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NASA NTRS1985

More than you want to know about maximum likelihood estimation

The maximum likelihood estimator has been used to extract stability and control derivatives from flight data for many years. Most of the literature on aircraft estimation concentrates on new developments and applications, assuming familiarity with basic estimation concepts. Some of these basic concepts are presented. The maximum likelihood estimator is briefly discussed and the aircraft equations of motion that the estimator uses. The basic concepts of minimization and estimation are examined for a simple computed aircraft example. The cost functions that are to be minimized during estimation are defined and discussed. Graphic representations of the cost functions are given to help illustrate the minimization process. Finally, the basic concepts are generalized, and estimation from flight data is discussed. Some of the major conclusions for the computed example are also developed for the analysis of flight data.

Iliff, K. W.↗

DOE OSTI2025

Maximum Likelihood Estimation: Some Basics

The maximum likelihood estimation is a general estimation procedure. It is often compared to estimation procedures like the ordinary least squares regression or generalized method of moments, to name a few. We discuss some basics about the maximum likelihood estimation, its advantages and disadvantages, and provide an example application to a gamma distribution function.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Maximum Likelihood Estimation: Some Basics

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1994

Digital combining-weight estimation for broadband sources using maximum-likelihood estimates

An algorithm described for estimating the optimum combining weights for the Ka-band (33.7-GHz) array feed compensation system is compared with the maximum-likelihood estimate. This provides some improvement in performance, with an increase in computational complexity. However, the maximum-likelihood algorithm is simple enough to allow implementation on a PC-based combining system.

Rodemich, E. R.↗

NASA NTRS1976

The numerical evaluation of the maximum-likelihood estimate of a subset of mixture proportions

Necessary and sufficient conditions are given for a maximum likelihood estimate of a subset of mixture proportions. From these conditions, likelihood equations are derived satisfied by the maximum-likelihood estimate and a successive-approximations procedure is discussed as suggested by equations for numerically evaluating the maximum-likelihood estimate. It is shown that, with probability one for large samples, this procedure converges locally to the maximum-likelihood estimate whenever a certain step-size lies between zero and two. Furthermore, optimal rates of local convergence are obtained for a step-size which is bounded below by a number between one and two.

Peters, B. C., Jr.↗

NASA NTRS1986

Determination of lift and drag characteristics of Space Shuttle Orbiter using maximum likelihood estimation technique

This paper presents the technique and results of maximum likelihood estimation used to determine lift and drag characteristics of the Space Shuttle Orbiter. Maximum likelihood estimation uses measurable parameters to estimate nonmeasurable parameters. The nonmeasurable parameters for this case are elements of a nonlinear, dynamic model of the orbiter. The estimated parameters are used to evaluate a cost function that computes the differences between the measured and estimated longitudinal parameters. The case presented is a dynamic analysis. This places less restriction on pitching motion and can provide additional information about the orbiter such as lift and drag characteristics at conditions other than trim, instrument biases, and pitching moment characteristics. In addition, an output of the analysis is an estimate of the values for the individual components of lift and drag that contribute to the total lift and drag. The results show that maximum likelihood estimation is a useful tool for analysis of Space Shuttle Orbiter performance and is also applicable to parameter analysis of other types of aircraft.

Trujillo, B. M.↗

NASA NTRS1977

The Maximum Likelihood Estimation of Signature Transformation /MLEST/ algorithm

The Maximum Likelihood Estimation of Signature Transformation (MLEST) algorithm is used to obtain maximum likelihood estimates (MLE) of affine transformation. The algorithm has been evaluated for three sets of data: simulated (training and recognition segment pairs), consecutive-day (data gathered from Landsat images), and geographical-extension (large-area crop inventory experiment) data sets. For each set, MLEST signature extension runs were made to determine MLE values and the affine-transformed training segment signatures were used to classify the recognition segments. The classification results were used to estimate wheat proportions at 0 and 1% threshold values.

Thadani, S. G.↗

NASA NTRS1985

Maximum Likelihood Estimation with Emphasis on Aircraft Flight Data

Accurate modeling of flexible space structures is an important field that is currently under investigation. Parameter estimation, using methods such as maximum likelihood, is one of the ways that the model can be improved. The maximum likelihood estimator has been used to extract stability and control derivatives from flight data for many years. Most of the literature on aircraft estimation concentrates on new developments and applications, assuming familiarity with basic estimation concepts. Some of these basic concepts are presented. The maximum likelihood estimator and the aircraft equations of motion that the estimator uses are briefly discussed. The basic concepts of minimization and estimation are examined for a simple computed aircraft example. The cost functions that are to be minimized during estimation are defined and discussed. Graphic representations of the cost functions are given to help illustrate the minimization process. Finally, the basic concepts are generalized, and estimation from flight data is discussed. Specific examples of estimation of structural dynamics are included. Some of the major conclusions for the computed example are also developed for the analysis of flight data.

Iliff, K. W.↗

NASA NTRS1976

Maximum likelihood estimates of lift and drag characteristics obtained from dynamic aircraft maneuvers

A maximum likelihood estimation method for obtaining lift and drag characteristics from dynamic flight maneuvers was investigated. This paper describes the method and compares the estimates of lift and drag obtained by using the method with estimates obtained from wind-tunnel tests and from established methods for obtaining estimates from flight data. In general, the lift and drag coefficients extracted from dynamic flight maneuvers by the maximum likelihood estimation technique are in good agreement with the estimates obtained from the wind-tunnel tests and the other methods. When maneuvers that met the requirements of both flight methods were analyzed, the results of each method were nearly the same. The maximum likelihood estimation technique showed promise in terms of estimating lift and drag characteristics from dynamic flight maneuvers. Further studies should be made to assess the best mathematical model and the most desirable type of dynamic maneuver to get the highest quality results from this technique.

Iliff, K. W.↗

NASA NTRS1974

Nonparametric maximum likelihood estimation of probability densities by penalty function methods

When it is known a priori exactly to which finite dimensional manifold the probability density function gives rise to a set of samples, the parametric maximum likelihood estimation procedure leads to poor estimates and is unstable; while the nonparametric maximum likelihood procedure is undefined. A very general theory of maximum penalized likelihood estimation which should avoid many of these difficulties is presented. It is demonstrated that each reproducing kernel Hilbert space leads, in a very natural way, to a maximum penalized likelihood estimator and that a well-known class of reproducing kernel Hilbert spaces gives polynomial splines as the nonparametric maximum penalized likelihood estimates.

Demontricher, G. F.↗

NASA NTRS1976

The numerical evaluation of maximum-likelihood estimates of the parameters for a mixture of normal distributions from partially identified samples

Likelihood equations determined by the two types of samples which are necessary conditions for a maximum-likelihood estimate are considered. These equations, suggest certain successive-approximations iterative procedures for obtaining maximum-likelihood estimates. These are generalized steepest ascent (deflected gradient) procedures. It is shown that, with probability 1 as N sub 0 approaches infinity (regardless of the relative sizes of N sub 0 and N sub 1, i=1,...,m), these procedures converge locally to the strongly consistent maximum-likelihood estimates whenever the step size is between 0 and 2. Furthermore, the value of the step size which yields optimal local convergence rates is bounded from below by a number which always lies between 1 and 2.

Walker, H. F.↗

NASA NTRS1976

An iterative procedure for obtaining maximum-likelihood estimates of the parameters for a mixture of normal distributions, 2

The problem of obtaining numerically maximum likelihood estimates of the parameters for a mixture of normal distributions is addressed. In recent literature, a certain successive approximations procedure, based on the likelihood equations, is shown empirically to be effective in numerically approximating such maximum-likelihood estimates; however, the reliability of this procedure was not established theoretically. Here, a general iterative procedure is introduced, of the generalized steepest-ascent (deflected-gradient) type, which is just the procedure known in the literature when the step-size is taken to be 1. With probability 1 as the sample size grows large, it is shown that this procedure converges locally to the strongly consistent maximum-likelihood estimate whenever the step-size lies between 0 and 2. The step-size which yields optimal local convergence rates for large samples is determined in a sense by the separation of the component normal densities and is bounded below by a number between 1 and 2.

Peters, B. C., Jr.↗

NASA NTRS1978

An iterative procedure for obtaining maximum-likelihood estimates of the parameters for a mixture of normal distributions

This paper addresses the problem of obtaining numerically maximum-likelihood estimates of the parameters for a mixture of normal distributions. In recent literature, a certain successive-approximations procedure, based on the likelihood equations, was shown empirically to be effective in numerically approximating such maximum-likelihood estimates; however, the reliability of this procedure was not established theoretically. Here, we introduce a general iterative procedure, of the generalized steepest-ascent (deflected-gradient) type, which is just the procedure known in the literature when the step-size is taken to be 1. We show that, with probability 1 as the sample size grows large, this procedure converges locally to the strongly consistent maximum-likelihood estimate whenever the step-size lies between 0 and 2. We also show that the step-size which yields optimal local convergence rates for large samples is determined in a sense by the 'separation' of the component normal densities and is bounded below by a number between 1 and 2.

Peters, B. C., Jr.↗

NASA NTRS1975

Practical aspects of using a maximum likelihood estimator

The application of a maximum likelihood estimator to flight data is discussed and procedures to facilitate routine analysis of a large amount of flight data are proposed. Flight data were used to demonstrate the proposed procedures. Modeling considerations are discussed for the system to be identified, including linear aerodynamics, instrumentation, and data time shifts, and aerodynamic biases for the specific types of maneuvers to be analyzed. Data editing to eliminate common data acquisition problems, and a method of identifying other problems are considered. The need for careful selection of the maneuver or portions of the maneuver to be analyzed is pointed out. Uncertaintly levels (analogous to Cramer-Rao bounds) are discussed as a way of recognizing significant new information.

Iliff, K. W.↗

NASA NTRS1976

The numerical evaluation of maximum-likelihood estimates of the parameters for a mixture of normal distributions from partially identified samples

Likelihood equations determined by the two types of samples which are necessary conditions for a maximum-likelihood estimate were considered. These equations suggest certain successive approximations iterative procedures for obtaining maximum likelihood estimates. The procedures, which are generalized steepest ascent (deflected gradient) procedures, contain those of Hosmer as a special case.

Walker, H. F.↗

NASA NTRS1974

Computational aspects of maximum likelihood estimation and reduction in sensitivity function calculations

This paper discusses numerical aspects of computing maximum likelihood estimates for linear dynamical systems in state-vector form. Different gradient-based nonlinear programming methods are discussed in a unified framework and their applicability to maximum likelihood estimation is examined. The problems due to singular Hessian or singular information matrix that are common in practice are discussed in detail and methods for their solution are proposed. New results on the calculation of state sensitivity functions via reduced order models are given. Several methods for speeding convergence and reducing computation time are also discussed.

Gupta, N. K.↗

NASA NTRS1979

Maximum likelihood estimation of label imperfections and its use in the identification of mislabeled patterns

The problem of estimating label imperfections and the use of the estimation in identifying mislabeled patterns is presented. Expressions for the maximum likelihood estimates of classification errors and a priori probabilities are derived from the classification of a set of labeled patterns. Expressions also are given 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 threshold on the discriminant functions for both two-class and multiclass cases. Expressions are derived for the probability that the imperfect label identification scheme will result in a wrong decision and are used in computing thresholds. The results of practical applications of these techniques in the processing of remotely sensed multispectral data are presented.

Chittineni, C. B.↗

NASA NTRS1972

Longitudinal stability and control derivatives of a jet fighter airplane extracted from flight test data by utilizing maximum likelihood estimation

A method of parameter extraction for stability and control derivatives of aircraft from flight test data, implementing maximum likelihood estimation, was developed and successfully applied to actual longitudinal flight test data from a modern sophisticated jet fighter. The results of this application establish the merits of the estimation technique and its computer implementation(allowing full analyst interaction with the program) as well as provide data for the validation of a portion of the differential maneuvering simulator (DMS). The results are presented for all flight test runs in tabular form and as time history comparisons between the estimated states and the actual flight test data. Comparisons between extracted and manufacturer's values for five major derivatives are presented and reveal good agreement for these principal derivatives with one exception. This particular derivative is extensively investigated by utilizing the interactive capabilities of the computer program. The results of this investigation verify the numbers extracted by maximum likelihood estimation.

Steinmetz, G. G.↗