A comparison of three estimators of the Weibull parameters

Using mean square error as the criterion, we compare two least squares estimates of the Weibull parameters based on non‐parametric estimates of the unreliability with the maximum likelihood estimates (MLEs). The two non‐parametric estimators are that of Herd–Johnson and one recently proposed by Zimmer. Data was generated using computer simulation with three small sample sizes (5, 10 and 15) with three multiply‐censored patterns for each sample size. Our results indicate that the MLE is a better estimator of the Weibull characteristic value, θ, than the least squares estimators considered. No firm conclusions may be made regarding the best estimate of the Weibull shape parameter, although the use of maximum likelihood is not recommended for small sample sizes. Whenever least squares estimation of both Weibull parameters is appropriate, we recommend the use of the Zimmer estimator of reliability. Copyright © 2001 John Wiley & Sons, Ltd.     

GLM profile monitoring using robust estimators

It is well known that performance of control scheme in phase II of statistical process control depends on the estimators utilized in phase I. Sometimes, outliers may be present in the data, which could seriously impact the performance of the estimators. In some practical situations, generalized linear models (GLMs) are used to model a wide class of response variables. This study deals with the robust estimation and monitoring of parameters in GLM profiles in the presence of outliers. In this study, robust estimators are used to estimate the parameters of logistic and Poisson profiles. The results are compared with the maximum likelihood estimators (MLEs). In a numerical example, the profile parameters are estimated by the MLE and robust estimators and the resulting test statistics are monitored by a control scheme. The phase II control charts are determined based on these two types of estimators and compared for different out-of-control conditions. The simulation results confirm that robust estimators in most cases lead to better estimates in comparison with the MLE estimator in terms of average run length criterion.     

Some Percentile Estimators for Weibull Parameters

A percentile estimator for the shape parameter of the Weibull distribution, based on the 17th and 97th sample percentiles, is proposed which is asymptotically about 66% efficient when compared with the MLE (maximum likelihood estimator). A two-observation percentile estimator, based on the 40th and 82nd sample percentiles, for the scale parameter when the shape parameter is unknown is asymptotically about 82y0 efficient when compared with the MLE. The 24th and 93rd sample percentiles yield asymptotically about 41ye jointly efficient percentile estimators for both the scale and shape parameters in a class of two-observation percentile estimators when compared with their MLEs. Some other simple percentile estimators for these parameters are also briefly discussed. Finally, asymptotic properties of these estimators are investigated and their application in statistical inference problems is mentioned.     

Optimal velocity estimation in ultrasound color flow imaging in presence of clutter

In color flow imaging (CFI), the rejection of tissue clutter signal is treated separately from blood velocity estimation by high-pass filtering the received Doppler signal. The complete suppression of clutter is then difficult to achieve without affecting the subsequent velocity estimates. In this work, a different approach to velocity estimation is investigated, based on a statistical model of the signal from both clutter and blood. An analytic expression for the Cramer-Rao lower bound (CRLB) is developed, and used to determine the existence of an efficient maximum likelihood estimator (MLE) of blood velocity in CFI when assuming full knowledge of the clutter statistics. We further simulate and compare the performance of the MLE to that of the autocorrelation method (ACM) using finite-impulse response (FIR) and polynomial regression clutter filters. Two signal scenarios are simulated, representing a central and peripheral vessel. Simulations showed that, by including 3-9 (independent) spatial points, the MLE variance approached the CRLB in both scenarios. The ACM was approximately unbiased only for the central scenario in the clutter filter pass band, then with a variance of up to four times the CRLB. The ACM suffered from a severe bias in the filter transition region, and a significant performance gain was achieved here using the MLE. For practical use, the clutter properties must be estimated. We finally replaced the known clutter statistics with an estimate obtained from low-rank approximations of the received sample correlation matrix. Used in the model-based framework, this method came close to the performance of the MLE, and it may be an important step toward a practical model-based estimator, including tissue clutter with optimal performance.     

Optical anemometry based on the temporal cross-correlation of angle-of-arrival fluctuations obtained from spatially separated light sources

The temporal cross-correlation function of the angle-of-arrival (AOA) fluctuations of two optical waves propagating through atmospheric turbulence carries information regarding the average wind velocity transverse to the propagation path. We present and discuss two estimators for the retrieval of the path-averaged beam-transverse horizontal wind velocity, v_t. Both methods retrieve v_t from the temporal cross-correlation function of AOA fluctuations obtained from two closely spaced light-emitting diodes (LEDs). The first method relies on the time delay of the peak (TDP) of the cross-correlation function, and the second method exploits its slope at zero lag (SZL). Over a 9 h period during which v_t varied between -1.3 ms^-1 and 2.0 ms^-1, the maximum rms difference between optically retrieved and in situ measured 10?s estimates of v_t was found to be 0.18 ms^-1 for the TDP estimator and 0.23 ms^-1 for the SZL estimator. Applicability and limitations of these two optical wind retrieval techniques are discussed.     

A new time-domain narrowband velocity estimation technique for Doppler ultrasound flow imaging. II. Comparative performance assessment

For pt.I see ibid., vol.45, no.4, pp.939-54 (1998). The statistical performance of the new 2-D narrowband time-domain root-MUSIC blood velocity estimator described previously is evaluated using both simulated and flow phantom wideband (50% fractional bandwidth) ultrasonic data. Comparisons are made with the standard 1-D Kasai estimator and two other wideband strategies: the time domain correlator and the wideband point maximum likelihood estimator. A special case of the root-MUSIC, the "spatial" Kasai, is also considered. Simulation and flow phantom results indicate that the root-MUSIC blood velocity estimator displays a superior ability to reconstruct spatial blood velocity information under a wide range of operating conditions. The root-MUSIC mode velocity estimator can be extended to effectively remove the clutter component from the sample volume data. A bimodal velocity estimator is formed by processing the signal subspace spanned by the eigenvectors corresponding to the two largest eigenvalues of the Doppler correlation matrix. To test this scheme, in vivo common carotid flow complex Doppler data was obtained from a commercially available color flow imaging system. Velocity estimates were made using a reduced form of this data corresponding to higher frame rates. The extended root-MUSIC approach was found to produce superior results when compared to both 1- and 2-D Kasai-type estimators that used initialized clutter filters. The results obtained using simulated, flow phantom, and in vivo data suggest that increased sensitivity as well as effective clutter suppression can be achieved using the root-MUSIC technique, and this may be particularly important for wideband high frame rate imaging applications.     

Estimating 2-D vector velocities using multidimensional spectrum analysis

Wilson (1991) presented an ultrasonic wideband estimator for axial blood flow velocity estimation through the use of the 2-D Fourier transform. It was shown how a single velocity component was concentrated along a line in the 2-D Fourier space, where the slope was given by the axial velocity. Later, it was shown that this approach could also be used for finding the lateral velocity component by also including a lateral sampling. A single velocity component would then be concentrated along a plane in the 3-D Fourier space, tilted according to the 2 velocity components. This paper presents 2 new velocity estimators for finding both the axial and lateral velocity components. The estimators essentially search for the plane in the 3- D Fourier space, where the integrated power spectrum is largest. The first uses the 3-D Fourier transform to find the power spectrum, while the second uses a minimum variance approach. Based on this plane, the axial and lateral velocity components are estimated. Several phantom measurements, for flow-to-depth angles of 60, 75, and 90 degrees, were performed. Multiple parallel lines were beamformed simultaneously, and 2 different receive apodization schemes were tried. The 2 estimators were then applied to the data. The axial velocity component was estimated with an average standard deviation below 2.8% of the peak velocity, while the average standard deviation of the lateral velocity estimates was between 2.0% and 16.4%. The 2 estimators were also tested on in vivo data from a transverse scan of the common carotid artery, showing the potential of the vector velocity estimation method under in vivo conditions.     

Estimating the Change Point of a Poisson Rate Parameter with a Linear Trend Disturbance

Knowing when a process changed would simplify the search and identification of the special cause. In this paper, we compare the maximum likelihood estimator (MLE) of the process change point designed for linear trends to the MLE of the process change point designed for step changes when a linear trend disturbance is present. We conclude that the MLE of the process change point designed for linear trends outperforms the MLE designed for step changes when a linear trend disturbance is present. We also present an approach based on the likelihood function for estimating a confidence set for the process change point. We study the performance of this estimator when it is used with a cumulative sum (CUSUM) control chart and make direct performance comparisons with the estimated confidence sets obtained from the MLE for step changes. The results show that better confidence can be obtained using the MLE for linear trends when a linear trend disturbance is present. Copyright © 2005 John Wiley & Sons, Ltd.     

Single-ensemble-based eigen-processing methods for color flow imaging--Part II. The matrix pencil estimator

Parametric spectral estimators can potentially be used to obtain flow estimates directly from raw slow-time ensembles whose clutter has not been suppressed. We present a new eigen-based parametric flow estimation method called the matrix pencil, whose principles are based on a matrix form under the same name. The presented method models the slow-time signal as a sum of dominant complex sinusoids in the slow-time ensemble, and it computes the principal Doppler frequencies by using a generalized eigen-value problem-formulation and matrix rank reduction principles. Both fixed rank (rank-one, rank-two) and adaptive-rank matrix pencil flow estimators are proposed, and their potential applicability to color flow signal processing is discussed. For the adaptive-rank estimator, the nominal rank was defined as the minimum eigen-structure rank that yields principal frequency estimates with a spread greater than a prescribed bandwidth. In our initial performance evaluation, the fixed-rank matrix pencil estimators were applied to raw color flow data (transmit frequency: 5 MHz; pulse repetition period: 0.175 ms; ensemble size: 14) acquired from a steady flow phantom (70 cm/s at centerline) that was surrounded by rigid-tissue-mimicking material. These fixed-rank estimators produced velocity maps that are well correlated with the theoretical flow profile (correlation coefficient: 0.964 to 0.975). To facilitate further evaluation, the matrix pencil estimators were applied to synthetic slow-time data (transmit frequency: 5 MHz; pulse repetition period: 1.0 ms; ensemble size: 10) modeling flow scenarios without and with tissue motion (up to 1 cm/s). The bias and root-mean-squared error of the estimators were computed as a function of blood-signal-to-noise ratio and blood velocity. The matrix pencil flow estimators showed that they are comparatively less biased than most of the existing frequency-based flow estimators like the lagone autocorrelator.     

The effect of frequency dependent scattering and attenuation on the estimation of blood velocity using ultrasound

A comprehensive theoretical performance comparison of the wideband maximum-likelihood (WMLE) and cross-correlation strategies, previously proposed and evaluated for the estimation of blood velocity using ultrasound is presented. It is based on evaluation of the bias, local and global accuracy, and signal-to-noise ratio (SNR) performance. The results show that the intervening medium does not bias either wideband estimation, due to the effect of tracking the scattering target. The presence of intervening tissue actually improves the global accuracy of both wideband estimators, without a significant change in the local accuracy of either wideband estimator. After the transmission of P pulses, a comparison of the performance of the two strategies shows that the cross-correlation estimator requires P(2 ) correlations to achieve performance similar to that of the WMLE with P operations. In addition, the WMLE can increase the effective SNR in comparison with cross correlation.     

Tissue strain imaging using a wavelet transform-based peak search algorithm

A new method is proposed to estimate the motion and relative local compression between two successive ultrasound RF signals under different compression states. The algorithm uses the continuous wavelet transform to locate the peaks in the RF signals. The estimated peaks in the pre- and post-compression signals are assigned to each other by a peak matching technique with the goal of minimizing the number of false matches. The method allows local shifts of the tissue to be estimated. The method has been tested in one-dimensional simulations and phantom experiments. The signal-to-noise ratio and the rms error are shown to be better than for the standard cross-correlation method (CC). The new estimator remains unbiased for up to 10% strain which is a larger range than that of CC. The maximum signal-to-noise ratio is 3 times as high as that of the CC method, showing higher sensitivity as well. The method is computationally efficient, achieving 0.7 msec/RF line on a standard personal computer.     

极大似然高分辨波达估计新方法

极大似然估计器是波达方向估计中公认的最佳估计器,但是计算量很大。为了解决极大似然估计器由于进行多维格形搜索而带来的计算量大的不足,将粒子滤波方法与极大似然估计相结合,提出了一种基于粒子滤波的极大似然波达方向估计器(Maximum Likelihood DOA Estimator Based on Particle Filtering,简称MLE-PF)。研究结果表明,MLE-PF不但保持了原极大似然估计方法的优良性能,大大减小了计算量,计算复杂度由O(LK)降至O(K×Ns),而且在低信噪比时也具有比MUSIC以及MiniNorm方法更加优越的估计性能。     

A robust correlation receiver for distance estimation

Many methods for distance estimation, such as the ultrasonic pulse-echo method, involve the estimation of a time-of-flight (TOF). In this paper, a signal model is developed that, apart from the TOF, accounts for an unknown, linear frequency dependent distortion as well as for additive noise. We derive a TOF estimator for this model based on the criteria of maximum likelihood. The resulting receiver can be seen as an extension or generalization of the well known cross-correlation, or “matched filter”, estimator described, e.g., by Nilsson. The novel receiver is found to be more robust against unknown pulse shape distortion than the cross-correlation estimator, giving less biased TOF estimates. Also, bias versus noise sensitivity can be controlled by proper model order selection     

A new wideband spread target maximum likelihood estimator for blood velocity estimation. I. Theory

The derivation and theoretical evaluation of new wideband maximum-likelihood strategies for the estimation of blood velocity using acoustic signals are presented. A model for the received signal from blood scatterers, using a train of short wideband pulses, is described. Evaluation of the autocorrelation of the signal based on this model shows that the magnitude, periodicity, and phase of the autocorrelation are affected by the mean scatterer velocity and the presence of a velocity spread target. New velocity estimators are then derived that exploit the effect of the scatterer velocity on both the signal delay and the shift in frequency. The wideband range spread estimator is derived using a statistical model of the target. Based on the point target assumption, a simpler wideband maximum-likelihood estimator is also obtained. These new estimation strategies are analyzed for their local and global performance. Evaluation of the Cramer-Rao bound shows that the bound on the estimator variance is reduced using these estimators, in comparison with narrowband strategies. In order to study global accuracy, the expected estimator output is evaluated, and it is determined that the width of the mainlobe is reduced. In addition, it is shown that the height of subsidiary velocity peaks is reduced through the use of these new estimators.     

Pseudo maximum likelihood estimates using ranked set sampling with applications to estimating correlation

Suppose θ is the parameter of interest and λ is the nuisance parameter. When obtaining the maximum likelihood estimator (MLE) of θ in the presence of λ requires intensive computation, the pseudo MLE of θ, based on a pseudo likelihood function, can be used. Gong and Samaniego (Ann. Stat. 9:861–869, 1981) proposed a pseudo MLE (PMLE) based on simple random samples. Ranked set sampling has been applied to the bivariate variables (X,Y) where measuring one of the variables is difficult or costly. In this paper, we obtain the pseudo MLE of the correlation coefficient from a bivariate normal distribution (X,Y) based on ranked set samples, assuming that Y is difficult or more expensive to measure and that the mean and variance of Y are the nuisance parameters. The PMLE is compared with three other estimators of the correlation coefficient. Simulations show that the PMLE is more (less) efficient than other estimators, depending on value of ρ. Testing of soil contamination provides an example of the use of the methods.     

Use of median-based estimator to construct Phase II exponential chart

The Shewhart-type exponential control chart is a popular and extensively used among all time-between-events control charts for its simplicity. When the parameter is unknown, Phase II control limits are constructed, and the success of its implementation depends to an extent on the estimated value of the parameter, obtained from Phase I dataset. However, when the Phase I data are contaminated with spurious observations/outliers, the performance of the chart is suspected to deviate from what is normally expected. Traditionally, maximum likelihood estimator (MLE) and minimum variance unbiased estimator (MVUE) are used to estimate the unknown process parameter. Both of estimators are the functions of sample mean. In this paper, the median-based estimator (MBE) that is a function of sample median is used to construct Phase II control limits. Moreover, performance of the proposed chart is examined when Phase I sample consists of contaminated observations/outliers. It is found that the proposed chart outperforms the existing charts whether the sample is contaminated or not.     

Experimental evaluation of velocity and power estimation forultrasound blood flow imaging, by means of a two-dimensionalautocorrelation approach

This paper evaluates experimentally the performance of a novel axial velocity estimator, the 2D autocorrelator, and its Doppler power estimation counterpart, the 2D zero-lag autocorrelator, in the context of ultrasound color flow mapping. The evaluation also encompasses the well-established 1D autocorrelation technique for velocity estimation and its corresponding power estimator (1D zero-lag autocorrelator), to allow performance comparisons under identical conditions. Clutter-suppressed in vitro data sets from a steady-flow system are used to document the effect of the range gate and ensemble length, noise level and angle of insonation on the precision of the velocity estimates. The same data sets are used to examine issues related to the estimation of the Doppler signal's power. The first-order statistics of power estimates from regions corresponding to flow and noise are determined experimentally and the ability of power-based thresholding to separate flow signals from noise is characterized by means of ROC analysis. In summary, the results of the in vitro evaluation show that the proposed 2D-autocorrelation form of processing is consistently better than the corresponding 1D-autocorrelation techniques, in terms of both velocity and power estimation. Therefore, given their relatively modest implementation requirements, the 2D-autocorrelation algorithms for velocity and power estimation appear to represent a superior, yet realistic, alternative to conventional Doppler processing for color flow mapping     

Coherent Doppler lidar measurements of winds in the weak signal regime

In the weak signal regime coherent Doppler lidar velocity estimates are characterized by a localized distribution around the true mean velocity and a uniform distribution of random outliers over the velocity search space. The performance of velocity estimators is defined by the standard deviation of the good estimates around the true mean velocity and the fraction of random outliers. The quality of velocity estimates is improved with pulse accumulation. The performance of velocity estimates from two different coherent Doppler lidars in the weak signal regime is compared with the predictions of computer simulations for pulse accumulation from 1 to 100 pulses.     

A new adaptive mean frequency estimator: application to constant variance color flow mapping

A class of adapted mean frequency estimators is proposed for color flow mapping. These estimators can be fitted to the specific characteristics of a given Doppler signal to optimize the compromise between the range of analysable frequencies and the variance of mean frequency estimation. A sub-optimal estimator is derived for real-time applications, and an adaptive criterion based on the Doppler signal variance is developed for color flow mapping applications. Its performance is compared to that of the usual correlation phase estimator on simulated Doppler signals and on synthetic Doppler images. An improvement in image quality is achieved, mainly for low signal-to-noise ratio Doppler signals.