3
$\begingroup$

Assume $X,Y,Z$ are three variables over a set of data (say, a finite set of data to avoid discussions of convergence). Suppose we know the Pearson correlation coefficient $r_{X,Y}$ and $r_{Y,Z}$: given these data, what bounds (ideally sharp) can we put on $r_{X,Z}$?

If I am not mistaken, the question is equivalent to the following: for a positive semidefinite matrix with a diagonal of $1$, if we know the coefficients $(i,j)$ and $(j,k)$ in the matrix, what (ideally sharp) bounds can we put on the coefficient $(i,k)$?

Surely this is a classical problem and has been considered before, but I don't know what terms to search for.

$\endgroup$
3
  • 1
    $\begingroup$ The key lemmas to look up are equivalent conditions for a matrix to be positive semi definite. The relevant one is the characterisation in terms of principal minors. $\endgroup$
    – Ragib Zaman
    Commented Jun 12 at 14:29
  • 1
    $\begingroup$ @RagibZaman Ah yes! Sylvester's criterion gives the condition $r_{XY}^2 + r_{YZ}^2 + r_{XZ}^2 - 2 r_{XY} \, r_{YZ} \, r_{XZ} - 1 \leq 0$ (this being the determinant of the relevant $3\times 3$ minor). Which in turn means $r_{XZ}$ must be between $r_{XY} \, r_{YZ} \pm \sqrt{(1-r_{XY}^2)(1-r_{YZ}^2)}$. $\endgroup$
    – Gro-Tsen
    Commented Jun 12 at 14:59
  • 1
    $\begingroup$ As was pointed to me somewhere else, the Pearson correlation coefficient can also be viewed as the cosine of the angle of the value vectors normalized and rescaled (i.e., a spherical distance). So the inequality is then obtained by taking the triangle inequality on the angles and adding adding cosines. Which (taking into account the formula for $\cos(u\pm v)$) yields the same inequality as in the previous comment. $\endgroup$
    – Gro-Tsen
    Commented Jun 12 at 16:42

1 Answer 1

Reset to default
0
$\begingroup$

The question has been mostly answered in the comments, the bound being $$ r_{XY}^2 + r_{YZ}^2 + r_{XZ}^2 - 2 r_{XY} \, r_{YZ} \, r_{XZ} \leq 1 $$ by Sylvester's criterion. However, there are a few additional comments one can make on the convex 3D shape (elliptical tetrahedron or “elliptotope”) defined by this inequality: its boundary contains the vertices and edges of a regular tetrahedron, yet all its cross-sections perpendicular to one of the axes are ellipses. For more about this, see P. J. Rousseeuw & G. Molenberghs, “The Shape of Correlation Matrices”, The American Statistician, 48 (1994) 276–279.

This web page by Ben Bolker further explores this shape and how to plot it.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .