Mixed Continuous-discrete Variables Research Articles

Semiparametric Bayesian multiple imputation for regression models with missing mixed continuous–discrete covariates

Issues regarding missing data are critical in observational and experimental research. Recently, for datasets with mixed continuous–discrete variables, multiple imputation by chained equation (MICE) has been widely used, although MICE may yield severely biased estimates. We propose a new semiparametric Bayes multiple imputation approach that can deal with continuous and discrete variables. This enables us to overcome the shortcomings of MICE; they must satisfy strong conditions (known as compatibility) to guarantee obtained estimators are consistent. Our simulation studies show the coverage probability of 95% interval calculated using MICE can be less than 1%, while the MSE of the proposed can be less than one-fiftieth. We applied our method to the Alzheimer’s Disease Neuroimaging Initiative (ADNI) dataset, and the results are consistent with those of the previous works that used panel data other than ADNI database, whereas the existing methods, such as MICE, resulted in inconsistent results.

A cooperative negotiation embedded NSGA-II for solving an integrated product family and supply chain design problem with remanufacturing consideration

Product family design is a popular approach adopted by manufacturers to increase their product varieties in order to satisfy the needs of various markets. In recent years, because of increasing environmental concerns in societies and strict regulations of environmental protection, quite a number of manufacturers adopted remanufacturing strategy in their product development in response to the challenges. Remanufacturing of used products unavoidably involves a closed-loop supply chain system. To achieve the best outcomes, the supply chain design should be considered in product family design process. In this research, a multi-objective optimization model of integrated product family and closed loop supply chain design is formulated based on a cooperative game model for minimizing manufacturer’s total cost and maximize suppliers’ total payoffs. Since the optimization problem could be a large- scale one and involves mixed continuous-discrete variables, a new version of nondominated sorting genetic algorithm-II (NSGA-II), namely cooperative negotiation embedded NSGA-II (NSGA-CO), is proposed to solve the optimization model. Simulation tests are conducted to validate the effectiveness of the proposed NSGA-CO. The test results indicate that the proposed NSGA-CO outperforms NSGA-II in solving various scale of multi-objective optimization problems in terms of convergence. With the formulated optimization model and the proposed NSGA-CO, a case study of integrated product family and supply chain design is conducted to investigate the effects of environmental penalty, quantity of demand and marginal cost of remanufacturing on used product return rate, manufacturers’ and suppliers’ profits and joint payoff.

Bayesian Dependence Tests for Continuous, Binary and Mixed Continuous-Binary Variables

Tests for dependence of continuous, discrete and mixed continuous-discrete variables are ubiquitous in science. The goal of this paper is to derive Bayesian alternatives to frequentist null hypothesis significance tests for dependence. In particular, we will present three Bayesian tests for dependence of binary, continuous and mixed variables. These tests are nonparametric and based on the Dirichlet Process, which allows us to use the same prior model for all of them. Therefore, the tests are “consistent” among each other, in the sense that the probabilities that variables are dependent computed with these tests are commensurable across the different types of variables being tested. By means of simulations with artificial data, we show the effectiveness of the new tests.

Multiobjective Optimization of Steering Mechanism for Rotary Steering System Using Modified NSGA-II and Fuzzy Set Theory

Due to the complicated design process of gear train, optimization is a significant approach to improve design efficiency. However, the design of gear train is a complex multiobjective optimization with mixed continuous-discrete variables under numerous nonlinear constraints, and conventional optimization algorithms are not suitable to deal with such optimization problems. In this paper, based on the established dynamic model of steering mechanism for rotary steering system, the key component of which is a planetary gear set with teeth number difference, the optimization problem of steering mechanism is formulated to achieve minimum dynamic responses and outer diameter by optimizing structural parameters under geometric, kinematic, and strength constraints. An optimization procedure based on modified NSGA-II by incorporating dynamic crowding distance strategies and fuzzy set theory is applied to the multiobjective optimization. For comparative purpose, NSGA-II is also employed to obtain Pareto optimal set, and dynamic responses of original and optimized designs are compared. The results show the optimized design has better dynamic responses with minimum outer diameter and the response decay decreases faster. The optimization procedure is feasible to the design of gear train, and this study can provide guidance for designer at the preliminary design phase of mechanical structures with gear train.

On the Performance of the PSP Method for Mixed-Variable Multi-Objective Design Optimization

Practical design optimization problems require use of computationally expensive “black-box” functions. The Pareto set pursuing (PSP) method, for solving multi-objective optimization problems with expensive black-box functions, was originally developed for continuous variables. In this paper, modifications are made to allow solution of problems with mixed continuous-discrete variables. A performance comparison strategy for nongradient-based multi-objective algorithms is discussed based on algorithm efficiency, robustness, and closeness to the true Pareto front with a limited number of function evaluations. Results using several methods, along with the modified PSP, are given for a suite of benchmark problems and two engineering design ones. The modified PSP is found to be competitive when the total number of function evaluations is limited, but faces an increased computational challenge when the number of design variables increases.

Maximum entropy inference for mixed continuous-discrete variables

We represent knowledge by probability distributions of mixed continuous and discrete variables. From the joint distribution of all items, one can compute arbitrary conditional distributions, which may be used for prediction. However, in many cases only some marginal distributions, inverse probabilities, or moments are known. Under these conditions, a principle is needed to determine the full joint distribution of all variables. The principle of maximum entropy (Jaynes, Phys Rev 1957;106:620–630 and 1957;108:171–190; Jaynes, Probability Theory—The Logic of Science, Cambridge, UK: Cambridge University Press, 2003; Haken, Synergetics, Berlin: Springer-Verlag, 1977; Guiasu and Shenitzer, Math Intell 1985;117:83–106) ensures an unbiased estimation of the full multivariate relationships by using only known facts. For the case of discrete variables, the expert shell SPIRIT implements this approach (cf. Rödder, Artif Intell 2000;117:83–106; Rödder and Meyer, in Proceedings of the 12th Conference on Uncertainty in Artificial Intelligence, San Francisco, CA, 2006; Rödder et al., Logical J IGPL 2006;14(3):483–500). In this paper, the approach is generalized to continuous and mixed continuous-discrete distributions and applied to the problem of credit scoring. © 2010 Wiley Periodicals, Inc.

Multiobjective Collaborative Robust Optimization With Interval Uncertainty and Interdisciplinary Uncertainty Propagation

We present a new solution approach for multidisciplinary design optimization (MDO) problems that, for the first time in literature, has all of the following characteristics: Each discipline has multiple objectives and constraints with mixed continuous-discrete variables; uncertainty exists in parameters and as a result, uncertainty propagation exists within and across disciplines; probability distributions of uncertain parameters are not available but their interval of uncertainty is known; and disciplines can be fully (two-way) coupled. The proposed multiobjective collaborative robust optimization (McRO) approach uses a multiobjective genetic algorithm as an optimizer. McRO obtains solutions that are as best as possible in a multiobjective and multidisciplinary sense. Moreover, for McRO solutions, the variation of objective and/or constraint functions can be kept within an acceptable range. McRO includes a technique for interdisciplinary uncertainty propagation. The approach can be used for robust optimization of MDO problems with multiple objectives, or constraints, or both together at system and subsystem levels. Results from an application of McRO to a numerical and an engineering example are presented. It is concluded that McRO can solve fully coupled MDO problems with interval uncertainty and obtain solutions that are comparable to a single-disciplinary robust optimization approach.

A Maximum Entropy Based Approach to Fault Diagnosis Using Discrete and Continuous Features

This paper presents a new maximum entropy (ME) based hybrid inference engine to improve the accuracy of diagnostic decisions using mixed continuous-discrete variables. By fusing the complementary fault information provided by discrete and continuous fault features, false alarms due to misclassification and modeling uncertainty can be significantly reduced. Simulation results using a three-tank benchmark system have clearly illustrated the advantages of diagnostics based on mixed continuous-discrete variables. Moreover, in the presence of significant measurement noise, simulation results show that the proposed ME method achieves better performance than the support vector machine classifier.

Optimum design of trusses with discrete sizing and shape variables

The objective here is to present a method for optimizing truss structures with discrete design variables. The design variables are considered to be sizing variables as well as coordinates of joints. Both types of variables can be discrete simultaneously. Mixed continuous-discrete variables can also be considered. To increase the efficiency of the method, the structural responses, such as forces and displacements are approximated in each design cycle. The approximation of responses is carried out with respect to the design variables or their reciprocals. By substituting these approximate functions of the responses into the original design problem, an explicit high quality approximation is achieved, the solution of which does not require the detailed finite element analysis of the structure in each sub-optimization iteration. First it is assumed that all the design variables are continuous and a continuous variable optimization is performed. With the results of this step, the branch and bound method is employed on the same approximate problem to achieve a discrete solution. The numerical results indicate that the method is efficient and robust in terms of the required number of structural analyses. Several examples are presented to show the efficiency of the method.