Least-Squares Estimation and Kalman Filtering
This chapter presents the estimation and filtering principles as used in global navigation satellite system (GNSS ) data processing. Estimation and filtering are concerned with retrieving or recovering parameters of interest from noisy measurements. The l
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Least-Square
22. Least-Squares Estimation and Kalman Filtering
Sandra Verhagen, Peter J.G. Teunissen
22.1 22.1.1 22.1.2 22.1.3 22.1.4
Linear Least-Squares Estimation........ Least-Squares Principle...................... Weighted Least-Squares ..................... Computation of LS Solution ................ Statistical Properties ..........................
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22.2 22.2.1 22.2.2 22.2.3
Optimal Estimation ........................... Best Linear Unbiased Estimation ......... Maximum Likelihood Estimation ......... Confidence Regions ...........................
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22.3 Special Forms of Least Squares .......... 22.3.1 Recursive Estimation.......................... 22.3.2 Estimation with Partitioned Parameter Vector ...... 22.3.3 Block Estimation ............................... 22.3.4 Constrained Least-Squares ................. 22.3.5 Rank-Defect Least Squares ................. 22.3.6 Non-Linear Least-Squares ..................
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22.4 Prediction and Filtering .................... 22.4.1 Prediction Problem ............................ 22.4.2 Minimum Mean Squared Error Prediction ......................................... 22.4.3 Properties of MMSE Prediction ............
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22.5 22.5.1 22.5.2 22.5.3 22.5.4 22.5.5
Kalman Filtering ............................... Model Assumptions ........................... The Kalman Filter Recursion ............... Kalman Filter Information Form .......... Extended Kalman Filter...................... Smoothing ........................................
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References...................................................
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22.1 Linear Least-Squares Estimation GNSS observations are, like all empirical data, subject to uncertainty – measurements are never perfect. Moreover, we generally have redundant measurements: There are more observations available than strictly needed for estimating the parameters of interest. In this section, we introduce the principle of least-squares (LS) estimation for solving overdetermined systems, that is, estimation problems with redundant measurements. Due to the uncertainty of the measurements, the redundancy generally leads to inconsistent systems
of equations; estimating the parameters with different subsets of the observations will then lead to different solutions. The LS method ensures that still a unique solution can be obtained by imposing additional criteria. This section explains the LS principle and its properties.
22.1.1 Least-Squares Principle The objective is to obtain estimates of n unknown parameters xj , j D 1; : : : ; n from a set of m measurements
Part D | 22.1
This chapter presents the estimation and filtering principles as used in global navigation satellite system (GNSS) data processing. Estimation and filtering are concerned with retrieving or recovering parameters of interest from noisy measurements. The least-squares (LS) principle is the standard approach for estimating unknown parameters from uncertain data. Various forms of LS
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