Debashis Mondal

Wavelet Variance Analysis for Random Fields on a Regular Lattice

There has been considerable recent interest in using wavelets to analyze time series and images that can be regarded as realizations of certain 1-D and 2-D stochastic processes on a regular lattice. Wavelets give rise to the concept of the wavelet variance (or wavelet power spectrum), which decomposes the variance of a stochastic process on a scale-by-scale basis. The wavelet variance has been applied to a variety of time series, and a statistical theory for estimators of this variance has been developed. While there have been applications of the wavelet variance in the 2-D context (in particular, in works by Unser in 1995 on wavelet-based texture analysis for images and by Lark and Webster in 2004 on analysis of soil properties), a formal statistical theory for such analysis has been lacking. In this paper, we develop the statistical theory by generalizing and extending some of the approaches developed for time series, thus leading to a large-sample theory for estimators of 2-D wavelet variances. We apply our theory to simulated data from Gaussian random fields with exponential covariances and from fractional Brownian surfaces. We demonstrate that the wavelet variance is potentially useful for texture discrimination. We also use our methodology to analyze images of four types of clouds observed over the southeast Pacific Ocean.

Exact Goodness-of-Fit Tests for Markov Chains

Goodness-of-fit tests are useful in assessing whether a statistical model is consistent with available data. However, the usual χ² asymptotics often fail, either because of the paucity of the data or because a nonstandard test statistic is of interest. In this article, we describe exact goodness-of-fit tests for first- and higher order Markov chains, with particular attention given to time-reversible ones. The tests are obtained by conditioning on the sufficient statistics for the transition probabilities and are implemented by simple Monte Carlo sampling or by Markov chain Monte Carlo. They apply both to single and to multiple sequences and allow a free choice of test statistic. Three examples are given. The first concerns multiple sequences of dry and wet January days for the years 1948-1983 at Snoqualmie Falls, Washington State, and suggests that standard analysis may be misleading. The second one is for a four-state DNA sequence and lends support to the original conclusion that a second-order Markov chain provides an adequate fit to the data. The last one is six-state atomistic data arising in molecular conformational dynamics simulation of solvated alanine dipeptide and points to strong evidence against a first-order reversible Markov chain at 6 picosecond time steps.

A Wavelet Variance Primer

Abstract The wavelet variance is a decomposition of the variance of a time series. Because of its scale-based nature, the wavelet variance offers insight into various time series, particularly in the physical sciences. This primer is a basic introduction to the wavelet variance, starting with its definition in terms of the discrete wavelet transform, proceeding with a discussion of the large-sample statistical properties of its basic estimators, and then continuing with an examination of estimators appropriate for time series with either missing values or contamination by discordant values. The discussion then moves to two uses of the wavelet variance involving its across-scale patterns, namely, estimation of exponents of power-law processes and estimations of characteristic scales. The primer closes with examples of the wavelet variance applied to time series involving atomic clocks, sea-ice thickness, the albedo of Arctic ice, X-ray fluctuations from binary stars, and coherent structures in river flow.

Slepian Wavelet Variances for Regularly and Irregularly Sampled Time Series

We discuss approximate scale-based analysis of variance for Gaussian time series based upon Slepian wavelets. These wavelets arise as eigenfunctions of an energy maximization problem in a pass band of frequencies. Unlike the commonly used Daubechies wavelets, Slepian wavelets have the ability to accommodate both regularly and irregularly sampled data. For regularly sampled Gaussian time series, we derive statistical theory for Slepian-based wavelet variances and show that it is comparable to Daubechies-based variances. For irregularly sampled time series data, we derive a corresponding statistical theory for Slepian-based wavelet variances. We demonstrate its use on X-ray fluctuations from a binary star system and on a light curve from the variable star Z UMa.

Evidence of Systematic Triggering at Teleseismic Distances Following Large Earthquakes

Earthquakes are part of a cycle of tectonic stress buildup and release. As fault zones near the end of this seismic cycle, tipping points may be reached whereby triggering occurs and small forces result in cascading failures. The extent of this effect on global seismicity is currently unknown. Here we present evidence of ongoing triggering of earthquakes at remote distances following large source events. The earthquakes used in this study had magnitudes ≥M5.0 and the time period analyzed following large events spans three days. Earthquake occurrences display increases over baseline rates as a function of arc distance away from the epicenters. The p-values deviate from a uniform distribution, with values for collective features commonly below 0.01. An average global forcing function of increased short term seismic risk is obtained along with an upper bound response. The highest magnitude source events trigger more events, and the average global response indicates initial increased earthquake counts followed by quiescence and recovery. Higher magnitude earthquakes also appear to be triggered more often than lower magnitude events. The region with the greatest chance of induced earthquakes following all source events is on the opposite side of the earth, within 30 degrees of the antipode.

Applying Dynkin's isomorphism: An alternative approach to understand the Markov property of the de Wijs process

Dynkins (Bull. Amer. Math. Soc. 3 (1980) 975-999) seminal work associates a multidimensional transient symmetric Markov process with a multidimensional Gaussian random field. This association, known as Dynkins isomorphism, has profoundly influenced the studies of Markov properties of generalized Gaussian random fields. Extending Dykins isomorphism, we study here a particular generalized Gaussian Markov random field, namely, the de Wijs process that originated in Georges Matherons pioneering work on mining geostatistics and, following McCullagh (Ann. Statist. 30 (2002) 1225-1310), is now receiving renewed attention in spatial statistics. This extension of Dynkins theory associates the de Wijs process with the (recurrent) Brownian motion on the two dimensional plane, grants us further insight into Matherons kriging formula for the de Wijs process and highlights previously unexplored relationships of the central Markov models in spatial statistics with Markov processes on the plane.

Rank Tests from Partially Ordered Data Using Importance and MCMC Sampling Methods

espanolExact tests, fuzzy p-values, Gibbs sampling, iterval censoring, linear extensions, linear rank statistics, perfect MCMC, proportional hazard model, topological sorting EnglishWe discuss distribution-free exact rank tests from partially ordered data that arise in various biological and other applications where the primary objective is to conduct testing of significance to assess the linear dependence or to compare different groups. The tests here are obtained by treating the usual rank statistics, based on the completely ordered data as “latent” or missing, and conceptualizing the “latent” p-value as the random probability under the null hypothesis of a test statistic that is as extreme, or more extreme, than the latent test statistics based on the completely ordered data. The latent p-value is then predicted by sampling linear extensions or the complete orderings that are consistent with the observed partially ordered data. The sampling methods explored here include importance sampling methods based on randomized topological sorting algorithms, Gibbs sampling methods, random-walk based Metropolis–Hasting sampling methods and random-walk based modern perfect Markov chain Monte Carlo sampling methods. We discuss running times of these sampling methods and their strength and weaknesses. A simulation experiment and three data examples are given. The simulation experiment illustrates how the exact rank tests from partially ordered data work when the desired result is known. The first data example concerns the light preference behavior of fruit flies and tests whether heterogeneity observed in average light-preference behavior can be explained by manipulations in serotonin signaling. The second one is a reanalysis of the lead absorption data in children of employees who worked in a lead battery factory and consolidates the results reported in Rosenbaum [Ann. Statist. 19 (1991) 1091–1097]. The third one reexamines the breast cosmesis data from Finkelstein [Biometrics 42 (1986) 845–854].