Fig. 1

Graphical illustration of the PB-transformation. Step 1: Estimate \(\hat \mu (\mathbf {Y})\) (i.e. the weighted mean of the original data), and subtract \(\hat \mu \) from all data. This process is an oblique (i.e. non-orthogonal) projection from \(\mathbb {R}^{n}\) to an (n−1)-dimensional subspace of \(\mathbb {R}^{n}\). The intermediate data from this step is Y⁽¹⁾, also called the centered data. If H₀ is true, Y⁽¹⁾ centers at the origin of the reduce space; otherwise, the data cloud Y⁽¹⁾ deviates from the origin. Step 2: Use eigen-decomposition to reshape the “elliptical” distribution to an “spherical” distribution. The intermediate data from this step is Y⁽²⁾. Step 3: Use QR-decomposition to find a unique rotation that transforms the original H-T problem to an equivalent problem. The equivalent problem tests for a constant deviation along the unit vector in the reduced space, thus it can be approached by existing parametric and rank-based methods. The final data from this step is \(\tilde {\mathbf {Y}}\)