On optimal completions of incomplete pairwise comparison matrices

An important variant of a key problem for multi-attribute decision making is considered. We study the extension of the pairwise comparison matrix to the case when only partial information is available: for some pairs no comparison is given. It is natural to define the inconsistency of a partially filled matrix as the inconsistency of its best, completely filled completion. We study here the uniqueness problem of the best completion for two weighting methods, the Eigen-vector Method and the Logarithmic Least Squares Method. In both settings we obtain the same simple graph theoretic characterization of the uniqueness. The optimal completion will be unique if and only if the graph associated with the partially defined matrix is connected. Some numerical experiences are discussed at the end of the paper

An Open Source C++ Implementation of Multi-Threaded Gaussian Mixture Models, k-Means and Expectation Maximisation

Modelling of multivariate densities is a core component in many signal processing, pattern recognition and machine learning applications. The modelling is often done via Gaussian mixture models (GMMs), which use computationally expensive and potentially unstable training algorithms. We provide an overview of a fast and robust implementation of GMMs in the C++ language, employing multi-threaded versions of the Expectation Maximisation (EM) and k-means training algorithms. Multi-threading is achieved through reformulation of the EM and k-means algorithms into a MapReduce-like framework. Furthermore, the implementation uses several techniques to improve numerical stability and modelling accuracy. We demonstrate that the multi-threaded implementation achieves a speedup of an order of magnitude on a recent 16 core machine, and that it can achieve higher modelling accuracy than a previously well-established publically accessible implementation. The multi-threaded implementation is included as a user-friendly class in recent releases of the open source Armadillo C++ linear algebra library. The library is provided under the permissive Apache~2.0 license, allowing unencumbered use in commercial products

Approximation of the scattering amplitude

The simultaneous solution of Ax=b and ATy=g is required in a number of situations. Darmofal and Lu have proposed a method based on the Quasi-Minimal residual algorithm (QMR). We will introduce a technique for the same purpose based on the LSQR method and show how its performance can be improved when using the Generalized LSQR method. We further show how preconditioners can be introduced to enhance the speed of convergence and discuss different preconditioners that can be used. The scattering amplitude gTx, a widely used quantity in signal processing for example, has a close connection to the above problem since x represents the solution of the forward problem and g is the right hand side of the adjoint system. We show how this quantity can be efficiently approximated using Gauss quadrature and introduce a Block-Lanczos process that approximates the scattering amplitude and which can also be used with preconditioners

The Estimation of Flows on Regional Labour Markets By Using the ADETON Procedure

An in-depth analysis of regional labour markets requires information not only on stocks but also on flows. As a tool for detailed flow analysis the Multi Account System (MAS) has been developed at the Institute for Employment Research (IAB) which uses fine grained transition matrices as basic information. To estimate these transition matrices from possibly incomplete and inconsistent statistical data the ADETON procedure has been worked out to compute matrices which are struc-turally similar to given reference matrix and at the same time satisfy certain linear con-straints. As main advantage of ADETON in comparison to conventional methods soft constraints may be specified which allow information of inherently fuzzy character about transition flows to be taken into account. By using soft constraints of this kind it is possible to in-clude data affected by sampling errors or are distorted by some kind of “noiseâ€. To obtain structural similarity to a reference matrix two different distance measures can be used: relative entropy and the chi-square distance function. They have been proven to give approximately identical results. Up to now ADETON has shown to be an efficient computation method even for complex problems with thousands of matrix elements and constraints. ADETON has been applied to estimate matrices of the Multi Account System. The results are used for the guidance of regional labour market policy.