Benjamin Kedem

Spectral analysis and discrimination by zero-crossings

We advance a coherent development of zero-crossing-based methods and theory appropriate for fast signal analysis. Quite a few ideas pertaining to zero-crossing counts found in the literature can be expressed and interpreted with the help of this more general setup. A central issue addressed in some detail is the fruitful connection which exists between zero-crossing counts and linear filtering. This connection is explored and interpreted with the help of a certain zero-crossing spectral representation, and is then applied in spectral analysis, detection, and discrimination. Zero-crossing counts in filtered time series are called higher order crossings. The theme of this work is that higher order crossings analysis provides a useful descriptive as well as an analytical tool that can in many respects match spectral analysis. To a great extent these two types of analysis are, in fact, equivalent, but each emphasizes a different point of view. Advantages offered by higher order crossings are great simplicity and a drastic data reduction.

Bayesian Prediction of Transformed Gaussian Random Fields

Abstract A model for prediction in some types of non-Gaussian random fields is presented. It extends the work of Handcock and Stein to prediction in transformed Gaussian random fields, where the transformation is known to belong to a parametric family of monotone transformations. The Bayesian transformed Gaussian model (BTG) provides an alternative to trans-Gaussian kriging taking into account the major sources of uncertainty, including uncertainty about the “normalizing transformation” itself, in the computation of the predictive density function. Unlike trans-Gaussian kriging, this approach mitigates the consequences of a misspecified transformation, giving in this sense a more robust predictive inference. Because the mean of the predictive distribution does not exist for some commonly used families of transformations, the median is used as the optimal predictor. The BTG model is applied in the spatial prediction of weekly rainfall amounts. Cross-validation shows the predicting performance of the BTG mo...

An Analysis of the Threshold Method for Measuring Area-Average Rainfall

Abstract Experomental evidence shows that the area-average rain rate and the fractional area covered by rain rate exceeding a fixed threshold are highly correlated; that is, are highly linearly related. A precise theoretical explanation of this fact is given. The explanation is based on the observation that rain rate has a mixed distribution, one that is a mixture of a discrete distribution and a continuous distribution. Under a homogeneity assumption, the slope of the linear relationship depends only on the continuous part of the distribution and as such is found to be markedly immune to parameter changes. This is illustrated by certain slope surfaces obtained from three specific distributions. The threshold level can be chosen in an optimal way by minimizing a certain distance function defined over the threshold range. In general, the threshold level should be not too far from the mean rain rate conditional on rain. The so-called threshold method advocates measuring rainfall from fractional area exploit...

Zero-crossing rates of functions of Gaussian processes

Formulas for the expected zero-crossing rates of random processes that are monotone transformations of Gaussian processes can be obtained by using two different techniques. The first technique involves derivation of the expected zero-crossing rate for discrete-time processes and extends the result of the continuous-time case by using an appropriate limiting argument. The second is a direct method that makes successive use of R. Prices (1958) theorem, the chain rule for derivatives, and S.O. Rices (1954) formula for the expected zero-crossing rate of a Gaussian process. A constant, which depends on the variance of the transformed process and a second-moment of its derivative, is derived. Multiplying Rices original expression by this constant yields the zero-crossing formula for the transformed process. The two methods can be used for the general level-crossing problem of random processes that are monotone functions of a Gaussian process. >

Iterative filtering for multiple frequency estimation

It is well-known that Pronys least-squares estimator gives inconsistent estimates for multiple frequency estimation. In a recent attempt to diminish this problem, Dragosevic and Stankovic (1989) couple the least-squares method of autoregressive (AR) estimation with an iterative filtering scheme discussed by Kay (1988) using an all-pole filter. But the inconsistency still persists. This paper attacks the chronic inconsistency with a general approach of parametric filtering that unifies and extends the previous work. It is shown that the inconsistency can be eliminated with an appropriately parametrized filter. The clue for the correct parametrization comes from a formula for the bias of the least squares AR estimator. The fact of the matter is that as long as a filter satisfies the parametrization requirement, consistent estimates can be obtained from the least-squares AR estimator on the basis of the filtered data. In particular, the all-pole filter considered by Dragosevic and Stankovic can be easily reparametrized so that it too satisfies the parametrization requirement and thus leads to a consistent estimator. Experimental results show that the modified method has a higher resolution than the discrete Fourier transform and that its overall performance is quite remarkable. >

On the Threshold Method for Rainfall Estimation: Choosing the Optimal Threshold Level

Abstract Real data experiments show that, for large areas, the area average rain rate and the fraction of the area covered by rain rate above a fixed threshold are highly correlated. For the right choice of the threshold level the correlation can easily exceed 95% and may even reach 99%. This remarkable fact observed in nature is the basis for the so-called threshold method for measuring rainfall from space via satellite. The method depends critically on the threshold level, showing significant improvement in performance for optimal and nearly optimal thresholds. Under the assumption that the continuous part of the distribution of rain rate is lognormal, two theoretically derived optimal levels agree closely with the best level obtained empirically from a well-known data set of rain rate.

Strong consistency of the contraction mapping method for frequency estimation

Some statistical properties in regard to the contraction mapping (CM) method are discussed. One of the requirements in this method is that the filter be parameterized to satisfy a certain fundamental property. The parameterization clearly depends on the normalized noise spectrum which theoretically has to be known or estimated a priori. If this information is available, one can first whiten the noise with a linear filter and then apply the CM method to the filtered data. In this way, the parameterization only needs to be done under the white noise assumption and filters like the AR(2) can be used by the CM method. In applications, however, prewhitening may not always be necessary. >

Optimal thresholds for the estimation of area rain-rate moments by the threshold method

Abstract Optimization of the threshold method, achieved by determination of the threshold that maximizes the correlation between an area-average rain-rate moment and the area coverage of rain rates exceeding the threshold, is demonstrated empirically and theoretically. Empirical results for a sequence of GATE radar snapshots show optimal thresholds of 5 and 27 mm h−1 for the first and second moments, respectively. Theoretical optimization of the threshold method by the maximum-likelihood approach of Kedem and Pavlopoulos predicts optimal thresholds near 5 and 26 mm h−1 for lognormally distributed rain rates with GATE-like parameters. The agreement between theory and observations suggests that the optimal threshold can be understood as arising due to sampling variations, from snapshot to snapshot, of a parent rain-rate distribution. Optimal thresholds for gamma and inverse Gaussian distributions are also derived and compared

Asymptotic normality of sample autocovariances with an application in frequency estimation

The asymptotic normality of sample autocovariances is proved for time series with mixed-spectra, which extends the classical results of Bartlett for linear processes. It is also shown that the asymptotic normality remains valid after linear filtering, if the filter is strictly stable so that the end-point effect of finite sample can be ignored. The developed theory is then employed to establish the asymptotic normality of a recently proposed fast frequency estimation procedure.

Estimation of the Parameters in Stationary Autoregressive Processes after Hard Limiting

Abstract Estimates of the parameters in normal autoregressive (AR(p)) processes may be obtained as functions of certain runs and subsequences in the associated clipped 0 − 1 processes. For example, the parameter in AR (1) is a function of the number of 1 runs only. Equivalently, this parameter can be estimated by counting only the number of axis crossings by the process. The estimates are obtained by a modification of the likelihood function of the clipped data. The loss of information because of hard limiting results in a loss of efficiency relative to the usual maximum likelihood estimates. Nevertheless, for large records our procedure yields quick estimates that do perform well.