Ex. 14.1

Weights for clustering. Show that weighted Euclidean distance

d_{e}^{(w)} (x_{i}, x_{i^{'}}) = \frac{\sum_{l = 1}^{p} w_{l} (x_{i l} - x_{i^{'} l})^{2}}{\sum_{l = 1}^{p} w_{l}}

satisfies

d_{e}^{(w)} (x_{i}, x_{i^{'}}) = d_{e} (z_{i}, z_{i^{'}}) = \sum_{l = 1}^{p} (z_{i l} - z_{i^{'} l})^{2},

where

z_{i l} = x_{i l} \cdot {(\frac{w_{l}}{\sum_{l = 1}^{p} w_{l}})}^{1 / 2} .

Thus weighted Euclidean distance based on $x$ is equivalent to unweighted Euclidean distance based on $z$ .

Soln. 14.1

By definition of $z_{i l}$ we have

\begin{array}{rcl} d_{e} (z_{i}, z_{i^{'}}) & = & \sum_{l = 1}^{p} (\frac{w_{l}}{\sum_{l = 1}^{p} w_{l}}) (x_{i l} - x_{i^{'} l})^{2} \\ = & \frac{\sum_{l = 1}^{p} w_{l} (x_{i l} - x_{i^{'} l})^{2}}{\sum_{l = 1}^{p} w_{l}} \\ = & d_{e}^{(w)} (x_{i}, x_{i^{'}}) . \end{array}