Effect of Objective Function on Data-Driven Greedy Sparse Sensor Optimization

TL;DR

This paper investigates how different objective functions influence greedy sensor selection for high-dimensional data, finding D-optimality offers the best balance of accuracy and efficiency, while A-optimality excels in eigenvalue metrics.

Contribution

It introduces a unified A-optimality formulation, proves its submodularity, and compares the effectiveness of D-, A-, and E-optimality in sensor selection.

Findings

D-optimality yields better determinant and reconstruction error.

A-optimality performs best for minimum eigenvalue.

E-optimality performs worse across all metrics.

Abstract

The selection problem of an optimal set of sensors estimating the snapshot of high-dimensional data is considered. The objective functions based on various criteria of optimal design are adopted to the greedy method: D-optimality, A-optimality, and E-optimality, which maximizes the determinant, minimize the trace of inverse, and maximize the minimum eigenvalue of the Fisher information matrix, respectively. First, the Fisher information matrix is derived depending on the numbers of latent state variables and sensors. Then, the unified formulation of the objective function based on A-optimality is introduced and proved to be submodular, which provides the lower bound on the performance of the greedy method. Next, the greedy methods based on D-, A-, and E-optimality are applied to randomly generated systems and a practical data set of global climates. The sensors selected by the D-optimality objective function works better than those by A- and E-optimality with regard to the determinant, trace of the inverse, and reconstruction error, while those by A-optimality works the best with regard to the minimum eigenvalue. On the other hand, the performance of sensors selected by the E-optimality objective function is worse for all indices and reconstruction error. This might be because of the lack of submodularity as proved in the paper. The results indicate that the greedy method based on D-optimality is the most suitable for high accurate reconstruction with low computational cost.