  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.
