On robust estimation of the location parameter

Data Reduction And Error Analysis For The Physical Sciences

▶ Gathers original articles on topics in earth and planetary sciences ▶ Coverage includes geomagnetism, aeronomy, space science, seismology, volcanology, geodesy and planetary science ▶ Official journal of the Society of Geomagnetism and Earth, Planetary and Space Sciences, The Seismological Society of Japan, The Volcanological Society of Japan, The Geodetic Society of Japan, and The Japanese Society for Planetary Sciences

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The commutation matrix $K$ is defined as a square matrix containing only zeroes and ones. Its main properties are that it transforms vecA into vecA', and that it reverses the order of a Kronecker product. An analytic expression for $K$ is given and many further properties are derived. Subsequently, these properties are applied to some problems connected with the normal distribution. The expectation is derived of $\varepsilon' A\varepsilon\cdot\varepsilon' B\varepsilon\cdot\varepsilon'C\varepsilon$, where $\varepsilon \sim N(0, V)$, and $A, B, C$ are symmetric. Further, the expectation and covariance matrix of $x \otimes y$ are found, where $x$ and $y$ are normally distributed dependent variables. Finally, the variance matrix of the (noncentral) Wishart distribution is derived.

The commutation matrix: Some properties and applications

Detecting, Extracting, and Monitoring Surface Water From Space Using Optical Sensors: A Review

A standard errors-in-variables (EIV) model refers to a Gauss–Markov model with an uncertain model matrix from a geodetic perspective. Least squares within the EIV model is usually called the total least squares (TLS) technique because of its symmetrical adjustment. However, the solutions and computational advantages of the weighted TLS problem with a general weight matrix (WTLS) are mostly unknown. In this study, the WTLS problem was solved using three different approaches: iterative methods based on the normal equation, the iteratively linearized Gauss–Helmert model with algebraic Jacobian matrices, and numerical analysis. Furthermore, sufficient conditions for WTLS optimization were investigated systematically as proposed solutions yield only necessary conditions for optimality. A WTLS solution was considered to treat random parameters within the EIV model. Last, applications to test these novel algorithms are presented.

Weighted total least squares: necessary and sufficient conditions, fixed and random parameters

Although the analytical solutions for total least-squares with multiple linear and single quadratic constraints were developed quite recently in different geodetic publications, these methods are restricted in number and type of constraints, and currently their computational efficiency and applications are mostly unknown. In this contribution, it is shown how the weighted total least-squares (WTLS) problem with arbitrary applicable constraints can be solved based on a Newton type methodology. This iterative process with quadratic convergence is expanded upon to become a compact solution for the WTLS with or without constraints. This compact solution is then further interpreted as a universal formula for the symmetrical adjustment of the errors-in-variables model which represents affine, similarity and rigid transformations in two- and three-dimensional space. Furthermore, statistical analysis of the constrained WTLS including the first-order approximation of precision and the bias was investigated. In order to substantiate our proposed method’s applicability, it was used to solve the affine, similarity and rigid transformation problem in two- and three-dimensional cases, where the structure of the coefficient matrix and multiple constraints were taken into account simultaneously.

Weighted total least-squares with constraints: a universal formula for geodetic symmetrical transformations

Water is one of the vital components for the ecological environment, which plays an important role in human survival and socioeconomic development. Water resources in urban areas are gradually decreasing due to the rapid urbanization, especially in developing countries. Therefore, the precise extraction and automatic identification of water bodies are of great significance and urgently required for urban planning. It should be noted that although some studies have been reported regarding the water-area extraction, to our knowledge, few papers concern the identification of urban water types (e.g., rivers, lakes, canals, and ponds). In this paper, a novel two-level machine-learning framework is proposed for identifying the water types from urban high-resolution remote-sensing images. The framework consists of two interpretation levels: 1) water bodies are extracted at the pixel level, where the water/shadow/vegetation indexes are considered and 2) water types are further identified at the object level, where a set of geometrical and textural features are used. Both levels employ machine learning for the image interpretation. The proposed framework is validated using the GeoEye-1 and WorldView-2 images, over two mega cities in China, i.e., Wuhan and Shenzhen, respectively. The experimental results show that the proposed method achieved satisfactory accuracies for both water extraction [95.4% (Shenzhen), 96.2% (Wuhan)], and water type classification [94.1% (Shenzhen), 95.9% (Wuhan)] in complex urban areas.

Combining Pixel- and Object-Based Machine Learning for Identification of Water-Body Types From Urban High-Resolution Remote-Sensing Imagery

A new proof is presented of the desirable property of the weighted total least-squares (WTLS) approach in preserving the structure of the coefficient matrix in terms of the functional independent elements. The WTLS considers the full covariance matrix of observed quantities in the observation vector and in the coefficient matrix; possible correlation between entries in the observation vector and the coefficient matrix are also considered. The WTLS approach is then equipped with constraints in order to produce the constrained structured TLS (CSTLS) solution. The proposed approach considers the correlation between the observation vector and the coefficient matrix of an Error-In-Variables model, which is not considered in other, recently proposed approaches. A rigid transformation problem is done by preservation of the structure and satisfying the constraints simultaneously.

A structured and constrained Total Least-Squares solution with cross-covariances

Observation systems known as errors-in-variables (EIV) models with model parameters estimated by total least squares (TLS) have been discussed for more than a century, though the terms EIV and TLS were coined much more recently. So far, it has only been shown that the inequality-constrained TLS (ICTLS) solution can be obtained by the combinatorial methods, assuming that the weight matrices of observations involved in the data vector and the data matrix are identity matrices. Although the previous works test all combinations of active sets or solution schemes in a clear way, some aspects have received little or no attention such as admissible weights, solution characteristics and numerical efficiency. Therefore, the aim of this study was to adjust the EIV model, subject to linear inequality constraints. In particular, (1) This work deals with a symmetrical positive-definite cofactor matrix that could otherwise be quite arbitrary. It also considers cross-correlations between cofactor matrices for the random coefficient matrix and the random observation vector. (2) From a theoretical perspective, we present first-order Karush–Kuhn–Tucker (KKT) necessary conditions and the second-order sufficient conditions of the inequality-constrained weighted TLS (ICWTLS) solution by analytical formulation. (3) From a numerical perspective, an active set method without combinatorial tests as well as a method based on sequential quadratic programming (SQP) is established. By way of applications, computational costs of the proposed algorithms are shown to be significantly lower than the currently existing ICTLS methods. It is also shown that the proposed methods can treat the ICWTLS problem in the case of more general weight matrices. Finally, we study the ICWTLS solution in terms of non-convex weighted TLS contours from a geometrical perspective.

Xing Fang

Papers

Weighted total least squares: necessary and sufficient conditions, fixed and random parameters

Weighted total least-squares with constraints: a universal formula for geodetic symmetrical transformations

Combining Pixel- and Object-Based Machine Learning for Identification of Water-Body Types From Urban High-Resolution Remote-Sensing Imagery

A structured and constrained Total Least-Squares solution with cross-covariances

On non-combinatorial weighted total least squares with inequality constraints