ZCA

PCA is a technique that decorrelates the components, but this does not prevent their variances from being different. If the principal components are rescaled so that they all have unit variance (see also 2.4.1), the distribution is said to be whitened, and the procedure is called whitening.

The whitening matrix $\mathbf W$ is a linear transformation such that

$$\cov (\mathbf{Y})=\mathbf{I}.$$ (2.76)

After whitening, the data have zero mean, uncorrelated components, and an identity covariance matrix.

Starting from the eigendecomposition

$$\boldsymbol\Sigma = \mathbf V \boldsymbol\Delta \mathbf V^\top$$ (2.77)

whitening in PCA space is obtained as

$$\mathbf X_{PCA} = \mathbf V^\top \mathbf X^\top$$ (2.78)

and subsequently

$$\mathbf X_{white} = \boldsymbol\Delta^{-1/2} \mathbf X_{PCA}.$$ (2.79)

Since

$$\boldsymbol\Delta = \frac1n\mathbf S^2,$$ (2.80)

the transformation can also be written, up to a constant scale factor, in terms of the singular values.

The ZCA transformation (Zero-Phase Component Analysis) finally maps the whitened data back to the original coordinate system:

$$\mathbf X_{ZCA} = \mathbf V \boldsymbol\Delta^{-1/2} \mathbf V^\top \mathbf X^\top.$$ (2.81)

Unlike the PCA-whitened representation, the ZCA transformation preserves the original orientation of the data as much as possible and maintains the same dimensionality as the initial space.