isotropy of the radial basis functions lowers the efficiency of the directional representation. This thesis proposes a new multiscale method that uses anisotropic radial basis functions to efficiently represent the direction from the noisy scattered data in two-dimensional Euclidean space. Basis functions are orthogonalized across the scales so that each scale can represent a global or local directional structure separately. It is shown that the proposed method is remarkable for representing directional scattered data through numerical experiments. Convergence property and practical issues in implementation are discussed as well.2차원 공간에서 관측되는 비정상 자료는 그 공간적 비동질성이 1차원 곡선을 따라 나타난다. 이러한 방향적 특이성을 표현하기 위한 다중척도 방법론으로는 Candes and Donoho (1999)가 처음 제시한 커브렛 변환이 널리 알려져 있지만 이는 자료가 일정한 간격으로 관측되어야 한다는 제약이 있다. 한편 산재된 자료에 내재된 함수를 근사하기 위해서는 방사기저함수를 이용한 내삽법이 흔히 이용되지만 등방성이 있는 방사기저함수로는 방향성을 효율적으로 표현할 수 없다. 본 학위논문에서는 2차원 유클리드 공간에서 잡음과 함께 산재되어 관측되는 방향성 자료의 효율적인 표현을 위해 비등방성 방사기저함수를 이용한 새로운 다중척도 방법론을 제안한다. 이때 각 스케일에서 전반적인 방향성 구조와 국소적인 방향성 구조를 분리하여 표현하기 위해 기저함수의 스케일 간 직교화가 이루어진다. 제안된 방법이 산재된 방향성 자료를 표현하는 데 있어 우수함을 보이기 위해 모의실험과 실제 자료에 대한 수치실험을 한 결과를 제시하였다. 한편 제안된 방법의 수렴성과 실제 구현 방법에 관한 사안들도 다루었다.1 Introduction 1 2 Multiscale Analysis 4 2.1 Classical wavelet transform 5 2.1.1 Continuous wavelet transform 5 2.1.2 Multiresolution analysis 7 2.1.3 Discrete wavelet transform 10 2.1.4 Two-dimensional wavelet transform 13 2.2 Wavelets for equally-spaced directional data 14 2.2.1 Ridgelets 15 2.2.2 Curvelets 16 2.3 Wavelets for scattered data 19 2.3.1 Lifting scheme 21 2.3.2 Spherical wavelets 23 3 Radial Basis Function Approximation 26 3.1 Radial basis function interpolation 27 3.1.1 Radial basis functions and scattered data interpolation 27 3.1.2 Compactly supported radial basis functions 29 3.1.3 Error bounds 32 3.2 Multiscale representation with radial basis functions 35 3.2.1 Multiscale approximation 35 3.2.2 Error bounds 37 4 Multiscale Representation of Directional Scattered Data 41 4.1 Anisotropic radial basis function approximation 41 4.1.1 Representation of a single linear directional structure 42 4.1.2 Representation of complex directional structure 46 4.1.3 Multiscale representation of the directional structure 46 4.2 Directional wavelets for scattered data 47 4.2.1 Directional wavelets 48 4.2.2 Estimation of coefficients 49 4.2.3 Practical issues in implementation 50 5 Numerical Experiments 57 5.1 Simulation study 57 5.1.1 Scattered observation sites 60 5.1.2 Equally-spaced observation sites 69 5.2 Real data analysis 70 5.2.1 Temperature data in South Korea 70 6 Concluding Remarks 74 6.1 Summary of results 74 6.2 Future research 74 Abstract (in Korean) 82박