Local bilinear multiple-output quantile/depth regression

A new quantile regression concept, based on a directional version of Koenker and Bassett's traditional single-output one, has been introduced in [Ann. Statist. (2010) 38 635-669] for multiple-output location/linear regression problems. The polyhedral contours provided by the empirical counterpart of that concept, however, cannot adapt to unknown nonlinear and/or heteroskedastic dependencies. This paper therefore introduces local constant and local linear (actually, bilinear) versions of those contours, which both allow to asymptotically recover the conditional halfspace depth contours that completely characterize the response's conditional distributions. Bahadur representation and asymptotic normality results are established. Illustrations are provided both on simulated and real data.Comment: Published at http://dx.doi.org/10.3150/14-BEJ610 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Support Vector Regression Based S-transform for Prediction of Single and Multiple Power Quality Disturbances

This paper presents a novel approach using Support Vector Regression (SVR) based S-transform to predict the classes of single and multiple power quality disturbances in a three-phase industrial power system. Most of the power quality disturbances recorded in an industrial power system are non-stationary and comprise of multiple power quality disturbances that coexist together for only a short duration in time due to the contribution of the network impedances and types of customers’ connected loads. The ability to detect and predict all the types of power quality disturbances encrypted in a voltage signal is vital in the analyses on the causes of the power quality disturbances and in the identification of incipient fault in the networks. In this paper, the performances of two types of SVR based S-transform, the non-linear radial basis function (RBF) SVR based S-transform and the multilayer perceptron (MLP) SVR based S-transform, were compared for their abilities in making prediction for the classes of single and multiple power quality disturbances. The results for the analyses of 651 numbers of single and multiple voltage disturbances gave prediction accuracies of 86.1% (MLP SVR) and 93.9% (RBF SVR) respectively. Keywords: Power Quality, Power Quality Prediction, S-transform, SVM, SV

Data clustering for circle detection

This paper considers a multiple-circle detection problem on the basis of given data. The problem is solved by application of the center-based clustering method. For the purpose of searching for a locally optimal partition modeled on the well-known k-means algorithm, the k-closest circles algorithm has been constructed. The method has been illustrated by several numerical examples

Orthogonal weighted linear L1 and L∞ approximation and applications

AbstractLet S={s1,s2,...,sn} be a set of sites in Ed, where every site si has a positive real weight ωi. This paper gives algorithms to find weighted orthogonal L∞ and L1 approximating hyperplanes for S. The algorithm for the weighted orthogonal L1 approximation is shown to require O(nd) worst-case time and O(n) space for d ≥ 2. The algorithm for the weighted orthogonal L∞ approximation is shown to require O(n log n) worst-case time and O(n) space for d = 2, and O(n⌊dl2 + 1⌋) worst-case time and O(n⌊(d+1)/2⌋) space for d > 2. In the latter case, the expected time complexity may be reduced to O(n⌊(d+1)/2⌋). The L∞ approximation algorithm can be modified to solve the problem of finding the width of a set of n points in Ed, and the problem of finding a stabbing hyperplane for a set of n hyperspheres in Ed with varying radii. The time and space complexities of the width and stabbing algorithms are seen to be the same as those of the L∞ approximation algorithm

Data clustering for circle detection

This paper considers a multiple-circle detection problem on the basis of given data. The problem is solved by application of the center-based clustering method. For the purpose of searching for a locally optimal partition modeled on the well-known k-means algorithm, the k-closest circles algorithm has been constructed. The method has been illustrated by several numerical examples

Fitting an Equation to Data Impartially

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY), https://creativecommons.org/licenses/by/4.0/We consider the problem of fitting a relationship (e.g., a potential scientific law) to data involving multiple variables. Ordinary (least squares) regression is not suitable for this because the estimated relationship will differ according to which variable is chosen as being dependent, and the dependent variable is unrealistically assumed to be the only variable which has any measurement error (noise). We present a very general method for estimating a linear functional relationship between multiple noisy variables, which are treated impartially, i.e., no distinction between dependent and independent variables. The data are not assumed to follow any distribution, but all variables are treated as being equally reliable. Our approach extends the geometric mean functional relationship to multiple dimensions. This is especially useful with variables measured in different units, as it is naturally scale invariant, whereas orthogonal regression is not. This is because our approach is not based on minimizing distances, but on the symmetric concept of correlation. The estimated coefficients are easily obtained from the covariances or correlations, and correspond to geometric means of associated least squares coefficients. The ease of calculation will hopefully allow widespread application of impartial fitting to estimate relationships in a neutral way.Peer reviewe