# Alternative to visualize hyper-ellipsoid defined by variance-covariance matrix

Let say at the beginning I have two variables $a_{1}$ and $a_{2}$ of the same type then I would have the variance-covariance matrix defined as:

$\sum = \begin{bmatrix} \sigma^{2}_{a_{1}} & \sigma_{a_{1},a_{2}} \\ \sigma_{a_{2},a_{1}} & \sigma^{2}_{a_{2}} \end{bmatrix}$,

where diagonals are indication of variances and off-diagonals as covariances. I know that it is possible to visualize the confidence region by an error ellipse which its parameters would be defined as:

• the center of the ellipse is the mean (or expected value) of two variables,
• the orientation computed as $\alpha = \frac{1}{2}\tan^{-1}(\frac{2\sigma_{{a_{1}},a_{2}}}{\sigma^{2}_{a_{1}}-\sigma^{2}_{a_{2}}})$
• the axes length computed as squared root of eigenvalues of the covariance matrix.
• etc.

Now if I have let say, $n$ variables, where in this way I will have variance-covariance matirx of size $n\times n$:

$\sum = \begin{bmatrix} \sigma^{2}_{a_{1}} & \sigma_{a_{1},a_{2}} &\dots & \sigma_{a_{1},a_{n}}\\ \sigma_{a_{2},a_{1}} & \sigma^{2}_{a_{2}} & \dots & \sigma_{a_{2},a_{n}} \\ \vdots & \vdots & \ddots & \vdots\\ \sigma_{a_{n},a_{1}} & \sigma_{a_{n},a_{2}} & \dots & \sigma^{2}_{a_{n}} \end{bmatrix}$,

the confidence region would be a hyper-ellipsoid (with dimension of $n$ ) which is not possible to visualize. My question: is there any alternative way to visualize or map this region in 2-dimension?

• the projection of the original observations (those used to compute that n by n covariance matrix) projected on the direction corresponding to the largest and smallest (positive) eigenvectors of Sigma. Otherwise the so called pca-distance-distance plot. – user603 Mar 10 '14 at 19:08
• see for example page 32-33 here – user603 Mar 10 '14 at 19:10
• what I understand is that, Sigma can be decomposed into loading matrix P(nxn) and diagonal matrix L(nxn) containing the eigenvalues where Sigma = PLP' and score elements as T = P'x (x as original observations). In matlab I've written the code which computes OD (orthogonal distance) and SD (score distance). but what I am not sure is that if I have observation vector (Nx1) and sigma(NxN) then I will have one pair of SD-OD ? is that right? – pasha Mar 11 '14 at 17:07
• yes! (the rest of this comment is because of the lower bound on the length of a comment) – user603 Mar 11 '14 at 17:26
• Do you think the procedure that I wrote the code is right ? because I am hesitating if pca distance-distance representation is useful for me. – pasha Mar 11 '14 at 17:54