Table 1 - uploaded by Man Sik Park
Content may be subject to copyright.
Source publication
Institute of Statistics Mimeo Series# 2584 SUMMARY Environmental spatial data often show complex spatial-temporal dependency structures that are difficult to model and estimate due to the lack of symmetry and other standard assumptions of the covariance function. In this study, we introduce certain types of symmetry in spatial-temporal processes: a...
Contexts in source publication
Context 1
... we explain the result from the analysis of PM 2.5 daily concentration dataset. The parameter (31) and the negative log-likelihood (− log(L)) are minimized using the routine nlminb in R. Table 1 shows the estimates of Θ from the WLS and the ML methods for each model (M.1-M.4). In terms of M.1 and M.2, the estimates are quite similar regardless of the estimation method except that the WLS estimate of partial sill (φ) is larger than the ML one. ...
Context 2
... we compare symmetric or asymmetric spatial-temporal coavariance models by means of the objective functions, WSSE and − log(L). Table 1 displays the criteria helpful to find out which model is best for this dataset. In terms of the objective functions, any asymmetric covariance model performs better than the symmetric one. ...
Context 3
... on the ML estimates for each model (see Table 1), we predict the PM 2.5 concentrations at the preassigned grids on the three consecutive dates, August 25, 2003 through August 27, 2003. Table 1 on September 1, 2003. ...
Citations
... Several versions of the Lagrangian spatio-temporal covariance functions have appeared since the seminal work of Cox and Isham (1988). Park and Fuentes (2006) adapted the modeling framework for axial symmetry in time, axial symmetry in space, and diagonal symmetry in space. Porcu et al. (2006) explored some anisotropic extensions and Christakos (2017) introduced an acceleration component. ...
When analyzing the spatio-temporal dependence in most environmental and earth sciences variables such as pollutant concentrations at different levels of the atmosphere, a special property is observed: the covariances and cross-covariances are stronger in certain directions. This property is attributed to the presence of natural forces, such as wind, which cause the transport and dispersion of these variables. This spatio-temporal dynamics prompted the use of the Lagrangian reference frame alongside any Gaussian spatio-temporal geostatistical model. Under this modeling framework, a whole new class was birthed and was known as the class of spatio-temporal covariance functions under the Lagrangian framework, with several developments already established in the univariate setting, in both stationary and nonstationary formulations, but less so in the multivariate case. Despite the many advances in this modeling approach, efforts have yet to be directed to probing the case for the use of multiple advections, especially when several variables are involved. Accounting for multiple advections would make the Lagrangian framework a more viable approach in modeling realistic multivariate transport scenarios. In this work, we establish a class of Lagrangian spatio-temporal cross-covariance functions with multiple advections, study its properties, and demonstrate its use on a bivariate pollutant dataset of particulate matter in Saudi Arabia.
... The limiting capabilities of separable models in capturing the complex correlation patterns in real-world processes motivated the geo-statistic community in the past decade to establish classes of non-separable, but still symmetric, models that capture spatio-temporal interactions [4], [10], as well as asymmetric non-separable models that, in addition to the interactions, account for the lack of spatio-temporal symmetry [6]- [8], [11]. By establishing an analogy between the atmospheric processes studied in the aforementioned literature and the wind field on a farm, one expects that the lack of symmetry is extended to the farm-level wind dynamics, and thus, anticipates a successful application of non-separable asymmetric models in short-term wind forecasting. ...
The massive amounts of spatio-temporal data collected in today's wind farms have created a necessity for accurate spatio-temporal models. Despite the growing recognition for non-separable spatio-temporal models, a significant reliance on separable, symmetric models is still the norm in today's renewable industry. We discover that the broad use of separable models is due to the handling of wind data in a setting that does not reveal their fine-scale spatio-temporal structure. The contribution of this research is two-fold. First, we devise a special pair of spatio-temporal “lens” that allows us to see the fine-scale spatio-temporal variations and interactions, and subsequently, we conclude that local wind fields exhibit strong signs of non-separability and asymmetry. Using one year of turbine-specific wind measurements, we show that asymmetry can in fact be detected in more than 93% of the time. Second, making use of the spatio-temporal lens, we propose an enhanced procedure for short-term wind speed forecast. Substantial improvements in forecast accuracy in both wind speed and wind power were observed. When combined with certain intelligent methods such as support vector machine, additional improvements are possible.
... For the purposes of this paper we are only concerned with the lower-triangle of the covariance matrix. For recent reviews of this topic see Stein (2005) and Park and Fuentes (2006). examples suggest three common models: connectivity that grows over time, hub and spoke growth, and connectivity that turns on in every other period. ...
We connect time varying spatial correlation patterns to examples in the theoretical and empirical literature. Then we use simulation experiments to compare the performance of estimation techniques that use spatial weights matrices (W) and those that do not. The results suggest that the pattern of time varying correlation does impact inference, but not as much as the W misspecification literature suggests. We find choosing the appropriate inferential method is less of a concern if the data generating process follows a hub-spoke correlation structure. Finally, we confirm earlier results that the cluster robust modifications proposed by Bester et al. (2011) perform well if the group sizes are chosen appropriately.
... Irish wind data (Gneiting et al., 2007). This motivates two lines of research: the development of parametric classes of covariance functions that allow space–time asymmetry and/or nonseparability (Cressie and Huang, 1999; Gneiting, 2002; Stein, 2005; Jun and Stein, 2007; Park and Fuentes, 2009) and formal statistical assessment of symmetry and separability for space–time data (Mitchell et al., 2005; Fuentes, 2006; Li et al., 2007, 2008; Park and Fuentes, 2008). ...
We propose a new nonparametric test to test for symmetry and separability of space-time covariance functions. Unlike the existing nonparametric tests, our test has the attractive convenience of being free of choosing any user-chosen number or smoothing parameter. The asymptotic null distributions of the test statistics are free of nuisance parameters and the critical values have been tabulated in the literature. From a practical point of view, our test is easy to implement and can be readily used by the practitioner. A Monte-Carlo experiment and real data analysis illustrate the finite sample performance of the new test.
... Their tests of symmetries are also based on certain ratios of spatial periodograms. However, these noteworthy studies are only applicable for spatial processes, not spatial-temporal ones and, therefore, no formal tests for lack of symmetry in spatialtemporal processes have been developed yet although the modeling of asymmetric spatialtemporal processes has been researched by Stein (2005), and Park and Fuentes (2006). ...
... If only one element in v is zero, then axial symmetry in space is satisfied. We call asymmetry in space and time, otherwise (see Park and Fuentes, 2006). Now we explain the fundamental simulation setup for realizing the tests. ...
... In reality, spatial or spatial-temporal processes may have strong evidence for lack of the symmetries introduced in this study. That may be a good motivation for taking asymmetric spatial-temporal covariance functions into account for explaining the given data better (see Park and Fuentes, 2006). ...
Symmetry and separability of spatial-temporal covariances are the main assumptions that are frequently taken for granted in most applications because of the simplicity of constructing covariance structure. However, many studies in environmental sciences show that real data have complex spatial-temporal dependency structures resulting from lack of symmetry or violation of other standard assumptions of the covariance function. In this study, we propose new formal tests for lack of symmetry by using spectral representations of the spatial-temporal covariance functions. The advantage of the proposed tests is that classical analysis of variance (ANOVA) models can be used for detecting lack of symmetry inherent in spatial-temporal processes. We evaluate the performance of the tests with simulation studies and we apply them to air pollution data.
When analyzing the spatio-temporal dependence in most environmental and earth sciences variables such as pollutant concentrations at different levels of the atmosphere, a special property is observed: the covariances and cross-covariances are stronger in certain directions. This property is attributed to the presence of natural forces, such as wind, which cause the transport and dispersion of these variables. This spatio-temporal dynamics prompted the use of the Lagrangian reference frame alongside any Gaussian spatio-temporal geostatistical model. Under this modeling framework, a whole new class was birthed and was known as the class of spatio-temporal covariance functions under the Lagrangian framework, with several developments already established in the univariate setting, in both stationary and nonstationary formulations, but less so in the multivariate case. Despite the many advances in this modeling approach, efforts have yet to be directed to probing the case for the use of multiple advections, especially when several variables are involved. Accounting for multiple advections would make the Lagrangian framework a more viable approach in modeling realistic multivariate transport scenarios. In this work, we establish a class of Lagrangian spatio-temporal cross-covariance functions with multiple advections, study its properties, and demonstrate its use on a bivariate pollutant dataset of particulate matter in Saudi Arabia. Supplementary materials for this article are available online.
