A new approach for data partitioning through high dimensional model representation

A multivariate function f(x ₁, …, x _N ) can be evaluated via interpolation if its values are given at a finite number nodes of a hyperprismatic grid in the space of independent variables x ₁, x ₂, …, x _N . Interpolation is a way to characterize an infinite data structure (function) by a finite number of data approximately. Hence it leaves an infinite arbitrariness unless a mathematical structure with finite number of flexibilities is imposed for the unknown function. Imposed structure has finite dimensionality. When the dimensionality increases unboundedly, the complexities grow rapidly in the standard methods. The main purpose here is to partition the given multivariate data into a set of low-variate data by using high dimensional model representation (HDMR) and then, to interpolate each individual data in the set via Lagrange interpolation formula. As a result, computational complexity of the given problem and needed CPU time to obtain the results through a series of programs in computers decrease.     

Finding better weight functions for generalized Shepard's operator on infinite intervals

In this paper, we investigate the difference of Shepard's generalized operators S _σ from the approximated set of data for various weight functions σ. Bounds are given for the sizes of the ‘bumps’ shown on the graph of S _σ for σ(d)=1/d in dimension N=1, and the best weight function σ for practical use is proposed.     

Some Systematic Supersaturated Designs

Supersaturated designs are factorial designs in which the number of factors exceeds the number of observations. Satterthwaite has suggested constructing such designs by a randomization procedure (random balance designs). In the present paper systematic designs are constructed which are, in a certain sense, m nearly orthogonal M possible. Examples of such designs are given constructed on an electronic computer and some properties of the designs are discussed.     

Determination of bone mineral volume fraction using impedance analysis and Bruggeman model

Measurements by impedance spectroscopy and Bruggeman effective medium approximation model were employed in order to determine the mineral volume fraction of dry bone. This approach assumes that two or more phases are present into the composite: the matrix (environment) and the other ones are inclusion phases. A fragment of femur diaphysis dense bone from a young pig was investigated in its dehydrated state. Measuring the dielectric properties of bone and its main components (hydroxyapatite and collagen) and using the Bruggeman approach, the mineral volume filling factor was determined. The computed volume fraction of the mineral volume fraction was confirmed by a histogram test analysis based on the SEM microstructures. In spite of its simplicity, the method provides a good approximation for the bone mineral volume fraction. The method which uses impedance spectroscopy and EMA modeling can be further developed by considering the conductive components of the bone tissue as a non-invasive in situ impedance technique for bone composition evaluation and monitoring.     

Least squares approximated stability boundaries of milling process

First and second order least squares methods are used in generating simple approximation polynomials for the state term of the model for regenerative chatter in the milling process. The least squares approximation of delayed state term and periodic term of the model does not go beyond first order. The resulting discrete maps are demonstrated to have same convergence rate as the discrete maps in other works that are based on the interpolation theory. The presented discrete maps are illustrated to be beneficial in terms of computational time (CT) savings that derive from reduction in the number of calculation needed for generation system monodromy matrix. This benefit is so much that computational time of second order least squares-based discrete map is noticeably shorter than that of first order interpolation-based discrete map. It is expected from analysis then verified numerically that savings in CT due to use of least squares theory relative to use of interpolation theory of same order rises with rise in order of approximation. The experimentally determined model parameters used for numerical calculations are extracted from literature.     

Approximation analysis of multi-class closed queueing maintenance networks with a parts inventory system and two-phase Coxian time distributions

We consider a maintenance network where a set of bases is supported by a replacement parts inventory system and a centrally located repair depot. The ordering policy for the parts is the (S, Q) inventory policy. We extended the previous results to the network, where processing times at each node follow a two-phase Coxian distribution. The proposed network was modeled as a multi-class closed queueing network with a synchronization station. To make the analysis of the network computationally tractable, we developed a two-phase approximation method. In the first phase of the method, the proposed network was analyzed with the previous algorithm based on a product-form approximation. In the second phase, a sub-network was again analyzed with the procedure of a product-form approximation method such that the state space of the sub-network was reduced. In the analysis of a sub-network, a recursive method was also used to solve balance equations by exploiting the special structure of the Markov chain. The new algorithm provided a good estimation of the performance measures of interest. In addition to being accurate, the new algorithm is simple and converges rapidly.     

A simulation‐based algorithm for optimal pricing policy under demand uncertainty

We propose a simulation‐based algorithm for computing the optimal pricing policy for a product under uncertain demand dynamics. We consider a parameterized stochastic differential equation (SDE) model for the uncertain demand dynamics of the product over the planning horizon. In particular, we consider a dynamic model that is an extension of the Bass model. The performance of our algorithm is compared to that of a myopic pricing policy and is shown to give better results. Two significant advantages with our algorithm are as follows: (a) it does not require information on the system model parameters if the SDE system state is known via either a simulation device or real data, and (b) as it works efficiently even for high‐dimensional parameters, it uses the efficient smoothed functional gradient estimator.     

Heat effects due to mixing (dilution) the mixed acid solutions—Application of neural networks to approximate and generalize experimental data

The feed-forward neural networks have been used to approximate the specific molar enthalpy and the specific molar heat capacity of the mixed acid solutions. The nets have been trained with experimental data taken from the literature, so the values of the specific molar enthalpy and the specific molar heat capacity at the reference temperature T = 0 °C could be successively estimated for any composition of the mixed acid. Two principal methods have been considered and tested. In the first method two independent neural nets have been employed: the net NN-H, which approximates separately the specific molar enthalpy and the net NN-C, to approximate the specific molar heat capacity, respectively. In the second method only one net is employed (the net NN-HC), which simultaneously approximates both the specific molar enthalpy and the specific molar heat capacity. Then following both mentioned methods, the trained neural nets have been used to model the heat effects due to dilution of mixed acid solutions, carried out at various conditions – i.e. at any temperature and composition. Using these nets, both, the integral and the differential enthalpy balance can be carried out, so the smart and accurate method to model the mixed acid dilution has been elaborated. The proposed methods and their prediction accuracy have been successfully verified with our own experimental data carried out in the RC1 reaction calorimeter.     

Disjoint convex shell and its applications in mesh unfolding

In this work, we study a geometric structure called disjoint convex shell or simply DC-shell. A DC-shell of a polyhedron is a set of pairwise interior disjoint convex objects that collectively approximate the given polyhedron. Preventing convex objects from overlapping enables faster and robust collision response and more realistic fracturing simulation. Without the disjointness constraint, a physical realization of the approximation becomes impossible. This paper investigates multiple approaches that construct DC-shells from shapes that are either composed of overlapping components or segmented into parts. We show theoretically that, even under this rather simplified setting, constructing DC-shell is difficult.To demonstrate the power of DC-shell, we studied how DC-shell can be used in mesh unfolding, an important computational method in manufacturing 3D shape from the 2D material. Approximating a given polyhedron model by DC-shells provides two major benefits. First, they are much easier to unfold using the existing unfolding methods. Second, they can be folded easily by both human folder or self-folding machines. Consequently, DC-shell makes paper craft creation and design more accessible to younger children and provides chances to enrich their education experiences.