# Rotational Rescaling and Disposable Dimensions

• Wayne L. Myers
• Ganapati P. Patil
Chapter
Part of the Environmental and Ecological Statistics book series (ENES, volume 6)

## Abstract

The scalings done thus far have treated the variates individually and have been of a relatively simple nature, i.e., either additive, multiplicative, or ranking. We now proceed to considering the data jointly among variates as a constellation of points in multidimensional space having as many perpendicular axes as there are variates (Manly 1998; Krzanowski 2000; Anderson 2003). A point along an axis one unit from the origin provides a basis case or basis vector for that axis. This is effectively an “artificial” point having a coordinate of 1.0 on that axis and 0.0 on all other axes. An additive rescaling shifts the constellation of points upward or downward relative to the respective axis. A multiplicative rescaling changes the magnitudes of values on an axis, thus effectively stretching or compressing the axis and expanding or shrinking the constellation of points relative to that axis. It is also possible (and useful) to consider rigidly rotating the axial framework itself, which does not change the overall multidimensional shape of the constellation and does not change the (Euclidian) distances between points but does give a different perspective view as seen along an axis. The rotational rescaling consists of determining the value along a repositioned axis at which a perpendicular projection of each data point will impinge on the new axis.

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