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NASA NTRS1982

Applications of parameter estimation in the study of spinning airplanes

Spinning airplanes offer challenges to estimating dynamic parameters because of the nonlinear nature of the dynamics. In this paper, parameter estimation techniques are applied to spin flight test data for estimating the error in measuring post-stall angles of attack, deriving Euler angles from angular velocity data, and estimating nonlinear aerodynamic characteristics. The value of the scale factor for post-stall angles of attack agrees closely with that obtained from special wind-tunnel tests. The independently derived Euler angles are seen to be valid in spite of steep pitch angles. Estimates of flight derived nonlinear aerodynamic parameters are evaluated in terms of the expected fit error.

W Taylor, L., Jr.↗

NASA NTRS1981

Contextual classification of multispectral image data - An unbiased estimator for the context distribution

Recent investigations have demonstrated the effectiveness of a contextual classifier that combines spatial and spectral information employing a general statistical approach. This statistical classification algorithm exploits the tendency of certain ground-cover classes to occur more frequently in some spatial contexts than in others. Indeed, a key input to this algorithm is a statistical characterization of the context: the context distribution. Here a discussion is given of an unbiased estimator of the context distribution which, besides having the advantage of statistical unbiasedness, has the additional advantage over other estimation techniques of being amenable to an adaptive implementation in which the context distribution estimate varies according to local contextual information. Results from applying the unbiased estimator to the contextual classification of three real Landsat data sets are presented and contrasted with results from noncontextual classifications and from contextual classifications utilizing other context distribution estimation techniques.

Tilton, J. C.↗

NASA NTRS1982

Evaluation of small area crop estimation techniques using LANDSAT- and ground-derived data

Studies completed in fiscal year 1981 in support of the clustering/classification and preprocessing activities of the Domestic Crops and Land Cover project. The theme throughout the study was the improvement of subanalysis district (usually county level) crop hectarage estimates, as reflected in the following three objectives: (1) to evaluate the current U.S. Department of Agriculture Statistical Reporting Service regression approach to crop area estimation as applied to the problem of obtaining subanalysis district estimates; (2) to develop and test alternative approaches to subanalysis district estimation; and (3) to develop and test preprocessing techniques for use in improving subanalysis district estimates.

Amis, M. L.↗

NASA NTRS1981

Contextual classification of multispectral image data: An unbiased estimator for the context distribution

A key input to a statistical classification algorithm, which exploits the tendency of certain ground cover classes to occur more frequently in some spatial context than in others, is a statistical characterization of the context: the context distribution. An unbiased estimator of the context distribution is discussed which, besides having the advantage of statistical unbiasedness, has the additional advantage over other estimation techniques of being amenable to an adaptive implementation in which the context distribution estimate varies according to local contextual information. Results from applying the unbiased estimator to the contextual classification of three real LANDSAT data sets are presented and contrasted with results from non-contextual classifications and from contextual classifications utilizing other context distribution estimation techniques.

Tilton, J. C.↗

NASA NTRS1983

On the error in crop acreage estimation using satellite (LANDSAT) data

The problem of crop acreage estimation using satellite data is discussed. Bias and variance of a crop proportion estimate in an area segment obtained from the classification of its multispectral sensor data are derived as functions of the means, variances, and covariance of error rates. The linear discriminant analysis and the class proportion estimation for the two class case are extended to include a third class of measurement units, where these units are mixed on ground. Special attention is given to the investigation of mislabeling in training samples and its effect on crop proportion estimation. It is shown that the bias and variance of the estimate of a specific crop acreage proportion increase as the disparity in mislabeling rates between two classes increases. Some interaction is shown to take place, causing the bias and the variance to decrease at first and then to increase, as the mixed unit class varies in size from 0 to 50 percent of the total area segment.

Chhikara, R.↗

NASA NTRS1982

A comparison of minimum distance and maximum likelihood techniques for proportion estimation

The estimation of mixing proportions P sub 1, P sub 2,...P sub m in the mixture density f(x) = the sum of the series P sub i F sub i(X) with i = 1 to M is often encountered in agricultural remote sensing problems in which case the p sub i's usually represent crop proportions. In these remote sensing applications, component densities f sub i(x) have typically been assumed to be normally distributed, and parameter estimation has been accomplished using maximum likelihood (ML) techniques. Minimum distance (MD) estimation is examined as an alternative to ML where, in this investigation, both procedures are based upon normal components. Results indicate that ML techniques are superior to MD when component distributions actually are normal, while MD estimation provides better estimates than ML under symmetric departures from normality. When component distributions are not symmetric, however, it is seen that neither of these normal based techniques provides satisfactory results.

Woodward, W. A.↗

NASA NTRS1983

State and model error estimation for elliptic systems: Applications to large antenna static shape determination

This paper outlines the application of various estimation approaches to the problem of static shape determination for large antenna systems. The problem consists of estimating the shape of an antenna surface from measurements of its static deflection. The estimation schemes are based on any one of the modeling options of a single PDE for early insight and understanding, coarse-resolution multiple-PDE models for parametric studies and fine-resolution piecewise-continuum models for detailed design. For any one of these three models, estimator design can be developed using an infinite-dimensional approach, where the necessary finite-element truncation and approximation is conducted after the analytical design has taken place, or it can be based on a finite-dimensional approach, where the model is truncated before the estimation problem is formulated. One of the main objectives of the paper is to develop both approaches while simultaneously investigating their differences and similarities. Simulation results of an application of the finite-dimensional approach to a large parabolic reflector are presented.

Rodriguez, G.↗

NASA NTRS1982

Estimation of context for statistical classification of multispectral image data

Recent investigations have demonstrated the effectiveness of a contextual classifier that combines spatial and spectral information employing a general statistical approach. This statistical classification algorithm exploits the tendency of certain ground cover classes to occur more frequently in some spatial contexts than in others. Indeed, a key input to this algorithm is a statistical characterization of the context: the context function. An unbiased estimator of the context function is discussed which, besides having the advantage of statistical unbiasedness, has the additional advantage over other estimation techniques of being amenable to an adaptive implementation in which the context-function estimate varies according to local contextual information. Results from applying the unbiased estimator to the contextual classification of three real Landsat data sets are presented and contrasted with results from noncontextual classifications and from contextual classifications utilizing other context-function estimation techniques.

Tilton, J. C.↗

NASA NTRS1981

A function space approach to smoothing with applications to model error estimation for flexible spacecraft control

A function space approach to smoothing is used to obtain a set of model error estimates inherent in a reduced-order model. By establishing knowledge of inevitable deficiencies in the truncated model, the error estimates provide a foundation for updating the model and thereby improving system performance. The function space smoothing solution leads to a specification of a method for computation of the model error estimates and development of model error analysis techniques for comparison between actual and estimated errors. The paper summarizes the model error estimation approach as well as an application arising in the area of modeling for spacecraft attitude control.

Rodriguez, G.↗

NASA NTRS1982

A new solution to parameter adaptive estimation of random processes

This paper is concerned with the development of an adaptive state estimator that is capable of tracking switched linear plants that undergo rapid configuration changes. The particular adaptive estimator developed here is called the Sliding Window Detector/Estimator (SWDE) algorithm. Unlike previous algorithms, the SWDE algorithm is designed specifically for the switched-linear plant problem. It uses a joint detection/estimation approach to give a very close approximation to the unrealizable optimum switched-linear estimator. An extremely reliable and accurate estimator can be constructed by combining a modified Parameter Adaptive Estimation (PAE) algorithm with SWDE. The algorithm has been fully verified by extensive computer simulation, and the implementation advantages afforded by this method make it suitable for use in a wide variety of applications.

Zwicke, P. E.↗

NASA NTRS1984

An approach to software cost estimation

A general procedure for software cost estimation in any environment is outlined. The basic concepts of work and effort estimation are explained, some popular resource estimation models are reviewed, and the accuracy of source estimates is discussed. A software cost prediction procedure based on the experiences of the Software Engineering Laboratory in the flight dynamics area and incorporating management expertise, cost models, and historical data is described. The sources of information and relevant parameters available during each phase of the software life cycle are identified. The methodology suggested incorporates these elements into a customized management tool for software cost prediction. Detailed guidelines for estimation in the flight dynamics environment developed using this methodology are presented.

Mcgarry, F.↗

NASA NTRS1983

Model error estimation for distributed systems described by elliptic equations

A function space approach is used to develop a theory for estimation of the errors inherent in an elliptic partial differential equation model for a distributed parameter system. By establishing knowledge of the inevitable deficiencies in the model, the error estimates provide a foundation for updating the model. The function space solution leads to a specification of a method for computation of the model error estimates and development of model error analysis techniques for comparison between actual and estimated errors. The paper summarizes the model error estimation approach as well as an application arising in the area of modeling for static shape determination of large flexible systems.

Rodriguez, G.↗

NASA NTRS1984

Center of mass estimation in closed vortices - A verification in principle and practice

The problem of tracking closed mesoscale vortices using center of mass estimation techniques is studied. Three estimators are evaluated using data from a warm core Gulf Stream ring. The comparisons show that a method based on the intersection of perpendicular bisectors and one using a least-squares fit of a conic section perform comparably. The perpendicular bisector algorithm is used in conjunction with a Gaussian ring model and a star-shaped survey pattern to produce an expected error curve as a function of vortex translation, survey speed and vortex size. For typical ring parameters, center estimation is usually possible to within + or - 5 km. The feasibility of using differing data sets to construct a history of ring motion based on a coordinate system moving with the ring is also investigated. In this way, the validity of using satellite-derived data and drifter trajectories to estimate the center of mass of a mesoscale feature is assessed. The results of the analysis demonstrate that the location of the deeper structure of the ring and the surface expression are sufficiently well correlated to permit dynamically relevant calculations based on surface measurements. It is shown that satellite-derived data can be used to approximate the center of mass trajectory to within the error in the individual center estimates for the period analyzed. The Lagrangian-drifter-derived centers are offset from the center of mass trajectory in a manner consistent with kinematic arguments.

Hooker, S. B.↗

NASA NTRS1985

Estimating the R-curve from residual strength data

A method is presented for estimating the crack-extension resistance curve (R-curve) from residual-strength (maximum load against original crack length) data for precracked fracture specimens. The method allows additional information to be inferred from simple test results, and that information can be used to estimate the failure loads of more complicated structures of the same material and thickness. The fundamentals of the R-curve concept are reviewed first. Then the analytical basis for the estimation method is presented. The estimation method has been verified in two ways. Data from the literature (involving several materials and different types of specimens) are used to show that the estimated R-curve is in good agreement with the measured R-curve. A recent predictive blind round-robin program offers a more crucial test. When the actual failure loads are disclosed, the predictions are found to be in good agreement.

Orange, T. W.↗

NASA NTRS1986

On realizations of least-squares estimation and Kalman filtering by systolic arrays

Least-squares (LS) estimation is a basic operation in many signal processing problems. Given y = Ax + v, where A is a m x n coefficient matrix, y is a m x 1 observation vector, and v is a m x 1 zero mean white noise vector, a simple least-squares solution is finding the estimated vector x which minimizes the norm of /Ax-y/. It is well known that for an ill-conditioned matrix A, solving least-squares problems by orthogonal triangular (QR) decomposition and back substitution has robust numerical properties under finite word length effect since 2-norm is preserved. Many fast algorithms have been proposed and applied to systolic arrays. Gentleman-Kung (1981) first presented the trianglular systolic array for a basic Givens reduction. McWhirter (1983) used this array structure to find the least-squares estimation errors. Then by geometric approach, several different systolic array realizations of the recursive least-squares estimation algorithms of Lee et al (1981) were derived by Kalson-Yao (1985). Basic QR decomposition algorithms are considered in this paper and it is found that under a one-row time updating situation, the Householder transformation degenerates to a simple Givens reduction. Next, an improved least-squares estimation algorithm is derived by considering a modified version of fast Givens reduction. From this approach, the basic relationship between Givens reduction and Modified-Gram-Schmidt transformation can easily be understood. This improved algorithm also has simpler computational and inter-cell connection complexities while compared with other known least-squares algorithms and is more realistic for systolic array implementation.

Chen, M. J.↗

NASA NTRS1985

Evaluation and interpretation of Thematic Mapper ratios in equations for estimating corn growth parameters

Reflectance and agronomic Thematic Mapper (TM) data were analyzed to determine possible data transformations for evaluating several plant parameters of corn. Three transformation forms were used: the ratio of two TM bands, logarithms of two-band ratios, and normalized differences of two bands. Normalized differences and logarithms of two-band ratios responsed similarly in the equations for estimating the plant growth parameters evaluated in this study. Two-term equations were required to obtain the maximum predictability of percent ground cover, canopy moisture content, and total wet phytomass. Standard error of estimate values were 15-26 percent lower for two-term estimates of these parameters than for one-term estimates. The terms log(TM4/TM2) and (TM4/TM5) produced the maximum predictability for leaf area and dry green leaf weight, respectively. The middle infrared bands TM5 and TM7 are essential for maximizing predictability for all measured plant parameters except leaf area index. The estimating models were evaluated over bare soil to discriminate between equations which are statistically similar. Qualitative interpretations of the resulting prediction equations are consistent with general agronomic and remote sensing theory.

Dardner, B. R.↗

NASA NTRS1986

Error analysis of leaf area estimates made from allometric regression models

Biological net productivity, measured in terms of the change in biomass with time, affects global productivity and the quality of life through biochemical and hydrological cycles and by its effect on the overall energy balance. Estimating leaf area for large ecosystems is one of the more important means of monitoring this productivity. For a particular forest plot, the leaf area is often estimated by a two-stage process. In the first stage, known as dimension analysis, a small number of trees are felled so that their areas can be measured as accurately as possible. These leaf areas are then related to non-destructive, easily-measured features such as bole diameter and tree height, by using a regression model. In the second stage, the non-destructive features are measured for all or for a sample of trees in the plots and then used as input into the regression model to estimate the total leaf area. Because both stages of the estimation process are subject to error, it is difficult to evaluate the accuracy of the final plot leaf area estimates. This paper illustrates how a complete error analysis can be made, using an example from a study made on aspen trees in northern Minnesota. The study was a joint effort by NASA and the University of California at Santa Barbara known as COVER (Characterization of Vegetation with Remote Sensing).

Feiveson, A. H.↗

NASA NTRS1987

On the representation and estimation of spatial uncertainty

This paper describes a general method for estimating the nominal relationship and expected error (covariance) between coordinate frames representing the relative locations of objects. The frames may be known only indirectly through a series of spatial relationships, each with its associated error, arising from diverse causes, including positioning errors, measurement errors, or tolerances in part dimensions. This estimation method can be used to answer such questions as whether a camera attached to a robot is likely to have a particular reference object in its field of view. The calculated estimates agree well with those from an independent Monte Carlo simulation. The method makes it possible to decide in advance whether an uncertain relationship is known accurately enough for some task and, if not, how much of an improvement in locational knowledge a proposed sensor will provide. The method presented can be generalized to six degrees of freedom and provides a practical means of estimating the relationships (position and orientation) among objects, as well as estimating the uncertainty associated with the relationships.

Smith, Randall C.↗