The Pathmox approach for PLS path modeling segmentation

Modeling has often failed to meet expectations, mostly because of the difficulty of comprehending relationships within phenomena and expressing them in mathematical models. Reality is frequently too complex to be reflected in a single model. This is often the case of marketing research, where variables relating to socioeconomics or psychographics constitute potential sources of heterogeneity. In such cases, the assumption of ‘one model fits all’ is unrealistic and may lead to inaccurate decisions. Thus, heterogeneity is a major issue in modeling. Once a model has been fitted to a complete data set that fulfills all validation criteria, it is difficult to establish whether it is valid for the whole population or it is merely an average artifact from several sub‐populations. The purpose of this paper is to present the Pathmox approach to deal with heterogeneity in partial least squares path modeling. The idea behind Pathmox is to build a tree of path models that have look‐alike structure as a binary decision tree, with different models for each of its nodes. The split criterion consists of an F statistic comparing two structural models. In order to ensure the suitability of the split criterion, a simulation study was conducted. Finally, we have applied Pathmox to a survey that measured Satisfaction among Spanish mobile phone operators. Results suggest that the Pathmox approach performs adequately in detecting partial least squares path modeling heterogeneity. Copyright © 2016 John Wiley & Sons, Ltd.     

“讨价还价”的“纳什解”的形成路径与结论拓展

建立了一个"讨价还价"的双边适应性预期模型,并利用这个模型的二元常系数线性差分方程组,揭示"纳什讨价还价解"的形成路径.对于不是"纳什讨价还价解"的一般讨价还价的成交结果,同样可以运用这样一个模型来揭示.基于这个模型,当买、卖双方希望成交,各方的出价通过逐次调整会最终收敛到同一个常数,因此能够从理论上保证讨价还价最终成交.双方当中有一方不肯让价而成交的情形,可以作为这个模型退化成为单边适应性预期模型的情形.举例计算说明了一个讨价还价问题的收敛过程与结果.     

A simple approach to fuzzy critical path analysis in project networks

This paper develops a simple approach to critical path analysis in a project network with activity times being fuzzy numbers. The idea is based on the linear programming (LP) formulation and fuzzy number ranking method. The fuzzy critical path problem is formulated as an LP model with fuzzy coefficients of the objective function, and then on the basis of properties of linearity and additivity, the Yager’s ranking method is adopted to transform the fuzzy LP formulation to the crisp one which can be solved by using the conventional streamlined solution methods. Consequently, the critical path and total duration time can be obtained from the derived optimal solution. Moreover, in this paper we also define the most critical path and the relative path degree of criticality, which are theoretically sound and easy to use in practice. An example discussed in some previous studies illustrates that the proposed approach is able to find the most critical path, which is proved to be the same as that derived from an exhausted comparison of all possible paths. The proposed approach is very simple to apply, and it is not require knowing the explicit form of the membership functions of the fuzzy activity times.     

On a generalized model of labeled graphs

Labeled graphs have applications in algorithms for reconstructing chains that have been split into smaller parts. Chain reconstruction is a common problem in biochemistry and bioinformatics, particularly for sequencing DNA or peptide chains. Labeled graphs (in the sense defined in this paper) have also the important structural property which allows to reduce the Hamiltonian path problem to Eulerian path problem. This work introduces a model and properties of a class of base-labeled graphs that unify the properties of labeled and free-labeled graphs (B?a?ewicz et al., 1999) [1]. It describes the basic relationships between those classes and some of their applications. It also introduces lexical graphs which are the superclass of de Bruijn graphs. Lexical graphs keep many properties of de Bruijn graphs which have a wide area of applications e.g. in mathematics, electronics and computing sciences.     

Estimation in the continuous time mover‐stayer model with an application to bond ratings migration

The usual tool for modelling bond ratings migration is a discrete, time‐homogeneous Markov chain. Such model assumes that all bonds are homogeneous with respect to their movement behaviour among rating categories and that the movement behaviour does not change over time. However, among recognized sources of heterogeneity in ratings migration is age of a bond (time elapsed since issuance). It has been observed that young bonds have a lower propensity to change ratings, and thus to default, than more seasoned bonds. The aim of this paper is to introduce a continuous, time‐non‐homogeneous model for bond ratings migration, which also incorporates a simple form of population heterogeneity. The specific form of heterogeneity postulated by the proposed model appears to be suitable for modelling the effect of age of a bond on its propensity to change ratings. This model, called a mover–stayer model, is an extension of a Markov chain. This paper derives the maximum likelihood estimators for the parameters of a continuous time mover–stayer model based on a sample of independent continuously monitored histories of the process, and develops the likelihood ratio statistic for discriminating between the Markov chain and the mover–stayer model. The methods are illustrated using a sample of rating histories of young corporate issuers. For these issuers the default probabilities predicted by the Markov chain and mover–stayer models are different. In particular for 1–4 years old bonds the mover–stayer model estimates substantially lower default probabilities from rating C than a Markov chain. Copyright © 2004 John Wiley & Sons, Ltd.     

OPTIMAL HARVESTING TIME OF FARMED AQUATIC POPULATIONS WITH NONLINEAR SIZE‐HETEROGENEOUS GROWTH

Abstract This paper examines the question of optimal harvesting time in a size‐heterogeneous farmed aquatic population, using a model reflecting the effect of population density on both overall mortality rate and individual growth. This analysis enables an optimal harvesting rule to be deduced. The results obtained are applied to shrimp culture in recirculation systems in Mexico. Numerical solutions are derived for different production scenarios. Assuming identical culture conditions, results are also obtained under the hypothesis of homogeneous population growth, the view traditionally taken in the relevant economic literature. The optimal harvesting times calculated tend to decrease with higher densities, although this rule fails under the size‐heterogeneous population model. In general, optimal harvesting times are overestimated when size‐homogeneity in the culture is assumed. Our analysis reveals that management predictions are significantly mistaken if the size‐heterogeneity phenomenon is not taken into account.     

Logistic Dirichlet problems with discontinuous coefficients

This paper is devoted to the study of the existence of positive solutions of semilinear Dirichlet eigenvalue problems for diffusive logistic equations with discontinuous coefficients which model population dynamics in environments with spatial heterogeneity. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of singular integral operators. Moreover, we make use of an L^p variant of an estimate for the Green operator of the Dirichlet problem introduced in the study of Feller semigroups.     

Linear programming with interval coefficients

In order to solve a linear programme, the model coefficients must be fixed at specific values, which implies that the coefficients are perfectly accurate. In practice, however, the coefficients are generally estimates. The only way to deal with uncertain coefficients is to test the sensitivity of the model to changes in their values, either singly or in very small groups. We propose a new approach in which some or all of the coefficients of the LP are specified as intervals. We then find the best optimum and the worst optimum for the model, and the point settings of the interval coefficients that yield these two extremes. This provides the range of the optimised objective function, and the coefficient settings give some insight into the likelihood of these extremes.     

An existence and uniqueness principle for a nonlinear version of the Lebowitz‐Rubinow model with infinite maximum cycle length

The present article deals with existence and uniqueness results for a nonlinear evolution initial‐boundary value problem, which originates in an age‐structured cell population model introduced by Lebowitz and Rubinow (1974) describing the growth of a cell population. Cells of this population are distinguished by age a and cycle length l. In our framework, daughter and mother cells are related by a general reproduction rule that covers all known biological ones. In this paper, the cycle length l is allowed to be infinite. This hypothesis introduces some mathematical difficulties. We consider both local and nonlocal boundary conditions.     

Haplotyping with missing data via perfect path phylogenies

Computational methods for inferring haplotype information from genotype data are used in studying the association between genomic variation and medical condition. Recently, Gusfield proposed a haplotype inference method that is based on perfect phylogeny principles. A fundamental problem arises when one tries to apply this approach in the presence of missing genotype data, which is common in practice. We show that the resulting theoretical problem is NP-hard even in very restricted cases. To cope with missing data, we introduce a variant of haplotyping via perfect phylogeny in which a path phylogeny is sought. Searching for perfect path phylogenies is strongly motivated by the characteristics of human genotype data: 70% of real instances that admit a perfect phylogeny also admit a perfect path phylogeny. Our main result is a fixed-parameter algorithm for haplotyping with missing data via perfect path phylogenies. We also present a simple linear-time algorithm for the problem on complete data.     

Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity

In this paper, we consider an initial boundary value problem for the 3‐dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity coefficients over a bounded smooth domain. Global in time unique strong solution is proved to exist when the L² norms of initial vorticity and current density are both suitably small with arbitrary large initial density, and the vacuum of initial density is also allowed. Finally, we revisit the Navier‐Stokes model without electromagnetic effect. We find that this initial boundary problem also admits a unique global strong solution under other conditions. In particular, we prove small kinetic‐energy strong solution exists globally in time, which extends the recent result of Huang and Wang.     

Effect of emergent carrying capacity in an eco‐epidemiological system

The paper explores an eco‐epidemiological model of a predator–prey type, where the prey population is subject to infection. The model is basically a combination of S‐I type model and a Rosenzweig–MacArthur predator–prey model. The novelty of this contribution is to consider different competition coefficients within the prey population, which leads to the emergent carrying capacity. We explicitly separate the competition between non‐infected and infected individuals. This emergent carrying capacity is markedly different to the explicit carrying capacities that have been considered in many eco‐epidemiological models. We observed that different intra‐class and inter‐class competition can facilitate the coexistence of susceptible prey‐infected prey–predator, which is impossible for the case of the explicit carrying capacity model. We also show that these findings are closely associated with bi‐stability. The present system undergoes bi‐stability in two different scenarios: (a) bi‐stability between the planner equilibria where susceptible prey co‐exists with predator or infected prey and (b) bi‐stability between co‐existence equilibrium and the planner equilibrium where susceptible prey coexists with infected prey; have been discussed. The conditions for which the system is to be permanent and the global stability of the system around disease‐free equilibrium are worked out. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence and non‐existence of steady states to a cross‐diffusion system arising in a Leslie predator–prey model

This paper is concerned with a cross‐diffusion system arising in a Leslie predator–prey population model in a bounded domain with no flux boundary condition. We investigate sufficient condition for the existence and the non‐existence of non‐constant positive solution. We obtain that if natural diffusion coefficient of predator is large enough and cross‐diffusion coefficients are fixed, then under some conditions there exists non‐constant positive solution. Furthermore, we show that if natural diffusion coefficients of predator and prey are both large enough, and cross‐diffusion coefficients are small enough, then there exists no non‐constant positive solution. Copyright © 2012 John Wiley & Sons, Ltd.     

Using matrices to link conflict evolution and resolution in a graph model

The graph model for conflict resolution provides a convenient and effective means to model and analyze a strategic conflict. Standard practice is to carry out a stability analysis of a graph model, and then to follow up with a post-stability analysis, an important component of which is status quo analysis. A graph model can be viewed as an edge-colored graph, but the fundamental problem of status quo analysis – to find a shortest colored path from the status quo node to a desired equilibrium – is different from the well-known network analysis problem of finding the shortest path between two nodes. The only matrix method that has been proposed cannot track all aspects of the evolution of a conflict from the status quo state. Our explicit algebraic approach is convenient for computer implementation and, as demonstrated with a real world case study, easy to use. It provides new insights into a graph model, not only identifying all equilibria reachable from the status quo, but also how to reach them. Moreover, this approach bridges the gap between stability analysis and status quo analysis in the graph model for conflict resolution.     

Heteroscedastic Laplace mixture of experts regression models and applications

Mixture of Experts(MoE) regression models are widely studied in statistics and machine learning for modeling heterogeneity in data for regression, clustering and classification.Laplace distribution is one of the most important statistical tools to analyze thick and tail data. Laplace Mixture of Linear Experts(LMoLE) regression models are based on the Laplace distribution which is more robust. Similar to modelling variance parameter in a homogeneous population, we propose and study a new novel class of models: heteroscedastic Laplace mixture of experts regression models to analyze the heteroscedastic data coming from a heterogeneous population in this paper. The issues of maximum likelihood estimation are addressed. In particular, Minorization-Maximization(MM) algorithm for estimating the regression parameters is developed. Properties of the estimators of the regression coefficients are evaluated through Monte Carlo simulations. Results from the analysis of two real data sets are presented.     

Modelling heterogeneity in manpower planning: dividing the personnel system into more homogeneous subgroups

Manpower planning is very useful for human resource management in large organizations. Most manpower models are concerned with the prediction of the future behaviour of the staff: they might leave the organization, get promoted or acquire more and new skills. This behaviour can vary a lot among different employees, what makes prediction difficult. It is common to tackle this problem by dividing the whole heterogeneous personnel system in several more homogeneous subgroups. This approach is often used to develop manpower planning models for prediction, control or optimization. Although the division in homogeneous subcategories is a fundamental and important step in the application of the models, up till now literature neglects to discuss a procedure to deal with this in practice. This paper suggests a general framework to find the distinguished homogeneous subcategories by determining and considering observable sources of personnel heterogeneity. Copyright © 2006 John Wiley & Sons, Ltd.     

New models for the robust shortest path problem: complexity, resolution and generalization

In optimization, it is common to deal with uncertain and inaccurate factors which make it difficult to assign a single value to each parameter in the model. It may be more suitable to assign a set of values to each uncertain parameter. A scenario is defined as a realization of the uncertain parameters. In this context, a robust solution has to be as good as possible on a majority of scenarios and never be too bad. Such characterization admits numerous possible interpretations and therefore gives rise to various approaches of robustness. These approaches differ from each other depending on models used to represent uncertain factors, on methodology used to measure robustness, and finally on analysis and design of solution methods. In this paper, we focus on the application of a recent criterion for the shortest path problem with uncertain arc lengths. We first present two usual uncertainty models: the interval model and the discrete scenario set model. For each model, we then apply a criterion, called bw-robustness (originally proposed by B. Roy) which defines a new measure of robustness. According to each uncertainty model, we propose a formulation in terms of large scale integer linear program. Furthermore, we analyze the theoretical complexity of the resulting problems. Our computational experiments perform on a set of large scale graphs. By observing the results, we can conclude that the approved solvers, e.g. Cplex, are able to solve the mathematical models proposed which are promising for robustness analysis. In the end, we show that our formulations can be applied to the general linear program in which the objective function includes uncertain coefficients.     

On the complexity of finding paths in a two‐dimensional domain I: Shortest paths

The computational complexity of finding a shortest path in a two‐dimensional domain is studied in the Turing machine‐based computational model and in the discrete complexity theory. This problem is studied with respect to two formulations of polynomial‐time computable two‐dimensional domains: (A) domains with polynomialtime computable boundaries, and (B) polynomial‐time recognizable domains with polynomial‐time computable distance functions. It is proved that the shortest path problem has the polynomial‐space upper bound for domains of both type (A) and type (B); and it has a polynomial‐space lower bound for the domains of type (B), and has a #P lower bound for the domains of type (A). (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A useful reparameterization for second‐order autoregressive processes

In time series analysis, autoregressive (a.r.) models are often fitted successfully to data, and such models are usually among the first to be taught. It is therefore important to understand why such models are useful in practice. An alternative parameterization for second order a.r. models, which assists understanding and interpretation, is discussed. The models are most plausible if certain restrictions are placed on the autoregressive parameters, which in turn lead to restrictions on the autocorrelation coefficients. Fitting the reparameterized model is straightforward, and the reparameterization may be extended, less usefully, to higher order a.r. models. The second‐order model is fitted to some meteorological data.     

On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids

We consider finite difference approximations of solutions of inverse Sturm‐Liouville problems in bounded intervals. Using three‐point finite difference schemes, we discretize the equations on so‐called optimal grids constructed as follows: For a staggered grid with 2 k points, we ask that the finite difference operator (a k × k Jacobi matrix) and the Sturm‐Liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval. This requirement determines uniquely the entries in the Jacobi matrix, which are grid cell averages of the coefficients in the continuum problem. If these coefficients are known, we can find the grid, which we call optimal because it gives, by design, a finite difference operator with a prescribed spectral measure. We focus attention on the inverse problem, where neither the coefficients nor the grid are known. A key question in inversion is how to parametrize the coefficients, i.e., how to choose the grid. It is clear that, to be successful, this grid must be close to the optimal one, which is unknown. Fortunately, as we show here, the grid dependence on the unknown coefficients is weak, so the inversion can be done on a precomputed grid for an a priori guess of the unknown coefficients. This observation leads to a simple yet efficient inversion algorithm, which gives coefficients that converge pointwise to the true solution as the number k of data points tends to infinity. The cornerstone of our convergence proof is showing that optimal grids provide an implicit, natural regularization of the inverse problem, by giving reconstructions with uniformly bounded total variation. The analysis is based on a novel, explicit perturbation analysis of Lanczos recursions and on a discrete Gel'fand‐Levitan formulation. © 2005 Wiley Periodicals, Inc.