A generalized convolution model and estimation for non-stationary random functions

Abstract

Standard geostatistical models assume second order stationarity of the underlying Random Function. In some instances, there is little reason to expect the spatial dependence structure to be stationary over the whole region of interest. In this paper, we introduce a new model for second order non-stationary Random Functions as a convolution of an orthogonal random measure with a spatially varying random weighting function. This new model is a generalization of the common convolution model where a non-random weighting function is used. The resulting class of non-stationary covariance functions is very general, flexible and allows to retrieve classes of closed-form nonstationary covariance functions known from the literature, for a suitable choices of the random weighting functions family. Under the framework of a single realization and local stationarity, we develop parameter inference procedure of these explicit classes of non-stationary covariance functions. From a local variogram non-parametric kernel estimator, a weighted local least-squares approach in combination with kernel smoothing method is developed to estimate the parameters. Performances are assessed on two real datasets: soil and rainfall data. It is shown in particular that the proposed approach outperforms the stationary one, according to several criteria. Beyond the spatial predictions, we also show how conditional simulations can be carried out in this non-stationary framework.

Fifty Years of Kriging

“…This locally stationary correlation function can be generalized by replacing expð − Q xx′ Þ by ρð − Q xx′ Þ where ρ is a stationary correlation function that is valid in every dimension. This class of nonstationary covariance functions can be fitted by using local variograms whose parameters are used to build local Σ x matrices (e.g., Fouedjio et al 2016). Emery and Arroyo (2018) describe a spectral algorithm for simulating such models.…”

Modeling PM2.5 Urban Pollution Using Machine Learning and Selected Meteorological Parameters

“…Convolutionbased models for spatial data have increased in popularity as a result of their flexibility in modeling spatial dependence and their ability to accommodate large datasets [27]. This generated Convolutional Generalization Model (CGM) [28] is an averaged value of the PM 2.5 pollution level (PL), in which the regional quantity of influence per data point is modeled as a 2D Gaussian matrix (see (2)). A Gaussian convolution is applied (i) to spatially interpolate data, in order to get a 2D representation from the points' coordinates calculated in (1) and (ii) to smooth the PL concentration values of this representation.…”

Predictive Geological Mapping Using Closed-Form Non-stationary Covariance Functions with Locally Varying Anisotropy: Case Study at El Teniente Mine (Chile)

“…(2) is that developed by Paciorek and Schervish (2006). Fouedjio et al (2016) shows that this class is obtained by convolving an orthogonal random measure with a spatially varying random weighting function. The intuition behind this class is that at each location x is assigned a matrix R x interpreted as a locally varying geometric anisotropy matrix.…”

“…Anderes and Stein (2011) propose a weighted local likelihood approach where the influence of faraway observations is smoothly down-weighted. Fouedjio et al (2016) propose a distribution-free approach involving a local stationary variogram kernel estimator, a weighted local least-squares method in combination with a kernel smoothing method.…”

“…Knowing the exact location of the pipe surface is important because it constitutes the internal limit of the deposit and the mining operation. To do this, a geostatistical approach based on closed-form nonstationary covariance functions with locally varying anisotropy proposed by Fouedjio et al (2016) is used. Previously, Fouedjio (2015) applied a nonstationary geostatistical approach based on space deformation to address this problem.…”