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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.
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.
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.
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.
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.
The problem of estimating the parameters which determine a mixture density is reviewed as well as maximum likelihood estimation for it. A particular iterative procedure for numerically approximating maximum likelihood estimates for mixture density problems is considered. This EM algorithm, is a specialization to the mixture density context of a general algorithm of the same name used to approximate maximum likelihood estimates for incomplete data problems. The formulation and theoretical and practical properties of the EM algorithm for mixture densities are discussed focussing in particular on mixtures of densities from exponential families.
Estimation of the integral scale and intensity of a generic turbulence record is treated as a statistical problem of parameter estimation. Properties of parameter estimators and the method of maximum likelihood are reviewed. Likelihood equations are derived for estimation of the integral scale and intensity applicable to a general class of turbulence spectra that includes the von Karman and Dryden transverse and longitudinal spectra as special cases. The method is extended to include the Bullen transverse and longitudinal spectra. Coefficients of variation are given for maximum likelihood estimates of the integral scale and intensity of the von Karman spectra. Application of the method is illustrated by estimating the integral scale and intensity of an atmospheric turbulence vertical velocity record assumed to be governed by the von Karman transverse spectrum.
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.
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.
In a likelihood alarm display (LAD) information about event likelihood is computed by an automated monitoring system and encoded into an alerting signal for the human operator. Operator performance within a dual-task paradigm was evaluated with two LADs: a color-coded visual alarm and a linguistically coded synthetic speech alarm. The operator's primary task was one of tracking; the secondary task was to monitor a four-element numerical display and determine whether the data arose from a 'signal' or 'no-signal' condition. A simulated 'intelligent' monitoring system alerted the operator to the likelihood of a signal. The results indicated that (1) automated monitoring systems can improve performance on primary and secondary tasks; (2) LADs can improve the allocation of attention among tasks and provide information integrated into operator decisions; and (3) LADs do not necessarily add to the operator's attentional load.
This paper describes the use of maximum likelihood parameter estimation unknown parameters appearing in a nonlinear vehicle motion filter. The filter uses the kinematic equations of motion of a rigid body in motion over a spherical earth. The nine states of the filter represent vehicle velocity, attitude, and position. The inputs to the filter are three components of translational acceleration and three components of angular rate. Measurements used to update states include air data, altitude, position, and attitude. Expressions are derived for the elements of filter matrices needed to use air data in a body-fixed frame with filter states expressed in a geographic frame. An expression for the likelihood functions of the data is given, along with accurate approximations for the function's gradient and Hessian with respect to unknown parameters. These are used by a numerical quasi-Newton algorithm for maximizing the likelihood function of the data in order to estimate the unknown parameters. The parameter estimation algorithm is useful for processing data from aircraft flight tests or for tuning inertial navigation systems.
A multistage classification that reduces the processing time substantially is proposed. This classification algorithm consists of several stages, and in each stage likelihood values of classes are calculated and compared. If a class has a likelihood value less than a threshold, the class if truncated at that stage as an unlikely class, thus reducing the number of classes for which likelihood values are to be calculated at the next stage. Thus a host of classes can be truncated using a small portion of the total features at early stages, resulting in substantial reduction of computing time. Several truncation criteria are developed, and the relationship between thresholds and the error caused by the truncation is investigated. Experiments show that the proposed algorithm reduces the processing time by the factor of 3-7, depending on the number of classes and features, while maintaining essentially the same accuracies.
The A* algorithm finds the path in a finite depth binary tree that optimizes a function. Here, it is applied to maximum-likelihood soft-decision decoding of block codes where the function optimized over the codewords is the likelihood function of the received sequence given each codeword. The algorithm considers codewords one bit at a time, making use of the most reliable received symbols first and pursuing only the partially expanded codewords that might be maximally likely. A version of the A* algorithm for maximum-likelihood decoding of block codes has been implemented for block codes up to 64 bits in length. The efficiency of this algorithm makes simulations of codes up to length 64 feasible. This article details the implementation currently in use, compares the decoding complexity with that of exhaustive search and Viterbi decoding algorithms, and presents performance curves obtained with this implementation of the A* algorithm for several codes.
The concept of function offers a high potential for thinking and reasoning about designs as well as providing a common thread for relating together other design information. This paper focuses specifically on the relation between function and risk by presenting a mathematical mapping from product function to risk likelihood. This risk information is composed of design parameters, failure modes, and likelihood values. A spacecraft orientation subsystem, subsystem used to guide science instruments, and a Bell 206 rotorcraft are used to test the mapping which continues research on these products relating function to failure. Finally, a case study is presented in which the risk element likelihood is calculated for a fuel cell which is in the conceptual design phase at NASA JPL.
Traditional hazard analysis techniques utilize a two-dimensional representation of the results determined by relative likelihood and severity of the residual risk. These matrices present a quick-look at the Likelihood (Y-axis) and Severity (X-axis) of the probable outcome of a hazardous event. A three-dimensional method, described herein, utilizes the traditional X and Y axes, while adding a new, third dimension, shown as the Z-axis, and referred to as the Level of Control. The elements of the Z-axis are modifications of the Hazard Elimination and Control steps (also known as the Hazard Reduction Precedence Sequence). These steps are: 1. Eliminate risk through design. 2. Substitute less risky materials for more hazardous materials. 3. Install safety devices. 4. Install caution and warning devices. 5. Develop administrative controls (to include special procedures and training.) 6. Provide protective clothing and equipment. When added to the two-dimensional models, the level of control adds a visual representation of the risk associated with the hazardous condition, creating a tall-pole for the leastwell-controlled failure while establishing the relative likelihood and severity of all causes and effects for an identified hazard. Computer modeling of the analytical results, using spreadsheets and three-dimensional charting gives a visual confirmation of the relationship between causes and their controls.
Traditional hazard analysis techniques utilize a two-dimensional representation of the results determined by relative likelihood and severity of the residual risk. These matrices present a quick-look at the Likelihood (Y-axis) and Severity (X-axis) of the probable outcome of a hazardous event. A three-dimensional method, described herein, utilizes the traditional X and Y axes, while adding a new, third dimension, shown as the Z-axis, and referred to as the Level of Control. The elements of the Z-axis are modifications of the Hazard Elimination and Control steps (also known as the Hazard Reduction Precedence Sequence). These steps are: 1. Eliminate risk through design. 2. Substitute less risky materials for more hazardous materials. 3. Install safety devices. 4. Install caution and warning devices. 5. Develop administrative controls (to include special procedures and training.) 6. Provide protective clothing and equipment. When added to the twodimensional models, the level of control adds a visual representation of the risk associated with the hazardous condition, creating a tall-pole for the least-well-controlled failure while establishing the relative likelihood and severity of all causes and effects for an identified hazard. Computer modeling of the analytical results, using spreadsheets and threedimensional charting gives a visual confirmation of the relationship between causes and their controls
Traditional hazard analysis techniques utilize a two-dimensional representation of the results determined by relative likelihood and severity of the residual risk. These matrices present a quick-look at the Likelihood (Y-axis) and Severity (X-axis) of the probable outcome of a hazardous event. A three-dimensional method, described herein, utilizes the traditional X and Y axes, while adding a new, third dimension, shown as the Z-axis, and referred to as the Level of Control. The elements of the Z-axis are modifications of the Hazard Elimination and Control steps (also known as the Hazard Reduction Precedence Sequence). These steps are: 1. Eliminate risk through design. 2. Substitute less risky materials for more hazardous materials. 3. Install safety devices. 4. Install caution and warning devices. 5. Develop administrative controls (to include special procedures and training.) 6. Provide protective clothing and equipment. When added to the two-dimensional models, the level of control adds a visual representation of the risk associated with the hazardous condition, creating a tall-pole for the least-well-controlled failure while establishing the relative likelihood and severity of all causes and effects for an identified hazard. Computer modeling of the analytical results, using spreadsheets and three-dimensional charting gives a visual confirmation of the relationship between causes and their controls.
Optical sensors aboard Earth orbiting satellites such as the next generation Visible/Infrared Imager/Radiometer Suite (VIIRS) assume that the sensors radiometric response in the Reflective Solar Bands (RSB) is described by a quadratic polynomial, in relating the aperture spectral radiance to the sensor Digital Number (DN) readout. For VIIRS Flight Unit 1, the coefficients are to be determined before launch by an attenuation method, although the linear coefficient will be further determined on-orbit through observing the Solar Diffuser. In determining the quadratic polynomial coefficients by the attenuation method, a Maximum Likelihood approach is applied in carrying out the least-squares procedure. Crucial to the Maximum Likelihood least-squares procedure is the computation of the weight. The weight not only has a contribution from the noise of the sensor s digital count, with an important contribution from digitization error, but also is affected heavily by the mathematical expression used to predict the value of the dependent variable, because both the independent and the dependent variables contain random noise. In addition, model errors have a major impact on the uncertainties of the coefficients. The Maximum Likelihood approach demonstrates the inadequacy of the attenuation method model with a quadratic polynomial for the retrieved spectral radiance. We show that using the inadequate model dramatically increases the uncertainties of the coefficients. We compute the coefficient values and their uncertainties, considering both measurement and model errors.