## Data Normalization

An application of the floating reference hypothesis is normalization of data prior to evaluation of the SPA for a fixed reference hypothesis. Suppose we would like to evaluate

 (2.16)

for arbitrary vectors . Let be an estimate of the scale of . In practice, is a sample variance, sample mean, or standard deviation estimate. Important is that as is scaled, all the elements of will vary in proportion to . Let be a reference hypothesis that depends on this scaling. As long as the feature contains , or can be computed from , then remains in the ROS of . Thus, equation (2.13) is theoretically independent of ,

If the elements of are linearly related to , we may write

where is the dimension of . Therefore,

 (2.17)

which provides a convenient way to normalize prior to calculating the SPA.

Baggenstoss 2017-05-19