Inside-out cross-covariance for spatial multivariate data

TL;DR

Inside-out cross-covariance (IOX) models for multivariate spatial likelihood-based inference are developed and superior performance of IOX is demonstrated on synthetic datasets as well as on colorectal cancer proteomics data.

Abstract

As the spatial features of multivariate data are increasingly central in researchers' applied problems, there is a growing demand for novel spatially-aware methods that are flexible, easily interpretable, and scalable to large data. We develop inside-out cross-covariance (IOX) models for multivariate spatial likelihood-based inference. IOX leads to valid cross-covariance matrix functions which we interpret as inducing spatial dependence on independent replicates of a correlated random vector. The resulting sample cross-covariance matrices are "inside-out" relative to the ubiquitous linear model of coregionalization (LMC). However, unlike LMCs, our methods offer direct marginal inference, easy prior elicitation of covariance parameters, the ability to model outcomes with unequal smoothness, and flexible dimension reduction. As a covariance model for a q-variate Gaussian process, IOX leads to scalable models for noisy vector data as well as flexible latent models. For large n cases, IOX complements Vecchia approximations and related process-based methods based on sparse graphical models. We demonstrate superior performance of IOX on synthetic datasets as well as on colorectal cancer proteomics data. An R package implementing the proposed methods is available at github.com/mkln/spiox.

Figures (15)

• Figure 1: Three spatially correlated outcomes generated via GP-IOX at $n=$ 40,000 gridded locations. We let $\sigma_{ii}=1$ for $i\in\{1,2,3\}$ and $\sigma_{12}=-0.9, \sigma_{13}=0.7, \sigma_{23}=-0.5$ and choose $\rho_i(\cdot, \cdot)$ as Matérn (Vecchia-approximated with $m=50$ neighbors) with range $1/5, 1/15, 1/30$ and smoothness $0.5, 1.2, 1.9$, respectively.
• Figure 2: Four spatially correlated outcomes generated via GP-IOX at $n=$ 40,000 gridded locations. Outcomes 1 and 2 have stationary marginal covariances, whereas 3 and 4 are nonstationary.
• Figure 3: We plot $C_{ij}(\boldsymbol{\ell}, \boldsymbol{\ell} + \boldsymbol{h})/\sigma_{ij} = \boldsymbol{h}_i(\boldsymbol{\ell}) \boldsymbol{L}_i \boldsymbol{L}_j^\top \boldsymbol{h}_j(\boldsymbol{\ell} + \boldsymbol{h})^\top$ at varying distances $\|\boldsymbol{h}\|$ and taking $\rho_i(\cdot, \cdot)$ as a Matérn correlation with $\nu_i=1$, $\phi_i=10$ and $\rho_j(\cdot, \cdot)$ as a Matérn with $\nu_j=1$ and varying the values of $\phi_j$ (left), or with $\phi_j=10$ and varying the values of $\nu_j$ (right), averaged over locations of a gridded set ${\cal S}$ of size 400.
• Figure 4: Maximum allowable cross-correlation at zero distance of a bivariate IOX on two Matérn margins (red horizontal lines), compared to a bivariate Matérn model, at different values of $\phi_1, \nu_1, \phi_2, \nu_2$ (subplots), $\phi_{12}$ (x-axes), and $\nu_{12}$ (by line color). IOX cross-correlations are averaged over locations of a gridded set ${\cal S}$ of size 400.
• Figure 5: The first of 20 datasets sampled from GP-IOX for the comparison of Section \ref{['sec:compare24']}.

Theorems & Definitions (25)

• Definition 2.1: Inside-out cross-covariance
• Proposition 2.1: Marginal covariance
• Proposition 2.2: Cross-covariance and $\sigma_{ij}$
• Proposition 2.3: IOX cross-covariance matrix function
• Corollary 2.1
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• Proposition 3.1
• Proposition 3.2: Conditional density