Classical Estimation Methods Research Articles

Local Enthalpy Measurements in a Supersonic Arcjet Facility

Results of the application of a mass-injection probe to estimate the local mass-specific enthalpy in a supersonic air plasma flow are presented. Up to now this probe has only been applied to subsonic flows. Numerical calculations to interpret the probe behavior are shown and compared to experimental results. Furthermore, to interpret the measured local mass-specific enthalpies, classical enthalpy estimation methods are used together with analytical calculations of the turbulent free plasma jet and fairly good agreement is found.

Probabilistic assessment on the basis of interval data

Uncertainties enter a complex analysis from a variety of sources: variability, lack of data, human errors, model simplification and lack of understanding of the underlying physics. However, for many important engineering applications insufficient data are available to justify the choice of a particular probability density function (PDF). Sometimes the only data available are in the form of interval estimates which represent, often conflicting, expert opinion. In this paper we demonstrate that Bayesian estimation techniques can successfully be used in applications where only vague interval measurements are available. The proposed approach is intended to fit within a probabilistic framework, which is established and widely accepted. To circumvent the problem of selecting a specific PDF when only little or vague data are available, a hierarchical model of a continuous family of PDF's is used. The classical Bayesian estimation methods are expanded to make use of imprecise interval data. Each of the expert opinions (interval data) are interpreted as random interval samples of a parent PDF. Consequently, a partial conflict between experts is automatically accounted for through the likelihood function. Our need for increased reliance on these types of simulations is changing the way in which we develop and use models. In the past, models were used to provide insight during the design phases. The models were first “calibrated” to match experimental data, and then exercised to identify problem areas (e.g., high stress), and to explore the effect of changes in properties, geometry, etc.

Multiple linear regression with some correlated errors: Classical and robust methods

In this paper we consider classical and robust methods of estimation and diagnostics for the multiple linear regression model when some of the errors are correlated. This work was motivated by the analysis of a medical data set, from an observational study aimed at identifying factors affecting the outcome of a surgical method for the correction of scoliosis (abnormal lateral spinal curvature). There are 392 observations but some of them are on the same patient (double curves). It seems adequate to consider a multiple linear regression model but, since it is not desirable to discard the double curves, the assumption of non-correlated errors is clearly violated, and this is indeed confirmed by related diagnostics on the residuals (Durbin-Watson test). A more appropriate model retains the linear structure but allows for non-null correlation between the errors on the same patient. We propose two different procedures for the estimation of the parameters of the linear model and the correlation parameters: maximum likelihood assuming normal errors and a robustified version obtained by plugging-in results from robust linear regression. The latter procedure is designed to be resistant to outlying observations or error distributions with heavy tails and has produced the most satisfactory results for the analysed data set.

Bilinear equation method for unbiased identification of linear FIR systems in the presence of input and output noises

The presence of contaminating noises in both the input and output of an FIR system usually results in a biased least squares (LS) parameter estimate. The total least squares (TLS) methods are known to be efficient in achieving unbiased estimation, if the ratio of the input noise variance to the output noise variance (NNR) is known. However, when the NNR is unknown, a simple analysis shows that the classical LS and TLS estimation methods usually have such insufficient degree of freedom that they can achieve the unbiased solution. In this paper, it is shown by analyzing the algebraic structure of the correlation matrix that the unbiased estimate of FIR parameters can be obtained by solving a special bilinear equation. Then we develop a bilinear equation method (BEM) for solving the bilinear equation associated with the unbiased solution of the FIR system or filtering under the unknown NNR. Unlike the available unbiased estimators, the main advantage is that the proposed method exploits much sufficiently the special structure of the correlation matrix and obtains much accurate estimation for FIR filtering in the presence of input and output noises. Two simulation examples are presented that show the good performance of the proposed method, including its superiority over the classical LS and TLS approaches, and the instrumental variable methods.

Estimation of polarization parameters using time–frequency representations and its application to waves separation

This paper deals with the detection of polarized seismic waves, the estimation of their polarization parameters, and the use of these parameters to apply waves separation. The data, containing several polarized waves together with some noise, are recorded by two-component sensors. After a review presenting the tools classically used to estimate the polarization parameters of a wave in the time domain and in the time–frequency domain, respectively, we present a new methodology to detect polarized waves, and estimate their polarization parameters automatically. The proposed method is based on the segmentation of a time–frequency representation of the data. In addition, after describing the proposed polarization estimation method, we present the oblique polarization filter (OPF) that enables the separation of two polarized waves using their polarization parameters, even if the corresponding patterns partially overlap in the time–frequency plane. The OPF consists in applying phase shifts, rotations, and amplifications in order to project one wave on one single component and the other wave on the other component. Being more efficient than classical polarization estimation methods, our approach greatly increases the separation performances of the OPF. Results are presented both on synthetic and real seismic data.

A robust design criterion for interpretable fuzzy models with uncertain data

We believe that nonlinear fuzzy filtering techniques may be turned out to give better robustness performance than the existing linear methods of estimation (H/sup 2/ and H/sup /spl infin// filtering techniques), because of the fact that not only linear parameters (consequents), but also the nonlinear parameters (membership functions) attempt to identify the uncertain behavior of the unknown system. However, the fuzzy identification methods must be robust to data uncertainties and modeling errors to ensure that the fuzzy approximation of unknown system's behavior is optimal in some sense. This study presents a deterministic approach to the robust design of fuzzy models in the presence of unknown but finite uncertainties in the identification data. We consider online identification of an interpretable fuzzy model, based on the robust solution of a regularized least-squares fuzzy parameters estimation problem. The aim is to resolve the difficulties associated with the robust fuzzy identification method due to lack of a priori knowledge about upper bounds on the data uncertainties. The study derives an optimal level of regularization that should be provided to ensure the robustness of fuzzy identification strategy by achieving an upper bound on the value of energy gain from data uncertainties and modeling errors to the estimation errors. A time-domain feedback analysis of the proposed identification approach is carried out with emphasis on stability, robustness, and steady-state issues. The simulation studies are provided to show the superiority of the proposed fuzzy estimation over the classical estimation methods.

Bayesian reliability analysis for fuzzy lifetime data

Lifetime data are important in reliability analysis. Classical reliability estimation is based on precise lifetime data. It is usually assumed that observed lifetime data are precise real numbers. However, some collected lifetime data might be imprecise and are represented in the form of fuzzy numbers. Thus, it is necessary to generalize classical statistical estimation methods for real numbers to fuzzy numbers. Bayesian methods have proved to be very useful when the sample size is small. There is little study on Bayesian reliability estimation based on fuzzy lifetime data. Most of the reported works in this area is limited to single parameter lifetime distributions. In this paper, we propose a new method to determine the membership function of the estimates of the parameters and the reliability function of multi-parameter lifetime distributions. An artificial neural network is used to approximate the calculation process of parameter estimation and reliability prediction. The genetic algorithm is used to find the boundary values of the membership function of the estimate of interest at any cut level. This method can be used to determine the membership functions of the Bayesian estimates of multi-parameter distributions. The effectiveness of the proposed method is illustrated with normal and Weibull distributions.

Bayesian Parameter Estimation of Convective Heat Transfer Coefficients of a Roof-Mounted Radiant Barrier System

AbstractThis paper deals with the application of Bayesian methods to the estimation of two convective heat-transfer coefficients of a roof-mounted radiant barrier system. As part of an empirical validation of the thermal model of the roofing complex, a parametric sensitivity analysis highlighted the importance of convective coefficients in the thermal behavior of a roofing complex. A parameter estimation method is then used in order to find the values of the coefficients that lead to an improvement of the thermal model. However, instead of using a classical parameter estimation method, we used a Bayesian inference approach to parameter estimation. The aim of the paper is to introduce the basic concepts of this powerful method in this simple two-parameter case. We show that Bayesian methods introduce an explicit treatment of uncertainty in modeling and a corresponding measure of reliability for the conclusions reached.

Using CRD method for quantification of groundwater recharge in the Gaza Strip, Palestine

Rainfall is the main source of groundwater recharge in the Gaza Strip area in Palestine. The area is located in the semi-arid zone and there is no source of recharge other than rainfall. Estimation of groundwater recharge from rainfall is not an easy task since it depends on many uncertain parameters. The cumulative rainfall departure (CRD) method, which depends on the water balance principle, was used in this study to estimate the net groundwater recharge from rainfall. This method does not require much data as is the case with other classical recharge estimation methods. The CRD method was carried out using optimisation approach to minimise the root mean square error (RMSE) between the measured and the simulated groundwater head. The results of this method were compared with the results of other recharge estimation methods from literature. It was found that the results of the CRD method are very close to the results of the other methods, but with less data requirements and greater ease of application. Based on the CRD method, the annual amount of groundwater recharge from rainfall in the Gaza Strip is about 43 million m3.

Localization of realistic cortical activity in MEG using current multipoles

We present a novel approach to MEG source estimation based on a regularized first-order multipole solution. The Gaussian regularizing prior is obtained by calculation of the sample mean and covariance matrix for the equivalent moments of realistic simulated cortical activity. We compare the regularized multipole localization framework to the classical dipole and general multipole source estimation methods by evaluating the ability of all three solutions to localize the centroids of physiologically plausible patches of activity simulated on the surface of a human cerebral cortex. The results, obtained with a realistic sensor configuration, a spherical head model, and given in terms of field and localization error, depict the performance of the dipolar and multipolar models as a function of variable source surface area (50–500 mm 2), noise conditions (20, 10, and 5 dB SNR), source orientation (0–90°), and source depth (3–11 cm). We show that as the sources increase in size, they become less accurately modeled as current dipoles. The regularized multipole systematically outperforms the single dipole model, increasingly so as the spatial extent of the sources increases. In addition, our simulations demonstrate that as the orientation of the sources becomes more radial, dipole localization accuracy decreases substantially, while the performance of the regularized multipole model is far less sensitive to orientation and even succeeds in localizing quasi-radial source configurations. Furthermore, our results show that the multipole model is able to localize superficial sources with higher accuracy than the current dipole. These results indicate that the regularized multipole solution may be an attractive alternative to current-dipole-based source estimation methods in MEG.

A new extreme quantile estimator for heavy-tailed distributions

The classical estimation method for extreme quantiles of heavy-tailed distributions was presented by Weissman (J. Amer. Statist. Assoc. 73 (1978) 812–815) and makes use of the Hill estimator (Ann. Statist. 3 (1975) 1163–1174) for the positive extreme value index. This index estimator can be interpreted as an estimator of the slope in the Pareto quantile plot in case one considers regression lines passing through a fixed anchor point. In this Note we propose a new extreme quantile estimator based on an unconstrained least squares estimator of the index, introduced by Kratz and Resnick (Comm. Statist. Stochastic Models 12 (1996) 699–724) and Schultze and Steinebach (Statist. Decisions 14 (1996) 353–372) and we study its asymptotic behavior. To cite this article: A. Fils, A. Guillou, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Motion estimation of transparent objects in the frequency domain

Motion transparency phenomena in image sequences are frequent, but classical methods of motion estimation are unable to deal with them. There is a need for more general techniques in order to solve this important problem. The method described here is based on an image sequence analysis in the frequency domain. It is mainly composed of a Stochastic-Expectation-Maximisation algorithm which provides a new statistical model for this problem. This method, despite its large execution time, offers some interesting results on artificial and natural image sequences.

A robust multiscale B-spline function decomposition for estimating motion transparency

Motion transparency phenomena in image sequences are frequent, but classical methods of motion estimation are unable to deal with them. This paper describes a method for estimating optical flow by a generalization of the brightness constancy assumption to additive transparencies. The brightness constancy assumption is obtained by setting constant velocity fields during three images of a sequence. Thus, by a Taylor development to its second order, we reach an extension of the optical flow constraint equation. Since the equation is nonlinear, the Levenberg-Marquardt algorithm is used. In order to suppress the unavoidable aperture problem, a global model based on B-spline basis functions is applied with the aim of constraining optical flows. This description of motion allows us to work on a coarse to fine estimation of artificial image sequences that shows good convergence properties. It is also applied to natural image sequences.

Time-Delay Estimation of a Managed River Reach from Supervisory Data

The problem addressed in this article is the estimation of the time-delay between the inflow rate and the downstream water level of a managed river reach from data collected in imposed experimental conditions. The modelling of the managed river reach shows that the feedforward control performed by the operator "hides" the reach time-delay in the transfer function of the closed-loop system. Therefore, most of the classical time-delay estimation methods are inappropriate. A time-delay estimation approach devoted to managed rivers is proposed, in which a time-day description, composed of estimated inlpulse responses, allows to clearly observe the evolution of the time-delays over one year. In practice, this approach is particulary interesting since it is based on production data and hence does not disturb the daily management of the process by any experimental protocol. Moreover, the use of the suggested approach does not require any synthesis parameter.

A Bayesian approach for time-delay estimation of a managed river reach in imposed experimental conditions

This article deals with the problem of estimating time-delays in the experimental modelling of river reaches managed by hydroelectric power plants. The purpose of the article is to estimate the reach time-delay from the data collected in imposed experimental conditions. The modelling of the managed river reach shows that the feedforward control performed by the operator “hides” the reach time-delay in the transfer function of the closed-loop system. So, classical time-delay estimation methods are inappropriate. The proposed solution considers the delay estimation as a problem of detecting a discontinuity in impulse response. A Bayesian approach is proposed to detect the delay and to estimate the impulse response from production data. This method is applied to one-day data sets supplied over one year in order to show its efficiency.

Trimmed L-moments

Classical estimation methods (least squares, the method of moments and maximum likelihood) work well in regular cases such as the exponential family, but outliers can have undue influence on these methods. We define population trimmed L-moments (TL-moments) and corresponding sample TL-moments as robust generalisations of population and sample L-moments. TL-moments assign zero weight to extreme observations, they are easy to compute, their sample variances and covariances can be obtained in closed form, and they are more robust than L-moments are to the presence of outliers. Moreover, a population TL-moment may be well defined where the corresponding population L-moment does not exist: for example, the first population TL-moment is well defined for a Cauchy distribution, but the first population L-moment, the population mean, does not exist. The sample TL-mean is compared with other robust estimators of location.

Synthesis inverse mapping: A robust method applicable to atmospheric remote sounding

We develop a new inversion method that may be advantageously applied to remote sounding in atmospheric physics. The technique is based on the synthesis of the inversion operator which can be used in noisy measurements where the success of classical regularization and optimal estimation methods strongly depends on the accurate knowledge of the a priori atmospheric state. We outline the benefits of the method: robustness, full control of validity range as well as easy assessment of the smoothing error. An example of spectral inversion is extensively analyzed in the context of aerosol retrievals from limb occultation experiments.

Evaluation of an autoregressive spectral estimation technique for determining horizontal wave-number content in shallow water

Modal-based techniques for geoacoustic inversion require estimates of discrete horizontal wave numbers corresponding to the propagating modes in a shallow-water waveguide. For range-dependent waveguides, local wave-number content can be extracted as a function of range using methods analogous to short-time Fourier transform techniques as applied to non-stationary time series data. A limiting factor in these techniques is the aperture length, L, required to resolve closely spaced horizontal wave-number content. Using classical spectral estimation methods, the limit of resolution is proportional to 1/L. In order to increase the spatial resolution of local horizontal wave-number estimates, the use of a high-resolution autoregressive spectral estimator was investigated for determining horizontal wave-number content. The performance of the estimator was evaluated numerically using both time-series and spatial data. Results are presented of estimator performance for various aperture lengths using signals with increasing levels of additive noise and signals with closely spaced wave-number components. Comparisons are made with results of classical methods as well as other high-resolution methods. The results show the AR method to be well suited for wave-number estimation in shallow-water waveguides. [Work supported by ONR.] a)Presently with the Acoustics Group, The Pennsylvania State Univ., Appl. Res. Lab., P.O. Box 30, State College, PA 16804, kmb166@psu.edu.

A Comparison of Modelling Strategies for Value-Added Analyses of Educational Data

Modelling strategies for value-added multilevel models are examined. These types of models typically include an endogenous variable and this causes difficulties for the standard estimation techniques that are commonly used to analyse multilevel models. Two alternative estimation strategies are proposed: one using an instrumental variable approach and the other using a Bayesian analysis as available through the BUGS software. We conclude that the approach offered by the BUGS software has advantages over more classical estimation methods

An Evaluation of Classical Steady-State Off-Line Linear Parameter Estimation Methods Applied to Chiller Performance Data

The objective of this paper is to evaluate different inverse methods with application to off-line model parameter estimation using data from a field-operated chiller. In HVAC&R data analysis, there is sometimes a need to evaluate and use estimation techniques that are more subtle than the ordinary least square (OLS) method. One example is in fault detection and diagnosis of HVAC&R equipment and systems using performance data obtained from field monitoring. By identifying a better performance model, the fault detection process is more likely to be refined and accurate. In this paper a number of exploratory, diagnostic, and classical estimation methods are reviewed to determine the circumstances in which they are likely to be superior to the OLS method. These methods are then evaluated using monitored data from a field-operated chiller. This study provides a reference on parameter estimation methods for the HVAC&R community.