Stable graphical model estimation with Random Forests for discrete, continuous, and mixed variables

A model for mixed continuous and discrete variables suggested by Chang and Afifi (1974) and Krzanowski (1975) is used to explore the bias in the discriminant function (DF) approach to estimation of the coefficients in the multiple1ogistic regression model. When the data come from this mixed variable model the DF estimator of the coefficients of the continuous variables are asymptotically unbiased. The DF estimator of the intercept and coefficients for the discrete variables may be severely biased. The magnitude of the bias is shown to depend in a systematic way on the true value of the coefficients and the underlying probabilities of the out-come of discrete variables. The implications for analysis are discussed.

A method to efficiently solve the problem of minimum weight design of plane and space trusses with discrete or mixed variables is developed. The method can also be applied to continuous variables. The original formulation leads to a non‐linear constrained minimization problem with inequality constraints, which is solved by means of a sequence of approximate problems using dual techniques. In the dual space, the objective function is to be maximized, depends on continuous variables, is concave and has first and second order discontinuities. In addition, the constraints deal simply with restricting the dual variables to be non‐negative. To solve the problem an ad hoc algorithm from mathematical programming has been adapted.Some examples have been developed to show the effectiveness of the method.

This paper discusses the integrated topology and sizing optimization of frame structures where beam elements are used in the structural analysis model. Based on the ground structure approach, discrete variables (0, 1) representing the absence or presence of beam members and continuous sizing variables for beam cross sections are involved in the problem. A two-level multipoint approximation strategy is introduced to address this mixed-variable problem. A first-level approximate problem is first constructed using the branched multipoint approximate function involving both discrete and continuous variables. For solving this problem, discrete variables are optimized through a genetic algorithm, and when calculating the fitness in the genetic algorithm, a second-level approximate problem solved by the dual method is established to optimize the cross sections for the retained beam members. Temporal deletion techniques are used for displacement and stress constraints. Meanwhile, by considering local vibration modes in frequency-constrained problems, material density is assigned with an extremely small value for the removed beams to avoid these local modes. Thence, the singularities encountered in stress, displacement, and frequency constraints are overcome. Typical examples and a satellite application are given to verify the feasibility and efficiency of this method in handling the integrated topology and sizing optimization for frame structures.

Thrombocytosis associated with malignant ovarian lesions within a pediatric and adolescent population

Bayesian networks (BNs) are being studied in recent years for system diagnosis, reliability analysis, and design of complex engineered systems. In several practical applications, BNs need to be learned from available data before being used for design or other purposes. Current BN learning algorithms are mainly developed for networks with only discrete variables. Engineering design problems often consist of both discrete and continuous variables. This paper develops a framework to handle continuous variables in BN learning by integrating learning algorithms of discrete BNs with Gaussian mixture models (GMMs). We first make the topology learning more robust by optimizing the number of Gaussian components in the univariate GMMs currently available in the literature. Based on the BN topology learning, a new multivariate Gaussian mixture (MGM) strategy is developed to improve the accuracy of conditional probability learning in the BN. A method is proposed to address this difficulty of MGM modeling with data of mixed discrete and continuous variables by mapping the data for discrete variables into data for a standard normal variable. The proposed framework is capable of learning BNs without discretizing the continuous variables or making assumptions about their conditional probability densities (CPDs). The applications of the learned BN to uncertainty quantification and model calibration are also investigated. The results of a mathematical example and an engineering application example demonstrate the effectiveness of the proposed framework.

This paper presents an optimal voltage control algorithm developed to increase the efficiency of operation of power systems and minimizing the reaction time for compensating voltage deviations due to load fluctuations. Because of the many variables of influence in the optimal voltage control (e.g. line power flow limits, bus voltage limits, etc.), the mathematical formulation is a constrained optimization problem with continuous and discrete variables, known as Optimal Power Flow (OPF), which is non-linear and non-convex. We propose a methodology to convexify the OPF problem based on Semidefinite Programming (SDP), with a novel formulation to include discrete decision variables such as shunt elements and tap changers positions as discrete variables. Finally, the SDP formulation of the case of a DC network is included.

Inference in hybrid Bayesian networks with large discrete and continuous domains

An improved mixed-integer particle swarm optimization (MI-PSO) algorithm is presented, which is called asynchronous updating particle swarm optimization (AU-PSO). The proposed algorithm speeds up convergence rate of PSO by handling discrete and continuous decision variables differently. The discrete variables have a possibility holding previous values, while the continuous variables are treated normally. This kind of treatment provides more chances to find a better solution. The proposed AU-PSO is then examined by test functions and successfully applied in a design of circular antenna array.

Engineering design problems often involve global optimization of functions that are supplied as ’black box‘ functions. These functions may be nonconvex, nondifferentiable and even discontinuous. In addition, the decision variables may be a combination of discrete and continuous variables. The functions are usually computationally expensive, and may involve finite element methods. An engineering example of this type of problem is to minimize the weight of a structure, while limiting strain to be below a certain threshold. This type of global optimization problem is very difficult to solve, yet design engineers must find some solution to their problem – even if it is a suboptimal one. Sometimes the most difficult part of the problem is finding any feasible solution. Stochastic methods, including sequential random search and simulated annealing, are finding many applications to this type of practical global optimization problem. Improving Hit-and-Run (IHR) is a sequential random search method that has been successfully used in several engineering design applications, such as the optimal design of composite structures. A motivation to IHR is discussed as well as several enhancements. The enhancements include allowing both continuous and discrete variables in the problem formulation. This has many practical advantages, because design variables often involve a mixture of continuous and discrete values. IHR and several variations have been applied to the composites design problem. Some of this practical experience is discussed.

If the row and column categories of a contingency table sequentially seen, as objects and variables, then they are object data with discrete variables. Often from objects obtained additional data in the form of continuous variables. Based on these discrete and continuous variables, the more accurate method is necessary to analyse the associations between these variables. Treating discrete data as continuous is wrong, so this article aims to analyse the association of data in the form of / x 2 contingency tables with additional data that is continuous. From the data in the form of / x 2 contingency table, it converted using the simplification of correspondence analysis (SoCA), so that continuous principal coordinates obtained. Furthermore, the association between continuous variables was analysed using the cosine value of the angle between the two vectors. Case studies use poverty data in Indonesia, which published by the Central Statistics Agency (BPS-Statistics Indonesia). Data in the form of contingency tables are people population lived in poverty based on province and area of residence (urban or rural). Additional variables are poverty depth index, severity index, Gini ratio, food poverty line and non-food poverty line. The results of the analysis obtained information that in urban areas tend to have high Gini ratio, food poverty lines and non-food poverty lines, for rural areas tend to have a high poverty depth and severity index.

Host area, potential pest impact and probability of pest presence are frequently displayed on maps by pest risk assessors. These variables can be mapped separately, but it is also important to map combinations of these variables in order to define the area of potential establishment and the endangered area to assist decision‐making processes. This paper presents different methods for combining maps, and discusses their advantages and disadvantages. Different methods are shown that can be used to combine maps depending on whether the individual maps were derived from continuous quantitative variables or from discrete variables. The authors suggest combining maps derived from continuous variables using simple mathematical equations in order to compute expected invaded areas and expected potential impacts. Maps derived from discrete variables (e.g. scores) can be combined using a risk matrix, but the results may be highly dependent on the chosen matrix. The practical interest of these methods is illustrated in a case study on Diabrotica virgifera virgifera. The authors recommend combining the original continuous variables when such variables are available. The combination of categories defined from continuous variables led to a loss of information and may decrease the values of the maps. Risk matrices should be used only if the individual variables are discrete and if the underlying continuous variables are not available.

A method for optimal design of automotive body assembly using multi-material construction

The Model Predictive Control (MPC) framework has shown great potential for the control of Variable Speed Limits (VSLs) and Ramp Metering (RM) installations. However, the implementation to large freeway networks remains challenging. One major reason is that, by considering the VSLs to be discrete decision variables, an extremely difficult Mixed Integer Nonlinear Programming (MINLP) optimization problem has to be solved within every controller sampling interval. Consequently, many related papers relax the MINLP problems by considering the VSLs to be continuous variables. This paper proposes two novel MPC algorithms for coordinated control of discrete VSLs and continuous RM rates that do not make this relaxation. The proposed algorithms use a distributed control architecture and an alternating optimization scheme to relax the MINLP optimization problems but still consider the VSLs as discrete variables and, hence, offer a trade-off between computational complexity and system performance. The performance of the proposed algorithms is evaluated in a case study. The case study shows that relaxing the VSLs to be continuous variables with a distributed architecture results in a significant performance loss. Furthermore, both proposed algorithms have a lower computational complexity than the more conventional centralized approach and, as a result, they do manage to solve all optimization problems within the sampling intervals. Moreover, one of the proposed algorithms has a system performance that is remarkably similar to the optimal performance of the centralized approach.

Quantum teleportation, a transfer protocol of quantum states, is the essence of many sophisticated quantum information protocols. There have been two complementary approaches to optical quantum teleportation: discrete variables (DVs) and continuous variables (CVs). However, both approaches have pros and cons. Here we take a “hybrid” approach to overcome the current limitations: CV quantum teleportation of DVs. This approach enabled the first realization of deterministic quantum teleportation of photonic qubits without post-selection. We also applied the hybrid scheme to several experiments, including entanglement swapping between DVs and CVs, conditional CV teleportation of single photons, and CV teleportation of qutrits. We are now aiming at universal, scalable, and fault-tolerant quantum computing based on these hybrid technologies.

A numerical solution procedure for optimizing the thermal-mechanical reliability of an IGBT module has been described in this paper. The procedure is robust and it can handle both discrete and continuous decision variables in the design of IGBT packaging. An example has been given to demonstrate its application. The objective functions in this example are the accumulated plastic work density in solder joints of a simplified IGBT module that is subject to cyclic temperature cycling and the maximum junction temperature when the model is subject to a constant thermal loading. The thickness of the chip mount-down solder, thickness of the substrate solder joint, and the location of the chip are used as continuous decision variables. The material selection for the baseplate is treated as a discrete variable. The objective function values are calculated using Finite Element Analysis method.