Under the section Understanding the Transformation Model there is an explanation of Variance and the Rotation. I’m unable to figure out where the following values came from :

Angle 45 degree

[0.166 0] Variance

x ~ U(1, 2)

I’m unable to relate these to the cell just above it.

In the second cell of the notebook, n is set to 1 and y to x.copy() * n. In other words, y = x.

The rotation required to arrive at uncorrelated features is 45 degrees, with one eigenvector [-0.70710678, -0.70710678] pointing along x = y, and one eigenvector perpendicular (i.e. uncorrelated) [-0.70710678, 0.70710678], pointing along x = -y.

x contains 1000 samples from a uniform random distribution with low value of 1 and high value of 2. This is notated as x ~ U(1, 2). The variance of such a distribution is 1/12*9((high value - low value)**2) as proven here, which in this case is 0.083333.

This variance is found in both x and y. And the value of 0.166465230e-01 indicates that this variance is captured by the first principal component, while the variance captured by the second component is near zero. This is to be expected because x = y.