"SPATIAL ERROR ANALYSIS is an all-in-one sourcebook on error measurements in one-, two-, and three-dimensional spaces. This book features exhaustive, systematic coverage of error measurement relationships, techniques, and solutions used to solve general, correlated cases. It is packed with 62 figures and 24 tables. MATLAB-based M-files* for practical applications created especially for this volume are available on the Web at ftp://ftp.mathworks.com/pub/books/hsu. Solutions to two- and three-dimensional problems are presented without relying on equal standard deviations from each channel. They also make no assumption that the random variables of interest are independent or uncorrelated. * MATLAB (developed by MathWorks, Inc.) must be purchased separately." Sponsored by: IEEE Aerospace and Electronic Systems Society.

About the Author David Y. Hsu is currently a member of the Senior Technical Staff with the Advanced Systems Analysis Group at Litton Guidance and Control Systems Division. Since 1982, his principal tasks have included gravity modeling/compensation, simulation, and analysis. He has authored over 20 scientific articles in the ION, IEEE Plans, NAECON proceedings, and other Mathematical/Engineering journals. Dr. Hsu received his B.S.E.E. from the National Taiwan University and his Ph.D. in Applied Mathematics from the University of Virginia. He has taught in the Electrical Engineering, Mathematics, and Computer Science departments of various universities in the United States as well as in the Taiwan province of China. He is a member of the Institute of Navigation and the Mathematical Association of America.

PREFACE

xi

(4)

LIST OF FIGURES

xv

(2)

LIST OF TABLES

xvii

CHAPTER 1 Introduction

1

(6)

1.1 Notation

2

(1)

1.2 Direct Problems

2

(1)

1.3 Inverse Problems

3

(1)

1.4 Use of Author-Generated M-files

4

(1)

1.5 Summary of M-files

5

(2)

CHAPTER 2 Parameter Estimation from Samples

7

(16)

2.1 Point Estimate of Population Parameters

8

(3)

2.2 Sampling Distribution

11

(1)

2.3 Interval Estimate of Population Mean

12

(2)

2.4 Interval Estimate of Population Standard Deviation

14

(5)

2.5 Summary of M-files

19

(4)

CHAPTER 3 One-Dimensional Error Analysis

23

(12)

3.1 Normal Distribution

23

(3)

3.2 One-Dimensional Error Measures

26

(2)

3.2.1 Standard Error

26

(1)

3.2.2 Root Mean Square (RMS) Error

26

(1)

3.2.3 Mean Absolute Error (MAE)

26

(1)

3.2.4 Linear Error Probable (LEP)

27

(1)

3.3 Direct Problems

28

(1)

3.4 Inverse Problems

29

(4)

3.5 Summary of M-files

33

(2)

CHAPTER 4 Two-Dimensional Error Analysis

35

(30)

4.1 Two-Dimensional Normal Distribution

35

(2)

4.2 Direct and Inverse Problems

37

(2)

4.3 Case 1, A = A(r), p =O, u epsilon [0,1]

39

(2)

4.3.1 Limiting Situation: u XXX 0

41

(1)

4.4 Case 2, A = A(c), p = 0, u = 1

41

(1)

4.5 Two-Dimensional Error Measures

42

(7)

4.5.1 Mean Radial Error (MRE)

42

(1)

4.5.2 Root Mean Square Radial Error (RMSR)

43

(1)

4.5.3 Distance Root Mean Square Error (DRMS)

43

(1)

4.5.4 Standard Radial Error (SRE)

43

(1)

4.5.5 Circular Error Probable (CEP)

43

(5)

4.5.6 CEP Rate and RPE Rate

48

(1)

4.6 Case 3, A = A(C), p = 0, u Epsilon [0,1]

49

(3)

4.6.1 Limiting Situation: u XXX 0

50

(2)

4.7 Case 4, A=A(c), Rho Epsilon [-1,1], u Epsilon [0,1]

52

(4)

4.7.1 Limiting Situation: [Rho]-1

55

(1)

4.8 Case 5, A=A(e), Rho=0, u Epsilon [0,1]

56

(3)

4.8.1 Limiting Situation: a/b =XXX(x)/XXX(y)

59

(1)

4.9 Case 6, A=A(e), Rho Epsilon [-1,1], u Epsilon [0,1]

59

(2)

4.9.1 Limiting Situation: Rho 1

60

(1)

4.10 PDF of the Angular Position of a Random Point

61

(3)

4.11 Summary of M-files

64

(1)

CHAPTER 5 Three-Dimensional Error Analysis

65

(22)

5.1 Three-Dimensional Normal Distribution

65

(1)

5.2 Direct and Inverse Problems

66

(2)

5.3 Case 1, V=V(b), Rho(xy)=Rho(yz)=Rho(xz)=0, u, v Epsilon [0,1]

68

(1)

5.4 Case 2, V=V(s), Rho(xy)=Rho(yz)=Rho(xz)=0, u=v=1

69

(2)

5.5 Three-Dimensional Error Measures

71

(5)

5.5.1 Mean Radial Error (MRF)

71

(1)

5.5.2 Root Mean Square Radial Error (RMSR)

71

(1)

5.5.3 Distance Root Mean Square Error (DRMS)

71

(1)

5.5.4 Standard Radial Error (SRE)

71

(1)

5.5.5 Spherical Error Probable (SEP)

72

(4)

5.6 Case 3, V=V(s), Rho(xy)=Rho(yz)=Rho(xz)=0, u, v Epsilon [0,1]

76

(4)

5.7 Case 4, V=V(s), Rho(xy), Rho(yz), Rho(xz) Epsilon [-1,1], u, v Epsilon [0,1]

80

(2)

5.8 Case 5, V=V(e), Rho(xy)=Rho(yz)=Rho(xz)=0, u, v Epsilon [0,1]

82

(2)

5.8.1 Limiting Situation: a/XXX(x) = b/XXX(y) = c/XXX(z)

83

(1)

5.9 Case 6, A=A(e), Rho(xy), Rho(yz), Rho(xz) Epsilon[-1,1], u, v Epsilon [0,1]

84

(1)

5.10 Summary of M-files

85

(2)

CHAPTER 6 Maximum Likelihood Estimation of Radial Error PDF

87

(10)

6.1 Basic Assumptions and General Approach

87

(1)

6.2 Maximum Likelihood Estimation of n, a

88

(1)

6.3 Dependence of GM/RMS on n

89

(3)

6.3.1 Psi(n/2) for n = Positive Even Integer

90

(1)

6.3.2 Psi(n/2) for n = Positive Odd Integer

90

(1)

6.3.3 Psi(n/2) for 0 Less Than n/2 Less Than 1

90

(2)

6.4 Dependency of R(p)/RMS on n

92

(1)

6.5 Relationship Between R(p)/RMS and GM/RMS

93

(2)

6.6 Summary of M-files

95

(2)

CHAPTER 7 Position Location Problems

97

(20)

7.1 Single Error-Ellipse Analysis

98

(7)

7.1.1 Approach 1. Fictitious Sigma and Cut Angle

101

(3)

7.1.2 Approach 2. New Sigma Along the Major and Minor Axes

104

(1)

7.2 Consideration of Geometrical Factors

105

(4)

7.3 Analysis of Multiple Error-Ellipses

109

(6)

7.3.1 Mutually Parallel Axes

110

(2)

7.3.2 Randomly Oriented Axes

112

(3)

7.4 Summary of M-files

115

(2)

CHAPTER 8 Risk Analysis

117

(12)

8.1 Definition of Notation

119

(1)

8.2 Seller's Risk

119

(2)

8.3 The Pass/Fail Criterion

121

(1)

8.4 Buyer's Risk

122

(1)

8.5 A Practical Example

123

(2)

8.6 Generalization

125

(1)

8.7 Summary of M-files

126

(3)

APPENDIX A Probability Density Functions

129

(4)

A.1 Relationship Between PDF and CDF

129

(1)

A.2 PDFs Used in This Book

129

(2)

A.2.1 Normal Distribution

129

(1)

A.2.2 Chi-Square Distribution

130

(1)

A.2.3 Student-t Distribution

130

(1)

A.2.4 Rayleigh Distribution

131

(1)

A.2.5 Maxwell Distribution

131

(1)

A.2.6 Cauchy Distribution

131

(1)

A.2.7 Uniform Distribution

131

(1)

A.3 Central Limit Theorem

131

(1)

A.4 Generation of Standard Normal Random Variable

131

(2)

APPENDIX B Method of Confidence Intervals

133

(2)

B.1 Confidence Interval and Confidence Limits

133

(1)

B.2 Determination of y-Confidence Interval

134

(1)

APPENDIX C Function of N Random Variables

135

(8)

C.1 Linear Combination of N Independent Random Variables

135

(1)

C.1.1 Sum of N Random Variables

136

(1)

C.1.2 Average of N Random Variables

136

(1)

C.1.3 Difference of Two Random Variables

136

(1)

C.2 Product of Two Random Variables

136

(1)

C.3 Sum and Difference of Two Correlated Random Variables

137

(1)

C.4 Independence and Uncorrelatedness

137

(1)

C.5 PDF of z=x+y

138

(2)

C.6 PDF of z=x-y

140

(1)

C.7 PDF of z=x/y

140

(3)

APPENDIX D GPS Dilution of Precisions

143

(2)

APPENDIX E Listing of Author-Generated M-files

145

(66)

E.1 M-files Used in Chapter 1

145

(3)

E.2 M-files Used in Chapter 2

148

(5)

E.3 M-files Used in Chapter 3

153

(5)

E.4 M-files Used in Chapter 4

158

(15)

E.5 M-files Used in Chapter 5

173

(17)

E.6 M-files Used in Chapter 6

190

(10)

E.7 M-files Used in Chapter 7

200

(7)

E.8 M-files Used in Chapter 8

207

(4)

BIBLIOGRAPHY

211

(2)

INDEX

213

(4)

ABOUT THE AUTHOR

217

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Spatial Error Analysis A Unified Application-Oriented Treatment

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