Generalized Convexity Properties and Shape-Based Approximation in Networks Reliability

Some properties of generalized convexity for sets and functions are identified in case of the reliability polynomials of two dual minimal networks. A method of approximating the reliability polynomials of two dual minimal network is developed based on their mutual complementarity properties. The approximating objects are from the class of quadratic spline functions, constructed based on both interpolation conditions and shape knowledge. It is proved that the approximant objects preserve both the high-order convexity and some extremum properties of the exact reliability polynomials. It leads to pointing out the area of the network where the maximum number of paths is achieved. Numerical examples and simulations show the performance of the algorithm, both in terms of low complexity, small error and shape preserving. Possibilities of increasing the accuracy of approximation are discussed.

Statistical Inference on Nonparametric Spline Models and its Applications

One of the components of the Human Development Index which is still a problem and concern in the world today is the Life Expectancy Rate (LER). United Nations Development Program (UNDP). United Nations Development Program (UNDP) uses the LER to measure community health status as well as a benchmark for development success. LER in Indonesia continues to increase almost throughout the year. That is, the hope of a newborn baby to be able to live longer is getting higher. LER data modelling with parametric regression is not necessarily suitable to be applied because the LER relationship pattern has a pattern that varies at certain age intervals. Spline regression is a regression method that can handle data whose pattern changes at certain intervals. Spline is one of the models in nonparametric regression that has a very special and very good visual statistical interpretation. In addition, splines are also able to handle data characters or functions that are smooth (smooth). This study aims to derive the form of the estimator and the shortest confidence interval for the quadratic spline model and model the LER data in Indonesia.

Implementation and Performance Evaluation of a Bivariate Cut-HDMR Metamodel for Semiconductor Packaging Design Problems with a Large Number of Input Variables

A metamodeling technique based on Bivariate Cut High Dimensional Model Representation (Bivariate Cut HDMR) is implemented for a semiconductor packaging design problem with 10 design variables. Bivariate Cut-HDMR constructs a metamodel by considering only up to second-order interactions. The implementation uses three uniformly distributed sample points (s = 3) with quadratic spline interpolation to construct the component functions of Bivariate Cut-HDMR, which can be used to make a direct comparison with a metamodel based on Central Composite Design (CCD). The performance of Bivariate Cut-HDMR is evaluated by two well-known error metrics: R-squared and Relative Average Absolute Error (RAAE). The results are compared with the performance of CCD. Bivariate Cut HDMR does not compromise the accuracy compared to CCD, although the former uses only one-fifth of sample points (201 sample points) required by the latter (1045 sample points). The sampling schemes and the predictions of cut-planes and boundary-planes are discussed to explain possible reasons for the outstanding performance of Bivariate Cut HDMR.

Approximate Solution of Volterra Integro-Fractional Differential Equations Using Quadratic Spline Function

In this paper, we suggest two new methods for approximating the solution to the Volterra integro-fractional differential equation (VIFDEs), based on the normal quadratic spline function and the second method used the Richardson Extrapolation technique the usage of discrete collocation points. The fractional derivatives are regarded in the Caputo perception. A new theorem for the Richardson Extrapolation points for using the finite difference approximation of Caputo derivative is introduced with their proof. New techniques using the first derivative at the initial point such that obtained by follow two cases the first using trapezoidal rule and the second using the first step of linear spline function using the Richardson Extrapolation method. Specifically, the program is given in examples analysis in Matlab (R2018b). Numerical examples are available to illuminate the productivity and trustworthiness of the methods, as well as, follow the Clenshaw Curtis rule for calculating the required integrals for those equations.