Formulation of Weighted Average Smoothing as a Projection of the Origin onto a Convex Polytope
This work addresses a specific optimization problem in signal processing or statistics, offering incremental improvements in smoothing techniques.
The paper tackles the problem of finding optimal weight windows for weighted moving average smoothing under squared loss, showing that the optimal window is symmetrical and can be formulated as projecting the origin onto a convex polytope, with analytical solutions provided under specific data conditions.
Our study focuses on determining the best weight windows for a weighted moving average smoother under squared loss. We show that there exists an optimal weight window that is symmetrical around its center. We study the class of tapered weight windows, which decrease in weight as they move away from the center. We formulate the corresponding least squares problem as a quadratic program and finally as a projection of the origin onto a convex polytope. Additionally, we provide some analytical solutions to the best window when some conditions are met on the input data.