Let $D = \begin{bmatrix} x_{11} & \cdot \cdot \cdot & x_{1d}\\ \cdot \cdot \cdot & \\ \cdot \cdot \cdot & \\ \cdot \cdot \cdot &\\ x_{N1} & \cdot \cdot \cdot & x_{Nd} \end{bmatrix} = \begin{bmatrix} X_{1}\\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ X_{N} \end{bmatrix} $

whose columns (or variables) are $\begin{bmatrix} x_{11}\\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ x_{N1} \end{bmatrix}, \cdot \cdot \cdot ,\begin{bmatrix} x_{1d}\\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ x_{Nd} \end{bmatrix}$

Let the mean associated with each variable in the data set D be given by the row vector $\bar{\mu} = \begin{bmatrix} \bar{\mu_{1}} &\cdot \cdot \cdot & \bar{\mu_{d}} \end{bmatrix}$

Suppose that the covariance matrix associated with the dataset $D$ yields a set $E$ of eigenvectors (column vectors) $E = {E_{1}, ..., E_{d}}$, all of which are mutually orthogonal since the symmetry of the covariance matrix guarantees the orthogonality of the set $E$ and whose corresponding eigenvectors are $\lambda_{1}, \cdot \cdot \cdot \lambda_{d}$

What does, for a data point $X$,

$\sum_{j = 1}^{d}\frac{|X - \bar{\mu}|\cdot\vec{E_{j}}}{\lambda_{j}}$

tells me? particularly, I am interested to know what the projection of $|X - \bar{\mu}|$ in the direction of an arbitrary $j^{th}$ eigenvector tells me.



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