Knight Lee comments on The purposeful drunkard

[Knight Lee]

I think there is a typo somewhere, probably because you switched whether the $x$ vectors were rows or columns.

Based on the dimensions of the matrices, it should be $X=M_{upd}\cdot S$

And $X_{cent}=M_{upd}SC$

And I think $X_{cent}{X}_{cent}^T=M_{upd}SC{C}^T{S}^T{M}_{upd}^T$

Instead of $X_{cent}^T{X}_{cent}=M_{upd}^T{S}^T{C}^{T}CS{M}_{upd}$

$S$ should still be upper triangular.

Though don’t trust me either, I often do math in a hand-wavy fashion.

My intuition was that PCA selects the “angle” you view the data from which stretches out the data as much as possible, forcing the random walk to appear relatively straighter.

But somehow the random walk is smooth on a over a few data points, but still turns back and forth over the duration of $T$. This contradicts my intuition and I have no idea what’s going on.