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A well-known property of the multiple linear regression is that its plane goes through the point of the mean values of all variables, and this feature can be used to find the model's intercept. This work shows that a regression by n predictors also passes via n additional points of the specificweighted mean values. Thus, the regression is uniquely defined by all these n+1multidimensional points ofmeans, and approximation of observations by the theoretical model collapses to the interpolation function going through the knots of the weighted means. This property is obtained from the normal system of equations which serves for finding the linear regression parameters in the ordinary least squares approach. The derived features can be applied in nonlinearmodeling for adjusting the model parameters so that the fitted values would go through the same reference points of means, that can be useful in applied regression analysis. Numerical examples are discussed. The found properties reveal the essence of regression function as hyperplane going through special points of mean values, which makes regression models more transparent and useful for solving and interpretation in various applied statistical problems. [ABSTRACT FROM AUTHOR]

This paper considers polynomial characteristics useful for a better understanding of the behaviour of these functions. Taylor series for the polynomials are described by the items with even and odd derivatives and powered changes in the argument, which leads to more specific studying of their properties. Connections between the derivative and antiderivative of the polynomial functions are defined. The structure of polynomial functions reveals their specific characteristic that the mean value of their roots equals the mean value of the locations of the critical points such as the extrema and inflection points. Derivatives of the quadratic exponent in relation to an interesting connection of two transcendental numbers are also described. The discussed properties of the polynomials can be helpful for practical implementations and educational purposes. [ABSTRACT FROM AUTHOR]

"Essential Mathematics for Engineers and Scientists" is a textbook written by Thomas J. Pence and Indrek S. Wichman. The book covers various topics in applied mathematics and is divided into three parts with a total of 13 chapters. Part I focuses on linear algebra, Part II covers complex variables, and Part III explores partial differential equations. The book includes numerous exercises with detailed solutions and additional exercises at the end of each chapter. It is designed to be a practical resource for students and professionals in academia and industry. [Extracted from the article]

"Quantitative Investing: from Theory to Industry" by Lingjie Ma is a comprehensive textbook that covers quantitative investment theory and practice. The author draws on their 15 years of experience in the investment industry and 6 years of teaching to provide explanations on financial markets and econometric modeling. The book is divided into nine chapters, each covering different investment topics, finance theories, portfolio performance, statistical models, and R programming for data analysis. It includes real-world examples, R codes, and references, making it a valuable resource for advanced students and practitioners in finance and investments. [Extracted from the article]

This study considers the statistical estimation of relations presented by implicit functions. Such structures define mutual interconnections of variables rather than outcome variable dependence by predictor variables considered in regular regression analysis. For a simple case of two variables, pairwise regression modeling produces two different lines of each variable dependence using another variable, but building an implicit relation yields one invertible model composed of two simple regressions. Modeling an implicit linear relation for multiple variables can be expressed as a generalized eigenproblem of the covariance matrix of the variables in the metric of the covariance matrix of their errors. For unknown errors, this work describes their estimation by the residual errors of each variable in its regression by the other predictors. Then, the generalized eigenproblem can be reduced to the diagonalization of a special matrix built from the variables' covariance matrix and its inversion. Numerical examples demonstrate the eigenvector solution's good properties for building a unique equation of the relations between all variables. The proposed approach can be useful in practical regression modeling with all variables containing unobserved errors, which is a common situation for the applied problems. [ABSTRACT FROM AUTHOR]

The Pareto 80/20 Rule, also known as the Pareto principle or law, states that a small number of causes (20%) is responsible for a large percentage (80%) of the effect. Although widely recognized as a heuristic rule, this proportion has not been theoretically based. The article considers derivation of this 80/20 rule and some other standard quotients from the mean and its interval estimation for the total value defined by the product of two variables in the random partitioning model. (Contains 2 figures.)

Benford's law of the "first digits" states that in spite of intuitively expected equal frequency of 1/9 of the decimal digits r = 1, ... , 9 appearance on the first place of any number, various empirical studies show another pattern of these frequencies distribution, which is log[subscript 10](1 + 1/r). The article considers this law and other designs of data generation for pair comparison matrices, and several estimation techniques for the priority vectors. Among those are Analytic Hierarchy Process (AHP), Simple Multi-Attribute Rating Technique (SMART), and several additional approaches of the multiple criteria decision making. The closeness of a priority vector evaluated by a data elicited from an expert to one or another pattern helps to identify a nature of the data comparison process. (Contains 1 table and 1 figure.)

A regular problem in regression analysis is estimating the comparative importance of the predictors in the model. This work considers the 'net effects', or shares of the predictors in the coefficient of the multiple determination, which is a widely used characteristic of the quality of a regression model. Estimation of the net effects can be a difficult task because multicollinearity among the regressors can produce negative inputs to multiple determination. This paper suggests estimating the incremental net effects as subsequent marginal inputs to the coefficient of multiple determination, and it is shown that the results coincide with estimation by cooperative game theory. This approach guarantees positive and interpretable net effects, which offers a better interpretation of the regression results.

Relations between pairwise correlations and the coefficient of multiple determination in regression analysis are considered. The conditions for the occurrence of enhance-synergism and suppression effects when multiple determination becomes bigger than the total of squared correlations of the dependent variable with the regressors are discussed. It is shown that such effects can occur just for stochastic relations among the variables that have non-transitive signs of pairwise correlations. Consideration of these problems facilitates better understanding the properties of regression.

The book belongs to the series I Institute of Mathematical Statistics Monographs i , and presents a contemporary manual on statistical inference methods and machine learning algorithms. Chapter 2 "Frequentist Inference" notes that if a probability model produces the observed data, the accuracy of an observed estimate can be defined by the theoretical estimator accuracy. A relation that the posterior odds ratio equals the prior odds ratio times the likelihood ratio is explained. [Extracted from the article]

On examples of the Chuk-a-luck three-dice game of chance and several card games, the probabilities, expectations, variances, and standard deviations of random variables and their sum are considered. Part I "Games of Chance" consists of 17 chapters on probability approach, about which the famous physicist R. Feynman said that "The theory of probability is a system for making better guesses" (p. 3). The book presents mathematical explanation of problems related to playing games of chance, combinatorial and strategic games, with descriptions of their historical perspectives and recreational aspects. [Extracted from the article]