### One-to-one (invertible) transformations

One-to-one transformations do not change the information content of the data but they are important for feature conditioning prior to PDF estimation. For 1:1 transformations, the J-function (2.3) reduces to the absolute value of the determinant of the Jacobian matrix (2.24),

The PDF projection theorem (2.2) may be thought of as a generalization of the well-known change of variables theorem from basic probability. Let , where is an invertible and differentiable multidimensional transformation. Then,

 (2.24)

where is the determinant of the Jacobian matrix of the transformation

Baggenstoss 2017-05-19