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A Robbins-Monro type multidimensional stochastic approximation algorithm which converges in mean square and with probability one to the fixed point of a locally contractive regression function is developed. The algorithm is applied to obtain maximum likelihood estimates of the parameters for a mixture of multivariate normal distributions.
A programmer guide for computing maximum likelihood estimates of mixture distributions from an unlabeled sample is presented. A subroutine provides graphic representation of mixture distributions.
The small laboratory computer is ideal for experimental control and data acquisition. Postexperimental data processing is often performed on large computers because of the availability of sophisticated programs, but costs and data compatibility are negative factors. Parameter optimization can be accomplished on the small computer, offering ease of programming, data compatibility, and low cost. A previously proposed random-search algorithm ('random creep') was found to be very slow in convergence. A method is proposed (the 'random leap' algorithm) which starts in a global search mode and automatically adjusts step size to speed convergence. A FORTRAN executive program for the random-leap algorithm is presented which calls a user-supplied function subroutine. An example of a function subroutine is given which calculates maximum-likelihood estimates of receiver operating-characteristic parameters from binary response data. Other applications in parameter estimation, generalized least squares, and matrix inversion are discussed.
An adaptive concept which combines gain-scheduled control laws with explicit maximum likelihood estimation (MLE) identification to provide the scheduling values is described. The MLE algorithm was improved by incorporating attitude data, estimating gust statistics for setting filter gains, and improving parameter tracking during changing flight conditions. A lateral MLE algorithm was designed to improve true air speed and angle of attack estimates during lateral maneuvers. Relationships between the pitch axis sensors inherent in the MLE design were examined and used for sensor failure detection. Design details and simulation performance are presented for each of the three areas investigated.
An explicit adaptive control algorithm based on maximum likelihood estimation of parameters has been designed for NASA's DFBW F-8 aircraft. To avoid iterative calculations, the algorithm uses parallel channels of Kalman filters operating at fixed locations in parameter space. This algorithm has been implemented in NASA/DFRC's Remotely Augmented Vehicle (RAV) facility. Real-time sensor outputs (rate gyro, accelerometer and surface position) are telemetered to a ground computer which sends new gain values to an on-board system. Ground test data and flight records were used to establish design values of noise statistics and to verify the ground-based adaptive software. The software and its performance evaluation based on flight data are described
A complete set of linear stability and control derivatives of the F-111A airplane was determined with a modified maximum likelihood estimator. The derivatives were determined at wing sweep angles of 26 deg, 35 deg, and 58 deg. The flight conditions included a Mach number range of 0.63 to 1.43 and an angle of attack range of 2 deg to 15 deg. Maneuvers were performed at normal accelerations from 0.9g to 3.8g during steady turns to assess the aeroelastic effects on the stability and control characteristics. The derivatives generally showed consistent trends and reasonable agreement with the wind tunnel estimates. Significant Mach effects were observed for Mach numbers as low as 0.82. No large effects attributable to aeroelasticity were noted.
The problem to be solved is formulated precisely and the introduction of quasi-Newton methods is motivated by considering the classical Newton and secant methods and their properties. Three highly successful quasi-Newton methods are surveyed: Broyden's method for the solution of general nonlinear equations, and the Davidon-Fletcher-Powell and Broyden-Fletcher-Goldfarb-Shanno procedures for unconstrained minimization. Finally, the properties of these methods are compared to those of Newton's method and UHMLE in potential applications to maximum-likelihood estimation of parameters in mixture distributions.
Papers are presented on a variety of techniques for the machine processing of remotely sensed data. Consideration is given to preprocessing methods such as the correction of Landsat data for the effects of haze, sun angle, and reflectance and to the maximum likelihood estimation of signature transformation algorithm. Several applications of machine processing to agriculture are identified. Various types of processing systems are discussed such as ground-data processing/support systems for sensor systems and the transfer of remotely sensed data to operational systems. The application of machine processing to hydrology, geology, and land-use mapping is outlined. Data analysis is considered with reference to several types of classification methods and systems.
A simplified aerodynamic force model based on the physical principle of Prandtl's lifting line theory and trailing vortex concept has been developed to account for unsteady aerodynamic effects in aircraft dynamics. Longitudinal equations of motion have been modified to include these effects. The presence of convolution integrals in the modified equations of motion led to a frequency domain analysis utilizing Fourier transforms. This reduces the integro-differential equations to relatively simple algebraic equations, thereby reducing computation time significantly. A parameter extraction program based on the maximum likelihood estimation technique is developed in the frequency domain. The extraction algorithm contains a new scheme for obtaining sensitivity functions by using numerical differentiation. The paper concludes with examples using computer generated and real flight data
This paper discusses the application of a maximum likelihood estimator to dynamic flight-test data. The information presented is based on the experience in the past twelve years at the NASA Dryden Flight Research Center of estimating stability and control derivatives from over 3,000 maneuvers from 32 aircraft. The overall approach to the analysis of dynamic flight-test data is discussed. Detailed requirements for data and instrumentation are discussed and several examples of the types of problems that may be encountered are presented.
Efforts to develop a technology for signature extension during LACIE phases 1 and 2 are described. A number of haze and Sun angle correction procedures were developed and tested. These included the ROOSTER and OSCAR cluster-matching algorithms and their modifications, the MLEST and UHMLE maximum likelihood estimation procedures, and the ATCOR procedure. All these algorithms were tested on simulated data and consecutive-day LANDSAT imagery. The ATCOR, OSCAR, and MLEST algorithms were also tested for their capability to geographically extend signatures using LANDSAT imagery.
The author has identified the following significant results. The statistical mapping approach to handling boundary pixels can be used as a standard for objectively comparing the cluster based technique, the maximum likelihood estimate based technique, and multicategory labeling.
The subsonic stability and control testing of the Space Shuttle Orbiter in its two test flights in the tailcone-off configuration is discussed, and test results are presented. Flight test maneuvers were designed to maximize the quality and quantity of stability and control data in the minimal time allotted using the Space Shuttle Functional Simulator and the Modified Maximum Likelihood Estimator (MMLE) programs, and coefficients were determined from standard sensor data sets using the MMLE, despite problems encountered in timing due to the different measurement systems used. Results are included for lateral directional and longitudinal maneuvers as well as the Space Shuttle aerodynamic data base obtained using the results of wind tunnel tests. The flight test data are found to permit greater confidence in the data base since the differences found are well within control system capability. It is suggested that the areas of major differences, including lateral directional data with open speedbrake, roll due to rudder and normal force due to elevon, be investigated in any further subsonic flight testing. Improvements in sensor data and data handling techniques for future orbital test flights are indicated.
An explicit adaptive control algorithm based on maximum likelihood estimation of parameters was designed. To avoid iterative calculations, the algorithm uses parallel channels of Kalman filters operating at fixed locations in parameter space. This algorithm was implemented in NASA/DFRC's Remotely Augmented Vehicle (RAV) facility. Real-time sensor outputs (rate gyro, accelerometer, surface position) are telemetered to a ground computer which sends new gain values to an on-board system. Ground test data and flight records were used to establish design values of noise statistics and to verify the ground-based adaptive software.
Methods for characterizing atmospheric turbulence are described. The methods illustrated include maximum likelihood estimation of the integral scale and intensity of records obeying the von Karman transverse power spectral form, constrained least-squares estimation of the parameters of a parametric representation of autocorrelation functions, estimation of the power spectra density of the instantaneous variance of a record with temporally fluctuating variance, and estimation of the probability density functions of various turbulence components. Descriptions of the computer programs used in the computations are given, and a full listing of these programs is included.
The filter produces the maximum likelihood estimates for average values of the spacecraft position and attitude errors during a single scene. The quality of the filter performance is characterized by the maximal cross and along track residual errors for which probability distributions can be calculated analytically for a given pattern of control points. The filter with an automatic selection of the best set of estimates provides geodetic correction at 90% of pixels with residual errors less than 40 m for four or more control points and the mean squared measurement errors of the order of 20-25 m. The same accuracy can be preserved for eight or more control points and measurement errors of 30-35 m.
The parameter identification problem of general discrete time, nonlinear, multiple input/multiple output dynamic systems with Gaussian white distributed measurement errors is considered. The knowledge of the system parameterization was assumed to be known. Concepts of local parameter identifiability and local constrained maximum likelihood parameter identifiability were established. A set of sufficient conditions for the existence of a region of parameter identifiability was derived. A computation procedure employing interval arithmetic was provided for finding the regions of parameter identifiability. If the vector of the true parameters is locally constrained maximum likelihood (CML) identifiable, then with probability one, the vector of true parameters is a unique maximal point of the maximum likelihood function in the region of parameter identifiability and the constrained maximum likelihood estimation sequence will converge to the vector of true parameters.
The effects of local flow, upwash, and sidewash on angle of attack and sideslip (measured with boom-mounted vanes) were determined for subsonic, transonic, and supersonic flight using a maximum likelihood estimator. The correction terms accounting for these effects were determined using a series of maneuvers flown at a large number of flight conditions in both augmented and unaugmented control modes. The correction terms provide improved angle-of-attack and sideslip values for use in the estimation of stability and control derivatives. In addition to detailing the procedure used to determine these correction terms, this paper discusses various effects, such as those related to Mach number, on the correction terms. The use of maneuvers flown in augmented and unaugmented control modes is also discussed.