lineNormalize

[snotskie]

Add an alternative to sphere normalization, which does not create an "umbrella" axis

One alternative is "line normalization". This normalizes units to the plane perp to 1, adjusting distances from 1 such that the new linear distance is equal to the original arc distance

As a proof of concept:

Before:

After:

Above, I've forced the x-axis to align moreorless with the "umbrella" axis. Notice the drop in variance explained as well as the change in shape

Here I've run a stock SVD rotation, before on the left, after on the right. Notice the change in the shape. The "bend" on the left was due to the normalization technique, not a meaningful feature of the data. The image on the right removes the bend, showing more meaningful variance

[snotskie]

staged in https://github.com/snotskie/EpistemicNetworkAnalysis.jl/commit/048af4777e72c37d899f2b6483680a342856a3a3

[snotskie]

Slight tweak, staged in https://github.com/snotskie/EpistemicNetworkAnalysis.jl/commit/063f9dcff2012bcebf82c7dcb57c8d6be5e0f5d6

Using the mean vector (normalized) instead of 1. Here we see a perfectly straight line, not because this makes things straighter, but because the method I was using to "show" the "umbrella" looked for the mean line, not 1

The impact on SVD was minimal in this case, but the choice of the mean vector instead of 1 might be more justifiable. The mean vector tends to be important for downstream steps, and there is no guarantee that 1 will be near any data

The impact on non-SVD rotations is shown below

Before

After