# Measurement Data Modeling and Parameter Estimation

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## Book Description

**Measurement Data Modeling and Parameter Estimation** integrates mathematical theory with engineering practice in the field of measurement data processing. Presenting the first-hand insights and experiences of the authors and their research group, it summarizes cutting-edge research to facilitate the application of mathematical theory in measurement and control engineering, particularly for those interested in aeronautics, astronautics, instrumentation, and economics.

Requiring a basic knowledge of linear algebra, computing, and probability and statistics, the book illustrates key lessons with tables, examples, and exercises. It emphasizes the mathematical processing methods of measurement data and avoids the derivation procedures of specific formulas to help readers grasp key points quickly and easily. Employing the theories and methods of parameter estimation as the fundamental analysis tool, this reference:

- Introduces the basic concepts of measurements and errors
- Applies ideas from mathematical branches, such as numerical analysis and statistics, to the modeling and processing of measurement data
- Examines methods of regression analysis that are closely related to the mathematical processing of dynamic measurement data
- Covers Kalman filtering with colored noises and its applications

Converting time series models into problems of parameter estimation, the authors discuss modeling methods for the true signals to be estimated as well as systematic errors. They provide comprehensive coverage that includes model establishment, parameter estimation, abnormal data detection, hypothesis tests, systematic errors, trajectory parameters, and modeling of radar measurement data. Although the book is based on the authors’ research and teaching experience in aeronautics and astronautics data processing, the theories and methods introduced are applicable to processing dynamic measurement data across a wide range of fields.

## Table of Contents

**Chapter 1: Error Theory **1.1 Measurement

1.1.1 Measurement Data

1.1.2 Classification of Measurement

1.1.2.1 Concept of Measurement

1.1.2.2 Methods of Measurement

1.1.2.3 Equal Precision and Unequal Precision Measurements

1.1.2.4 Measurements of Static and Dynamic Objects

1.2 Measurement Error

1.2.1 Concept of Error

1.2.2 Source of Errors

1.2.3 Error Classification

1.2.4 Quality of Measurement Data

1.2.5 Summary

1.3 Random Error in Independent Measurements with Equal Precision

1.3.1 Postulate of Random Error and Gaussian Law of Error

1.3.2 Numerical Characteristics of a Random Error

1.3.2.1 Mean

1.3.2.2 Standard Deviation

1.3.2.3 Estimation of Standard Deviation

1.3.2.4 Estimation of Mean and Standard Deviation

1.3.3 Distributions and Precision Indices of Random Errors

1.3.3.1 Distributions of Random Errors

1.3.3.2 Precision Index of Measurement

1.4 Systematic Errors

1.4.1 Causes of Systematic Errors

1.4.2 Variation Rules of Systematic Errors

1.4.3 Identification of Systematic Errors

1.4.4 Reduction and Elimination of Systematic Errors

1.5 Negligent Errors

1.5.1 Causes and Avoidance of Negligent Errors

1.5.1.1 Causes of Negligent Errors

1.5.1.2 Avoidance of Negligent Errors

1.5.2 Negligent Errors in Measurement Data of Static Objects

1.5.2.1 Romannovschi Criterion

1.5.2.2 Grubbs Criterion

1.5.2.3 Summary of Identification Criteria

1.6 Synthesis of Errors

1.6.1 Uncertainty of Measurement

1.6.1.1 Estimation of Measurement Uncertainty

1.6.1.2 Propagation of Uncertainties

1.6.2 Functional Errors

1.6.2.1 Functional Systematic Errors

1.6.2.2 Functional Random Errors

1.7 Steps of Data Processing: Static Measurement Data

References

**Chapter 2: Parametric Representations of Functions to Be Estimated**2.1 Introduction

2.2 Polynomial Representations of Functions to Be Estimated

2.2.1 Weierstrass Theorem

2.2.2 Best Approximation Polynomials

2.2.3 Best Approximation of Induced Functions

2.2.4 Degrees of Best Approximation Polynomials

2.2.5 Bases of Polynomial Representations of Functions to Be Estimated

2.2.5.1 Significance of Basis Selection

2.2.5.2 Chebyshev Polynomials

2.2.5.3 Bases of Interpolation Polynomials of Order n

2.2.5.4 Chebyshev Polynomial Bases

2.2.5.5 Bases and Coefficients

2.3 Spline Representations of Functions to Be Estimated

2.3.1 Basic Concept of Spline Functions

2.3.2 Properties of Cubic Spline Functions

2.3.3 Standard B Splines

2.3.4 Bases of Spline Representations of Functions to Be Estimated

2.4 Using General Solutions of Ordinary Differential Equations to Represent Functions to Be Estimated

2.4.1 Introduction

2.4.2 General Solutions of Linear Ordinary Differential Equations

2.4.3 General Solutions of Nonlinear Equation or Equations

2.5 Empirical Formulas

2.5.1 Empirical Formulas from Scientific Laws

2.5.2 Empirical Formula from Experience

2.5.3 Empirical Formulas of Mechanical Type

2.5.4 Empirical Formulas of Progressive Type

References

**Chapter 3: Methods of Modern Regression Analysis **3.1 Introduction

3.2 Basic Methods of Linear Regression Analysis

3.2.1 Point Estimates of Parameters

3.2.2 Hypothesis Tests on Regression Coefficients

3.2.3 Interval Estimates of Parameters

3.2.4 Least Squares Estimates and Multicollinearity

3.3 Optimization of Regression Models

3.3.1 Dynamic Measurement Data and Regression Models

3.3.2 Compound Models for Signals and Systematic Errors

3.4 Variable Selection

3.4.1 Consequences of Variable Selection

3.4.2 Criteria of Variable Selection

3.4.3 Fast Algorithms to Select Optimal Reduced Regression Model

3.4.4 Summary

3.5 Biased Estimation in Linear Regression Models

3.5.1 Introduction

3.5.2 Biased Estimates of Compression Type

3.5.3 A New Method to Determine Ridge Parameters

3.5.4 Scale Factors

3.5.5 Numerical Examples

3.6 The Method of Point-by-Point Elimination for Outliers

3.6.1 Introduction

3.6.2 Derivation of Criteria

3.6.3 Numerical Examples

3.7 Efficiency of Parameter Estimation in Linear Regression Models

3.7.1 Introduction

3.7.2 Efficiency of Parameter Estimation in Linear Regression Models with One Variable

3.7.3 Efficiency of Parameter Estimation in Multiple Linear Regression Models

3.8 Methods of Nonlinear Regression Analysis

3.8.1 Models of Nonlinear Regression Analysis

3.8.2 Methods of Parameter Estimation

3.9 Additional Information

3.9.1 Sources of Additional Information

3.9.2 Applications of Additional Information

References

**Chapter 4: Methods of Time Series Analysis **4.1 Introduction to Time Series

4.1.1 Time Series and Random Process

4.1.2 Time Series Analysis

4.2 Stationary Time Series Models

4.2.1 Stationary Random Processes

4.2.2 Autoregressive Models

4.2.3 Moving Average Model

4.2.4 ARMA(p,q) Model

4.2.5 Partial Correlation Function of a Stationary Model

4.3 Parameter Estimation of Stationary Time Series Models

4.3.1 Estimation of Autocovariance Functions and Autocorrelation Functions

4.3.2 Parameter Estimation of AR(p) Models

4.3.2.1 Moment Estimation of Parameters in AR Models

4.3.2.2 Least Squares Estimation of Parameters in AR Models

4.3.3 Parameter Estimation of MA(q) Models

4.3.3.1 Linear Iteration Method

4.3.3.2 Newton–Raphson Algorithm

4.3.4 Parameter Estimation of ARMA(p,q) Models

4.3.4.1 Moment Estimation

4.3.4.2 Nonlinear Least Squares Estimation

4.4 Tests of Observational Data from a Time Series

4.4.1 Normality Test

4.4.2 Independence Test

4.4.3 Stationarity Test: Reverse Method

4.4.3.1 Testing the Mean Stationarity

4.4.3.2 Testing the Variance Stationarity

4.5 Modeling Stationary Time Series

4.5.1 Model Selection: Box–Jenkins Method

4.5.2 AIC Criterion for Model Order Determination

4.5.2.1 AIC for AR Models

4.5.2.2 AIC for MA and ARMA Models

4.5.3 Model Testing

4.5.3.1 AR Models Testing

4.5.3.2 MA Models Testing

4.5.3.3 ARMA Models Testing

4.6 Nonstationary Time Series

4.6.1 Nonstationarity of Time Series

4.6.1.1 Processing Variance Nonstationarity

4.6.1.2 Processing Mean Nonstationarity

4.6.2 ARIMA Model

4.6.2.1 Definition of ARIMA Model

4.6.2.2 ARIMA Model Fitting for Time Series Data

4.6.3 RARMA Model

4.6.4 PAR Model

4.6.4.1 Model and Parameter Estimation

4.6.4.2 PAR Model Fitting

4.6.4.3 Further Discussions

4.6.5 Parameter Estimation of RAR Model

4.6.6 Parameter Estimation of RMA Model

4.6.7 Parameter Estimation of RARMA Model

4.7 Mathematical Modeling of CW Radar Measurement Noise References

**Chapter 5: Discrete-Time Kalman Filter **5.1 Introduction

5.2 Random Vector and Estimation

5.2.1 Random Vector and Its Process

5.2.1.1 Mean Vector and Variance Matrix

5.2.1.2 Conditional Mean Vector and Conditional Variance Matrix

5.2.1.3 Vector Random Process

5.2.2 Estimate of the State Vector

5.2.2.1 Minimum Mean Square Error Estimate

5.2.2.2 Linear Minimum Mean Square Error Estimate (LMMSEE)

5.2.2.3 The Relation between MMSEE and LMMSEE

5.3 Discrete-Time Kalman Filter

5.3.1 Orthogonal Projection

5.3.2 The Formula of Kalman Filter

5.3.3 Examples

5.4 Kalman Filter with Colored Noise

5.4.1 Kalman Filter with Colored State Noise

5.4.2 Kalman Filtering with Colored Measurement Noise

5.4.3 Kalman Filtering with Both Colored State Noise and Measurement Noise

5.5 Divergence of Kalman Filter

5.6 Kalman Filter with Noises of Unknown Statistical Characteristics

5.6.1 Selection of Correlation Matrix Qk of the Dynamic Noise

5.6.2 Extracting Statistical Features of Measurement Noises

References

**Chapter 6: Processing Data from Radar Measurements **6.1 Introduction

6.1.1 Space Measurements

6.1.2 Tracking Measurements and Trajectory Determination Principle

6.1.2.1 Optical Measurements

6.1.2.2 Radar Measurements

6.1.3 Precision Appraisal and Calibration of Measurement Equipments

6.1.3.1 Precision Appraisal

6.1.3.2 Precision Calibration

6.1.4 Systematic Error Model of CW Radar

6.1.5 Mathematical Processing for Radar Measurement Data

6.2 Parametric Representation of the Trajectory

6.2.1 Equation Representation of Trajectory

6.2.2 Polynomial Representation of Trajectory

6.2.3 Matching Principle

6.2.4 Spline Representation of Trajectory

6.3 Trajectory Calculation

6.3.1 Mathematical Method for MISTRAM System Trajectory Determination

6.3.1.1 Problem Introduction

6.3.1.2 Mathematical Model for the MISTRAM System Measurement Data

6.3.1.3 Mathematical Method for Trajectory Determination

6.3.1.4 Error Propagation Relationship

6.3.2 Nonlinear Regression Analysis Method for Trajectory Determination

6.3.2.1 Introduction

6.3.2.2 Mathematical Model Establishment

6.3.2.3 Algorithm and Error Analysis

6.3.3.4 Simulation Calculation Results

6.4 Composite Model of Systematic Error and Trajectory Parameters

6.4.1 Measurement Data Models

6.4.2 Matched Systematic Error and Unmatched Systematic Error

6.4.3 Summary

6.5 Time Alignment of CW Radar Multistation Tracking Data

6.5.1 Introduction

6.5.5 Time Alignment between the Distance Sum and Its Change Rate

6.6 Estimation for Constant Systematic Error of CW Radars

6.6.1 Mathematical Model of Measurement Data

6.6.2 EMBET Method Analysis

6.6.3 Nonlinear Modeling Method

6.6.4 Algorithm and Numerical Examples

6.6.5 Conclusions

6.7 Systematic Error Estimation for the Free Flight Phase

6.7.1 Trajectory Equations in the Free Flight Phase

6.7.2 Nonlinear Model of the Measurement Data

6.7.3 Parameter Estimation Method

6.7.4 Numerical Example and Analysis

6.8 Estimation of Slow Drift Error in Range Rate Measurement

6.8.1 Mathematical Model of Measurement Data

6.8.2 Selection of the Spline Nodes

6.8.3 Estimation of the Slow Drift Errors

6.9 Summary of Radar Measurement Data Processing

6.9.1 Data Processing Procedures

6.9.1.1 Analysis of Abnormal Data

6.9.1.2 Analysis of the Measurement Principle and the Measurement Data

6.9.1.3 Measurement Data Modeling

6.9.1.4 Estimation of Statistical Features of Random Errors

6.9.1.5 Estimation of True Signal and Systematic Error

6.9.1.6 Engineering Analysis for Data Processing Results

6.9.2 Basic Conclusions

References

**Chapter 7: Precise Orbit Determination of LEO Satellites Based on Dual-Frequency GPS **7.1 Introduction

7.2 Spaceborne Dual-Frequency GPS Data Preprocessing

7.2.1 Basic Observation Equations

7.2.2 Pseudocode Outliers Removal

7.2.2.1 Threshold Method of Signal-to-Noise Ratio

7.2.2.2 Threshold Method of Ionospheric Delay

7.2.2.3 Fitting Residual Method of Ionospheric Delay

7.2.2.4 Method of Monitoring Receiver Autonomous Integrity

7.2.3 Carrier Phase Outliers Removal and Cycle Slip Detection

7.2.3.1 M–W Combination Epoch Difference Method

7.2.3.2 Ionosphere-Free Ambiguity Epoch Difference Method

7.2.3.3 Cumulative Sum Method

7.2.4 Data Preprocessing Flow

7.3 Orbit Determination by Zero-Difference Reduced Dynamics

7.3.1 Observational Equations and Error Correction

7.3.1.2 Antenna Offset Corrections for GPS Satellites

7.3.1.3 Antenna Offsets for LEO Satellites

7.3.2 Parameter Estimation of Orbit Models

7.3.3 Dynamic Orbit Models and Parameter Selections

7.3.3.1 Earth Nonspherical Perturbation

7.3.3.2 Third Body Gravitational Perturbations

7.3.3.3 Tide Perturbations

7.3.3.4 Atmospheric Drag Forces

7.3.3.5 Solar Radiation Pressures

7.3.3.6 Relativity Perturbations

7.3.3.7 Empirical Forces

7.3.3.8 Dynamic Orbit Models and Parameter Selections

7.3.4 Re-editing Observational Data

7.3.4.1 Re-editing Pseudocode Data

7.3.4.2 Re-editing Phase Data

7.3.5 The Flow of Zero-Difference Reduced Dynamic Orbit Determination

7.3.6 Analysis of Results from Orbit Determination

References

**Appendices: **

**A1.1 Trace of a Matrix**

Matrix Formulas in Common Use

Matrix Formulas in Common Use

A1.2 Inverse of a Block Matrix

A1.3 Positive Definite Character of a Matrix

A1.4 Idempotent Matrix

A1.5 Derivative of a Quadratic Form

**A2.1 χ2-Distribution**

Distributions in Common Use

Distributions in Common Use

A2.2 Noncentral χ2-Distribution

A2.3 t-Distribution

A2.4 F-Distribution

Index

## Author(s)

### Biography

**Dr. Zhengming Wang** received his BS and MS degrees in applied mathematics and a PhD degree in system engineering in 1982, 1986, and 1998, respectively. Currently, he is a professor in applied mathematics. He is also Standing Director of the Chinese Association for Quality Assurance Agencies in Higher Education, Director of Chinese Mathematical Society, Chairman of the Hunan Institute of Computational Mathematics and Application Software, and Associate Provost of National University of Defense Technology. He has completed four projects funded by the National Science Foundation of China. He has won five State Awards of Science and Technology Progress. He has co-published four monographs (all ranked first) and well over 80 papers, including 50 SCI or EI-indexed ones. His research interests cover areas such as mathematical modeling in tracking data, image processing, experiment evaluation, and data fusion.

**Dr. Dongyun Yi** received his BS and MS degrees in applied mathematics and a PhD degree in system engineering in 1985, 1992, and 2003, respectively. Currently, he is a professor in Systems Analysis and Integration. He is now the Director of the Department of Mathematics and Systems Science, College of Science, National University of Defense Technology. He has been engaged in data intelligent processing research for over twenty years. He is in charge of the National Foundation Research Project "The Structural Properties of Resource Aggregation—Analysis and Applications" and also participates in the National Science Foundation of China "Pattern Recognition Research Based on High-Dimensional Data Structure" as a deputy chair. He has co-published two monographs and published more than 60 papers. His research interests include data fusion, parameter estimation of satellite positioning, mathematical modeling, and analysis of financial data.**Dr. Xiaojun Duan** received her BS and MS degrees in applied mathematics and a PhD degree in system engineering in 1997, 2000, and 2003, respectively. She also had one year of visiting scholar experience at Ohio State University during 2007–2008. Currently, she is an associate professor in Systems Analysis and Integration. She teaches data analysis, systems science, linear algebra, probability and statistics, and mathematical modeling and trains undergraduates as a faculty advisor for participation in the Mathematical Contest in Modeling, which is held by the Society for Industry and Applied Mathematics in the United States. By teaching courses in data analysis, she gained valuable experience and also received suggestions from students on how to better organize materials so as to impart knowledge of data analysis. Her research is funded by the Natural Science Foundation of China, Spaceflight Science Foundation in China. She has published about 30 SCI or EI-indexed research papers. Her research interests cover areas such as data analysis, mathematical positioning and geodesy, complex system test, and evaluation.**Dr. Jing Yao** received her BS and MS degrees in applied mathematics and a PhD degree in systems analysis and integration in 2001, 2003, and 2008, respectively. Currently, she is a lecturer at the Department of Mathematics and Systems Science, College of Science, National University of Defense Technology. She teaches probability and statistics for the undergraduate level and time series analysis with applications for the graduate level. Some of her research is funded by the National Science Foundation of China and Spaceflight Science Foundation in China. She has published more than 20 research papers. Her research interests include mathematical geodesy, data analysis, and processing in navigation systems.

**Dr. Defeng Gu** received his BS degree in applied mathematics and a PhD degree in systems analysis and integration in 2003 and 2009, respectively. Currently, he is a lecturer at the Department of Mathematics and Systems Science, College of Science, National University of Defense Technology. He has published more than 20 research papers. His research interests are in mathematical modeling, data analysis, and spaceborne Global Positioning System data processing. The GPS processing software that is being maintained by Dr. Gu has achieved success in real satellite orbit determination.