Solved – First principal component of 2D data forming a rectangle

What is the first principal component of points that form a "filled" rectangle in the 2D space?

Is it one of the diagonals? Or are the first two principal components basically the sides of the rectangle?

Imagine data points filling a 2D rectangle in the center of the coordinate system, with its sides oriented along the coordinate axes: from $-a$ to $a$ along the $x$-axis, and from $-b$ to $b$ along the $y$-axis.

The projection on $x$ is a uniform distribution with variance $a^2/3$. The projection on $y$ is also a uniform distribution with variance $b^2/3$. Since $x$ and $y$ are obviously not correlated (if this is not obvious, ask yourself whether the correlation should be positive or negative?.. due to symmetry it can only be zero), the covariance between them is zero. This yields the covariance matrix $$left(begin{array}{c}a^2/3&0\0&b^2/3end{array}right).$$ The task of PCA is to diagonalize the covariance matrix. But this one is already diagonal! This means that no rotation is necessary, and $x$-axis and $y$-axis are themselves principal axes. If e.g. $a>b$, then the $x$-axis is the first PC.

This might be a bit counter-intuitive: it might seem that a projection on the diagonal should have larger variance than the projection on the longer side; but it is in fact not so.

Bonus: Dzhanibekov effect

You seem to have meant a 3D rectangular parallelepiped instead of 2D rectangle. The arguments of course stay the same: covariance matrix is $3times 3$ but still diagonal with principal axes being the coordinate axes.

Incidentally, there is a curious effect in mechanics concerning rotating solid body with three different moments of inertia (which is a mechanics analog of variance). It turns out that rotations around the axes with the largest and the smallest moment of inertia are stable, but rotation around the axis with the middle moment of inertia is unstable. Moreover, a rotating body will experience sudden "flips", which is known as Dzhanibekov effect — after a Russian cosmonaut who observed it in space. One can easily observe it when spinning a book or a table tennis racket. See the following great threads on mathoverflow and on physics.SE and these videos:

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