Improvement of Stability in Cluster Analysis and Principal Components Analysis by Special Weighting the Variables
Hans‐Joachim Mucha · 1992
Often the Euclidean metric applied to raw or standardized data leads to a bad result in cluster analysis (as well as in principal components analysis). A better one can be obtained in almost every case by using specific or adaptive weights. For instance the weights q jj =1/x̄ j 2 can be used for nonnegative values x ij (instead of q jj = 1/s j 2 ) in the squared weighted Euclidean distance $$d_Q^2({x_i},{x_{i'}}) = ({x_i} - {x_{i'}})'Q({x_i} - {x_{i'}}) = \left\| {{x_i} - {x_{i'}}} \right\|_Q^2$$ between two observations x i and x i ′. Here Qis diagonal, x̄ j and s j 2 are the mean value and the variance of variable j, respectively. In consequence these specific metrics should be used in principal components analysis too. In that way the principal components plot gives a good support in the interpretation of the results of cluster analysis.