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Don Percival

Senior Principal Mathematician

Professor, Statistics





Research Interests

Statistics, Spectral Analysis, Wavelets


Dr. Percival is interested in the application of statistical methodology in the physical sciences. His background includes teaching and research in time series and spectral analysis, simulation of stochastic processes, computational environments for interactive time series and signal analysis, statistical analysis of biomedical time series and underwater turbulence, and wavelets.

He is the co-author of the textbooks Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques (1993) and Wavelet Methods for Time Series Analysis (2000), both published by Cambridge University Press. Dr. Percival serves an Associate Editor of the Journal of Computational and Graphical Statistics. He has been with the Laboratory since 1983.


B.A. Astronomy, University of Pennsylvania, 1968

M.A. Mathematical Statistics, George Washington University, 1976

Ph.D. Mathematical Statistics, University of Washington, 1983


2000-present and while at APL-UW

Exact simulation of noncircular or improper complex-valued stationary Gaussian processes using circulant embedding

Sykulski, A.M., and D.B. Percival, "Exact simulation of noncircular or improper complex-valued stationary Gaussian processes using circulant embedding," Proc., IEEE International Workshop on Machine Learning for Signal Processing, 13-16 September, Salerno, Italy (IEEE, 2016).

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13 Sep 2016

This paper provides an algorithm for simulating improper (or noncircular) complex-valued stationary Gaussian processes. The technique utilizes recently developed methods for multivariate Gaussian processes from the circulant embedding literature. The method can be performed in Ο(nlog2n) operations, where n is the length of the desired sequence. The method is exact, except when eigenvalues of prescribed circulant matrices are negative. We evaluate the performance of the algorithm empirically, and provide a practical example where the method is guaranteed to be exact for all n, with an improper fractional Gaussian noise process.

A wavelet perspective on the Allan variance

Percival, D.B., "A wavelet perspective on the Allan variance," IEEE Trans. Ultrason., Ferroelect., Freq. Control, 63, 538-554, doi:10.1109/TUFFC.2015.2495012, 2016.

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1 Apr 2016

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.

Detiding DART® buoy data for real-time extraction of source coefficients for operational tsunami forecasting

Percival, D.B., and 8 others, "Detiding DART® buoy data for real-time extraction of source coefficients for operational tsunami forecasting," Pure Appl. Geophys., 172, 1653-1678, doi:10.1007/s00024-014-0962-0, 2015.

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1 Jun 2015

U.S. Tsunami Warning Centers use real-time bottom pressure (BP) data transmitted from a network of buoys deployed in the Pacific and Atlantic Oceans to tune source coefficients of tsunami forecast models. For accurate coefficients and therefore forecasts, tides and background noise at the buoys must be accounted for through detiding. In this study, five methods for coefficient estimation are compared, each of which handles detiding differently. The first three subtract off a tidal prediction based on (1) a localized harmonic analysis involving 29 days of data immediately preceding the tsunami event, (2) 68 preexisting harmonic constituents specific to each buoy, and (3) an empirical orthogonal function fit to the previous 25 h of data. Method (4) is a Kalman smoother that uses method (1) as its input. These four methods estimate source coefficients after detiding. Method (5) estimates the coefficients simultaneously with a two-component harmonic model that accounts for the tides. The five methods are evaluated using archived data from 11 DART® buoys, to which selected artificial tsunami signals are superimposed. These buoys represent a full range of observed tidal conditions and background BP noise in the Pacific and Atlantic, and the artificial signals have a variety of patterns and induce varying signal-to-noise ratios. The root-mean-square errors (RMSEs) of least squares estimates of source coefficients using varying amounts of data are used to compare the five detiding methods. The RMSE varies over two orders of magnitude among detiding methods, generally decreasing in the order listed, with method (5) yielding the most accurate estimate of the source coefficient. The RMSE is substantially reduced by waiting for the first full wave of the tsunami signal to arrive. As a case study, the five methods are compared using data recorded from the devastating 2011 Japan tsunami.

More Publications

Acoustics Air-Sea Interaction & Remote Sensing Center for Environmental & Information Systems Center for Industrial & Medical Ultrasound Electronic & Photonic Systems Ocean Engineering Ocean Physics Polar Science Center