Reading Multiple Regression from an Ellipsoid Inscribed in the Unit Hypercube
Abstract
Consider the linear regression of a response Y on explanatory variables X1,…,Xp with a positive definite joint correlation matrix. Let R be the correlation matrix of the explanatory variables and r the vector of their correlations with Y. The ellipsoid E={x : x⊤R−1x=1} is inscribed in the cube [−1,1]p and touches the face xj=1 at Tj, the jth column of R. The point P=r lies inside E, and three kinds of regression quantities are read off this diagram by elementary geometric constructions. The multiple correlation is $|OP|/|OP'|$, where O is the origin and P' is the point where the ray from O through P meets E (for P≠ O). The standardized regression coefficients are the coordinates of P in the basis T1,…,Tp. The partial correlation between Xi and Y given the other explanatory variables is the position of P on the chord of E through P parallel to the xi-axis, on a linear scale from −1 to $1$ increasing with xi; furthermore, the parallel chord through the origin has half-length 1/√(VIFi). Examples illustrate sign reversal and standardized coefficients outside the interval [−1,1].
Related
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BibTeX
@misc{shimono2026reading,
title = {{Reading Multiple Regression from an Ellipsoid Inscribed in the Unit Hypercube}},
author = {Toshiyuki Shimono},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06536},
eprint = {2610.06536},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06536},
}
RIS
TY - UNPB
TI - Reading Multiple Regression from an Ellipsoid Inscribed in the Unit Hypercube
AU - Toshiyuki Shimono
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06536
UR - https://arxiv.org/abs/2610.06536
AB - Consider the linear regression of a response Y on explanatory variables X_1,…,X_p with a positive definite joint correlation matrix. Let R be the correlation matrix of the explanatory variables and r the vector of their correlations with Y. The ellipsoid E={x : x^⊤R^−1x=1} is inscribed in the cube [−1,1]^p and touches the face x_j=1 at T_j, the jth column of R. The point P=r lies inside E, and three kinds of regression quantities are read off this diagram by elementary geometric constructions. The multiple correlation is $|OP|/|OP'|$, where O is the origin and P' is the point where the ray from O through P meets E (for P≠ O). The standardized regression coefficients are the coordinates of P in the basis T_1,…,T_p. The partial correlation between X_i and Y given the other explanatory variables is the position of P on the chord of E through P parallel to the x_i-axis, on a linear scale from −1 to $1$ increasing with x_i; furthermore, the parallel chord through the origin has half-length 1/√(VIF_i). Examples illustrate sign reversal and standardized coefficients outside the interval [−1,1].
ER -