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Star0 4 Dec121985 1986004609 1 , STAR0 4 DEC121985 _-- NASA Contractor Report 171904 1 _";_ ic33 NASA/ASEE Summer Faculty Fellowship Program Rc3earch Reports _ _'don B. J _.nson Space Cenr.er •_- e T_as A&M U_iversity _, • .... E _T,_,,_ _-_ ," - PAC_/Ty_N{_SA-C_-I7190_)EELLO_SBI_TBE1983 NAS&/ASEE SUMMEB N86-I_078 _ _' i_ REPORTS Final _e_ortsRES_A'_CB P£OGRAM RESEARCH THR0 HC A18/t_F AO1 N86-1_I05 ii _'_ CSCI 051 Uncles _ G3/85 02928 _ Editors : • _ Dr. Walter J. Horn ,. Professor of Aerospace Engineering ._ { < Texas A&M University .,," i College Station, Texas ;_-' [ •% Dr. Michael B. Duke ;. Office of University Affairs ', Lyndon B. Johnson Space Center " Houston, Texas ,_- I ; 1 "! t k # < September 1983 L \ ' ' ,,"r Prepared for .'.2 i ,,_. NASA Lyndon B. Johnson Space Center -._ Houston, Texas 77056 ,c ]986004609-002 . i •"i PREFACE - The 1983 NASA/ASEE Stmner Faculty Fellowship Research Program was conducted _. by Texas A&M University and the Lyudon B. Johnson Space Center (JSC). The lO-week program was operated under the auspices of the American Society for : Engineering Education (ASEE) These programs, conducted by JSC and other "_ NASA Centers, began in 1964. They are sponsored and funded by the Office oi -_ University Affairs, NASA Headquarters, Washington, D.C. The objectives of _ the programs are the followlng: • • : a. To further the professional knowledge of qualified engineering and science faculty members b. To st_nulate an exch_ge of ideas between participants and NASA ¢ c. To enrich and refresh the research and teaching activities of participants' institutions d. To contribute to the research objectives of the NASA Centers _, .- The faculty fellows spent i0 weeks at JSC engaged in a research project commensurate with their interests and background. They worked in co _ collaboration with a NASA/JSC colleague. This document is a compilation of the final reports on their research during the summer of 1983. Texas A&M Research Foundation Report No. 4194-83 is the Co-Directors' report on ! the administrative operations of the Summer Faculty Fellowship Program. 1 : I F I 1 • 1 f %- _ , 1986004609-003 , CONTENTS 1. Anderson, Charles: Estimation of Finite Mixtures Using the Empirlcal • ' Characteristic Function. ,_ 2. Barrett, Roy A.: The Scanning Electron Microscope as a Tool in Space Biology. \ 3. Blanford t George E.: Electron Microscopic Observations of Hydrogen • Implantation in Ilmenites. ° • 4. Bustin, Roberta: A Pyrolysis Technique for Determining Microamounts of Hydrogen in Lunar Soil Using the Helium Ionization Detector. 5. Durgun, Kanat: Finite Element or Calerkin Type Semidiscrete Schemes. ! 6. Ghorai, Susanta K.: Determination of Neutron Flux Distribution by Using Anisn, A One-Dimensional Discrete S Ordinates Transport Code with | n 0 Anisotropic Scattering. ": - 7. Hall, Minard L.: Paper Unavailable _ _ 8. Hanania, Jack I.: A Study of Some Features of AC and DC Electric Power _ Systems for a Space Station. ¢ 9. Ito, Takeru: Nutritional Characteristics of Moon Dust for Soil Microorganisms. i I0. Jackson, A. A.: Some Applications of Lie Groups in Astrodynamics. i ) Ii. Johansson, Karl R.: Interaction Between Escherichia coli and Lunar Fines. } 12. Kiesewetter, Carl H.: Spectral Reflectance of Surface Soils - Relationships _ i with Soma Soll Properties. i 13. Lee, Kwang Y.: Control of Space Stations. ' 14. Marrero, Thomas R._ Solid Waste Treatment Processes for Space Station. 15. Measel, John N.: Paper Unavailable 16. Morgan, Thomas H.: Lunar Luminescence Measurements. 17. Palms, Russell L.: Development of a Mass Spectrometer System for the Measurement of Inert Gases in Meteorites. 18. Presley, Bobby Joe: Paper Unavailable • 19. Ra_ow,(400Alleanre: a)AnandAnalysRecoismmendaof tCryionsotrapforHeaa tSysExchantem togerHandlePerforAmpollanceo RCTesSt Engines.Data • _ 20. Ridley, Esther J.: Comparative Effect of Lunar Fines and Terrestrial Ash _' on the Growth of a Blue-Green Al_a and Germinating Radish Seeds. ! ' 1986004609-004 21, Runnels, Robert C. : Variability of Raiufa11 Over Small Areas. 22. Sartain, Robert L.: T_ansformation Matrices Between Non-Linear and ; Linear Differential Equations. ' 23. Schulze, Paul D.- Ion Bombardment and Adsorption Studies on Ilmenite (FeTi03) by X-Ray Photoelectron Spectroscopy. : 24. Szebehely, Victor: Advanced Algorithm for Orbit Computation. 25. Thomas, Richard G.: Solar Concentrator Degradation in Low Earth Orbit (LEO). 26. Williams, Raymond: Fortran Plotting Subroutines for the Space Plasma _' Laboratory. • j 27. Wyzkoskl, Joan: Computer Graphics Appllcatlonb to Crew Displays. I ._ 28. Zimmermann, Wayne J.: Sisnal Processing of Anthropometric Data. i =- { ( ! ! : ! ! @ _t | .2 i i -_, iv ] 986004609-005 _t _ I ESTIMATION OF FINITE MIXTURES USING THE : EMPIRICAL CHARACTERISTIC FUNCTION N86- 1407 "_' Charles Anderson, Associate Professor ," DepartmenThtomasof MBoatuhemallion,ticsProandfessorStatistics :; University of Southwestern Louisiana " ' Lafayette, Louisiana 70504 _' ABSTRACT _ A problem which occurs in analyzing Landsat scenes is the problem of separating the components of a finite mixture of several distinct e probability distributions. A review of the literature indicates this is a problem which occurs in many disciplines, such as engineering, biology, l ._ phy3iology and economics. Many approaches to this problem have appeared - in the literature; however, most are very restrictive in their assump- == tions or have met with only a limited degree of success when applied to realistic situations. 1' We have been investigating a procedure which combines the "k-L ! _ procedure" of [Feurverger and McDunnough, 1981] with the "MAICE" procedure i of [Akaike, 1974]. The feasibility of this approach is being investigated I numerically via the development of a computer software package enabling r! a simulation study and comparison with other procedures. k_ } Center Research Advisor: M. C. Trichel " 1-1 i ] 986004609-006 INTRODUCTION ! A Problem which occurs in many disciplines is that of separating the components of a probability distribution which is a finite mixture of several distinct dist,,ibutions. See, for instance, [Yakowitz, 1970], [Bhattacharya, 1976], and [Day, 1969]. This problem is encountered in the Remote Sensing Research Branch of NASA Johnson Space Center in analyzlng Landsat data. A number of different approaches have been taken to resolve this problem, each enjoying a rather limited degree of success or being too i restrictive to be widely applicable. Since the likelihood function corresponding to finite mixtures of normal distributions is unbounded, % maximum likelihood estimation frequently breaks down in practice. The estimator which minimizes the sum of squares of differences between the I : theoretical and sample moment generating functions, given by [Quandt and J Ramsey, 1978], seems to suffer from inefficiency and some arbitrariness < 3 in the choice of weights given to the moments. Estimating the mixing i proportions of a mixture of known distributions [Bryant and Paulson, i 1983], using the distance between characteristic functions is too restrictive, since it assumes the parameters in the component distribu- _"4 tions are completely known. tL A recent approach by [Heydorn and Basu, 1983] makes use of a • t constructive proof of a theorem of Caratheodory on a trigonometric ! Q moment problem, as discussed in [Grenander and Szego, 1958], to determine identifiable mixtures for certain special cases of families of distributions. When only sample da_a is available, this approach does not seem to be immediately applicable. I 1-2 i ]986004609-007 _' PROCEDURE AND JUSTIFICATION _} The "k-L procedure" introduced by [Feurverger and McDunnough, 1981], J_ _ refers generally to approximate maximum likelihood estimation based o_ _. the asymptotic distribution at k points of the empirical characteristic _;, function (e.c.f.). Since the e.c.f, contains all the information in the i_.. sample, and for other reasons given later, it _eems to be a promising • technique. See Figure I. N Let _ be a column vector composed of the real and imaginary parts of the e.c.f, at points d, 2d,..., kd. The probability distribution of _ \- / _ is approximately multivariate normal, even for fairly small sample _. L - sizes, because of the smoothness and boundedness of the trigonometric functions. The covariance matrix a E(_-E(_)) (_-E(_)) T is by the values of the true characteristic function O(t) at t=d, 2d,..., 2kd and can be estimated from the values of the e.c.f, at these points. Since this estimate _ is consistent, the following estimation criteria are asymptotically equivalent: (I) maximize the likelihood given _, (2) T ^ minimize (_-E(_))T_-E(_)), (3) minimize (_-E(_)) _-_-E(_)). M The hypothesis Ho: _(t)=_Pi1 _l(t) lSil ,''') with certain of the parameters Pi, @ij specified, _an be tested against an alternative '_ i hypothesis specifying the same form of the model but with parameters not all as specified by Ho, using the approximate chi-square distribution of L=n_-E(_)}_-I(_-E_ )). The hypothesis H o is rejected if L is greater than the (1.._) point of its null distribution, which is approximately the chi-square with degrees of freedom equal to 2k minus the number of i functionally independent unspecified parameters under Ho. 1-3 ,, )_.................... .. ®. 1986004609-008 When there are several competing models, the MAICE procedure, intT,oduced by [Akaike, 1974], selects the model which gives the minimum value of AIC = (-2) log (maximum likelihood) + 2(number of independently adjusted parameters within the model). This procedure has been investigated by [Redner, Kitagawa, and _b Coberly, 1981], working directly with the mixed distributions. Our procedure applies the MAICE method to data reduced to a few carefully selected points of the e.c.f. Also, maximum likelihood is computed J approximately using the approximate normality of the e.c.f. FURTHER INVESTIGATIONS Although the above procedure is based on sound theoretical arguments, no numerical results have appeared indicating its efficiency of implementa- tion on a computer or its accuracy in determining the best model and estimates of the parameters.
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