Wavelets in statistics: beyond the standard assumptions
Wavelets in statistics: beyond the standard assumptions
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统计学中的小波：超出标准假设
DOI：
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
Bernard Walter Silverman
中科院分区：
文献类型：
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作者：
Bernard Walter Silverman
小波在统计学中的最初应用是在2J规则间隔的点上给出曲线的观测加上白噪声来估计曲线。简要回顾了在此背景下使用小波方法的基本原理。讨论了标准统计方法的各种扩展。这些包括在存在相关和非平稳噪声的情况下的曲线估计，(0−1)函数的估计，不规则间隔数据和带有重尾噪声的数据的处理，以及图像和形状分析中的可变形模板。重要的工具是贝叶斯方法，其中在小波展开上放置适当的先验，封装了大多数小波系数为零的概念；使用非抽取或平移不变的小波变换；以及在具有任意带限协方差结构的序列的小波系数表内寻找所有级别内协方差的快速算法。介绍了神经生理学、气象学和古病理学的实际应用。最后，对未来可能的研究方向进行了展望。
The original application of wavelets in statistics was to the estimation of a curve given observations of the curve plus white noise at 2J regularly spaced points. The rationale for the use of wavelet methods in this context is reviewed briefly. Various extensions of the standard statistical methodology are discussed. These include curve estimation in the presence of correlated and non–stationary noise, the estimation of (0−1) functions, the handling of irregularly spaced data and data with heavy–tailed noise, and deformable templates in image and shape analysis. Important tools are a Bayesian approach, where a suitable prior is placed on the wavelet expansion, encapsulating the notion that most of the wavelet coefficients are zero; the use of the non–decimated, or translation–invariant, wavelet transform; and a fast algorithm for finding all the within–level covariances within the table of wavelet coefficients of a sequence with arbitrary band–limited covariance structure. Practical applications drawn from neurophysiology, meteorology and palaeopathology are presented. Finally, some directions for possible future research are outlined.