Percival Donald B
IEEE Trans Ultrason Ferroelectr Freq Control. 2016 Apr;63(4):538-54. doi: 10.1109/TUFFC.2015.2495012. Epub 2015 Oct 27.
The origins of the Allan variance trace back 50 years ago to two seminal papers, one by Allan (1966) and the other by Barnes (1966). Since then, the Allan variance has played a leading role in the characterization of high-performance time and frequency standards. Wavelets first arose in the early 1980s in the geophysical literature, and the discrete wavelet transform (DWT) became prominent in the late 1980s in the signal processing literature. Flandrin (1992) briefly documented a connection between the Allan variance and a wavelet transform based upon the Haar wavelet. Percival and Guttorp (1994) noted that one popular estimator of the Allan variance-the maximal overlap estimator-can be interpreted in terms of a version of the DWT now widely referred to as the maximal overlap DWT (MODWT). In particular, when the MODWT is based on the Haar wavelet, the variance of the resulting wavelet coefficients-the wavelet variance-is identical to the Allan variance when the latter is multiplied by one-half. The theory behind the wavelet variance can thus deepen our understanding of the Allan variance. In this paper, we review basic wavelet variance theory with an emphasis on the Haar-based wavelet variance and its connection to the Allan variance. We then note that estimation theory for the wavelet variance offers a means of constructing asymptotically correct confidence intervals (CIs) for the Allan variance without reverting to the common practice of specifying a power-law noise type a priori. We also review recent work on specialized estimators of the wavelet variance that are of interest when some observations are missing (gappy data) or in the presence of contamination (rogue observations or outliers). It is a simple matter to adapt these estimators to become estimators of the Allan variance. Finally we note that wavelet variances based upon wavelets other than the Haar offer interesting generalizations of the Allan variance.
阿仑方差的起源可追溯到50年前的两篇开创性论文，一篇由阿仑（1966年）撰写，另一篇由巴恩斯（1966年）撰写。从那时起，阿仑方差在高性能时间和频率标准的表征中发挥了主导作用。小波最早出现在20世纪80年代初的地球物理文献中，离散小波变换（DWT）在20世纪80年代末的信号处理文献中变得突出。弗兰德林（1992年）简要记录了阿仑方差与基于哈尔小波的小波变换之间的联系。珀西瓦尔和古托普（1994年）指出，阿仑方差的一种流行估计器——最大重叠估计器——可以根据现在被广泛称为最大重叠DWT（MODWT）的DWT版本来解释。特别是，当MODWT基于哈尔小波时，所得小波系数的方差——小波方差——与阿仑方差乘以二分之一时相同。因此，小波方差背后的理论可以加深我们对阿仑方差的理解。在本文中，我们回顾基本的小波方差理论，重点是基于哈尔的小波方差及其与阿仑方差的联系。然后我们指出，小波方差的估计理论提供了一种为阿仑方差构建渐近正确置信区间（CIs）的方法，而无需诉诸事先指定幂律噪声类型的常见做法。我们还回顾了关于小波方差的专门估计器的最新工作，这些估计器在某些观测值缺失（间隙数据）或存在污染（异常观测值或离群值）时很有用。将这些估计器改编为阿仑方差的估计器很简单。最后我们指出，基于除哈尔小波之外的小波的小波方差提供了阿仑方差有趣的推广。
