Irregularly-spaced Samples Research Articles

Linking physics and spatial statistics: A new family of Boltzmann-Gibbs random fields

We investigate a connection between spatial statistics and statistical physics to obtain new covariance functions with direct physical interpretation for spatial random fields. These covariance functions are based on the exponential Boltzmann-Gibbs representation and use an energy functional to represent interactions between the values of the random field at different points in space. This formulation results in closed-form generalized covariance functions, which display infinite variance in Euclidean spaces of dimension larger than one. We propose regularization schemes in real and reciprocal (spectral) space that lead to well-behaved covariance structures. The real-space regularization parameter allows a continuous interpolation between the Boltzmann-Gibbs covariance and the exponential covariance. We also propose discretized approximations on regular grids, and we show that they represent reparametrized versions of the well-known Besag and Leroux lattice models. We then discuss parameter estimation and spatial prediction for the regularized Boltzmann-Gibbs covariance model in two dimensions. We recommend using the pairwise difference likelihood that combines satisfactory estimation performance and good scalability with many observation points. The predictive performance of the regularized covariance function is assessed by means of cross-validation statistics. Irregularly-spaced samples from the Walker Lake dataset are used, and spatial prediction is conducted by means of ordinary kriging. The regularized Boltzmann-Gibbs covariance yields improved predictive performance compared to the exponential covariance model.

Driving Factors Toward Accurate Mobile Opportunistic Sensing in Urban Environments

The dramatic increase in the number and sensing capabilities of mobile devices is fostering opportunistic sensing as a paramount data collection paradigm in smart cities. According to this paradigm, sensing of large-scale phenomena is autonomously performed by mobile devices that provide irregular samples in time and space. The collected data is then transferred to a central controller, and processed so as to obtain a representation of the phenomenon. In this paper, we investigate the factors that impact the accuracy of mobile opportunistic sensing. Specifically, we characterize the accuracy of a phenomenon representation obtained from samples collected by mobile devices and processed through the popular LMMSE filter. We do so by drawing on random matrix theory, which allows us to deal with irregularly spaced samples. Our analytical expressions capture the fundamental relationships existing between the accuracy and the parameters of mobile opportunistic sensing. We apply our analytical results to a realistic scenario where atmospheric pollution samples are collected by vehicular and pedestrian users. We validate the proposed analytical framework, and then exploit the model to investigate the impact on mobile sensing accuracy of a number of parameters. These include the pedestrian and vehicle density, the participation ratio to the sensing application, the type of phenomenon to be sensed, and the level of noise and position errors affecting the collected samples.

On Optimum Single-Tone Frequency Estimation Using Non-uniform Samples

Frequency estimation of a complex single-tone in additive white Gaussian noise from irregularly-spaced samples is addressed. In this Letter, we study the periodogram and weighted phase averager, which are standard solutions in the uniform sampling scenarios, for tackling the problem. It is shown that the estimation performance of both approaches can attain the optimum benchmark of the Cramer-Rao lower bound, although the former technique has a smaller threshold signal-to-noise ratio.

Maximum Likelihood Parameter Estimator for a Nonuniformly-Sampled Real Sinusoid

In this Letter, the maximum likelihood (ML) estimator for the parameters of a real sinusoid in additive white Gaussian noise using irregularly-spaced samples is derived. The ML frequency estimate is first determined by a one-dimensional search, from which optimum amplitude and phase estimates are then computed. It is shown that the estimation performance of the ML method can attain Cramer-Rao lower bound when the signal-to-noise ratio is sufficiently large.

Analytic Properties and Covariance Functions for a New Class of Generalized Gibbs Random Fields

Spartan Spatial Random Fields (SSRFs) are generalized Gibbs random fields, equipped with a coarse-graining kernel that acts as a low-pass filter for the fluctuations. SSRFs are defined by means of physically motivated spatial interactions and a small set of free parameters (interaction couplings). This paper focuses on the FGC-SSRF model, which is defined on the Euclidean space $\mathbb{R}^{d}$ by means of interactions proportional to the squares of the field realizations, as well as their gradient and curvature. The permissibility criteria of FGC-SSRFs are extended by considering the impact of a finite-bandwidth kernel. It is proved that the FGC-SSRFs are almost surely differentiable in the case of finite bandwidth. Asymptotic explicit expressions for the Spartan covariance function are derived for $d=1$ and $d=3$; both known and new covariance functions are obtained depending on the value of the FGC-SSRF shape parameter. Nonlinear dependence of the covariance integral scale on the FGC-SSRF characteristic length is established, and it is shown that the relation becomes linear asymptotically. The results presented in this paper are useful in random field parameter inference, as well as in spatial interpolation of irregularly-spaced samples.