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• W2339458425 abstract "—The theory of generalized least-squares is reformulated here using the notion of generalized means. The generalized least-squares problem seeks a line which minimizes the average generalized mean of the square deviations in x and y. The notion of a generalized mean is equivalent to the generating function concept of the previous papers but allows for a more robust understanding and has an already existing literature. Generalized means are applied to the task of constructing more examples, simplifying the theory, and further classifying generalized least-squares regressions. Keywords—Linear regression, least-squares, orthogonal regression, geometric mean regression, generalized least-squares, generalized mean square regression. I. OVERVIEW Ordinary least-squares regression suffers from a fundamental lack of symmetry: the regression line of y given x and the regression line of x given y are not inverses of each other. Ordinary least-squares yjx regression minimizes the average square deviation between the data and the line in the y variable and ordinary least-squares xjy minimizes the average square deviation between the data and the line in the x variable. A theory of generalized least-squares was described by this author for minimizing the average of a symmetric function of the square deviations in both x and y variables [7,8,9]. The symmetric function was referred to as a generating function for the particular regression method. This paper continues the development of the theory of generalized least-squares, reformulated using the notion of generalized means. The generalized least-squares problem described here seeks a line which minimizes the average generalized mean of the square deviations in x and y. The notion of a generalized mean is equivalent to the generating function concept of the previous papers but allows for a more robust understanding and has an already existing literature. It is clear from the name that geometric mean regression (GMR) seeks a line which minimizes the average geometric mean of the square deviations in x and y. Orthogonal regression seeks a line which minimizes the average harmonic mean of the square deviations in x and y. Therefore it is also called harmonic mean regression (HMR). Arithmetic mean regression (AMR) seeks a line which minimizes the average arithmetic mean of the square deviations in x and y and was called Pythagorean regression previously. Here, logarithmic, Heronian, centroidal, identric, Lorentz, and root mean square regressions are described for the rst time. Ordinary leastsquares regression is shown here to be equivalent to minimum or maximum mean regression. Regressions based on weighted arithmetic means of order and weighted geometric means of order are explored. The weights and parameterize all generalized mean square regression lines lying between the two ordinary least-squares lines. Power mean regression of order p offers a particularly simple framework for parameterizing all the generalized mean square regressions previously described. The p-scale has xed numerical values corresponding to many known special means. All the symmetric regressions discussed in the previous papers are power mean regressions for some value of p. Ordinary least-squares corresponds to p = 1. The power mean is one example of a generalized mean whose free parameter unites a variety of special means as subcases. Other generalized means which do the same include: the DietelGordon mean of order r, Stolarsky's mean of order s, and Gini's mean of order t. There are also two-parameter means due to Stolarsky and Gini. Regression formulas based on all these generalized means are worked out here for the rst time. II. REGRESSIONS BASED ON GENERALIZED MEANS Generalized means are applied to the task of constructing more examples, simplifying the theory, and further classifying generalized least-squares regressions. A. Axioms of a Generalized Mean The axioms of a generalized mean presented here are drawn from Mays [13] and also from Chen [2]. De nition 1: A function M (x; y) de nes a generalized mean for x; y > 0 if it satis es Properties 1-5 below. If it satis es Property 6 it is called a homogenous generalized mean. The properties are: 1. (Continuity) M (x; y) is continuous in each variable. 2. (Monotonicity) M (x; y) is non-decreasing in each variable. Mathematics and Computers in Science and Industry ISBN: 978-1-61804-247-7 19 3. (Symmetry) M (x; y) =M (y; x) : 4. (Identity) M (x; x) = x: 5. (Intermediacy) min (x; y) M (x; y) max (x; y) : 6. (Homogeneity) M (tx; ty) = tM (x; y) for all t > 0: All known means are included in this de nition. All the means discussed in this paper are homogeneous. The reader can verify that the weighted arithmetic mean or convex combination of any two generalized means is a generalized mean. The weighted geometric mean of any two generalized means is a generalized mean. More generally, the generalized mean of any two generalized means is itself a generalized mean. The equivalence of generalized means and generating functions is now demonstrated. Theorem 1: Let M (x; y) be any generalized mean, then (x; y) = M x; y is the generating function for a corresponding generalized symmetric least-squares regression. Let (x; y) be any generating function, then M (x; y) = p jxj; p jyj de nes a generalized mean. The weight function is given by g (b) = 1; 1b =M 1; 1 b2 : From here it is clear that the theory of generalized leastsquares can be reformulated using generalized means. The general symmetric least-squares problem is re-stated as follows. De nition 2: (The General Symmetric Least-Squares Problem) Values of a and b are sought which minimize an error function de ned by" @default.
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• W2339458425 title "Generalized Least-Squares Regressions IV: Theory and Classification Using Generalized Means" @default.
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