Identifying Mixtures of Mixtures Using Bayesian Estimation

The use of a finite mixture of normal distributions in model-based clustering allows us to capture non-Gaussian data clusters. However, identifying the clusters from the normal components is challenging and in general either achieved by imposing constraints on the model or by using post-processing procedures. Within the Bayesian framework, we propose a different approach based on sparse finite mixtures to achieve identifiability. We specify a hierarchical prior, where the hyperparameters are carefully selected such that they are reflective of the cluster structure aimed at. In addition, this prior allows us to estimate the model using standard MCMC sampling methods. In combination with a post-processing approach which resolves the label switching issue and results in an identified model, our approach allows us to simultaneously (1) determine the number of clusters, (2) flexibly approximate the cluster distributions in a semiparametric way using finite mixtures of normals and (3) identify cluster-specific parameters and classify observations. The proposed approach is illustrated in two simulation studies and on benchmark datasets. Supplementary materials for this article are available online.     

Simulation techniques for the sensitivity analysis of multi-criteria decision models

This paper presents a simulation approach for high dimensional sensitivity analysis of the weights of multi-criteria decision models. This approach allows simultaneous changes of the weights and generates results that can easily be analyzed statistically to provide insights into multi-criteria model recommendations. In this study we consider three cases: no information, order information, and partial information regarding the weights. Our approach also allows investigation of sensitivity to the form of multi-criteria decision models. The simulation procedures we propose can also be used to aide in the actual decision process, particularly when the task is to select a subset of superior alternatives.     

A method for generating a well-distributed Pareto set in nonlinear multiobjective optimization

A method is presented for generating a well-distributed Pareto set in nonlinear multiobjective optimization. The approach shares conceptual similarity with the Physical Programming-based method, the Normal-Boundary Intersection and the Normal Constraint methods, in its systematic approach investigating the objective space in order to obtain a well-distributed Pareto set. The proposed approach is based on the generalization of the class functions which allows the orientation of the search domain to be conducted in the objective space. It is shown that the proposed modification allows the method to generate an even representation of the entire Pareto surface. The generation is performed for both convex and nonconvex Pareto frontiers. A simple algorithm has been proposed to remove local Pareto solutions. The suggested approach has been verified by several test cases, including the generation of both convex and concave Pareto frontiers.     

Stochastic Programming with Recourse: A Modification from an Applications Viewpoint

The paper describes a modified "wait-and-see" approach to solving two-stage stochastic programming problems. The approach, which involves a detailed sensitivity analysis in the classical sense, is described within the frameworks of decision theory and probabilistic programming. Although optimality in the mathematical sense cannot be guaranteed by using the approach, it is suggested that the managerial benefits weigh heavily in its favour. The approach allows management to consider a wide variety of objectives in making the choice between alternatives and facilities detection of the cause of any infeasibility due to management policy constraints. In addition, it allows much simpler programming calculations and provides an upper bound on the benefits that can be obtained by solving the full "here-and-now" problem and thus a judgement of the worth of the added computational burden can easily be made.     

Valuing risky debt: A new model combining structural information with the reduced-form approach

A new model of credit risk is proposed in which the intensity of default is described by an additional stochastic differential equation coupled with the process of the obligor’s asset value. Such an approach allows us to incorporate structural information as well as to capture the effect of external factors (e.g. macroeconomic factors) in a both parsimonious and economically consistent way. From the practical standpoint, the proposed model offers great flexibility and allows us to obtain credit spread curves of many different shapes, including double humped term structures. Furthermore, an approximate closed-form solution is derived, which is accurate, easy to implement, and allows for an efficient calibration to realized credit spreads. Numerical experiments are presented showing that the novel approach provides a very satisfactory fitting to market data and outperforms the model developed by Madan and Unal (2000).     

On subparacompactness and related properties

An approach to the theory of subparacompactness is presented here. This approach allows one to understand the notions of subexpandability and to generalize a theorem from [6]. We also give an answer to a question from [5].     

Analyzing the potential of a firm: an operations research approach

An approach to analyzing the potential of a firm, which is understood as the firm's ability to provide goods or (and) services to be supplied to a marketplace under restrictions imposed by a business environment in which the firm functions, is proposed. The approach is based on using linear inequalities and, generally, mixed variables in modelling this ability for a broad spectrum of industrial, transportation, agricultural, and other types of firms and allows one to formulate problems of analyzing the potential of a firm as linear programming problems or mixed programming problems with linear constraints. This approach generalizes the one proposed by the author earlier for a more narrow class of models and allows one to effectively employ a widely available software for solving practical problems of the considered kind, especially for firms described by large scale models of mathematical programming.     

Generalized Smooth Monotonic Regression in Additive Modeling

Common approaches to monotonic regression focus on the case of a unidimensional covariate and continuous response variable. Here a general approach is proposed that allows for additive structures where one or more variables have monotone influence on the response variable. In addition the approach allows for response variables from an exponential family, including binary and Poisson distributed response variables. Flexibility of the smooth estimate is gained by expanding the unknown function in monotonic basis functions. For the estimation of coefficients and the selection of basis functions a likelihood-based boosting algorithm is proposed which is simple to implement. Stopping criteria and inference are based on AIC-type measures. The method is applied to several datasets.     

Geometric Models of the Statistical Theory of Fragmentation

We propose an approach to describing a medium fragmentation process based on studying the stochastic geometry of the medium states. This approach allows accounting for the interrelation of the produced fragments relative to their positions and, in particular, allows taking the size of the fragmenting object into account. We use this approach to analyze a one-dimensional model—a stochastic process with discrete time and a phase space consisting of partitions into fragments of the real axis. We derive the driving equation for the partition function with respect to sizes and prove the existence of a limit distribution.     

On Isomorphisms and Invariants of Finite Dimensional Complex Filiform Leibniz Algebras

In this article, we propose an approach classifying a class of filiform Leibniz algebras. The approach is based on algebraic invariants. The method allows to classify all filiform Leibniz algebras (including filiform Lie algebras) in a given fixed dimensional case.     

Solving some vector subset problems by Voronoi diagrams

We propose a general approach to solving some vector subset problems in a Euclidean space that is based on higher-order Voronoi diagrams. In the case of a fixed space dimension, this approach allows us to find optimal solutions to these problems in polynomial time which is better than the runtime of available algorithms.     

Finite-Temperature Coarse-Graining of One-Dimensional Models: Mathematical Analysis and Computational Approaches

We present a possible approach for the computation of free energies and ensemble averages of one-dimensional coarse-grained models in materials science. The approach is based upon a thermodynamic limit process, and makes use of ergodic theorems and large deviations theory. In addition to providing a possible efficient computational strategy for ensemble averages, the approach allows for assessing the accuracy of approximations commonly used in practice.     

Lagrange approach to the optimal control of diffusions

A new approach to the optimal control of diffusion processes based on Lagrange functionals is presented. The method is conceptually and technically simpler than existing ones. A first class of functionals allows to obtain optimality conditions without any resort to stochastic calculus and functional analysis. A second class, which requires Ito's rule, allows to establish optimality in a larger class of problems. Calculations in these two methods are sometimes akin to those in minimum principles and in dynamic programming, but the thinking behind them is new. A few examples are worked out to illustrate the power and simplicity of this approach.Research performed at the Mathematisches Seminar der Universität Kiel with support provided by an Alexander von Humboldt Foundation fellowship.     

A primal conical linear programming algorithm

This paper completes the treatment of the conical approach to linear programming, introducing a conical primal algorithm of linear programming. After some recalls and improvements of a previous paper dealing with such an approach, the algorithm is defined. A first convergence result is then proved, which, along with a series of lemmata, allows to prove that the algorithm terminates in a finite number of steps     

Linear and nonlinear dynamics of a circular cylindrical shell connected to a rigid disk

The dynamics of a circular cylindrical shell carrying a rigid disk on the top and clamped at the base is investigated. The Sanders–Koiter theory is considered to develop a nonlinear analytical model for moderately large shell vibration. A reduced order dynamical system is obtained using Lagrange equations: radial and in-plane displacement fields are expanded by using trial functions that respect the geometric boundary conditions.The theoretical model is compared with experiments and with a finite element model developed with commercial software: comparisons are carried out on linear dynamics.The dynamic stability of the system is studied, when a periodic vertical motion of the base is imposed. Both a perturbation approach and a direct numerical technique are used. The perturbation method allows to obtain instability boundaries by means of elementary formulae; the numerical approach allows to perform a complete analysis of the linear and nonlinear response.     

A general extrapolation procedure revisited

TheE-algorithm is the most general extrapolation algorithm actually known. The aim of this paper is to provide a new approach to this algorithm. This approach gives a deeper insight into theE-algorithm, and allows one to obtain new properties and to relate it to other algorithms. Some extensions of the procedure are discussed.     

Generalized compound matrix method

In this work we demonstrate how the extension of the Evans function method using the compound matrix approach can be implemented to undertake the stability analysis (normally done through numerical means) of nonlinear travelling waves. The main advantage of this approach is that it can easily overcome the stiffness which is normally associated with these kinds of problems. We present a general approach which allows this method to be used for a general class of nonlinear travelling wave problems.     

Initial‐to‐Interface Maps for the Heat Equation on Composite Domains

A map from the initial conditions to the function and its first spatial derivative evaluated at the interface is constructed for the heat equation on finite and infinite domains with n interfaces. The existence of this map allows changing the problem at hand from an interface problem to a boundary value problem which allows for an alternative to the approach of finding a closed‐form solution to the interface problem.     

Estimation of potential gains from mergers in multiple periods: a comparison of stochastic frontier analysis and Data Envelopment Analysis

A new dynamic Data Envelopment Analysis (DEA) approach is created to provide valuable managerial insights when assessing the merger performance. This new approach allows us to dynamically evaluate the pre-merger firms and the post-merger firm in a multi-period situation. A case study of bank branch merger is conducted to illustrate and validate the proposed approach. Both stochastic frontier analysis and data envelopment analysis are used and compared leading to highly correlated results. The computation show that merger results in an overall efficiency achievement in a banking industry.     

Estimating the Hausdorff dimensions of univoque sets for self-similar sets

An approach is given for estimating the Hausdorff dimension of the univoque set of a self-similar set. This sometimes allows us to get the exact Hausdorff dimensions of the univoque sets.