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NASA NTRS1988

Two biased estimation techniques in linear regression: Application to aircraft

Several ways for detection and assessment of collinearity in measured data are discussed. Because data collinearity usually results in poor least squares estimates, two estimation techniques which can limit a damaging effect of collinearity are presented. These two techniques, the principal components regression and mixed estimation, belong to a class of biased estimation techniques. Detection and assessment of data collinearity and the two biased estimation techniques are demonstrated in two examples using flight test data from longitudinal maneuvers of an experimental aircraft. The eigensystem analysis and parameter variance decomposition appeared to be a promising tool for collinearity evaluation. The biased estimators had far better accuracy than the results from the ordinary least squares technique.

Klein, Vladislav↗

NASA NTRS1989

Aerodynamic parameters of an advanced fighter aircraft estimated from flight data. Preliminary results

Preliminary estimates of aerodynamic parameters of an advanced fighter aircraft were obtained from flight data of different values of the angle of attack from 8 to 54 deg. The data were analyzed by a stepwise regression with the ordinary least squares technique. The estimated stability and control derivatives are plotted against the angle of attack and compared with wind tunnel measurement and previous flight results. Also included is the data compatibility check of measured data. The effect of various input forms on the estimates is demonstrated in two examples using simulated data.

Klein, Vladislav↗

NASA NTRS1998

Aerodynamic Parameters of High Performance Aircraft Estimated from Wind Tunnel and Flight Test Data

A concept of system identification applied to high performance aircraft is introduced followed by a discussion on the identification methodology. Special emphasis is given to model postulation using time invariant and time dependent aerodynamic parameters, model structure determination and parameter estimation using ordinary least squares an mixed estimation methods, At the same time problems of data collinearity detection and its assessment are discussed. These parts of methodology are demonstrated in examples using flight data of the X-29A and X-31A aircraft. In the third example wind tunnel oscillatory data of the F-16XL model are used. A strong dependence of these data on frequency led to the development of models with unsteady aerodynamic terms in the form of indicial functions. The paper is completed by concluding remarks.

Klein, Vladislav↗

NASA NTRS1999

Aerodynamic Parameters of High Performance Aircraft Estimated from Wind Tunnel and Flight Test Data

A concept of system identification applied to high performance aircraft is introduced followed by a discussion on the identification methodology. Special emphasis is given to model postulation using time invariant and time dependent aerodynamic parameters, model structure determination and parameter estimation using ordinary least squares and mixed estimation methods. At the same time problems of data collinearity detection and its assessment are discussed. These parts of methodology are demonstrated in examples using flight data of the X-29A and X-31A aircraft. In the third example wind tunnel oscillatory data of the F-16XL model are used. A strong dependence of these data on frequency led to the development of models with unsteady aerodynamic terms in the form of indicial functions. The paper is completed by concluding remarks.

Klein, Vladislav↗

NASA NTRS1998

Baseline Observations of Hemispheric Sea Ice with the Nimbus 7 Scanning Multichannel Microwave Radiometer

The Scanning Multichannel Microwave Radiometer (SMMR) on board the NASA Nimbus 7 satellite was designed to obtain data for sea surface temperatures (SSTs), near-surface wind speeds, sea ice coverage and type, rainfall rates over the oceans, cloud water content, snow water equivalent, and soil moisture. In this paper, I shall emphasize the sea ice observations and mention briefly some important SST observations. A prime factor contributing to the importance of SMMR sea ice observations lies in their successful integration into a long-term time series, presently being extended by observations from the series of Special Sensor Microwave/Imager (SSMI) on board the DOD/DMSP F8, Fl1, and F12 satellites. This currently constitutes a 19-year data set. Almost half of this was provided by the SMMR. Unfortunately, the 4-year data set produced earlier by the single-channel Electrically Scanned Microwave Radiometer (ESMR) was not successfully integrated into the SMMR/SSMI data set. This resulted primarily from the lack of an overlap period to provide intersensor adjustment, but also because of the large difference between the algorithms to produce ice concentrations and large temporal gaps in the ESMR data. The lack of overlap between the SeaSat and Nimbus 7 SMMR data sets was an important consideration for also excluding the SeatSat one, but the spatial gaps especially in the Southern Hemisphere daily SeaSat observations was another. The sea ice observations will continue into the future by means of the Advanced Microwave Scanning Radiometer (AMSR) on board the ADEOS II and EOS satellites due to be launched in mid- and late-2000, respectively. Analysis of the sea ice data has been carried out by a number of different techniques. Long-term trends have been examined by means of ordinary least squares and band-limited regression. Oscillations in the data have been examined by band-limited Fourier analysis. Here, I shall present results from a novel combination of Principal Component analysis and the recently-developed Empirical Mode Decomposition (EMD). In this method, the data are first separated into spatial and temporal parts, and then the temporal parts of the first few PCs are broken into intrinsic modes by the EMD method.

Gloersen, Per↗

NASA NTRS2003

An Airline-Based Multilevel Analysis of Airfare Elasticity for Passenger Demand

Price elasticity of passenger demand for a specific airline is estimated. The main drivers affecting passenger demand for air transportation are identified. First, an Ordinary Least Squares regression analysis is performed. Then, a multilevel analysis-based methodology to investigate the pattern of variation of price elasticity of demand among the various routes of the airline under study is proposed. The experienced daily passenger demands on each fare-class are grouped for each considered route. 9 routes were studied for the months of February and May in years from 1999 to 2002, and two fare-classes were defined (business and economy). The analysis has revealed that the airfare elasticity of passenger demand significantly varies among the different routes of the airline.

Castelli, Lorenzo↗

NASA NTRS2006

Classes of Split-Plot Response Surface Designs for Equivalent Estimation

When planning an experimental investigation, we are frequently faced with factors that are difficult or time consuming to manipulate, thereby making complete randomization impractical. A split-plot structure differentiates between the experimental units associated with these hard-to-change factors and others that are relatively easy-to-change and provides an efficient strategy that integrates the restrictions imposed by the experimental apparatus. Several industrial and scientific examples are presented to illustrate design considerations encountered in the restricted randomization context. In this paper, we propose classes of split-plot response designs that provide an intuitive and natural extension from the completely randomized context. For these designs, the ordinary least squares estimates of the model are equivalent to the generalized least squares estimates. This property provides best linear unbiased estimators and simplifies model estimation. The design conditions that allow for equivalent estimation are presented enabling design construction strategies to transform completely randomized Box-Behnken, equiradial, and small composite designs into a split-plot structure.

Parker, Peter A.↗

NASA NTRS2012

Measurement System Characterization in the Presence of Measurement Errors

In the calibration of a measurement system, data are collected in order to estimate a mathematical model between one or more factors of interest and a response. Ordinary least squares is a method employed to estimate the regression coefficients in the model. The method assumes that the factors are known without error; yet, it is implicitly known that the factors contain some uncertainty. In the literature, this uncertainty is known as measurement error. The measurement error affects both the estimates of the model coefficients and the prediction, or residual, errors. There are some methods, such as orthogonal least squares, that are employed in situations where measurement errors exist, but these methods do not directly incorporate the magnitude of the measurement errors. This research proposes a new method, known as modified least squares, that combines the principles of least squares with knowledge about the measurement errors. This knowledge is expressed in terms of the variance ratio - the ratio of response error variance to measurement error variance.

Commo, Sean A.↗

NASA NTRS2020

Recommendations for Using Noise Monitors to Estimate Noise Exposure During X-59 Community Tests

A low fidelity simulation approach is used to explore how to place and use noise monitors during X-59 QueSST community tests, where people’s annoyance to the noise produced by the X-59 aircraft will be gathered. Several recommendations are provided including: 1) the desired number of sparsely spaced noise monitor sites within the survey area, 2) whether to group and average measurements across multiple noise monitors located at a site, 3) what spacing should be used if grouped noise monitors are used, 4) an approach to mitigate ambient noise contamination at the measurement sites, 5) a method to combine empirical and predicted dose estimates to provide a single dose estimate for respondents, and 6) assessing how changes in turbulence intensity and array configuration affect dose uncertainty. To make these recommendations, the error that is expected when fitting contrived, smoothly varying sonic boom “reference exposure surfaces” is studied when a spatially sparse and scattered set of samples is used as responses for the fit. The reference exposure surfaces mimic the sonic boom exposure at ground level that might be expected in the X-59 survey area in the absence of atmospheric turbulence, ambient noise, and other localized effects. The spatial extent of these surfaces varies and is representative of the different survey area sizes that might be expected during future X-59 community overflight tests. These contrived reference surfaces are sampled, and those reference samples are then perturbed to mimic atmospheric turbulence, ambient noise and other localized effects that might affect noise monitor measurements within overflown communities. Two different surface fitting methods are investigated when fitting these perturbed samples to approximate the reference surface. The first method uses interpolation between the perturbed data at the scattered sites to compute the fit. The second method fits a polynomial surface model to the perturbed data using ordinary least squares regression analysis. For both fitting methods, the root mean square fit error is computed from the pointwise difference between the fit surface and the reference surface as the count and configuration of the sites is varied while also averaging the error across many different realizations of both the smooth variation of the reference exposure surface and the random, localized perturbations at the sample sites. Different site configurations are compared using this error statistic to make the recommendations noted above. Additionally, the two fitting approaches (interpolation vs linear regression) are compared based on the fit error observed in these simulations. These analyses, comparisons, and recommendations should inform future decisions on the noise monitor placement and the methods used to analyze the noise monitor data that is collected during X-59 community overflights.

sonic boom↗

NASA NTRS2022

Aboveground biomass density models for NASA’s Global Ecosystem Dynamics Investigation (GEDI) lidar mission

NASA’s Global Ecosystem Dynamics Investigation (GEDI) is collecting spaceborne full waveform lidar data with a primary science goal of producing accurate estimates of forest aboveground biomass density (AGBD). This paper presents the development of the models used to create GEDI’s footprint-level (~25 m) AGBD (GEDI04_A) product, including a description of the datasets used and the procedure for final model selection. The data used to fit our models are from a compilation of globally distributed spatially and temporally coincident field and airborne lidar datasets, whereby we simulated GEDI-like waveforms from airborne lidar to build a calibration database. We used this database to expand the geographic extent of past waveform lidar studies, and divided the globe into four broad strata by Plant Functional Type (PFT) and six geographic regions. GEDI’s waveform-to-biomass models take the form of parametric Ordinary Least Squares (OLS) models with simulated Relative Height (RH) metrics as predictor variables. From an exhaustive set of candidate models, we selected the best input predictor variables, and data transformations for each geographic stratum in the GEDI domain to produce a set of comprehensive predictive footprint-level models. We found that model selection frequently favored combinations of RH metrics at the 98th, 90th, 50th, and 10th height above ground-level percentiles (RH98, RH90, RH50, and RH10, respectively), but that inclusion of lower RH metrics (e.g. RH10) did not markedly improve model performance. Second, forced inclusion of RH98 in all models was important and did not degrade model performance, and the best performing models were parsimonious, typically having only 1-3 predictors. Third, stratification by geographic domain (PFT, geographic region) improved model performance in comparison to global models without stratification. Fourth, for the vast majority of strata, the best performing models were fit using square root transformation of field AGBD and/or height metrics. There was considerable variability in model performance across geographic strata, and areas with sparse training data and/or high AGBD values had the poorest performance. These models are used to produce global predictions of AGBD, but will be improved in the future as more and better training data become available.

Laura Duncanson↗

NASA NTRSDate not supplied

Using Distributed Fiber-optic Strain Sensing to Estimate Generalized Modal Coordinates from Flight-test Data

A method for estimating the generalized modal coordinates of an aircraft during flight has been developed. The Fiber-optic Sensing System (FOSS) offers an efficient and cost-effective method of measuring the strain at thousands of points along the wings. The estimation of modal coordinates was implemented as a two-step process. First, a maximum likelihood method is used to estimate the statistical properties of the sensors and generalized modal coordinates. Second, the strain mode shapes from the finite element model are used along with the statistical properties from the first step to estimate the generalized modal coordinates over time. Using simulated data from the X-56A Multi-Utility Technology Testbed (MUTT), different methods of modal coordinate estimation were compared to demonstrate the benefits and weaknesses of each. These were compared against the exact solution and an ordinary least squares (a more traditional method) estimate. Modal coordinate estimation methods were then applied to flight-test data from the X-56A aircraft to show that the method continues to work well with actual test data. The new estimation method provides insights unavailable from more classical approaches.

Jeffrey Ouellette↗

NASA NTRSDate not supplied

Using Distributed Fiber-Optic Strain Sensing to Estimate Modal Generalized Modal Coordinates from Flight-Test Data

Background and Motivation - Problem - Contemporary aircraft carry around structural mass so that the flutter instabilities lie well outside of the operational envelope. - Better methods of measuring the structural state could allow reduction of the extra structural weight. - Modal filtering - Standard method in structural analysis - Deformations are a combination of mode shapes - Modal filtering is estimating these modal coordinates from data - Often ordinary least squares methods - Often applied to simpler test articles - Factor Analysis - Classic (in psychology) analysis method - Measurements are a combination of small number of unmeasurable variables - Lessons from this factor can be adapted to improve the modal filtering methods

Jeffrey Ouellette↗

NASA NTRSDate not supplied

Using Distributed Fiber-optic Strain Sensing to Estimate Generalized Modal Coordinates from Flight-test Data

Background and Motivation - Problem - Contemporary aircraft carry around structural mass so that the flutter instabilities lie well outside of the operational envelope. - Better methods of measuring the structural state could allow reduction of the extra structural weight - Modal filtering - Standard method in structural analysis - Deformations are a combination of mode shapes - Modal filtering is estimating these modal coordinates from data - Often ordinary least squares methods - Often applied to simpler test articles - Factor Analysis - Analysis method from psychology - Measurements are a combination of small number of unmeasurable variables. - Lessons from factor analysis can be adapted to improve the modal filtering methods

Jeffrey Ouellette↗

NASA NTRS1990

Ridge regression processing

Current navigation requirements depend on a geometric dilution of precision (GDOP) criterion. As long as the GDOP stays below a specific value, navigation requirements are met. The GDOP will exceed the specified value when the measurement geometry becomes too collinear. A new signal processing technique, called Ridge Regression Processing, can reduce the effects of nearly collinear measurement geometry; thereby reducing the inflation of the measurement errors. It is shown that the Ridge signal processor gives a consistently better mean squared error (MSE) in position than the Ordinary Least Mean Squares (OLS) estimator. The applicability of this technique is currently being investigated to improve the following areas: receiver autonomous integrity monitoring (RAIM), coverage requirements, availability requirements, and precision approaches.

Kuhl, Mark R.↗

NASA NTRS1975

Comparison of some biased estimation methods (including ordinary subset regression) in the linear model

Ridge, Marquardt's generalized inverse, shrunken, and principal components estimators are discussed in terms of the objectives of point estimation of parameters, estimation of the predictive regression function, and hypothesis testing. It is found that as the normal equations approach singularity, more consideration must be given to estimable functions of the parameters as opposed to estimation of the full parameter vector; that biased estimators all introduce constraints on the parameter space; that adoption of mean squared error as a criterion of goodness should be independent of the degree of singularity; and that ordinary least-squares subset regression is the best overall method.

Sidik, S. M.↗

NASA NTRS1990

Linear regression in astronomy. I

Five methods for obtaining linear regression fits to bivariate data with unknown or insignificant measurement errors are discussed: ordinary least-squares (OLS) regression of Y on X, OLS regression of X on Y, the bisector of the two OLS lines, orthogonal regression, and 'reduced major-axis' regression. These methods have been used by various researchers in observational astronomy, most importantly in cosmic distance scale applications. Formulas for calculating the slope and intercept coefficients and their uncertainties are given for all the methods, including a new general form of the OLS variance estimates. The accuracy of the formulas was confirmed using numerical simulations. The applicability of the procedures is discussed with respect to their mathematical properties, the nature of the astronomical data under consideration, and the scientific purpose of the regression. It is found that, for problems needing symmetrical treatment of the variables, the OLS bisector performs significantly better than orthogonal or reduced major-axis regression.

Isobe, Takashi↗

NASA NTRS2017

Heuristic Area Cost Estimation for Observational Coverage Schedulers

This paper presents a comparison of heuris- tics used to estimate the amount of time it would take for a spacecraft to image an area using Boustrophedon decomposition (Choset and Pignon 1998). Machine learning tech- niques are used to characterize algorithmic performance of coverage algorithms. It is shown that an ordinary least-squares linear model is among the most accurate in a set of constant and linear order regression models both in terms of memory consumption and schedule duration. These are demonstrated using the ASPEN planning system (Fukunaga et al. 1997) on the Eagle Eye domain.

Knight, Russell↗

NASA NTRS1985

A Galerkin method for the estimation of parameters in hybrid systems governing the vibration of flexible beams with tip bodies

An approximation scheme is developed for the identification of hybrid systems describing the transverse vibrations of flexible beams with attached tip bodies. In particular, problems involving the estimation of functional parameters are considered. The identification problem is formulated as a least squares fit to data subject to the coupled system of partial and ordinary differential equations describing the transverse displacement of the beam and the motion of the tip bodies respectively. A cubic spline-based Galerkin method applied to the state equations in weak form and the discretization of the admissible parameter space yield a sequence of approximating finite dimensional identification problems. It is shown that each of the approximating problems admits a solution and that from the resulting sequence of optimal solutions a convergent subsequence can be extracted, the limit of which is a solution to the original identification problem. The approximating identification problems can be solved using standard techniques and readily available software.

Banks, H. T.↗