I'm trying to get some insight into how a "body" of 2D points is laid out. I represent my points as two vectors, $X$ and $Y$.
I'd like to be able to define the axis along which the points are most "spread out", e.g.:
I can easily get $varX$ and $varY$, which tell me something about how spread out the points are with respect to the $x$ and $y$ directions. And I could also get $var(.5X+.5Y)$ or $var(.5X-.5Y)$, which tell me the same thing for the diagonals.
I'm interested in finding $max_{(a,b)}var(aX + bY)$ where $a+b = 1$.
My goal is to use $a$ and $b$ to create an ellipse centered at the mean location of the group of points, with its major axis having slope $\frac{a}{b}$.
What's the best way to find $a$ and $b$?
Maybe your answer is PCA?
$(a,b)$ is calculated straightfordward from the first component of the PCA.