In this paper, the sparse sensor placement problem for least-squares estimation is considered, and the previous novel approach of the sparse sensor selection algorithm is extended. The maximization of the determinant of the matrix which appears in pseudo-inverse matrix operations is employed as an objective function of the problem in the present extended approach. The procedure for the maximization of the determinant of the corresponding matrix is proved to be mathematically the same as that of the previously proposed QR method when the number of sensors is less than that of state variables (undersampling). On the other hand, the authors have developed a new algorithm for when the number of sensors is greater than that of state variables (oversampling). Then, a unified formulation of the two algorithms is derived, and the lower bound of the objective function given by this algorithm is shown using the monotone submodularity of the objective function. The effectiveness of the proposed algorithm on the problem using real datasets is demonstrated by comparing with the results of other algorithms. The numerical results show that the proposed algorithm improves the estimation error by approximately 10% compared with the conventional methods in the oversampling case, where the estimation error is defined as the ratio of the difference between the reconstructed data and the full observation data to the full observation. For the NOAA-SST sensor problem, which has more than ten thousand sensor candidate points, the proposed algorithm selects the sensor positions in few seconds, which required several hours with the other algorithms in the oversampling case on a 3.40 GHz computer.

The problem of selecting an optimal set of sensors estimating a high-dimensional data is considered. Objective functions based on D-, A-, and E-optimality criteria of optimal design are adopted to greedy methods, that maximize the determinant, minimize the trace of the 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, a 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, greedy methods based on D-, A-, and E-optimality are applied to randomly generated systems and a practical dataset concerning the global climate; these correspond to an almost ideal and a practical case in terms of statistics, respectively. The D- and A-optimality-based greedy methods select better sensors. The E-optimality-based greedy method does not select better sensors in terms of the index of E-optimality in the oversample case, while the A-optimality-based greedy method unexpectedly does so in terms of the index of E-optimality. The poor performance of the E-optimality-based greedy method is due to the lack of submodularity in the E-optimality index and the better performance of the A-optimality-based greedy method is due to the relation between A- and E-optimality. Indices of D- and A-optimality seem to be important in the ideal case where the statistics for the system are well known, and therefore, the D- and A-optimality-based greedy methods are suitable for accurate reconstruction. On the other hand, the index of E-optimality seems to be critical in the more practical case where the statistics for the system are not well known, and therefore, the A-optimality-based greedy method performs best because of its superiority in terms of the index of E-optimality.

The present study proposes a sensor selection method based on the proximal splitting algorithm and the A-optimal design of experiment using the alternating direction method of multipliers (ADMM) algorithm. The performance of the proposed method was evaluated with a random sensor problem and compared with previously proposed methods, such as the greedy and convex relaxation methods. The performance of the proposed method is better than the existing greedy and convex relaxation methods in terms of the A-optimality criterion. Although, the proposed method requires a longer computational time than the greedy method, it is quite shorter than that of convex relaxation method in large-scale problems. Then the proposed method was applied to the data-driven sparse-sensor-selection problem. The dataset adopted was the National Oceanic and Atmospheric Administration optimum interpolation sea surface temperature dataset. At a number of sensors larger than that of the latent variables, the proposed method showed similar and better performance compared with previously proposed methods in terms of the A-optimality criterion and reconstruction error.

The present paper proposes a data-driven sensor selection method for a high-dimensional nondynamical system with strongly correlated measurement noise. The proposed method is based on proximal optimization and determines sensor locations by minimizing the trace of the inverse of the Fisher information matrix under a block-sparsity hard constraint. The proposed method can avoid the difficulty of sensor selection with strongly correlated measurement noise, in which the possible sensor locations must be known in advance for calculating the precision matrix for selecting sensor locations. The problem can be efficiently solved by the alternating direction method of multipliers, and the computational complexity of the proposed method is proportional to the number of potential sensor locations when it is used in combination with a low-rank expression of the measurement noise model. The advantage of the proposed method over existing sensor selection methods is demonstrated through experiments using artificial and real datasets.

Optimization of sensor selection has been studied to monitor complex and large-scale systems with data-driven linear reduced-order modeling. An algorithm for greedy sensor selection is presented under the assumption of correlated noise in the sensor signals. A noise model is given using truncated modes in reduced-order modeling, and sensor positions that are optimal for generalized least squares estimation are selected. The determinant of the covariance matrix of the estimation error is minimized by efficient one-rank computations in both underdetermined and overdetermined problems. The present study also reveals that the objective function with correlated noise is neither submodular nor supermodular. Several numerical experiments are conducted using randomly generated data and real-world data. The results show the effectiveness of the selection algorithm in terms of accuracy in the estimation of the states of large-dimensional measurement data.