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Asymmetric Space-Time Covariance Functions via Hierarchical Mixtures

Pulong Ma

TL;DR

This work introduces a hierarchical mixture framework to construct asymmetric space-time covariance functions with closed-form expressions by layering location and scale mixtures over a base spatial covariance. The two perspectives—space-time process representation and covariance function representation—facilitate interpretable, tractable models that incorporate Lagrangian-type asymmetry and tunable smoothness and tail behavior. The authors develop concrete models (Lagrangian Matérn, Lagrangian CH) and a general Lag form with positive definiteness guarantees, along with GL-Matérn/GL-CH generalizations, and demonstrate their advantages on Irish wind and U.S. air-temperature data. Through theoretical results and real-data applications, the paper shows hierarchical mixtures bridge existing models and provide flexible, physically meaningful covariances capable of capturing complex space-time dependence, including long-range effects in time and space.

Abstract

This work is focused on constructing space-time covariance functions through a hierarchical mixture approach that can serve as building blocks for capturing complex dependency structures. This hierarchical mixture approach provides a unified modeling framework that not only constructs a new class of asymmetric space-time covariance functions with closed-form expressions, but also provides corresponding space-time process representations, which further unify constructions for many existing space-time covariance models. This hierarchical mixture framework decomposes the complexity of model specification at different levels of hierarchy, for which parsimonious covariance models can be specified with simple mixing measures to yield flexible properties and closed-form derivation. A characterization theorem is provided for the hierarchical mixture approach on how the mixing measures determine the statistical properties of covariance functions. Several new covariance models resulting from this hierarchical mixture approach are discussed in terms of their practical usefulness. A theorem is also provided to construct a general class of valid asymmetric space-time covariance functions with arbitrary and possibly different degrees of smoothness in space and in time and flexible long-range dependence. The proposed covariance class also bridges a theoretical gap in using the Lagrangian reference framework. The superior performance of several new parsimonious covariance models over existing models is verified with the well-known Irish wind data and the U.S. air temperature data.

Asymmetric Space-Time Covariance Functions via Hierarchical Mixtures

TL;DR

This work introduces a hierarchical mixture framework to construct asymmetric space-time covariance functions with closed-form expressions by layering location and scale mixtures over a base spatial covariance. The two perspectives—space-time process representation and covariance function representation—facilitate interpretable, tractable models that incorporate Lagrangian-type asymmetry and tunable smoothness and tail behavior. The authors develop concrete models (Lagrangian Matérn, Lagrangian CH) and a general Lag form with positive definiteness guarantees, along with GL-Matérn/GL-CH generalizations, and demonstrate their advantages on Irish wind and U.S. air-temperature data. Through theoretical results and real-data applications, the paper shows hierarchical mixtures bridge existing models and provide flexible, physically meaningful covariances capable of capturing complex space-time dependence, including long-range effects in time and space.

Abstract

This work is focused on constructing space-time covariance functions through a hierarchical mixture approach that can serve as building blocks for capturing complex dependency structures. This hierarchical mixture approach provides a unified modeling framework that not only constructs a new class of asymmetric space-time covariance functions with closed-form expressions, but also provides corresponding space-time process representations, which further unify constructions for many existing space-time covariance models. This hierarchical mixture framework decomposes the complexity of model specification at different levels of hierarchy, for which parsimonious covariance models can be specified with simple mixing measures to yield flexible properties and closed-form derivation. A characterization theorem is provided for the hierarchical mixture approach on how the mixing measures determine the statistical properties of covariance functions. Several new covariance models resulting from this hierarchical mixture approach are discussed in terms of their practical usefulness. A theorem is also provided to construct a general class of valid asymmetric space-time covariance functions with arbitrary and possibly different degrees of smoothness in space and in time and flexible long-range dependence. The proposed covariance class also bridges a theoretical gap in using the Lagrangian reference framework. The superior performance of several new parsimonious covariance models over existing models is verified with the well-known Irish wind data and the U.S. air temperature data.

Paper Structure

This paper contains 26 sections, 5 theorems, 46 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. A Hierarchical Mixture Modeling Framework
  3. Space-Time Process Representation
  4. Main Statistical Properties
  5. Validity of Covariance Models via Hierarchical Mixtures
  6. Smoothness and Tail Behaviors
  7. A New Class of Asymmetric Space-Time Covariance Models
  8. A Lagrangian Matérn Model
  9. A Lagrangian CH Model
  10. A General Form of Asymmetric Covariance Models
  11. Hierarchical Mixtures for Existing Covariance Models
  12. Applications
  13. Irish Wind Data
  14. Air Temperature Data
  15. Discussion
  16. ...and 11 more sections

Key Result

Theorem 1

Suppose that $\mu$ is a nonnegative measure on $\mathbb{R}^d$, $\nu$ is a nonnegative measure on $\mathbb{R}^p$, and $C_s$ is a stationary covariance function in $\mathbb{R}^d$ that is positive definite. For any fixed $\boldsymbol \ell:=(\mathbf{s}, t) \in \mathbb{R}^d\times \mathbb{R}$, let $f(\cdo where $\mathbf{h}:=\mathbf{s}_1 - \mathbf{s}_2$, $u:=t_1 - t_2$, $\boldsymbol \zeta$ denotes the ve

Figures (4)

  • Figure 1: Contour plots of asymmetric space-time Matérn models over the space-time domain $[-2,2]\times [-2, 2]$. Horizontal axis represents spatial lag and vertical axis represents temporal lag. The first column corresponds to the frozen asymmetric Matérn models with different parameter settings. The second and third columns correspond to the (non-frozen) Lagrangian Matérn models with different parameter settings for $\boldsymbol \lambda$ and $\boldsymbol \Lambda$. All panels share the same Matérn parameters $\sigma^2=1, \nu=0.5$. The range parameter $\phi$ in each row are the same. The other parameters are specified in the title of each panel.
  • Figure 2: Contour plots of asymmetric CH models over the space-time domain $[-2,2]\times [-2, 2]$. The first column corresponds to the frozen asymmetric CH models. The second and third columns correspond to the (non-frozen) Lagrangian CH models with different parameter settings for $\boldsymbol \lambda$ and $\boldsymbol \Lambda$. All panels share the same parameters $\sigma^2=1, \nu=0.5, \alpha=0.2$. Each row shares the same range parameter $\beta$ and tail decay parameter $\alpha$. All other parameter settings are specified in the title of each panels.
  • Figure 3: Contour plots of GL-Matérn (first row) and GL-CH models (second row) over the space-time domain $[-2,2]\times [-2, 2]$. All panels share the same parameters $\sigma^2=1, \nu=0.5, \boldsymbol \Lambda = 1$. The GL-Matérn model has asymmetric parameters $\rho=0.2$ and the GL-CH model has asymmetric parameters $\rho=-0.5$. All other parameter settings are specified in the title of each panels.
  • Figure 4: Empirical space-time semivariogram differences at temporal lags $k=1, 2, 3$. Both first row and second row show $\delta(\mathbf{s}_j, \mathbf{s}_1; k)$ on the vertical axis against longitudinal difference in $\mathbf{s}_j - \mathbf{s}_1$ over all locations $j=1,\ldots, n_s$. In the first row, $\mathbf{s}_1=(-122.80, 41.31)$ which is in the northwest direction of the $\mathbf{s}_1=(-121.75, 37.98)$ in the second row. The third shows $\bar{\delta}(\mathbf{s}_j; k)$ across all locations $j=1,\ldots, n_s$.

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Example 1: The Matérn class
  • Example 2: The CH class
  • Proposition 1
  • Example 3: Space-Time Matérn with $\nu=1/2$
  • Proposition 2
  • Theorem 3
  • Example 4: The Lagrangian framework
  • Example 5: Mixture models
  • ...and 9 more