Smoothing with the Best Rectangle Window is Optimal for All Tapered Rectangle Windows

Abstract

We investigate the optimal selection of weight windows for the problem of weighted least squares. We show that weight windows should be symmetric around its center, which is also its peak. We consider the class of tapered rectangle window weights, which are nonincreasing away from the center. We show that the best rectangle window is optimal for such window definitions. We also extend our results to the least absolutes and more general case of arbitrary loss functions to find similar results.

An Auto-Regressive Formulation for Smoothing and Moving Mean with Exponentially Tapered Windows

“…Proof. G(x, y, α, β) is linear in the weights θ = (α, β), i.e., G(x, y, σα, σβ) = σG(x, y, α, β), (39) for ∀σ ∈ ℜ, and J(y, θ) is its minimization with respect to x; which concludes the proof from [58].…”

“…Traditionally, the most popular smoothing technique is the weighted moving average smoothing [58], where x is created by passing y through a weighted moving average (weighted mean) filter w = {w k } M k=−M of window size 2M + 1 ≤ N , where w is in a probability simplex, hence, the weights are positive, i.e.,…”

“…Moreover, in kernel smoothing [57], the estimates are created from the weighted averages of nearby data points using the appropriate kernel, where the relevant weighting is a tapered window, i.e., it diminishes from the middle peak. While the traditional problem of moving mean tapering window design has been studied [58], we focus on its extension where the moving mean can be expressed as an auto-regressive model.…”

Blind Source Separation for Mixture of Sinusoids with Near-Linear Computational Complexity

“…Hence, there is a need to jointly optimize the multi-tone parameters [68]. The reduction of interference have been studied extensively [23], [24], [28], [69], [70], where the observation is analyzed after multiplying with a tapering window [71], [72]. However, the incorporation of non-rectangle windows may be detrimental to the frequency estimation accuracy [68].…”

Robust, Nonparametric, Efficient Decomposition of Spectral Peaks under Distortion and Interference

“…To this end, joint optimization techniques have also been proposed [59], [67]. Moreover, the interference reducing methods are also a rich field of study [1], [19], [20], [68], [69], where the general approach is to apply a windowing function [70], [71] to limit the leakage. However, such windowing functions can result in decreased accuracy in frequency estimation because of the picket fence effect [67].…”