Projection and Correlation Approximation (PCA)

In the PCA method, a transformation $x \to y$ is used [1]. The monotonic function, $y_i=y(x_i)$ transforms a variable $x_i$ that has a distribution function $p(x_i)$ to the variable $y_i$ that follows a normal Gaussian distribution function (of mean 0 and variance 1). Thus

$$y(x_i) =\sqrt{2} {\rm erf}^{-1}\left[2F(x_i)-1 \right] .$$

Here $F(x_i)$ is the cumulative function of $x_i$ and erf$^{-1}$ is the inverse error function. The PCA joint PDF is

$$P(\vec{x})={e^{-{1\over2} \vec{y}^T(U^{-1}-I)\vec{y}} \over \vert U \vert^{1/2}} \prod_{i=1}^N p(x_i) ,$$ (2)

where $U$ is the covariance matrix for $\vec{y}$ and $I$ is the identity matrix.