Maximum Likelihood Properties
Maximum likelihood properties and asymptotic distributions for limiting distribution of likelihood ratio statistic under class of local alternatives
Lever, W. E.↗
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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.↗
Estimation of bias errors in measured airplane responses using maximum likelihood method
A maximum likelihood method is used for estimation of unknown bias errors in measured airplane responses. The mathematical model of an airplane is represented by six-degrees-of-freedom kinematic equations. In these equations the input variables are replaced by their measured values which are assumed to be without random errors. The resulting algorithm is verified with a simulation and flight test data. The maximum likelihood estimates from in-flight measured data are compared with those obtained by using a nonlinear-fixed-interval-smoother and an extended Kalmar filter.
Klein, Vladiaslav↗
Maximum likelihood signature estimation
Maximum-likelihood estimates are discussed which are based on an unlabeled sample of observations, of unknown parameters in a mixture of normal distributions. Several successive approximation procedures for obtaining such maximum-likelihood estimates are described. These procedures, which are theoretically justified by the local contractibility of certain maps, are designed to take advantage of good initial estimates of the unknown parameters. They can be applied to the signature extension problem, in which good initial estimates of the unknown parameters are obtained from segments which are geographically near the segments from which the unlabeled samples are taken. Additional problems to which these methods are applicable include: estimation of proportions and adaptive classification (estimation of mean signatures and covariances).
Walker, H. F.↗
Determination of instrumentation errors from measured data using maximum likelihood method
The maximum likelihood method is used for estimation of unknown initial conditions, constant bias and scale factor errors in measured flight data. The model for the system to be identified consists of the airplane six-degree-of-freedom kinematic equations, and the output equations specifying the measured variables. The estimation problem is formulated in a general way and then, for practical use, simplified by ignoring the effect of process noise. The algorithm developed is first applied to computer generated data having different levels of process noise for the demonstration of the robustness of the method. Then the real flight data are analyzed and the results compared with those obtained by the extended Kalman filter algorithm.
Keskar, D. A.↗
Convolutional codes. II - Maximum-likelihood decoding. III - Sequential decoding
Maximum-likelihood decoding is characterized as the determination of the shortest path through a topological structure called a trellis. Aspects of code structure are discussed along with questions regarding maximum-likelihood decoding on memoryless channels. A general bounding technique is introduced. The technique is used to obtain asymptotic bounds on the probability of error for maximum-likelihood decoding and list-of-2 decoding. The basic features of sequential algorithms are discussed along with a stack algorithm, questions of computational distribution, and the martingale approach to computational bounds.
Forney, G. D., Jr.↗
Acreage estimation, feature selection, and signature extension dependent upon the maximum likelihood decision rule
A maximum likelihood estimation technique is used for the analysis of agricultural remote sensor data. The m-class probability of misclassification is estimated using unlabeled test samples and labeled training samples. A bound on the variance of a proposed unbiased estimator of the m-class probability of error is derived. The particular case in which each class density is assumed to be a mixture of multivariate normal densities is considered. The extension of spectral signatures in space and time is discussed.
Quirein, J. A.↗
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.↗
Maximum likelihood analysis of balanced incomplete block models
Maximum likelihood analysis of balanced incomplete blocks, and amall-sample properties of maximum likelihood estimates
Kutner, M. H.↗
Procedure for estimating stability and control parameters from flight test data by using maximum likelihood methods employing a real-time digital system
A maximum likelihood parameter estimation procedure and program were developed for the extraction of the stability and control derivatives of aircraft from flight test data. Nonlinear six-degree-of-freedom equations describing aircraft dynamics were used to derive sensitivity equations for quasilinearization. The maximum likelihood function with quasilinearization was used to derive the parameter change equations, the covariance matrices for the parameters and measurement noise, and the performance index function. The maximum likelihood estimator was mechanized into an iterative estimation procedure utilizing a real time digital computer and graphic display system. This program was developed for 8 measured state variables and 40 parameters. Test cases were conducted with simulated data for validation of the estimation procedure and program. The program was applied to a V/STOL tilt wing aircraft, a military fighter airplane, and a light single engine airplane. The particular nonlinear equations of motion, derivation of the sensitivity equations, addition of accelerations into the algorithm, operational features of the real time digital system, and test cases are described.
Grove, R. D.↗
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.↗
Maximum likelihood classification by thresholding
The standard maximum-likelihood classifier is reformulated so that, in most cases, only a small number of density functions need be computed each time a data point is to be classified. The technique relies upon class thresholds which are obtained at the beginning of the classification process and which remain fixed thereafter. The result of the reformulation is that a significant reduction in classification processing time is obtained while retaining complete consistency with the standard maximum-likelihood classifier.
Minter, T. C.↗
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.↗
Effect of radiance-to-reflectance transformation and atmosphere removal on maximum likelihood classification accuracy of high-dimensional remote sensing data
Many analysis algorithms for high-dimensional remote sensing data require that the remotely sensed radiance spectra be transformed to approximate reflectance to allow comparison with a library of laboratory reflectance spectra. In maximum likelihood classification, however, the remotely sensed spectra are compared to training samples, thus a transformation to reflectance may or may not be helpful. The effect of several radiance-to-reflectance transformations on maximum likelihood classification accuracy is investigated in this paper. We show that the empirical line approach, LOWTRAN7, flat-field correction, single spectrum method, and internal average reflectance are all non-singular affine transformations, and that non-singular affine transformations have no effect on discriminant analysis feature extraction and maximum likelihood classification accuracy. (An affine transformation is a linear transformation with an optional offset.) Since the Atmosphere Removal Program (ATREM) and the log residue method are not affine transformations, experiments with Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) data were conducted to determine the effect of these transformations on maximum likelihood classification accuracy. The average classification accuracy of the data transformed by ATREM and the log residue method was slightly less than the accuracy of the original radiance data. Since the radiance-to-reflectance transformations allow direct comparison of remotely sensed spectra with laboratory reflectance spectra, they can be quite useful in labeling the training samples required by maximum likelihood classification, but these transformations have only a slight effect or no effect at all on discriminant analysis and maximum likelihood classification accuracy.
Hoffbeck, Joseph P.↗
A multinomial maximum likelihood program /MUNOML/
A multinomial maximum likelihood program (MUNOML) for signal detection and for behavior models is discussed. It is found to be useful in day to day operation since it provides maximum flexibility with minimum duplicated effort. It has excellent convergence qualities and rarely goes beyond 10 iterations. A library of subroutines is being collected for use with MUNOML, including subroutines for a successive categories model and for signal detectability models.
Curry, R. E.↗
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.↗
Determining the accuracy of maximum likelihood parameter estimates with colored residuals
An important part of building high fidelity mathematical models based on measured data is calculating the accuracy associated with statistical estimates of the model parameters. Indeed, without some idea of the accuracy of parameter estimates, the estimates themselves have limited value. In this work, an expression based on theoretical analysis was developed to properly compute parameter accuracy measures for maximum likelihood estimates with colored residuals. This result is important because experience from the analysis of measured data reveals that the residuals from maximum likelihood estimation are almost always colored. The calculations involved can be appended to conventional maximum likelihood estimation algorithms. Simulated data runs were used to show that the parameter accuracy measures computed with this technique accurately reflect the quality of the parameter estimates from maximum likelihood estimation without the need for analysis of the output residuals in the frequency domain or heuristically determined multiplication factors. The result is general, although the application studied here is maximum likelihood estimation of aerodynamic model parameters from flight test data.
Morelli, Eugene A.↗
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.↗