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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

Evaluating the effectiveness of DART® Buoy networks based on forecast accuracy

Percival, D.B., D.W. Denbo, E. Gica, P.Y. Huang, H.O. Mofjeld, M.C. Spillane, and V.V. Titov, "Evaluating the effectiveness of DART® Buoy networks based on forecast accuracy," Pure Appl. Geophys., 175, 1445-1471, doi:10.1007/s00024-018-1824-y, 2018.

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1 Apr 2018

A performance measure for a DART® tsunami buoy network has been developed. DART® buoys are used to detect tsunamis, but the full potential of the data they collect is realized through accurate forecasts of inundations caused by the tsunamis. The performance measure assesses how well the network achieves its full potential through a statistical analysis of simulated forecasts of wave amplitudes outside an impact site and a consideration of how much the forecasts are degraded in accuracy when one or more buoys are inoperative. The analysis uses simulated tsunami amplitude time series collected at each buoy from selected source segments in the Short-term Inundation Forecast for Tsunamis database and involves a set for 1000 forecasts for each buoy/segment pair at sites just offshore of selected impact communities. Random error-producing scatter in the time series is induced by uncertainties in the source location, addition of real oceanic noise, and imperfect tidal removal. Comparison with an error-free standard leads to root-mean-square errors (RMSEs) for DART® buoys located near a subduction zone. The RMSEs indicate which buoy provides the best forecast (lowest RMSE) for sections of the zone, under a warning-time constraint for the forecasts of 3 h. The analysis also shows how the forecasts are degraded (larger minimum RMSE among the remaining buoys) when one or more buoys become inoperative. The RMSEs provide a way to assess array augmentation or redesign such as moving buoys to more optimal locations. Examples are shown for buoys off the Aleutian Islands and off the West Coast of South America for impact sites at Hilo HI and along the US West Coast (Crescent City CA and Port San Luis CA, USA). A simple measure (coded green, yellow or red) of the current status of the network’s ability to deliver accurate forecasts is proposed to flag the urgency of buoy repair.

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.

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