Dual and Feller-Reuter-Riley transition functions

In this paper, we investigate duality and Feller-Reuter-Riley (FRR) property of continuous-time Markov chains (CTMCs). A criterion of dual q-functions is given in terms of their q-matrices. For a dual q-matrix Q, a necessary and sufficient conditions for the minimal Q-function to be a FRR transition function are also given. Finally, by using dual technique, we give a criterion of FRR Q-functions when Q is monotone.     

Column continuous transition functions

A column continuous transition function is by definition a standard transition function P(t) whose every column is continuous for t?0 in the norm topology of bounded sequence space l_∞. We will prove that it has a stable q-matrix and that there exists a one-to-one relationship between column continuous transition functions and increasing integrated semigroups on l_∞. Using the theory of integrated semigroups, we give some necessary and sufficient conditions under which the minimal q-function is column continuous, in terms of its generator (of the Markov semigroup) as well as its q-matrix. Furthermore, we will construct all column continuous Q-functions for a conservative, single-exit and column bounded q-matrix Q. As applications, we find that many interesting continuous-time Markov chains (CTMCs), say Feller-Reuter-Riley processes, monotone processes, birth-death processes and branching processes, etc., have column continuity.     

Existence theory and Q-matrix characterization for the generalized linear complementarity problem: Revisited

We provide conditions under which a vertical block matrix is a Q-matrix if one or all representative sub-matrices are Q-matrices and vice versa. It is also shown, by means of counterexamples, that Eq. (3) of [A.A. Ebiefung, Existence theory and Q-matrix characterization for the generalized linear complementarity problem, Linear Algebra Appl. 223/224 (1995) 155-169] is incorrect.     

Decay Properties of <Emphasis Type="Italic">n</Emphasis>-type Markov Branching Processes with Disasters

In this paper, we consider the decay properties of n-type Markov branching processes with disasters, including the decay parameter and invariant measures. It is first proved that the corresponding q-matrix Q is always regular and, thus, that the Feller minimal Q-process is honest. Then, the exact value of the decay parameter \(\lambda _C\) is obtained. We prove that the decay parameter can be easily expressed explicitly. Furthermore, we prove that the Markov branching process with disasters is always \(\lambda _C\)-positive. The invariant vectors and the invariant measures are given explicitly.     

On the equivalence of μ-invariant measures for the minimal process and its q-matrix

In this paper we obtain necessary and sufficient conditions for a measure or vector that is μ-invariant for a q-matrix, Q, to be μ-invariant for the family of transition matrices, {P(t)}, of the minimal process it generates. Sufficient conditions are provided in the case when Q is regular and these are shown not to be necessary. When μ-invariant measures and vectors can be identified, they may be used, in certain cases, to determine quasistationary distributions for the process.     

Interactive Optimization Aspects of Electronic Spreadsheet Models for Design and Planning

This paper extends the discussion of the relationship between optimization and electronic spreadsheet modelling. Previous authors have described two connections. These are, first, the use of spreadsheets to facilitate optimization projects and, second, the use of optimization within the spreadsheet model itself such as for cell value computation. In this paper we argue that use of the spreadsheet model for planning and design problems often may actually be viewed as interactive multi-criterion optimization per se. This provides a third connection between optimization and spreadsheet models. We base our arguments on a simplified example arising in insurance consulting work and note other examples as well. We also propose a method for incorporating this observation into existing spreadsheet models and indicate desirable innovations in spreadsheet software.     

High-Dimensional Mixed Graphical Models

While graphical models for continuous data (Gaussian graphical models) and discrete data (Ising models) have been extensively studied, there is little work on graphical models for datasets with both continuous and discrete variables (mixed data), which are common in many scientific applications. We propose a novel graphical model for mixed data, which is simple enough to be suitable for high-dimensional data, yet flexible enough to represent all possible graph structures. We develop a computationally efficient regression-based algorithm for fitting the model by focusing on the conditional log-likelihood of each variable given the rest. The parameters have a natural group structure, and sparsity in the fitted graph is attained by incorporating a group lasso penalty, approximated by a weighted lasso penalty for computational efficiency. We demonstrate the effectiveness of our method through an extensive simulation study and apply it to a music annotation dataset (CAL500), obtaining a sparse and interpretable graphical model relating the continuous features of the audio signal to binary variables such as genre, emotions, and usage associated with particular songs. While we focus on binary discrete variables for the main presentation, we also show that the proposed methodology can be easily extended to general discrete variables.     

Inverse problems for ergodicity of Markov chains

For continuous-time Markov chains, we provide criteria for non-ergodicity, non-algebraic ergodicity, non-exponential ergodicity, and non-strong ergodicity. For discrete-time Markov chains, criteria for non-ergodicity, non-algebraic ergodicity, and non-strong ergodicity are given. Our criteria are in terms of the existence of solutions to inequalities involving the Q-matrix (or transition matrix P in time-discrete case) of the chain. Meanwhile, these practical criteria are applied to some examples, including a special class of single birth processes and several multi-dimensional models.     

Optimizing information using the EM algorithm in item response theory

Latent trait models such as item response theory (IRT) hypothesize a functional relationship between an unobservable, or latent, variable and an observable outcome variable. In educational measurement, a discrete item response is usually the observable outcome variable, and the latent variable is associated with an examinee’s trait level (e.g., skill, proficiency). The link between the two variables is called an item response function. This function, defined by a set of item parameters, models the probability of observing a given item response, conditional on a specific trait level. Typically in a measurement setting, neither the item parameters nor the trait levels are known, and so must be estimated from the pattern of observed item responses. Although a maximum likelihood approach can be taken in estimating these parameters, it usually cannot be employed directly. Instead, a method of marginal maximum likelihood (MML) is utilized, via the expectation-maximization (EM) algorithm. Alternating between an expectation (E) step and a maximization (M) step, the EM algorithm assures that the marginal log likelihood function will not decrease after each EM cycle, and will converge to a local maximum. Interestingly, the negative of this marginal log likelihood function is equal to the relative entropy, or Kullback-Leibler divergence, between the conditional distribution of the latent variables given the observable variables and the joint likelihood of the latent and observable variables. With an unconstrained optimization for the M-step proposed here, the EM algorithm as minimization of Kullback-Leibler divergence admits the convergence results due to Csiszár and Tusnády (Statistics & Decisions, 1:205–237, 1984), a consequence of the binomial likelihood common to latent trait models with dichotomous response variables. For this unconstrained optimization, the EM algorithm converges to a global maximum of the marginal log likelihood function, yielding an information bound that permits a fixed point of reference against which models may be tested. A likelihood ratio test between marginal log likelihood functions obtained through constrained and unconstrained M-steps is provided as a means for testing models against this bound. Empirical examples demonstrate the approach.     

OnQ-matrices

In a recent paper [1], Aganagic and Cottle have established a constructive characterization for aP ₀-matrix to be aQ-matrix. Among the principal results in this paper, we show that the same characterization holds for anL-matrix as well, and that the symmetric copositive-plusQ-matrices are precisely those which are strictly copositive.     

Modelling and inference with Conditional Gaussian Probabilistic Decision Graphs

Probabilistic Decision Graphs (PDGs) are probabilistic graphical models that represent a factorisation of a discrete joint probability distribution using a “decision graph”-like structure over local marginal parameters. The structure of a PDG enables the model to capture some context specific independence relations that are not representable in the structure of more commonly used graphical models such as Bayesian networks and Markov networks. This sometimes makes operations in PDGs more efficient than in alternative models. PDGs have previously been defined only in the discrete case, assuming a multinomial joint distribution over the variables in the model. We extend PDGs to incorporate continuous variables, by assuming a Conditional Gaussian (CG) joint distribution. We also show how inference can be carried out in an efficient way.     

Error bounds for linear complementarity problems of DB-matrices

Doubly B-matrices (DB-matrices), which properly contain B-matrices, are introduced by Peña (2003) [2]. In this paper we present error bounds for the linear complementarity problem when the matrix involved is a DB-matrix and a new bound for linear complementarity problem of a B-matrix. The numerical examples show that the bounds are sharp.     

One-dimensional stochastic Burgers equation driven by Lévy processes

In this paper, we proved the global existence and uniqueness of the strong, weak and mild solutions for one-dimensional Burgers equation perturbed by a Poisson form process, a Poisson form and Q-Wiener process with the Dirichlet bounded condition. We also proved the existence of the invariant measure of these models.     

Graphical Calculus for the Double Affine Q-Dependent Braid Group

In this paper, we present a straightforward pictorial representation of the double affine Hecke algebra (DAHA) which enables us to translate the abstract algebraic structure of a DAHA into an intuitive graphical calculus suitable for a physics audience. Initially, we define the larger double affine Q-dependent braid group. This group is constructed by appending to the braid group a set of operators Q _i , before extending it to an affine Q-dependent braid group. We show specifically that the elliptic braid group and the DAHA can be obtained as quotient groups. Complementing this, we present a pictorial representation of the double affine Q-dependent braid group based on ribbons living in a toroid. We show that in this pictorial representation, we can fully describe any DAHA. Specifically, we graphically describe the parameter q upon which this algebra is dependent and show that in this particular representation q corresponds to a twist in the ribbon.     

Quantile regression metamodeling: Toward improved responsiveness in the high-tech electronics manufacturing industry

Both technology and market demands within the high-tech electronics manufacturing industry change rapidly. Accurate and efficient estimation of cycle-time (CT) distribution remains a critical driver of on-time delivery and associated customer satisfaction metrics in these complex manufacturing systems. Simulation models are often used to emulate these systems in order to estimate parameters of the CT distribution. However, execution time of such simulation models can be excessively long limiting the number of simulation runs that can be executed for quantifying the impact of potential future operational changes. One solution is the use of simulation metamodeling which is to build a closed-form mathematical expression to approximate the input–output relationship implied by the simulation model based on simulation experiments run at selected design points in advance. Metamodels can be easily evaluated in a spreadsheet environment “on demand” to answer what-if questions without needing to run lengthy simulations. The majority of previous simulation metamodeling approaches have focused on estimating mean CT as a function of a single input variable (i.e., throughput). In this paper, we demonstrate the feasibility of a quantile regression based metamodeling approach. This method allows estimation of CT quantiles as a function of multiple input variables (e.g., throughput, product mix, and various distributional parameters of time-between-failures, repair time, setup time, loading and unloading times). Empirical results are provided to demonstrate the efficacy of the approach in a realistic simulation model representative of a semiconductor manufacturing system.     

Recursive algorithm for the reliability of a connected-(1, 2)-or-(2, 1)-out-of-(m, n):F lattice system

The connected-(1, 2)-or-(2, 1)-out-of-(m, n):F lattice system is included by the connected-X-out-of-(m, n):F lattice system defined by Boehme et al. [Boehme, T.K., Kossow, A., Preuss, W., 1992. A generalization of consecutive-k-out-of-n:F system. IEEE Transactions on Reliability 41, 451–457]. This system fails if and only if at least one subset of connected failed components occurs which includes at least a (1, 2)-matrix (that is, a row and two columns) or a (2, 1)-matrix(that is, two rows and a column) of failed components. This system is applied to two-dimensional network problems with adjacent constraints, and various systems, for example, a supervision system, etc.     

Characterization theorems for <Emphasis Type="Italic">Q</Emphasis>-independent random variables with values in a locally compact Abelian group

Let X be a locally compact Abelian group, Y be its character group. Following A. Kagan and G. Székely we introduce a notion of Q-independence for random variables with values in X. We prove group analogues of the Cramér, Kac–Bernstein, Skitovich–Darmois and Heyde theorems for Q-independent random variables with values in X. The proofs of these theorems are reduced to solving some functional equations on the group Y.     

The Emergence of Mathematical Modeling Competencies: An Investigation of Prospective Secondary Mathematics Teachers

A study with prospective teachers without prior mathematical modeling experience sheds light on how their newly developed conceptual understanding of modeling manifested itself in their work on the final task of a modeling module within a pedagogy course in secondary mathematics curriculum and assessment. The main purpose of the module was to provide opportunity for the prospective teachers to experience the Common Core Mathematical Practice Model with Mathematics and begin to develop competency in modeling. Their work and reflections displayed a range of proficiency in several competencies associated with the modeling process. Examples of their work illustrating these ranges are provided. The prospective teachers expressed both struggle and rewards during the process, and reflected on challenges for teaching modeling. The results suggest that infusing modules in existing courses can be an effective way to elevate prospective teachers from unfamiliarity with modeling to noticeable levels of proficiency in various modeling sub-competencies.     

LU decomposition of M-matrices by elimination without pivoting

It is shown that if A or ?A is a singular M-matrix satisfying the generalized diagonal dominance condition y^TA?0 for some vector y? 0, then A can be factored into A = LU by a certain elimination algorithm, where L is a lower triangular M-matrix with unit diagonal and U is an upper triangular M-matrix. The existence of LU decomposition of symmetric permutations of A and for irreducible M-matrices and symmetric M-matrices follow as colollaries. This work is motivated by applications to the solution of homogeneous systems of linear equations Ax = 0, where A or ?A is an M-matrix. These applications arise, e.g., in the analysis of Markov chains, input-output economic models, and compartmental systems. A converse of the theorem metioned above can be established by considering the reduced normal form of A.     

Concentration-compactness phenomena in the higher order Liouville's equation

We investigate different concentration-compactness and blow-up phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in R2m, then that of a closed manifold and, finally, the particular case of the sphere S2m. In all cases we allow the sign of the Q-curvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in R2m, blow-up phenomena can occur only at points of positive Q-curvature. As a consequence, on a locally conformally flat manifold of non-positive Euler characteristic we always have compactness.