Computing Principal Components Dynamically

In this paper we present closed-form solutions for efficiently updating the prin-cipal components of a set of n points, when m points are added or deleted fromthe point set. For both operations performed on a discrete point set in R, we cancompute the new principal components in O(m) time for fixed d. This is a signifi-cant improvement over the commonly used approach of recomputing the principalcomponents from scratch, which takes O(n+m) time. An important application ofthe above result is the dynamical computation of bounding boxes based on princi-pal component analysis. PCA bounding boxes are very often used in many fields,among others in computer graphics for collision detection and fast rendering. Wehave implemented and evaluated few algorithms for computing dynamically PCAbounding boxes in R3. In addition, we present closed-form solutions for comput-ing dynamically principal components of continuous point sets in R2 and R3. Inboth cases, discrete and continuous, to compute the new principal components, noadditional data structures or storage are needed.