A Wavelet Perspective on the Allan Variance.

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.

A multiscale wavelet-based test for isotropy of random fields on a regular lattice.

A test for isotropy of images modeled as stationary or intrinsically stationary random fields on a lattice is developed. The test is based on the wavelet theory, and can operate on the horizontal and vertical scale of choice, or on any combination of scales. Scale is introduced through the wavelet variances (sometimes called as the wavelet power spectrum), which decompose the variance over different horizontal and vertical spatial scales. The method is more general than existing tests for isotropy, since it handles intrinsically stationary random fields as well as second-order stationary fields. The performance of the method is demonstrated on samples from different random fields, and compared with three existing methods. It is competitive with or outperforms existing methods since it consistently rejects close to the nominal level for isotropic fields while having a rejection rate for anisotropic fields comparable with the existing methods in the stationary case, and superior in the intrinsic case. As practical examples, paper density images of handsheets and mammogram images are analyzed.

A wavelet-based multiscale ensemble time-scale algorithm.

The wide-spread availability of ensembles of high-performance clocks has motivated interest in time-scale algorithms. There are many such algorithms in use today in applications ranging from scientific to commercial. Although these algorithms differ in key aspects and are sometimes tailored for specific applications and mixtures of clocks, they all share the goal of combining measured time differences between clocks to form a reference time scale that is more stable than any of the clocks in the ensemble. A new approach to forming time scales is presented here, the multiscale ensemble timescale (METS) algorithm. This approach is based on a multiresolution analysis afforded by the discrete wavelet transform. The algorithm does not assume a specific parametric model for the clocks involved and hence is well-suited for an ensemble of highly disparate clocks. The approach is based on an appealing optimality criterion which yields a reference time scale that is more stable than the constituent clocks over all averaging intervals (scales). The METS algorithm is presented here in detail and is shown in a simulation study to compare favorably with a time-scale algorithm based on Kalman filtering.

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.

Depressed mood during the menopausal transition and early postmenopause: observations from the Seattle Midlife Women's Health Study.

OBJECTIVE: To characterize patterns of depressed mood during the menopausal transition (MT) in relation to age and MT-related factors and to assess the contribution of factors related to depressed mood at earlier points in the life span. DESIGN: Women (N = 508) were recruited from 1990 to 1992 from multiethnic neighborhoods and followed annually through 2005: 302 met the eligibility criteria for analyses reported here. The Center for Epidemiologic Studies Depression scale (CES-D) and a menstrual calendar were completed annually throughout the study. A subset of women provided a first morning voided urine specimen from 1997 through 2005. Urine samples were assayed for estrone glucuronide, follicle-stimulating hormone, testosterone, and cortisol. Mixed effects modeling was used to identify changes in CES-D scores over time, including the relationship to age, MT-related factors, and factors related to depression at other points in the life span (postpartum depression/blues, life stress, or family history of clinical depression). RESULTS: Age was modestly and negatively related to CES-D scores, but MT stage alone was not, except that the late MT stage was significantly related to depressed mood. Hot flash activity, life stress, family history of depression, history of "postpartum blues," sexual abuse history, body mass index, and use of antidepressants were also individually related to depressed mood; the hormonal assays and age of entry into and duration of late MT stage were unrelated. CONCLUSIONS: Although women in the late MT stage are vulnerable to depressed mood, factors that account for depressed mood earlier in the life span continue to have an important influence and should be considered in studies of etiology and therapeutics.

Hot flash severity in hormone therapy users/nonusers across the menopausal transition.

OBJECTIVES: The purpose of this study was to examine the pattern of and factors that influence hot flash severity across the menopausal transition (MT) and early postmenopause (PM). METHODS: Women from the Seattle Midlife Women's Health Study (N=302) provided data for these analyses: at least one annual health questionnaire and a menstrual calendar. A subset of women provided a first morning voided urine specimen from 1997 through 2005. Urine samples were assayed for estrone glucuronide and FSH. Linear mixed effects modeling was used to identify change in hot flash severity scores over time, including the relationship to age, MT-related, psychosocial and lifestyle factors. RESULTS: Increases in hot flash severity were associated with late transition stage, early postmenopause, use of HRT, duration of early transition stage, age of entry into early PM and level of FSH. Age of entry into early transition and estrone levels were associated with decreased hot flash severity. Not associated with hot flash severity were being in early transition stage, age of entry into or duration of late transition stage and all of the psychosocial (anxiety, stress, depressed mood) and lifestyle variables (BMI, activity level, sleep, alcohol use). CONCLUSIONS: Variables associated with reproductive aging independently predicted changes in hot flash severity; psychosocial and lifestyle variables did not. The effect of age dropped out when factors associated with reproductive aging were considered. Use of HRT ameliorated but did not eliminate severe hot flashes suggesting that there is room for alternative approaches less likely to cause harm.

Symptoms during the menopausal transition and early postmenopause and their relation to endocrine levels over time: observations from the Seattle Midlife Women's Health Study.

OBJECTIVE: To determine whether hot flashes, depressed mood, sleep, cognitive and sexual symptoms correlate with urinary follicle-stimulating hormone (FSH), estrone (E(1)G), and testosterone (T) and with each other during the menopausal transition and early postmenopause (PM). METHODS: Forty-one women who transitioned from middle or late transition stage to PM rated symptoms and provided monthly urine specimens as part of a longitudinal study of the menopausal transition. RESULTS: Correlations between endocrine levels and symptom severity ratings over time revealed that hot flash severity was significantly and positively related to FSH and negatively to E1 G. Vaginal dryness was positively correlated with FSH and negatively correlated with T. Decreased sexual desire was correlated negatively with E(1)G levels. Forgetfulness was positively correlated with FSH; difficulty concentrating was negatively correlated with T. Severity of sleep symptoms and depressed mood were not correlated with E(1)G, FSH, or T. Correlations among the symptoms revealed that severity of hot flashes was associated with sleep disruption and forgetfulness. Depressed mood was correlated with sleep disruption, difficulty concentrating, and decreased sexual desire but not with hot flashes or vaginal dryness. Awakening during the night was correlated with decreased sexual desire and vaginal dryness, as well as hot flashes. Forgetfulness was associated with hot flashes and difficulty concentrating, whereas difficulty concentrating was associated with depressed mood and early awakening. CONCLUSIONS: Symptoms many women experience during the menopausal transition and early PM are related to different endocrine levels (FSH, E(1)G, and T).

Evaluating scaled windowed variance methods for estimating the Hurst coefficient of time series.

Three-scaled windowed variance methods (standard, linear regression detrended, and brdge detrended) for estimating the Hurst coefficient (H) are evaluated. The Hurst coefficient, with 0 < H < 1, characterizes self-similar decay in the time-series autocorrelation function. The scaled windowed variance methods estimate H for fractional Brownian motion (fBm) signals which are cumulative sums of fractional Gaussian noise (fGn) signals. For all three methods both the bias and standard deviation of estimates are less than 0.05 for series having N ≥ 2(9) points. Estimates for short series (N < 2(8)) are unreliable. To have a 0.95 probability of distinguishing between two signals with true H differing by 0.1, more than 2(15) points are needed. All three methods proved more reliable (based on bias and variance of estimates) than Hurst's rescaled range analysis, periodogram analysis, and autocorrelation analysis, and as reliable as dispersional analysis. The latter methods can only be applied to fGn or differences of fBm, while the scaled windowed variance methods must be applied to fBm or cumulative sums of fGn.