Maximum Likelihood Estimation Scheme Research Articles

Towed array shape estimation using frequency-wavenumber data

Towed array beamforming algorithms require accurate array shape information in order to perform properly. Very often, these algorithms assume the array is linear. Unfortunately, the mechanical forces on the array due to ship motion and sea dynamics can change the shape of the array, which degrades the performance of the beamforming algorithm. A data-driven approach to estimating the relative shape of a nominally linear array is presented. The algorithm is robust in that it optimally combines information contained in a wide band of frequencies and source bearings. At the heart of the algorithm is a maximum-likelihood (ML) estimation scheme. The Cramer-Rao lower bound is derived and compared to the performance of the ML estimator. The utility of the algorithm is verified using both simulated and actual towed array data experiments. >

Fast maximum likelihood estimation of signal parameters using the shape of the compressed likelihood function

A computationally efficient fast maximum-likelihood (FML) estimation scheme, which makes use of the shape of the surface of the compressed likelihood function (CLF), is proposed. The scheme uses only multiple one-dimensional searches oriented along appropriate ridges on the surface of the CLF. Simulations indicate that the performances of the proposed estimators match those of the corresponding maximum-likelihood estimators with very high probability. The approach is demonstrated by applying it to two different problems. The first problem involves the estimation of time of arrival and Doppler compression of a wideband hyperbolic frequency modulated (HFM) active sonar signal buried in reverberation. The second problem deals with estimating the frequencies of sinusoids. A threshold analysis of the proposed scheme is carried out to predict the signal-to-noise ratio (SNR) at which large estimation errors begin to occur, i.e., the threshold SNR, and its computational complexity is discussed. >

Properties of modelling the error distribution with an extra shape parameter

A robust extension of normal theory regression is to add an extra parameter to model the kurtosis of the error distribution, for example by using the T-family or the power-exponential family of distributions. The statistical properties of maximum likelihood estimation schemes for both families of models are considered. This article extends the work of Lange et al. (1989) in which the usefulness of the multivariate T-family for modelling data was demonstrated. The cost of adding the extra kurtosis parameter to the standard normal model is considered. This cost is measured by comparing the variance of the quantity of interest when the estimation of the extra parameter is taken into account and the variance when the estimated value of the extra parameter is treated as if it were known. In Lange et al. (1989), it is shown in a general setting that asymptotically there is no cost due to the estimation of the extra parameter, where the quantity of interest is the location or regression parameters. Whether this finding remains valid in small samples for the T and power-exponential families is investigated in this paper using Monte Carlo simulations and a real dataset. The efficiency of parameter estimates and coverage rates are also considered under three different scenarios: when the extra kurtosis parameter is estimated from the data, when the extra parameter is fixed at the true value, and when it is fixed at a wrong value. The expected information matrix is used to estimate the confidence intervals, and the comparisons are based on asymptotic calculations and Monte Carlo simulations. It is found that for the T-family there is very little cost due to estimation of the extra parameter, except for small sample sizes. The inflation in variance due to the estimation of the extra parameter increases as the sample size decreases. In the Monte Carlo simulations of simple regression settings the inflation in variance is found to be at most 14% for the T-family and at most 62% for the power-exponential family. For the T-family, there is a considerable loss of efficiency in fitting a normal model when the true degrees of freedom is small, but only a small loss of efficiency if a model with a low number of degrees of freedom is fit to normal observations. In contrast, the coverage rates of confidence intervals are close to the nominal level if a normal model is fit to the data whatever the true degrees of freedom, but the coverage rates can be too low if a model with a low number of degrees of freedom is fit to normal data. The coverage rates of confidence intervals when the degrees of freedom is estimated from the data are satisfactory, except at small sample size. Similar results are obtained for the power-exponential family. In addition to the larger inflation in variance due to estimation of the extra parameter, the power-exponential family is less satisfactory then the T-family because the extra kurtosis parameter is frequently estimated to be at the boundary of its range in small samples. The findings in this article support the notion that the extra kurtosis

Maximum likelihood estimation of magnitude-squared multiple and ordinary coherence

The probability density function (pdf) of estimated magnitude-squared ordinary or multiple coherence has been known for some time, and considerable use has been made of it in the ordinary coherence case in producing unbiased estimates. Apparently only recently has the pdf been used—and only for ordinary coherence—to formulate a maximum likelihood estimation scheme. This paper improves and corrects this scheme, and extends it to the multiple coherence case. Computational procedures are described. It is shown that the first and second derivatives of the probability density function can be written very simply in terms of the probability density function itself. As a result the iterative scheme for maximum likelihood estimation can be expressed in much simpler terms than done heretofore. A way of treating small maximum likelihood ordinary coherence estimates stated in the recent study is shown to be false. Illustrative plots are given showing the relationship between the standard multiple and ordinary coherence estimates, and maximum likelihood counterparts. An expression for the mean of the standard multiple coherence estimate is given in terms of generalized hypergeometric series, and is shown to reduce to the correct form for ordinary coherence. A simple recursive formula is given for computing the mean. An interpretation of the maximum likelihood estimate of multiple coherence derived from a given value of the standard estimate is that it is the value of the multiple coherence which makes the standard estimate into approximately the mean of the probability density function.

Algorithms for removing recovery-related distortion from auditory-nerve discharge patterns.

The probability that a cochlear nerve fiber discharges during some specified time interval depends on both the acoustic stimulus and on refractory effects due to earlier spike discharges. With the objective of separating the stimulus-related effects from refractory-related effects seen in poststimulus-time histograms, various maximum-likelihood estimation schemes have been developed. By modeling auditory-nerve fiber discharges as a self-exciting point process in which the intensity depends on both the time during stimulation and the history of discharge, we have been able to independently verify the likelihood estimates of Gaumond et al. [J. Acoust. Soc. Am. 74, 1392-1398 (1983)] under conditions when the recovery function is known. Secondly by maximizing the likelihood function of the neural event process subject to periodic stimulus constraints we have derived estimates which as the number of stimulus presentations and or stimulus periods increase are free from the effects of both absolute and relative recovery-related distortion. Under conditions when the recovery function is unknown, a recursive algorithm is proposed that yields the joint maximum-likelihood estimates of both the stimulus-and recovery-related components in the response histograms. We state the conditions under which unique maximum-likelihood estimates exist and prove that the recursive algorithm converges to those unique estimates.

Maximum Likelihood Estimation of Label Imperfection Probabilities and Its Use in the Identification of Mislabeled Patterns

This paper presents the problem of estimating label imperfections and the use of the estimation in identifying mislabeled patterns. Expressions for the maximum likelihood estimates of classification errors and a priori probabilities are derived from the classification of a set of labeled and unlabeled patterns. Expressions also are presented for the asymptotic variances of probability of correct classification and proportions. Simple models are developed for imperfections in the labels and for classification errors and are used in the formulation of a maximum likelihood estimation scheme. Schemes are presented for the identification of mislabeled patterns in terms of thresholds on the discriminant functions for both two-class and multiclass cases. Expressions are derived for the probability that the imperfect label identifi'cation scheme will result in a wrong decision and are used in computing thresholds. Furthermore, the results of practical applications of these techniques in the processing of remotely sensed multispectral data are presented.