Introduction to adjustment calculus

P. VANICEK

September 1973

LECTURE NOTES 35

Introduction to adjustment calculus (Third Corrected Edition)

Petr Vanicek

Department of Geodesy & Geomatics Engineering University of New Brunswick P.O. Box 4400 Fredericton, N.B. Canada ЕЗВ 5A3

February, 1980 Latest Reprinting October 1995

PREFACE

In order to make our extensive series of lecture notes more readily available, we have scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. The images have not been converted to searchable text.

FOREWORD

It has long been the author^T s conviction that most of the existing courses tend to"slide over the fundamentals and treat the adjustment purely as a technique without giving the student a deeper insight without answering a good many questions beginning with "why". This course is a result of a humble attempt to present the adjustment as a discipline with its own rights, with a firm basis and internal structure; simply as an adjustment calculus. Evidently, when one tries to take an unconventional approach, one is only too liable to make mistakes. It is hoped that the student will hence display some patience and understanding.

These notes have evolved from the first rushed edition - termed as preliminary - of the Introduction to Adjustment Calculus, written for course SE 3101 in 1971. Many people have kindly communicated their comments and remarks to the author. To all these, the author is heavily indebted. In particular, Dr. L. Hradilek, Professor at the Charles University in Prague, and Dr. B. Lund, Assistant Professor at the Math­ematics Dept. ШВ, made very extensive reviews that helped in clarifying many points. Mr. M. Nassar, a Ph.D. student in this department, carried most of the burden connected with rewriting the notes on his shoulders. Many of the improvements in formulations as well as most of the examples and exercise problems contained herein originated from him.

None of the contributors should however, be held responsible for any errors and misconception still present. Any comment or critism com­municated to the author will be highly appreciated.

P. Vanlсек

October 7, 197^

CONTENTS

Introduction 1

1. Fundamentals of the Intuitive Theory of Sets

1. Sets, Elements and Subsets 6

2. Progression and Definition Set ........... 7

3. Cartesian Product of Sets 8

1. k Intersection of Sets 9

5. Union of Sets 10

5. Mapping of Sets ................... 12

6. Exercise 1 ..................... 13

2. Fundamentals of the Mathematical Theory of Probability

1. Probability Space ₅ Probability Function and Probabil­ity ... ............... 15

1. Conditional Probability . l6

2. Combined Probability 16

2. h Exercise 2 18

3 Fundamentals of Statistics

3.1 Statistics of an Actual Sample

1. Definition of a Random Sample 20

2. Actual (Experimental) PDF and CDF . 22

3. Mean of a Sample 26

3.1. h Variance of a Sample . 29

5. Other Characteristics of a Sample 32

6. Histograms and Polygons ............ 3^4

3.2 Statistics of a Random Variable

1. Random Function and Random Variable ...... ^7

2. PDF and CDF of a Random Variable Vfb

3. Mean and Variance of a Random Variable .... 51 3.2.1+ Basic Postulate (Hypothesis) of Statistics,

Testing 55

3.2.5 Two Examples of a Random Variable . 56

3.3 Random Multivariate

1. Multivariate, its PDF and CDF 66

2. Statistical Dependence and Independence .... 69

3. Mean and Variance of' a Multivariate 70

3.3.к Covariance and Variance-Covariance Matrix ... 72

5. Random Multisample, its PDF and CDF 76

5. Mean and Variance-Covariance Matrix of

a Mult is ample . 76

3.3.7 Correlation 8l

Fundamentals of the Theory of Errors

4.1 Basic Definitions 89

k.2 Random (Accidental) Errors 91

k. 3 Gaussian PDF, Gauss Law of Errors 92

k.k Mean and Variance of the Gaussian PDF . 9^

k. 5 Generalized or Normal Gaussian PDF 97

k.6 Standard Normal PDF 98

4.7 Basic Hypothesis (Postulate) of the Theory of Errors,

Testing ...... 106

8. Residuals, Corrections and Discrepancies 109

9. Other Possibilities Regarding the Postulated PDF 112

10. Other measures of Dispersion ............... 113

11. Exercise 4 • . • '. 118

Least-Squares Principle

1. The Sample Mean as "The Least Squares Estimator.' 123

2. The Sample Mean as "The Maximum Probability Estimator" . . 125

3. Least Squares Principle . . 128

4. Least-Squares Principle for Random Multivariate ...... 130

5. Exercise 5 ....................... . 132

Fundamentals of Adjustment Calculus

1. Primary and Derived Random Samples . 133

2. Statistical Transformation, Mathematical Model ...... 133

3. Propagation of Errors .

6.3.1 Propagation of Variance-Covariance Matrix, Covariance

Law ..... 137

2. Propagation of Errors, Uncorrelated Case • . . . .. 1U5

3. Propagation of Non-Random Errors, Propagation of

Total Errors .... 153

4. Truncation and Rounding 157

5. Tolerance Limits, Specifications and Preanalysis . . l62

6.4 Problem of Adjustment

6.4.1 Formulation of the Problem 166

6.U.2 Mean of a Sample as an Instructive Adjustment

Problem, Weights 167

6.H.3 Variance of the Sample Mean 170

6.h.k Variance -Covariance Matrix of the Mean of a

Multisample . • .. 17^-

6.4.5 The Method of Least-Squares, Weight Matrix . . . .. 176

6.k.6 Parametric Adjustment . 179

6.U.7 Variance-Co variance Matrix of the Parametric

Adjustment Solution Vector, Variance Factor and

Weight Coefficient Matrix 193

6.П.8 Some Properties of the Parametric Adjustment

Solution Vector 201

6.U.9 Relative Weights, Statistical Significance of a Priori

and a Posteriori Variance Factors 202

6Л.10 Conditional Adjustment 20k

6J4.ll Variance-Cоvariance Matrix of the Conditional Adjust- ment Solution 213

6 = 5 Exercise 6 220

Appendix I Assumptions for and Derivation of the Gaussian PDF . . 233

Appendix II Tables ...... 238

Bibliography ......... 2^1