Search

In Gaussian measurement models, the measurements are given by a known function of the unknown parameter vector, contaminated by additive zero-mean Gaussian noise. When the function is linear, the resulting maximum likelihood estimate (MLE) is well-known to be efficient [unbiased, with a mean square estimation error (MSE) matrix attaining the Cramér–Rao lower bound (CRLB)]. However, when the function is nonlinear, the MLE is only asymptotically efficient. The classical derivation of its asymptotic efficiency uses a first-order perturbation analysis, relying on a “small-errors” assumption, which under subasymptotic conditions turns inaccurate, rendering the MLE generally biased and inefficient. Although a more accurate (higher-order) performance analysis for such cases is of considerable interest, the associated derivations are rather involved, requiring cumbersome notations and indexing. Building on the recent assimilation of tensor computations into signal processing literature, we exploit the tensor formulation of higher-order derivatives to derive a tractable formulation of a higher (up to third-) order perturbation analysis, predicting the bias and MSE matrix of the MLE of parameter vectors in general nonlinear models under subasymptotic conditions. We provide explicit expressions depending on the first three derivatives of the nonlinear measurement function, and demonstrate the resulting ability to predict the “efficiency gap” (relative excess MSE beyond the CRLB) in simulation experiments. We also provide MATLAB code for easy computation of our resulting expressions. [ABSTRACT FROM AUTHOR]

Adaptive sampling theory has shown that, with proper assumptions on the signal class, algorithms exist to reconstruct a signal in \mathbb R^d with an optimal number of samples. We generalize this problem to the case of spatial signals, where the sampling cost is a function of both the number of samples taken and the distance traveled during estimation. This is motivated by our work studying regions of low oxygen concentration in the Great Lakes. We show that for one-dimensional threshold classifiers, a tradeoff between the number of samples taken and distance traveled can be achieved using a generalization of binary search, which we refer to as quantile search. We characterize both the estimation error after a fixed number of samples and the distance traveled in the noiseless case, as well as the estimation error in the case of noisy measurements. We illustrate our results in both simulations and experiments and show that our method outperforms existing algorithms in a large range of sampling scenarios. [ABSTRACT FROM AUTHOR]