Generalized Least-Powers Regressions I: Bivariate Regressions

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1 ITERATIOAL JOURAL OF MATHEMATICAL MODELS AD METHODS I APPLIED SCIECES Volume, Generalized Least-Powers Reressions I: Bivariate Reressions ataniel Greene Abstract The bivariate theory of eneralized least-squares is etended here to least-powers. The bivariate eneralized leastpowers problem of order p seeks a line which minimizes the averae eneralized mean of the absolute pth power deviations between the data and the line. Least-squares reressions utilize second order moments of the data to construct the reression line whereas leastpowers reressions use moments of order p to construct the line. The focus is on even values of p, since this case admits analytic solution methods for the reression coefcients. A numerical eample shows eneralized least-powers methods performin comparably to eneralized least-squares methods, but with a wider rane of slope values. Keywords Least-powers, eneralized least-powers, least-squares, eneralized least-squares, eometric mean reression, orthoonal reression, least-quartic reression. I. OVERVIEW FOR two variables and y ordinary least-squares yj reression suffers from a fundamental lack of symmetry. It minimizes the distance between the data and the reression line in the dependent variable y alone. To predict the value of the independent variable one cannot simply solve for this variable usin the reression equation. Instead one must derive a new reression equation treatin as the dependent variable. This is called ordinary least-squares jy reression. The fact that there are two ordinary least-squares lines to model a sinle set of data is problematic. One wishes to have a sinle linear model for the data, for which it is valid to solve for either variable for prediction purposes. A theory of eneralized least-squares was developed by this author to overcome this problem by minimizin the averae eneralized mean of the square deviations in both and y variables [] [8]. For the resultin reression equation, one can solve for in terms of y in order to predict the value of : This theory was subsequently etended to multiple variables [9]. In this paper, the etension of the bivariate theory of eneralized least-squares to least-powers is beun. The bivariate eneralized least-powers problem of order p seeks a line which minimizes the averae eneralized mean of the absolute pth power deviations between the data and the line. Unlike leastsquares reressions which utilize second order moments of the data to construct the reression line, least-pth powers reressions utilize moments of order p to construct the line. In the interest of enerality, the denitions here are formulated usin arbitrary powers p. evertheless, the focus. Greene is with the Department of Mathematics and Computer Science, Kinsborouh Community Collee, City University of ew York, Oriental Boulevard, Brooklyn, Y USA (phone: ; nreene.math@mail.com). of this paper is on even values of p, since this case allows for the derivation of analytic formulas to nd the reression coefcients, analoous to what was done for least-squares. The numerical eample presented illustrates that bivariate eneralized least-powers methods perform comparably to eneralized least-squares methods but have a reater rane of slope values. II. BIVARIATE ORDIARY AD GEERALIZED LEAST-POWERS REGRESSIO A. Bivariate Ordinary Least-Powers Reression OLP p The eneralization of ordinary least-squares reression (OLS) to arbitrary powers is called ordinary least-powers reression of order p and is denoted here by OLP p. OLS is the same thin as OLP. The case of OLP, called leastquartic reression, is described in a paper by Arbia []. Denition : (The Ordinary Least-Powers Problem) Values of a and b are souht which minimize an error function dened by E = ja + b i y i j p : () The resultin line y = a + b is called the ordinary leastpowers yj reression line. The eplicit bivariate formula for the ordinary least-squares error described by Ehrenber [] is eneralized now to hiherorder reressions usin eneralized product-moments. Denition : Dene the eneralized bivariate productmoment of order p = m + n as m;n = ( i ) m n y i y () for whole numbers m and n. Theorem : (Eplicit Bivariate Error Formula) Let p be an even whole number and let F = Ej a=y b : Then E = ( ) s p r; s; t r;s b r t a y b () or E = where r+s+t=p r+s+t=p;t= F = ( ) s p r; s; t r;s b r t a y b + F p ( ) p r p r r;p r b r : () r= ISS: 998-

2 ITERATIOAL JOURAL OF MATHEMATICAL MODELS AD METHODS I APPLIED SCIECES Volume, Proof: Assume p is even and omit absolute values. Bein with the error epression and manipulate as follows: E = = = = = (a + b i y i ) p b ( i ) y i y + a y b p r+s+t=p ( ) s p b r r; s; t ( i ) r s t y i y a y b ( ) s p b r r; s; t r+s+t=p ( ) ( i ) r s t y i y a y b ( ) s p r; s; t r;s b r t a y b : r+s+t=p ow separate out the terms with t = and obtain E = ( ) s p r; s; t r;s b r t a y b + F where r+s+t=p;t= F = ( ) s p r; s; r;s b r r+s=p p = ( ) p r p r r;p r b r : r= Observe that applyin the trinomial epansion theorem to the error epression E and then settin a = y b produces the same result F as rst settin a = y b and then applyin the binomial epansion: F = p b ( i ) y i y = p p b r ( i r ) r p r p r y i y ( ) r= p = ( ) p r p r r= ( ) ( i ) r p r y i y b r p = ( ) p r p r r;p r b r : r= Eample : For p =, which is least-squares, In more familiar notation this is F = ; b ; b + ; : () F = b y b + y: () For p =, For p =, F = ; b ; b + ; b ; b + ; : (7) F = ; b ; b + ; b ; b + ; b ; b + ; : (8) Theorem : (Bivariate OLP Reression) The OLP p yj reression line y = a + b is obtained by solvin p F (b) = ( ) p r p r r r;p r b r = (9) for b and settin r= a = y b : () Eample : For p =, which is least-squares, one solves For p = one solves For p = one solves = ; b ; : () = ; b ; b + ; b ; : () = ; b ; b + ; b ; b + ; b ; : () ow that the analo of OLS, called OLP, has been described, the correspondin bivariate theory of eneralized least-powers can be described as well. B. Bivariate Generalized Means and MR p otation The aioms of a eneralized mean were stated by us previously [8], [9] drawin from the work of Mays [] and also from Chen []. They are stated here aain for convenience. Denition 7: A function M (; y) denes a eneralized mean for ; y > if it satises Properties - below. If it satises Property it is called a homoeneous eneralized mean. The properties are:. (Continuity) M (; y) is continuous in each variable.. (Monotonicity) M (; y) is non-decreasin in each variable.. (Symmetry) M (; y) = M (y; ) :. (Identity) M (; ) = :. (Intermediacy) min (; y) M (; y) ma (; y) :. (Homoeneity) M (t; ty) = tm (; y) for all t > : All known means are included in this denition. All the means discussed in this paper are homoeneous. The eneralized mean of any two eneralized means is itself a eneralized mean. MR p notation is used here to name eneralized reressions: if `' is the letter used to denote a iven eneralized mean, then MR p is the correspondin eneralized least-pth power reression. MR is the same thin as MR without a ISS: 998-

3 ITERATIOAL JOURAL OF MATHEMATICAL MODELS AD METHODS I APPLIED SCIECES Volume, subscript. For eample, `G' is the letter usually used to denote the eometric mean and GMR p is least-pth power eometric mean reression. AMR p is least-pth power arithmetic mean reression. The eneralization of orthoonal least-squares reression to least-powers is HMR p since orthoonal reression is the same as harmonic mean reression. C. The Two Generalized Least-Powers Problems and the Equivalence Theorem The eneral symmetric least-powers problem is stated as follows. Denition 8: (The General Symmetric Least-Powers Problem) Values of a and b are souht which minimize an error function dened by E = M ja + b i y i j p ; a b + i b y p i () where M (; y) is any eneralized mean. Denition 9: (The General Weihted Ordinary Least- Powers Problem) Values of a and b are souht which minimize an error function dened by E = (b) ja + b i y i j p : () where (b) is a positive even function that is non-decreasin for b < and non-increasin for b > The net theorem states that every eneralized least-powers reression problem is equivalent to a weihted ordinary leastpowers problem with weiht function (b). Theorem : Every eneral symmetric least-powers error function can be written equivalently as where E = (b) ja + b i y i j p () (b) = M ; jbj p : Proof: Substitute a b + i b y i with b (a + b i then use the homoeneity property: E = Dene = (7) y i ) and M ja + b i y i j p ; ja + b i y i j p jbj p ja + b i (b) = M y i j p M ; ; jbj p ; and factor (b) outside of the summation. jbj p : D. How to Find the Reression Coefcients The fundamental practical question of bivariate eneralized least-powers reression is how to nd the coefcients a and b for the reression line y = a + b. The case of leasteven-power reressions can be solved analytically, analoous to how eneralized least-squares reressions are solved. For p even, to nd the reression coefcients a and b; take the rst order partial derivatives of the error function E a and E b and set them equal to zero. Solvin E a = yields a = y b. Solvin E = and then settin a = y b is equivalent to settin a = y b rst and then solvin df a=y = : The latter procedure is employed b here because it is simpler. Theorem : (Solvin for the Generalized Reression Coefcients) For p even, the eneralized least-powers slope b is found by solvin: where F = d f (b) F (b) = (8) db p ( ) p r p r r;p r b r (9) r= and the y-intercept a is iven by E. The Hessian Matri a = y b : () In order for the reression coefcients (a; b) to minimize the error function and be admissible, the Hessian matri of second order partial derivatives must be positive denite when evaluated at (a; b). The eneral Hessian matri is calculated net. As in the case of eneralized least-squares, certain combinations of and its rst and second partial derivatives appear in the matri. One combination is denoted here by J and another is denoted by G. They are called indicative functions. Denition : Dene the indicative functions J (b) = (b) (b) (b) () (b) and G (b) = (b) (b) (b) (b) : () The two indicative functions are related by the equation F =F = J=G. The latter differential equation can be solved eplicitly for (b). One obtains [] (b) = c + k R ep R : () G (b) db db Theorem : (Hessian matri) The Hessian matri H of second order partial derivatives of the error function iven by H H H = : () H H ISS: 998-

4 ITERATIOAL JOURAL OF MATHEMATICAL MODELS AD METHODS I APPLIED SCIECES Volume, is computed eplicitly as follows: H = E aa = p (p ) F p () H = H = E ab = E ba = p F p + Fp () H = E bb = (JF + F ) : (7) Alternatively, H = (GF + F ) : (8) Proof: Take second order partial derivatives of the error and evaluate at the solution (a; b). Let E = E OLP for purposes of this proof and assume p is even. ( E + E b ) = E + E b + E bb : Since E + E b = at the solution, substitute E b = into the middle term, simplify, and obtain!! H = E + E bb E which is the rst form of the Hessian. ow substitute E = E b and obtain the second form H = E b + E bb : As before, upon substitutin for a, E = F, E b = F and E bb = F. For H ; (a + b i y i ) p = p (p ) = p (p ) (a + b i y i ) p + a y b p = p (p ) = p (p ) F p : For H = H, (E) b ( i ) y i y b ( i ) y i y p! (a + b i y i ) p = p F p + F p : Theorem : The Hessian matri H is positive denite at the solution point (a; b) provided that det H > : Proof: H is positive denite provided that H > and det H > In our case H = (b) p (p ) F p (b) > for all b. Therefore it sufces to evaluate the determinant numerically at the slope b. III. REGRESSIO EAMPLES BASED O KOW SPECIAL MEAS A. Arithmetic Mean Reression (AMR p ) The arithmetic mean is iven by A (; y) = ( + y) : (9) This mean enerates arithmetic mean reression AMR p. The weiht function is iven by (b) = + jbj p : () The eneral AMR p slope equation for p even is iven by The AMR slope equation is b p+ + b F (b) pf (b) = : () = ; b ; b + ; b ; : () The AMR slope equation is = ; b 8 ; b 7 + ; b ; b The AMR slope equation is + ; b ; b + ; b ; : = ; b ; b + ; b ; b 9 + ; b 8 ; b 7 + ; b ; b () + ; b ; b + ; b ; : () B. Geometric Mean Reression (GMR p ) The eometric mean is iven by G (; y) = (y) = : () This mean enerates eometric mean reression GMR p. The weiht function is iven by (b) = jbj p= : () The eneral GMR p slope equation for p even is The GMR slope equation is The GMR slope equation is bf (b) pf (b) = : (7) = ; b ; : (8) = ; b ; b + ; b ; : (9) The GMR slope equation is = ; b ; b + ; b ; b + ; b ; : () ISS: 998-

5 ITERATIOAL JOURAL OF MATHEMATICAL MODELS AD METHODS I APPLIED SCIECES Volume, C. Harmonic Mean (Orthoonal) Reression (HMR p ) The harmonic mean is iven by H (; y) = y + y : () This mean enerates harmonic mean reression HMR p. The weiht function is iven by (b) = + jbj p : () The eneral HMR p slope equation for p even is (b p + ) F (b) pb p F (b) = : () The HMR slope equation is = ; b + ; ; b ; : () The HMR slope equation is = ; b ; b + ; b The HMR slope equation is + ; ; b ; b + ; b ; : () = ; b ; b 9 + ; b 8 ; b 7 + ; b + ; ; b ; b + ; b ; b + ; b ; : () Harmonic mean reression is the same thin as orthoonal reression. This is because of the Reciprocal Pythaorean Theorem [], [] which says that the diaonal deviation between a data point and the reression line is half the harmonic mean of the horizontal and vertical deviations. D. Ordinary Least-Powers jy Reression (OLP p jy) The selection mean iven by S (; y) = (7) enerates OLP p yj reression. The selection mean iven by S y (; y) = y (8) enerates OLP p jy reression. The weiht function correspondin to OLP p yj is iven by (b) = : (9) The weiht function correspondin to OLP p jy is iven by (b) = jbj p : () The eneral OLP p yj slope equation for p even is F (b) = () The specic equations for p = ; ; and were already described. The eneral OLP p jy slope equation for p even is bf (b) pf (b) = () The OLP jy slope equation is The OLP jy slope equation is = ; b ; : () = ; b ; b + ; b ; : () The OLP jy slope equation is = ; b ; b + ; b ; b + ; b ; : () IV. REGRESSIO EAMPLES BASED O GEERALIZED MEAS The eneralized means in the net eamples have free parameters which can be used to parameterize these and other known special cases. A. Weihted Arithmetic Mean Reression The weihted arithmetic mean with weiht in [; ] is iven by M (; y) = ( ) + y () for y. The weiht function correspondin to the weihted arithmetic mean is (b) = ( ) + jbj p : (7) For weihted AMR p; the eneral slope equation for p even is ( ) b p+ + b F (b) pf (b) = : (8) The weihted AMR ; slope equation is = ( ) ; b ( ) ; b + ; b ; (9) The weihted AMR ; slope equation is = ( ) ; b 8 ( ) ; b 7 + ( ) ; b ( ) ; b + ; b ; b + ; b ; : () The weihted AMR ; slope equation is = ( ) ; b ( ) ; b + ( ) ; b ( ) ; b 9 + ( ) ; b 8 ( ) ; b 7 + ; b ; b + ; b ; b + ; b ; () ISS: 998-

6 ITERATIOAL JOURAL OF MATHEMATICAL MODELS AD METHODS I APPLIED SCIECES Volume, B. Weihted Geometric Mean Reression The weihted eometric mean with weiht in [; ] is iven by M (; y) = y () for y. The weiht function correspondin to the weihted eometric mean is iven by (b) = jbj p : () For weihted GMR p; the eneral slope equation for p even is bf (b) pf (b) = : () The weihted GMR ; slope equation is = ( ) ; b + ( ) ; b ; () The weihted GMR ; slope equation is = ( ) ; b ( ) ; b + ( ) ; b ( ) ; b ; : The weihted GMR ; slope equation is = ( ) ; b ( ) ; b + ( ) ; b ( ) ; b + ( ) ; b ( ) ; b ; : C. Power Mean Reression () (7) The power mean of order q, for q =, is iven by =q M q (; y) = (q + y q ) (8) with M (; y) = G (; y), M (; y) = min (; y) and M (; y) = ma (; y). Many other special means are specic cases as well: q = is the harmonic mean, q = was the basis for squared harmonic mean reression (SHR), q = was the basis for square perimeter reression (SPR), q = is the arithmetic mean and q = is called the root-mean-square [], [8]. Many other special means are approimated well by power means: M = (; y) approimates the second loarithmic mean L (; y) and HG = well, M= (; y) ; called the Lorentz mean, approimates the rst loarithmic mean L (; y) well, and M = (; y) approimates both the Heronian mean (; y) and the identric mean I (; y) well. This is proven in our earlier paper [8]. The weiht function is iven by (b) = + jbj pq =q : (9) The eneral PMR p;q slope equation for p even is (b pq + ) bf (b) pf (b) = : (7) The PMR ;q slope equation is = ; b q+ ; b q+ + ; b ; (7) The PMR ;q slope equation is = ; b q+ ; b q+ The PMR ;q slope equation is + ; b q+ b q+ ; + ; b ; b + ; b ; : (7) = ; b q+ ; b q+ + ; b q+ ; b q+ + ; b q+ ; b q+ + ; b ; b + ; b ; b + ; b ; : (7) D. Reressions Based on Other Generalized Means As was done in our previous paper [8] for p =, reression formulas for p > based on other eneralized means can be worked out. The Dietel-Gordon Mean of order r, the Stolarsky mean of order s; the two-parameter Stolarsky mean of order (r; s), the Gini mean of order t, the two-parameter Gini mean of order (r; s) are alternative eneralized means which parameterize the known specic cases. Details on these means and the correspondin references can be found in that paper. We leave the detailed slope equations in these cases for a future work in this series. V. EQUIVALECE THEOREMS A. Solvin for the Generalized Mean Parameter as a Function of the Slope For the case of weihted AMR, weihted GMR, and PMR the free parameter in these cases can solved for eplicitly in terms of the slope. The result of doin this yields an equivalence theorem. Theorem : (Weihted Arithmetic Mean Equivalence Theorem) Let b be the slope of a eneralized least-powers reression line with p even. Then the line can be enerated by an equivalent weihted arithmetic mean reression with weiht iven by with = b p+ F (b) (b p+ b) F (b) + pf (b) b = b OLP yj +! b OLP jy b OLP yj (7) (7) and! in [; ] : Proof: Solve the weihted AMR p; slope equation for. Theorem : (Weihted Geometric Mean Equivalence Theorem) Let b be the slope of a eneralized least-powers reression line with p even. Then the line can be enerated by an equivalent weihted eometric mean reression with weiht iven by = bf (b) (7) pf (b) ISS: 998-7

7 ITERATIOAL JOURAL OF MATHEMATICAL MODELS AD METHODS I APPLIED SCIECES Volume, with b = b OLP yj +! b OLP jy b OLP yj (77) and! in [; ] : Proof: Solve the weihted GMR p; slope equation for. Theorem 7: (Power Mean Equivalence Theorem) Let b be the slope of a eneralized least-powers reression line with p even. Then the line can be enerated by an equivalent power mean reression of order q with q = pf (b) bf p ln (b) bf = ln b: (78) (b) Proof: Solve the weihted PMR p;q slope equation for q: For b >, if the interval bolp yj ; b OLP jy does not contain, or for b <, if the interval b OLP jy ; b OLP yj does not contain, then every reression line lyin between the two ordinary least-powers lines is enerated by a power mean of order q for some q in ( ; ) with the ordinary least-powers lines correspondin to q =. This was shown in detail for the case of least-squares [8]. B. The Eponential Equivalence Theorem and the Fundamental Formula for Generalized Least-Powers Reression In the bivariate case of eneralized least-squares it was shown [7], [8] that every weihted ordinary least-squares reression line can be enerated by an equivalent eponentially weihted reression with weiht function (b) = ep ( P jbj) for in [; ]. This theorem eneralizes to least-powers reressions as well. Theorem 8: (The Etremal Line) For eponentially weihted ordinary least-powers reression with weiht function (b) = ep ( P jbj), the reression line enerated by the maimum value of P is called the etremal line. The slope of the etremal line b ET is computed by solvin F (b) F (b) (F (b)) = (79) and the maimum value of P, called P, is computed by P = sn (b ET ) F (b ET ) F (b ET ) : (8) As the parameter P varies over the interval [; P ] all eponentially weihted least-powers reressions are enerated. Proof: Consider the weiht function (b) = ep ( P jbj). To nd the slope b, one must solve or d fep ( P jbj) F (b) = db ep ( P jbj) ( P ) (sn b) F (b) + ep ( P jbj) F (b) = which is equivalent to writin P (b) = sn b F (b) F (b) : To nd the maimum value of P one must solve P (b) = sn b F (b) F (b) (F (b)) (F (b)) = or simply F (b) F (b) (F (b)) = for b. Call the resultin value b ET. To insure that the value is a maimum, one must also verify that P (b ET ) < : The correspondin maimum value of P is P = P (b ET ). Eample 9: For p = one obtains ;b ; ; b + ; ; ; = (8) which in covariance notation is b y b + y y = : (8) This can be solved eplicitly to obtain the slope of the etremal line. It can also be epressed usin covariances or standard deviations and correlation coefcients as in the previous papers [7], [8]: q ; ; ; ; b = (8) ; q y y y = (8) For p = one obtains For p = one obtains = y y p : (8) = ;b ; ; b + ; + ; ; b = ;b ; ; b 9 ; ; ; ; b ; ; 8 ; b ; ; 8 ; ; b ; ; ; : (8) + ; + ; ; b 8 ; ; b 7 ; ; 8 ; ; ; b + ; ; ; ; + 8 ; ; b + 7 ; ; ; ; ; ; + ; b + ; ; ; ; + ; ; b ; ; 7 ; b + ; ; ; ; b ; ; ; : (87) Since one cannot solve these equations eplicitly when p > one solves the equations numerically for b instead. One selects the real root that shares the same sin as b OLP and is reater than b OLP in absolute value. ISS: 998-8

8 ITERATIOAL JOURAL OF MATHEMATICAL MODELS AD METHODS I APPLIED SCIECES Volume, Theorem : (Eponential Equivalence Theorem) Dene the normalized eponential parameter = P=P so that (b) = ep ( P jbj) for in [; ]. Then every weihted eneralized least-powers reression line with slope b is also enerated by an equivalent eponentially weihted reression with normalized eponential parameter = P sn b F (b) F (b) : (88) The case = is OLP p yj reression. The case = is the etremal line. Proof: For an arbitrary weiht function (b), the slope b is found by solvin d f (b) F (b) =db = which is equivalent to (b) = (b) = F (b) =F (b) at the solution. However, it is already known that for a ed slope b there eists a constant P in [; P ] such that P sn b = F (b) =F (b) : Thus P = (sn b) (b) = (b) and for every weiht function and slope b there is a correspondin value for the eponential parameters P and = P=P. Since b always lies in the interval between b OLP and b ET ; the fundamental formula of eneralized least-powers reression follows. Theorem : (Fundamental Formula of Generalized Least- Powers Reression) Every weihted eneralized least-powers reression line y = a + b has the form b = b OLP + (b ET b OLP ) (89) a = y b (9) for some = () in [; ]. Once the slope b of a reression line is known, the correspondin value of can be determined numerically usin = b b OLP b ET b OLP : (9) The function = () can be determined eplicitly by composin = (b) with b = b (). The result is = P sn b F (b OLP + (b ET b OLP )) F (b OLP + (b ET b OLP )) : (9) For p = only, the parameters and are related by the simple formulas = sin tan and = tan sin [8]. VI. UMERICAL EAMPLE This section revisits an eample eplored in the previous work. Reressions correspondin to OLP p, HMR p, GMR p, AMR p and the etremal line are computed for p = ; ; and. The correspondin effective weihted arithmetic and eometric mean parameters and are computed. The effective eponential parameters and are also computed. Eample : This eample appears in our previous papers and is oriinally from Martin []. Si data values are iven: (; ), (; ), (; ), (; ), (; ), and (; ). The reader can verify that = :97, = :, y = :, = :778, and y = :98. The second order product-moments are: ; = :, ; = :, ; = :97: The fourth order productmoments are: ; = :9, ; = :8, ; = :9, ; = :, ; = :79. The sith order product-moments are: ; = 87:79, ; = 8:8, ; = 8:8, ; = 8:98, ; = 8:7, ; = 8:9, ; = 8:8: The eneralized reression lines are plotted toether with the etremal line thereby displayin the reion containin all admissible eneralized reression lines. y 7 p = y 7 p = y 7 p = The equation of each line is presented alon with the eponential parameters and, the weihted arithmetic mean parameter, and the weihted eometric mean parameter. In all cases one uses = P (sn b) F (b) =F (b) and = (b b OLP ) = (b ET b OLP ) : Accordin to the eponential equivalence theorem, all reression lines are enerated for p = by (b) = ep ( :8 jbj), for p = by (b) = ep ( :7 jbj), and for p = by (b) = ep ( :9 jbj). p = OLP y HMR GMR AMR OLP y ET p = OLP y HMR GMR AMR OLP y ET p = OLP y HMR GMR AMR OLP y ET y = a + b γ λ α β y =.7 87 y = y = y =.8 99 y = y =.... y = a + b γ λ α β y = y = y = y = y = y = y = a + b γ λ α β y =.9 7 y = y = y =.77 9 y = y = It is readily observed that for p =, the slope values lie in the interval [ :; :87] with a rane of :7. For p =, the slope values lie in the interval [ :98; :78] ISS: 998-9

9 ITERATIOAL JOURAL OF MATHEMATICAL MODELS AD METHODS I APPLIED SCIECES Volume, with a rane of :78. The rane for p = is approimately.9 times as lon as the rane for p = : For p =, the slope values lie in the interval [ :9; :7] with a rane of :89. The rane for p = is approimately. times as lon as the rane for p = : For p =, the interval between the two OLP slopes is [ :; :87] with a rane of :. For p =, the interval between the two OLP slopes is [ :77; :78] with a rane of :9 The rane for p = is approimately. times as lon as the rane for p = : For p =, the interval between the two OLP slopes is [ :; :7] with a rane of :99. The rane for p = is approimately. times as lon as the rane for p = : VII. SUMMARY Least-powers reressions minimizin the averae eneralized mean of the absolute pth power deviations between the data and the reression line are described in this paper. Particular attention is paid to the case of p even, since this case admits analytic solution methods for the reression coefcients. Ordinary least-squares reression eneralizes to ordinary least-powers reression. The case p = corresponds to the eneralized least-squares reressions of our previous works. The specic cases of arithmetic, eometric and harmonic mean (orthoonal) reression are worked out in detail for the case of p =, and. Reressions based on weihted arithmetic means of order and weihted eometric means of order are also worked out. The weihts and continuously parameterize all eneralized reression lines lyin between the two ordinary least-powers lines. Power mean reression of order q has ed values of q correspondin to many known special means and offers another way to parameterize the eneralized mean reressions previously described. Every eneralized mean reression with error function iven by E = M ja + b i y i j p ; a b + i b y p i (9) is equivalent to a weihted ordinary least-powers reression with error function E = (b) ja + b i y i j p (9) and weiht function (b) = M ; jbj p (9) where M (; y) is any eneralized mean. The eponential equivalence theorem states that every weihted ordinary least-powers reression line can be enerated by an equivalent eponentially weihted reression with weiht function (b) = ep ( P jbj) for in [; ]. The case = corresponds to OLP p yj and the case = corresponds to the etremal line. It follows that every eneralized least-powers line has slope iven by b = b OLP + (b ET b OLP ) for = () in [; ] and y-intercept iven by a = y b. This is referred to here as the fundamental formula of eneralized least-powers reression, since it characterizes all possible reression lines in a simple way. A simple numerical eample shows eneralized leastpowers reressions performin comparably to eneralized least-squares but with a wider rane of slope values. The application of bivariate eneralized least-powers to non-normally distributed data and the potential advantae of these methods over eneralized least-squares is a subject of the net paper in this series. The etension of this theory to multiple variables is also a subject of the net paper in this series. REFERECES [] G. Arbia, "Least Quartic Reression Criterion with Application to Finance," ariv:.7 [q-n.st],. [] H. Chen. "Means Generated by an Interal." Mathematics Maazine, vol. 8, p. 7, Dec.. [] S. C. Ehrenber. "Derivin the Least-Squares Reression Equation." The American Statistician, vol. 7, p., Au. 98. [] V. Ferlini, Mathematics without (many) words, Collee Math Journal, o., p.7,. []. Greene, "Generalized Least-Squares Reressions I: Efcient Derivations," in Proceedins of the st International Conference on Computational Science and Enineerin (CSE'), Valencia, Spain,, pp. -8. []. Greene, "Generalized Least-Squares Reressions II: Theory and Classication," in Proceedins of the st International Conference on Computational Science and Enineerin (CSE '), Valencia, Spain,, pp. 9-. [7]. Greene, "Generalized Least-Squares Reressions III: Further Theory and Classication," in Proceedins of the th International Conference on Applied Mathematics and Informatics (AMATHI '), Cambride, MA,, pp. -8. [8]. Greene, "Generalized Least-Squares Reressions IV: Theory and Classication Usin Generalized Means," in Mathematics and Computers in Science and Industry, Varna, Bularia,, pp. 9-. [9]. Greene, "Generalized Least-Squares Reressions V: Multiple Variables," in ew Developments in Pure and Applied Mathematics, Vienna, Austria,, pp. 7-. [] S. B. Martin, Less than the Least: An Alternative Method of Least- Squares Linear Reression, Underraduate Honors Thesis, Department of Mathematics, McMurry University, Abilene, Teas, 998. [] M. E. Mays. "Functions Which Parametrize Means." American Mathematical Monthly, vol 9, pp. 77-8, 98. [] R. B. elson. Proof Without Words: A Reciprocal Pythaorean Theorem, Mathematics Maazine, Vol. 8, o., p. 7, Dec. 9. [] P. A. Samuelson, A ote on Alternative Reressions, Econometrica, Vol., o., pp. 8-8, Jan. 9. [] R. Taaepera, Makin Social Sciences More Scientic: The eed for Predictive Models, Oford University Press, ew York, 8. ISS: 998-