On approximating the distribution of indefinite quadratic expressions in singular normal vectors

General representations of quadratic forms and quadratic expressions in singular normal vectors are given in terms of the difference of two positive definite quadratic forms and an independentlydistributed linear combination of normal random variables. Up to now, only special cases have been treated in the statistical literature. The densities of the quadratic forms are then approximated with gamma and generalized gamma density functions. A moment-based technique whereby the initial approximations are adjusted by means of polynomials is presented. Closed form and integral formulae are provided for the approximate density functions of the quadratic forms and quadratic expressions. A detailed step-by-step algorithm for implementing the proposed density approximation technique is also provided. Two numerical examples illustrate the methodology

Polynomial expansions via embedded Pascal’s triangles

An expansion is given for polynomials of the form (ω + λ1) · · ·(ω +λn). The coefficients of the resulting polynomials are related to their roots, and a system of equations that enables one to numericallydetermine the roots in terms of the coefficients is specified. The case where all the roots λi are equal is considered as well. A multinomial extension to polynomials of the form (x1+ · · · + xI )n is then provided. As it turns out, the coefficients of the monomials contained in the resulting polynomial expansion can be determined in terms of the coefficients of the monomials included in the expansion of (x1+ · · · + xI-1 )n  and the rows of embedded Pascal’s triangles of successive orders. An algorithm is provided for generating and concatenating these rows, with the particulars of its implementation by means of the symbolic computation software Mathematica being discussed as well. Potential applications of such expansions to combinatorics and genomics are also suggested

Generalized r-Modes of the Maclaurin Spheroids

Analytical solutions are presented for a class of generalized r-modes of rigidly rotating uniform density stars---the Maclaurin spheroids---with arbitrary values of the angular velocity. Our analysis is based on the work of Bryan; however, we derive the solutions using slightly different coordinates that give purely real representations of the r-modes. The class of generalized r-modes is much larger than the previously studied `classical' r-modes. In particular, for each l and m we find l-m (or l-1 for the m=0 case) distinct r-modes. Many of these previously unstudied r-modes (about 30% of those examined) are subject to a secular instability driven by gravitational radiation. The eigenfunctions of the `classical' r-modes, the l=m+1 case here, are found to have particularly simple analytical representations. These r-modes provide an interesting mathematical example of solutions to a hyperbolic eigenvalue problem.Comment: 12 pages, 3 figures; minor changes and additions as will appear in the version to be published in Physical Review D, January 199

Asymptotic and measured large frequency separations

With the space-borne missions CoRoT and Kepler, a large amount of asteroseismic data is now available. So-called global oscillation parameters are inferred to characterize the large sets of stars, to perform ensemble asteroseismology, and to derive scaling relations. The mean large separation is such a key parameter. It is therefore crucial to measure it with the highest accuracy. As the conditions of measurement of the large separation do not coincide with its theoretical definition, we revisit the asymptotic expressions used for analysing the observed oscillation spectra. Then, we examine the consequence of the difference between the observed and asymptotic values of the mean large separation. The analysis is focused on radial modes. We use series of radial-mode frequencies to compare the asymptotic and observational values of the large separation. We propose a simple formulation to correct the observed value of the large separation and then derive its asymptotic counterpart. We prove that, apart from glitches due to stellar structure discontinuities, the asymptotic expansion is valid from main-sequence stars to red giants. Our model shows that the asymptotic offset is close to 1/4, as in the theoretical development. High-quality solar-like oscillation spectra derived from precise photometric measurements are definitely better described with the second-order asymptotic expansion. The second-order term is responsible for the curvature observed in the \'echelle diagrams used for analysing the oscillation spectra and this curvature is responsible for the difference between the observed and asymptotic values of the large separation. Taking it into account yields a revision of the scaling relations providing more accurate asteroseismic estimates of the stellar mass and radius.Comment: accepted in A&

Multivariate Statistical Analysis in the Real and Complex Domains

This book explores topics in multivariate statistical analysis, relevant in the real and complex domains. It utilizes simplified and unified notations to render the complex subject matter both accessible and enjoyable, drawing from clear exposition and numerous illustrative examples. The book features an in-depth treatment of theory with a fair balance of applied coverage, and a classroom lecture style so that the learning process feels organic. It also contains original results, with the goal of driving research conversations forward. This will be particularly useful for researchers working in machine learning, biomedical signal processing, and other fields that increasingly rely on complex random variables to model complex-valued data. It can also be used in advanced courses on multivariate analysis. Numerous exercises are included throughout

Multivariate time series modeling of monthly rainfall amounts

This paper discusses the tting of suitable models to rainfall observations.Daily rainfall amounts were aggregated to monthly data using the Thiessenpolygons method and multivariate seasonal vector integrated autoregressivemoving average models (sVARIMA) were tted to the monthly cumulative rainfall volume. The data were obtained from the 12 Palestinian meteorological gauge stations located across the 5 governorates of the Gaza Strip and incorporated 42 years (from 1973 to 2014) of irregular daily precipitation. It can be concluded that the use of sVARIMAmodels in the environmental science provide a useful method to forecast rainfall data as a preliminary guideline for short and long-term sustainable water resources management

Ensembles of probability estimation trees for customer churn prediction

Customer churn prediction is one of the most, important elements tents of a company's Customer Relationship Management, (CRM) strategy In tins study, two strategies are investigated to increase the lift. performance of ensemble classification models, i.e (1) using probability estimation trees (PETs) instead of standard decision trees as base classifiers; and (n) implementing alternative fusion rules based on lift weights lot the combination of ensemble member's outputs Experiments ale conducted lot font popular ensemble strategics on five real-life chin n data sets In general, the results demonstrate how lift performance can be substantially improved by using alternative base classifiers and fusion tides However: the effect vanes lot the (Idol cut ensemble strategies lit particular, the results indicate an increase of lift performance of (1) Bagging by implementing C4 4 base classifiets. (n) the Random Subspace Method (RSM) by using lift-weighted fusion rules, and (in) AdaBoost, by implementing both