A modified PNN algorithm with optimal PD modeling using the orthogonal least squares method

In this paper a modified algorithm is suggested for developing polynomial neural network (PNN) models. Optimal partial description (PD) modeling is introduced at each layer of the PNN expansion, a task accomplished using the orthogonal least squares (OLS) method. Based on the initial PD models determined by the polynomial order and the number of PD inputs, OLS selects the most significant regressor terms reducing the output error variance. The method produces PNN models exhibiting a high level of accuracy and superior generalization capabilities. Additionally, parsimonious models are obtained comprising a considerably smaller number of parameters compared to the ones generated by means of the conventional PNN algorithm. Three benchmark examples are elaborated, including modeling of the gas furnace process as well as the iris and wine classification problems. Extensive simulation results and comparison with other methods in the literature, demonstrate the effectiveness of the suggested modeling approach.

Application of a Bayesian/Generalised Least-Squares method to generate correlations between independent neutron fission yield data

Fission product yields are fundamental parameters for several nuclear engineering calculations and in particular for burn-up/activation problems. The impact of their uncertainties was widely studied in the past and valuations were released, although still incomplete. Recently, the nuclear community expressed the need for full fission yield covariance matrices to produce inventory calculation results that take into account the complete uncertainty data. In this work, we studied and applied a Bayesian/generalised least-squares method for covariance generation, and compared the generated uncertainties to the original data stored in the JEFF-3.1.2 library. Then, we focused on the effect of fission yield covariance information on fission pulse decay heat results for thermal fission of 235U. Calculations were carried out using different codes (ACAB and ALEPH-2) after introducing the new covariance values. Results were compared with those obtained with the uncertainty data currently provided by the library. The uncertainty quantification was performed with the Monte Carlo sampling technique. Indeed, correlations between fission yields strongly affect the statistics of decay heat. Introduction Nowadays, any engineering calculation performed in the nuclear field should be accompanied by an uncertainty analysis. In such an analysis, different sources of uncertainties are taken into account. Works such as those performed under the UAM project (Ivanov, et al., 2013) treat nuclear data as a source of uncertainty, in particular cross-section data for which uncertainties given in the form of covariance matrices are already provided in the major nuclear data libraries. Meanwhile, fission yield uncertainties were often neglected or treated shallowly, because their effects were considered of second order compared to cross-sections (Garcia-Herranz, et al., 2010). However, the Working Party on International Nuclear Data Evaluation Co-operation (WPEC)

Solving the Least Squares Method problem in the AHP for 3 X 3 and 4 X 4 matrices

The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM) are of the possible tools for computing the priorities of the alternatives. A method for generating all the solutions of the LSM problem for 3 × 3 and 4 × 4 matrices is discussed in the paper. Our algorithms are based on the theory of resultants.

Solution of the Least Squares Method problem of pairwise comparison matrices

The aim of the paper is to present a new global optimization method for determining all the optima of the Least Squares Method (LSM) problem of pairwise comparison matrices. Such matrices are used, e.g., in the Analytic Hierarchy Process (AHP). Unlike some other distance minimizing methods, LSM is usually hard to solve because of the corresponding nonlinear and non-convex objective function. It is found that the optimization problem can be reduced to solve a system of polynomial equations. Homotopy method is applied which is an efficient technique for solving nonlinear systems. The paper ends by two numerical example having multiple global and local minima.

Parameter estimation in water-distribution systems by least squares

The weighted-least-squares method using sensitivity-analysis technique is proposed for the estimation of parameters in water-distribution systems. The parameters considered are the Hazen-Williams coefficients for the pipes. The objective function used is the sum of the weighted squares of the differences between the computed and the observed values of the variables. The weighted-least-squares method can elegantly handle multiple loading conditions with mixed types of measurements such as heads and consumptions, different sets and number of measurements for each loading condition, and modifications in the network configuration due to inclusion or exclusion of some pipes affected by valve operations in each loading condition. Uncertainty in parameter estimates can also be obtained. The method is applied for the estimation of parameters in a metropolitan urban water-distribution system in India.

Weighted least squares with expectation-maximization algorithm for burst detection in U.K. water distribution systems

Flow measurement data at the district meter area (DMA) level has the potential for burst detection in the water distribution systems. This work investigates using a polynomial function fitted to the historic flow measurements based on a weighted least-squares method for automatic burst detection in the U.K. water distribution networks. This approach, when used in conjunction with an expectationmaximization (EM) algorithm, can automatically select useful data from the historic flow measurements, which may contain normal and abnormal operating conditions in the distribution network, e.g., water burst. Thus, the model can estimate the normal water flow (nonburst condition), and hence the burst size on the water distribution system can be calculated from the difference between the measured flow and the estimated flow. The distinguishing feature of this method is that the burst detection is fully unsupervised, and the burst events that have occurred in the historic data do not affect the procedure and bias the burst detection algorithm. Experimental validation of the method has been carried out using a series of flushing events that simulate burst conditions to confirm that the simulated burst sizes are capable of being estimated correctly. This method was also applied to eight DMAs with known real burst events, and the results of burst detections are shown to relate to the water company's records of pipeline reparation work. © 2014 American Society of Civil Engineers.

On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

Simple orbit determination using GPS based on a least-squares algorithm employing sequential givens rotations

A low-cost computer procedure to determine the orbit of an artificial satellite by using short arc data from an onboard GPS receiver is proposed. Pseudoranges are used as measurements to estimate the orbit via recursive least squares method. The algorithm applies orthogonal Givens rotations for solving recursive and sequential orbit determination problems. To assess the procedure, it was applied to the TOPEX/POSEIDON satellite for data batches of one orbital period (approximately two hours), and force modelling, due to the full JGM-2 gravity field model, was considered. When compared with the reference Precision Orbit Ephemeris (POE) of JPL/NASA, the results have indicated that precision better than 9 m is easily obtained, even when short batches of data are used. Copyright (c) 2007.

A brief look at the least-squares approach as a classifier applied to restricted-vocabulary speech recognition

In this letter, a speech recognition algorithm based on the least-squares method is presented. Particularly, the intention is to exemplify how such a traditional numerical technique can be applied to solve a signal processing problem that is usually treated by using more elaborated formulations.

A simplified implementation of the least squares solution for pairwise comparisons matrices

This is a follow up to "Solution of the least squares method problem of pairwise comparisons matrix" by Bozóki published by this journal in 2008. Familiarity with this paper is essential and assumed. For lower inconsistency and decreased accuracy, our proposed solutions run in seconds instead of days. As such, they may be useful for researchers willing to use the least squares method (LSM) instead of the geometric means (GM) method.

A least squares based finite volume method for the Cahn-Hilliard and Cahn-Hilliard-reaction equations

A vertex-centred finite volume method (FVM) for the Cahn-Hilliard (CH) and recently proposed Cahn-Hilliard-reaction (CHR) equations is presented. Information at control volume faces is computed using a high-order least-squares approach based on Taylor series approximations. This least-squares problem explicitly includes the variational boundary condition (VBC) that ensures that the discrete equations satisfy all of the boundary conditions. We use this approach to solve the CH and CHR equations in one and two dimensions and show that our scheme satisfies the VBC to at least second order. For the CH equation we show evidence of conservative, gradient stable solutions, however for the CHR equation, strict gradient-stability is more challenging to achieve.