On the optimal stacking of noisy observations

TL;DR

This paper analyzes how different methods of stacking noisy observations affect the variance of spectrum estimators, finding that square-shaped stacking minimizes variance and that asymptotic differences diminish with more observations.

Contribution

It provides a comparative analysis of various stacking strategies for noisy observations and identifies optimal stacking configurations for minimizing estimator variance.

Findings

Square stacking minimizes variance regardless of observation count.

Asymptotic variance differences between stacking methods become negligible with many observations.

Vertical and horizontal stacking have higher asymptotic variance than square stacking.

Abstract

Observations where additive noise is present can for many models be grouped into a compound observation matrix, adhering to the same type of model. There are many ways the observations can be stacked, for instance vertically, horizontally, or quadratically. An estimator for the spectrum of the underlying model can be formulated for each stacking scenario in the case of Gaussian noise. We compare these spectrum estimators for the different stacking scenarios, and show that all kinds of stacking actually decreases the variance when compared to just taking an average of the observations. We show that, regardless of the number of observations, the variance of the estimator is smallest when the compound observation matrix is made as square as possible. When the number of observations grow, however, it is shown that the difference between the estimators is marginal: Two stacking scenarios where the number of columns and rows grow to infinity are shown to have the same variance asymptotically, even if the asymptotic matrix aspect ratios differ. Only the cases of vertical and horizontal stackings display different behaviour, giving a higher variance asymptotically. Models where not all kinds of stackings are possible are also discussed.