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(a) Perspective plot of the covariance function C(h, u) in (1.2); (b) Crosssectional plots of C(h, u) vs. u for fixed values h = 0, 0.7, 1.5, 3.0, shown from top to bottom, respectively.
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Gneiting (2002) proposed a nonseparable covariance model for spatial-temporal data. In the present paper we show that in certain circumstances his model possesses a counterintuitive dimple. In some cases, the magnitude of the dimple can be nontrivial. Copyright 2011, Oxford University Press.
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... dimple in Gneiting's model does not seem to have been explicitly recognized before in the literature, though there are visual hints in some of the published figures, e.g. Cressie and Huang (1999, Fig. 2), and Gneiting (2002, Fig. 1), which are based on the model (1.2). More recently Gneiting's covariance function has used as building block in more complicated models; see e.g. Kolovos et al. (2004) and Porcu et al. ...
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... Therefore, taking the motion into account, the Eulerian version of spatiotemporal correlation of radar fields depends on the motion of the storm and changes in structure that occur within it, while the Lagrangian variant is defined relative to the storm coordinates and therefore is independent of the storm motion, which is indeed filtered out in Equation 5 (Zawadzki, 1973). An important consequence of the equivalence in Equation 6 is that a Lagrangian STCS does not peak at 0 for 0. A first description of this characteristic of Lagrangian covariance functions, which is called dimple effect, is due to Kent et al. (2011), while Cuevas et al. (2017 provided a discussion for the Gneiting class of STCFs. In essence, a STCS has a dimple if the rv ( , ) s here now t is more correlated with ( , ) s there then t than with ( , ) s there now t . ...
Realistic stochastic simulation of hydro-environmental fluxes in space and time, such as rainfall, is challenging yet of paramount importance to inform environmental risk analysis and decision making under uncertainty. Here, we advance random fields simulation by introducing the concepts of general velocity fields and general anisotropy transformations. This expands the capabilities of the so-called Complete Stochastic Modeling Solution (CoSMoS) framework enabling the simulation of random fields (RF's) preserving: (a) any non-Gaussian marginal distribution, (b) any spatiotemporal correlation structure (STCS), (c) general advection expressed by velocity fields with locally varying speed and direction, and (d) locally varying anisotropy. We also introduce new copula-based STCS's and provide conditions guaranteeing their positive definiteness. To illustrate the potential of CoSMoS, we simulate RF's with complex patterns and motion mimicking rainfall storms moving across an area, spiraling fields resembling weather cyclones, fields converging to (or diverging from) a point, and colliding air masses. The proposed methodology is implemented in the freely available CoSMoS R package.
... Section 6 provides a short discussion and concluding remarks. Kent et al. (2011) first defined the dimple property for scalar-valued space-time covariance functions when referring to the fact that, for a fixed spatial lag, the temporal margin of the covariance function might have a local maximum away from the origin. Subsequent contributions about dimples in the univariate context can be found in Mosammam (2015), Alegría and Porcu (2017) and Cuevas et al. (2017). ...
Modeling the spatial correlation structure of coregionalized data is a frequent task in numerous fields of the natural sciences. Even in the isotropic case, experimental covariances may exhibit complex features, such as a maximum cross-correlation attained at non-collocated locations (dimple or hole effect). Current construction principles for multivariate covariance models on Euclidean spaces do not allow accounting for such a property. We propose a spectral approach to modify cross-covariance functions of the isotropic bivariate Matérn model in order to obtain a cross-dimple. Our model admits analytic expressions in terms of special functions. Our findings are illustrated through applications to data sets from the fields of mining and geochemistry.
... This fact can be explained by following arguments in Lantuéjoul (1994): sample variograms of spectral realizations for a given space-time lag (h, u) have a higher variance than those obtained with the substitution approach, although their expectations are the same and equal to γ (h, u). Also note the dimple (hole effect) of the temporal variogram associated with the space lag h = (10, 10), a well-known property of the Gneiting model that arises even when the function ψ is monotonic (Kent et al. 2011;Cuevas et al. 2017). ...
Two algorithms are proposed to simulate space-time Gaussian random fields with a covariance function belonging to an extended Gneiting class, the definition of which depends on a completely monotone function associated with the spatial structure and a conditionally negative definite function associated with the temporal structure. In both cases, the simulated random field is constructed as a weighted sum of cosine waves, with a Gaussian spatial frequency vector and a uniform phase. The difference lies in the way to handle the temporal component. The first algorithm relies on a spectral decomposition in order to simulate a temporal frequency conditional upon the spatial one, while in the second algorithm the temporal frequency is replaced by an intrinsic random field whose variogram is proportional to the conditionally negative definite function associated with the temporal structure. Both algorithms are scalable as their computational cost is proportional to the number of space-time locations that may be irregular in space and time. They are illustrated and validated through synthetic examples.
... For isotropic covariance functions, this implies that, for a fixed r o > 0, the functions φ(r o , Á) in (2) and ψ(r o , Á) in (3) are no longer monotonically decreasing (for this last case, we obviously require r o ≤ π), thus resulting in a possibly counterintuitive property. A first description of a dimple is due to Kent, Mohammadzadeh, and Mosammam (2011). More recently, a description of dimples through contour curves has been provided by Cuevas, Porcu, and Bevilacqua (2017). ...
... According to Rodrigues and Diggle (2010), the Gneiting class is always negative nonseparable. Also, Kent et al. (2011) and more recently Cuevas et al. (2017) show conditions on the functions f and h such that a dimple can happen. In particular, Cuevas et al. (2017) offer a dual view of the dimple problem related to space-time correlation functions in terms of their contours. ...
In this article, we provide a comprehensive review of space–time covariance functions. As for the spatial domain, we focus on either the d‐dimensional Euclidean space or on the unit d‐dimensional sphere. We start by providing background information about (spatial) covariance functions and their properties along with different types of covariance functions. While we focus primarily on Gaussian processes, many of the results are independent of the underlying distribution, as the covariance only depends on second‐moment relationships. We discuss properties of space–time covariance functions along with the relevant results associated with spectral representations. Special attention is given to the Gneiting class of covariance functions, which has been especially popular in space–time geostatistical modeling. We then discuss some techniques that are useful for constructing new classes of space–time covariance functions. Separate treatment is reserved for spectral models, as well as to what are termed models with special features. We also discuss the problem of estimation of parametric classes of space–time covariance functions. An outlook concludes the paper.
This article is categorized under:
• Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data
• Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods
• Statistical and Graphical Methods of Data Analysis > Multivariate Analysis
Abstract
A separable covariance function (left) and a space‐time covariance function with dynamical compact support.
... A cross-dimple is present if Z 1 (s here ) is more correlated with Z 2 (s there ) than with Z 2 (s here ). Although the occurrence of a dimple-like behaviour was originally studied for the temporal margins of space-time covariance functions (Kent et al., 2011;Mosammam, 2015;Alegría and Porcu, 2017;Cuevas et al., 2017), we have noticed that the cross-covariance functions of (purely spatial) multivariate random fields can also possess this property. For anisotropic random fields, this behaviour can be attributed to a spatial delay between the components (Li and Zhang, 2011;Alegría et al., 2018). ...
Multivariate random fields allow to simultaneously model multiple spatially indexed variables, playing a fundamental role in geophysical, environmental and climate disciplines. This paper introduces the concept of cross-dimple for bivariate isotropic random fields on spheres, and proposes an approach to build parametric models that possess this attribute. Our findings are based on the spectral representation of the matrix-valued covariance function. We show that our construction is compatible with both the negative binomial and circular-Matérn bivariate families of covariance functions. We illustrate through simulation experiments that the models proposed in this work allow to achieve improvements in terms of predictive performance when a dimple-like intrinsic structure is present.
... A random field simulated from the frozen field model can exhibit a dimple effect. A dimple effect points to a phenomenon where the covariance between Z (s 1 , t 1 ) and Z (s 2 , t 2 ) is stronger than that of Z (s 1 , t 1 ) and Z (s 2 , t 1 ), where t 2 = t 1 + 1; see Kent et al. (2011). The frozen and non-frozen models, as well as the dimple effect, are physically justifiable by observations influenced by transport phenomena that are usually caused by predominant winds, waves, and flows, to name but a few. ...
In multivariate spatio-temporal analysis, we are faced with the formidable challenge of specifying a valid spatio-temporal cross-covariance function, either directly or through the construction of processes. This task is difficult as these functions should yield positive definite covariance matrices. In recent years, we have seen a flourishing of methods and theories on constructing spatio-temporal cross-covariance functions satisfying the positive definiteness requirement. A subset of those techniques produced spatio-temporal cross-covariance functions possessing the additional feature of nonstationarity. Here we provide a review of the state-of-the-art methods and technical progress regarding model construction. In addition, we introduce a rich class of multivariate spatio-temporal asymmetric nonstationary models stemming from the Lagrangian framework. We demonstrate the capabilities of the proposed models on a bivariate reanalysis climate model output dataset previously analyzed using purely spatial models. Furthermore, we carry out a cross-validation study to examine the advantages of using spatio-temporal models over purely spatial models. Finally, we outline future research directions and open problems.
... property of the Gneiting model that arises even when the function ψ is monotonic (Kent et al., 2011;Cuevas et al., 2017). ...
Two algorithms are proposed to simulate space-time Gaussian random fields with a covariance function belonging to an extended Gneiting class, the definition of which depends on a completely monotone function associated with the spatial structure and a conditionally negative definite function associated with the temporal structure. In both cases, the simulated random field is constructed as a weighted sum of cosine waves, with a Gaussian spatial frequency vector and a uniform phase. The difference lies in the way to handle the temporal component. The first algorithm relies on a spectral decomposition in order to simulate a temporal frequency conditional upon the spatial one, while in the second algorithm the temporal frequency is replaced by an intrinsic random field whose variogram is proportional to the conditionally negative definite function associated with the temporal structure. Both algorithms are scalable as their computational cost is proportional to the number of space-time locations, which may be unevenly spaced in space and/or in time. They are illustrated and validated through synthetic examples.
... For instance, Model 3 has an interesting behaviour, known as dimple effect, which means that for certain predetermined distance θ 0 ∈ [0, π], the covariance is not a monotonically decreasing function of the temporal lag u. This property has been studied on Euclidean spaces by Kent et al. (2011). They have proved that under certain hypothesis the Gneiting class presents this effect. ...
... They have proved that under certain hypothesis the Gneiting class presents this effect. Kent et al. (2011) argue that in some cases this property can be counterintuitive, but we believe that in many processes influenced by prevailing winds or ocean currents, this effect is consistent. Dimple effects have been studied also for random fields on spheres across time by Alegria and Porcu (2016), for covariances arising from transport phenomena. ...
The construction of valid and flexible cross-covariance functions is a fundamental task for modeling multivariate space-time data arising from climatological and oceanographical phenomena. Indeed, a suitable specification of the covariance structure allows to capture both the space-time dependencies between the observations and the development of accurate predictions. For data observed over large portions of planet Earth it is necessary to take into account the curvature of the planet. Hence the need for random field models defined over spheres across time. In particular, the associated covariance function should depend on the geodesic distance, which is the most natural metric over the spherical surface. In this work, we propose a flexible parametric family of matrix-valued covariance functions, with both marginal and cross structure being of the Gneiting type. We additionally introduce a different multivariate Gneiting model based on the adaptation of the latent dimension approach to the spherical context. Finally, we assess the performance of our models through the study of a bivariate space-time data set of surface air temperatures and precipitations.
... We offer a dual view of the dimple problem related to space-time correlation functions in terms of their contours. We find that the dimple property (Kent et al., 2011) in the Gneiting class of correlations is in oneto-one correspondence with nonmonotonicity of the parametric curve describing the associated contour lines. Further, we show that given such a nonmonotonic parametric curve associated with a given level set, all the other parametric curves at smaller levels inherit the nonmonotonicity. ...
... Throughout the paper, δ is fixed. We follow Kent et al. (2011) and assume without loss of generality that ψ(0) = ϕ(0) = 1. Also, we assume that ϕ(t) → 0 as t → ∞ and that ψ(t) → ∞ as t → ∞. ...
... Moreover, ϕ −1 will be strictly decreasing in its domain. Kent et al. (2011) describes the dimple property in a space-time correlation as follows: Z(x here , u now ) is more correlated with Z(x there , u tomorrow ) than with Z(x there , u now ). Depending on the choice of ϕ and ψ in (2), the Gneiting class can have a dimple. ...
We offer a dual view of the dimple problem related to space-time correlation functions in terms of their contours.We find that the dimple property (Kent et al., 2011) in the Gneiting class of correlations is in oneto-one correspondence with nonmonotonicity of the parametric curve describing the associated contour lines. Further, we show that given such a nonmonotonic parametric curve associated with a given level set, all the other parametric curves at smaller levels inherit the nonmonotonicity. We propose a modified Gneiting class of correlations having monotonically decreasing parametric curves and no dimple along the temporal axis.
... 1331). A general concern about the absence of interaction has lead to substantial effort being devoted to developing flexible and parametric stochastic processes with non-separable covariance functions (see, e.g., Cressie and Huang, 1999;Iaco et al., 2002;Gneiting, 2002;Stein, 2005;Kent et al., 2011). Note, to avoid confusion, that this type of interaction is different from that modelled in decompositions of the type ...
Many applications require stochastic processes specified on two-or higher-dimensional domains; spatial or spatial-temporal modelling, for exam-ple. In these applications it is attractive, for conceptual simplicity and computational tractability, to propose a covariance function that is sep-arable; e.g. the product of a covariance function in space and one in time. This paper presents a representation theorem for such a proposal, and shows that all processes with continuous separable covariance functions are second-order identical to the product of second-order uncorrelated processes. It discusses the implications of separable or nearly separable prior covariances for the statistical emulation of complicated functions such as computer codes, and critically reexamines the conventional wis-dom concerning emulator structure, and size of design.
