Vecchia Approximations and Optimization for Multivariate Mat ern Models Youssef Fahmy Joseph Guinness

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Vecchia Approximations and Optimization for

Multivariate Mat´ern Models

Youssef Fahmy, Joseph Guinness

Cornell University, Department of Statistics and Data Science

Abstract

We describe our implementation of the multivariate Mat´ern model for multivariate spatial datasets,

using Vecchia’s approximation and a Fisher scoring optimization algorithm. We consider various

pararameterizations for the multivariate Mat´ern that have been proposed in the literature for en-

suring model validity, as well as an unconstrained model. A strength of our study is that the code

is tested on many real-world multivariate spatial datasets. We use it to study the eﬀect of ordering

and conditioning in Vecchia’s approximation and the restrictions imposed by the various parame-

terizations. We also consider a model in which co-located nuggets are correlated across components

and ﬁnd that forcing this cross-component nugget correlation to be zero can have a serious impact

on the other model parameters, so we suggest allowing cross-component correlation in co-located

nugget terms.

1 Introduction

We attempt to understand the complexities of the Earth system by measuring and modeling many

variables that interact on a continuum of spatial-temporal scales. For example, modern climate

models include dozens of variables evolving in concert over space and time. In this paper, we

analyze elemental data from a soil monitoring network in France (Saby et al., 2009), elemental data

from a well-water monitoring program in Bangladesh (Kinniburgh and Smedley, 2001), multispectral

data from the GOES-16 satellite (maintained by NASA and NOAA), and the diﬀerence between

forecasted and actual pressures and temperatures in the Paciﬁc Northwest (Eckel and Mass, 2005).

Gaussian processes have become a workhorse for statistical analysis of spatial and spatial-

temporal data. A multivariate Gaussian process with p-components is a random vector func-

tion Z(x)=(Z1(x), . . . , Zp(x))>∈Rpindexed by a set D⊂Rd, d ≥1,such that for any

x1, . . . , xn∈D, the random vector (Z(x1)>, . . . , Z(xn)>)>∈Rnp has a multivariate normal distri-

bution. The process is completely speciﬁed by its mean function E[Z(x)] and covariance function

{Cov(Zi(x), Zj(x0))}p

i,j=1. When the covariances depend only on the separation lag h–that is, the

process is covariance-stationary–we write Cij (h) for Cov(Zi(x+h), Zj(x)), which is referred to as a

cross covariance function. Genton and Kleiber (2015) provide a thorough review of cross covariance

functions.

Following the popularity and success of the Mat´ern model for univariate spatial data, a mul-

tivariate version has been proposed and studied (Gneiting et al., 2010; Apanasovich et al., 2012;

Emery et al., 2022). The general form of the multivariate Mat´ern model is given by

Cij (h) = σij M(h|νij , αij )h∈Rd, i, j = 1, . . . , p

where

M(h|ν, α) = 1

2ν−1Γ(ν)(khk/α)νKν(khk/α)

with Kνbeing the modiﬁed Bessel function of the second kind of order ν. The parameter σij is the

covariance between co-located observations from components iand j. We refer to the parameters

arXiv:2210.09376v2 [stat.ME] 19 Oct 2022

using the following terminology: σij is a cross covariance parameter, αij is a cross range parameter,

and νij is a cross smoothness parameter. To be consistent with how the Mat´ern is parameterized in

our existing software, our αij parameters are ranges, whereas they are inverse ranges in Gneiting et al.

(2010), Apanasovich et al. (2012), and Emery et al. (2022). When i=j, we refer to them as marginal

parameters. Note that this model implies the symmetry Cij (h) = Cji(h) which need not hold in

general. Li and Zhang (2011) and Qadir et al. (2021) proposed methods for modeling asymmetries.

The univariate Mat´ern model Cii(h) = σiiM(h|νii, αii) provides a valid (i.e nonnegative deﬁnite)

second-order structure for the marginal process Zi(x) as long as the marginal parameters σii, αii, νii

are positive. Additional conditions are needed on the cross parameters σij , αij and νij to ensure

that the multivariate Mat´ern model is valid for the multivariate process Z(x). We discuss these in

the next section.

Kleiber (2017) studied the properties of various multivariate spatial models, including separa-

ble models, kernel convolution models, the linear model of coregionalization, and the multivariate

Mat´ern. He found that the multivariate Mat´ern is suﬃciently ﬂexible in that it allows the high

frequency coherence to exhibit a range of behaviors, depending on the parameter settings. Loosely

speaking, the coherence between two components is the correlation between the linear combination

of each component and a sinusoidal function; it measures correlation between components at a par-

ticular frequency of variation. Qadir and Sun (2021) demonstrated that further improvements in

ﬂexibility can be achieved with semiparametric models.

Over the past decades, spatial statisticians have produced a mountain of literature on the topic

of estimating covariance parameters, especially on the problem of computing estimates when the

dataset is very large. This work has led to various software packages, including INLA (Rue et al.,

2009; Lindgren et al., 2011), fields (Nychka et al., 2021), RandomFields (Schlather et al., 2022),

spBayes (Finley et al., 2015), spNNGP (Finley et al., 2022), LatticeKrig (Nychka et al., 2016), FRK

(Zammit-Mangion and Cressie, 2021), exageostat (Abdulah et al., 2018), GpGp (Guinness et al.,

2021), GPvecchia (Katzfuss et al., 2020), and GeoModels (Bevilacqua et al., 2018), to name a few.

Most of the research and software development is focused on the univariate case, with the exception

of RandomFields,exageostat, and spBayes, which are capable of ﬁtting bivariate Mat´ern models,

and GeoModels, which has a number of bivariate spatial models.

Clearly, there is a need for reliable software capable of ﬁtting multivariate spatial models with

two or more components to large datasets. In this work, we report on extending the GpGp R package

(Guinness et al., 2021) to handle the multivariate Mat´ern model, demonstrate its capabilities on sev-

eral datasets, and explore the implications of various proposed suﬃcient conditions on multivariate

Mat´ern parameters. GpGp implements Vecchia’s Gaussian process approximation (Vecchia, 1988),

along with improvements to the approximation (Guinness, 2018), and likelihood optimization proce-

dures that eﬃciently compute and make use of the gradient and Fisher information (Guinness, 2021).

The application of Vecchia’s approximation is agnostic to the covariance function, which has allowed

for the implementation of more than 25 covariance models in GpGp at the time of writing (package

version 0.4.0), including isotropic, geometrically anisotropic, nonstationary, and spatial-temporal

models on Euclidean spaces and spheres. GpGp has enjoyed success in spatial data competitions,

including being used by the winning team in the ﬁrst KAUST spatial data analysis competition

(Huang et al., 2021), and by the winners of the multivariate spatial data analysis section of the

second competition (Abdulah et al., 2022).

Adding the multivariate Mat´ern model to GpGp presents diﬃculties that go far beyond the normal

challenges of implementing a typical univariate covariance function. As opposed to the univariate

Mat´ern, whose parameters must simply be positive in order for the model to be valid, known suﬃcient

conditions are more complicated, so some care must be taken when enforcing them. The parameter

space is also large; our formulation of the model, which allows for correlated nuggets, has 2p(p+ 1)

parameters. Depending on the dataset, many of the parameters–or combinations thereof–are not well

identiﬁed. In short, it is a nasty optimization problem. In order to quickly maximize the likelihood,

one has to take large steps through a high dimensional space fraught with Errors, Infs, and NaNs.

R’s optim function does not cut it. In addition, as with all Vecchia approximations, decisions must

be made about how to order the observations and select neighbors, which is complicated by the fact

that the multivariate component of the data is usually categorical rather than numeric.

Our major contribution is the software we provide for ﬁtting multivariate Mat´ern models using

Vecchia’s approximation and a Fisher scoring algorithm. However, the software allows us to explore

the behavior of the multivariate Mat´ern model on various datasets. Our ﬁndings are generally

consistent with those of other authors; more ﬂexible conditions on the parameters tend to give

better ﬁts, but there are diminishing returns on added ﬂexibility. Perhaps our most interesting

modeling ﬁnding is that allowing the nugget term to be correlated across components can have a

large impact on the estimates of the other covariance parameters.

Section 2 reviews the multivariate Mat´ern model and its various parameterizations. Section 3

outlines Vecchia’s approximation for multivariate spatial data. Section 4 includes some notes on

the optimization procedures. Section 5 describes the datasets. Section 6 presents the results. We

conclude with a discussion.

2 Multivariate Mat´ern Parameterizations

We model the responses as

Yi(x) = µi+Zi(x) + εi(x),

where µiis component-speciﬁc mean, Ziis a multivariate Mat´ern process, and εi(x) is a nugget

term with covariances

Cov(εi(x+h), εj(x)) = τij 1(h= 0).

In other words, we assume constant mean within each component, and we add a nugget term but

allow the nugget to be correlated across components. The p×pmatrix formed by the τij parameters

must be positive deﬁnite. We parameterize the cross nugget variances as τij = (τiiτjj )1/2Sij ,where

Sis a correlation matrix.

Gneiting et al. (2010) provided necessary and suﬃcient validity conditions for the bivariate

Mat´ern model parameters. These conditions deﬁne the full bivariate model. For three or more

components, necessary and suﬃcient conditions on the parameters are not known, though various

authors have proposed suﬃcient conditions, some of which we explore here.

2.1 Parsimonious Model

Gneiting et al. (2010) proved that the multivariate Mat´ern model is valid for p≥2 if the following

conditions hold for every iand j:

1. αij =α(common marginal and cross ranges)

2. νij = (νii +νjj )/2

3. σij = (σiiσjj )1/2Vij Γ(νii+d/2)1/2

Γ(νii)1/2

Γ(νjj +d/2)1/2

Γ(νjj )1/2

Γ{(νii+νjj )/2}

Γ{(νii+νjj )/2+d/2}where Vis a correlation matrix.

These conditions deﬁne their parsimonious model. If we deﬁne ρij := σij /(σiiσjj )1/2,then condition

3 implies

|ρij | ≤ Γ(νii +d/2)1/2

Γ(νii)1/2

Γ(νjj +d/2)1/2

Γ(νjj )1/2

Γ{(νii +νjj )/2}

Γ{(νii +νjj )/2 + d/2}

which reduces to |ρij | ≤ (νiiνjj )1/2

(νii+νjj )/2when d= 2. In the bivariate case, condition 3 provides a complete

characterization of ρij when 1 and 2 hold. In our software, all correlation matrices, such as Vhere,

use a Cholesky-based parameterization (Pinheiro and Bates, 1996). All parameters that must be

positive use an exponential/log link.

2.2 Flexible-A Model

Apanasovich et al. (2012) provide a diﬀerent set of suﬃcient conditions in the p≥2 case which do

not require a common range parameter or the restriction that νij = (νii +νjj )/2:

1. νij =νii+νjj

2+ ∆A(1 −Aij ) where ∆A≥0 and Ais a positive correlation matrix,

2. (α−2

ij )p

i,j=1 is conditionally negative semideﬁnite,

3. σij = (σiiσjj )1/2Vij (uiiujj )−1/2uij , where

uij =α2∆A+νii+νjj

ij Γ(νij )Γ{(νii +νjj )/2 + d/2}/Γ(νij +d/2),

and Vis a correlation matrix.

We will refer to the model deﬁned by these conditions as the Flexible-Amodel. As suggested by

Apanasovich et al. (2012), we ensure condition 2 holds by parameterizing

α−2

ij =α−2

ii +α−2

2+ ∆B(1 −Bij )

where ∆B≥0 and Bis a positive correlation matrix. In the bivariate case, Aand Bbecome

redundant.

The conditions of the Flexible-A model are not necessary, so in the bivariate case the full model

of Gneiting et al. (2010) is less restrictive. However, the conditions of the full model are more

complicated to enforce, so it is still worthwhile to evaluate the performance of the Flexible-A model

on bivariate datasets. Apanasovich et al. (2012) illustrate on the same bivariate weather dataset

considered by Gneiting et al. (2010) that both models obtain similar ﬁts.

2.3 Flexible-E Model

Recently Emery et al. (2022) (Theorem 3B) gave another set of suﬃcient conditions with the goal

of alleviating the restriction imposed on |σij /(σiiσjj )1/2|by the Flexible-A model:

1. (νij )p

i,j=1 is conditionally negative semideﬁnite,

2. For β > 0, (α−2

ij −βνij )p

i,j=1 is conditionally negative semideﬁnite,

3. σij = (σiiσjj )1/2Vij (uiiujj )−1/2uij where

uij =α2νij

ij βνij exp(νij )Γ(νij )

and Vis a correlation matrix.

Emery et al. (2022) discuss in some detail how the conditions of the two models are related. A

practical way to ensure that condition 1 of the Flexible-E model holds is to parameterize νij exactly

as in condition 1 of the Flexible-A model. We use this parameterization in our software. Similarly,

we ensure 2 holds by deﬁning

α−2

ij =α−2

ii +α−2

2+ ∆B(1 −Bij ) + βνij −νii +νjj

2,

where ∆B≥0, B is a positive correlation matrix, and β > 0. We will refer to the model deﬁned by

these conditions as the Flexible-Emodel.

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VecchiaApproximationsandOptimizationforMultivariateMaternModelsYoussefFahmy,JosephGuinnessCornellUniversity,DepartmentofStatisticsandDataScienceAbstractWedescribeourimplementationofthemultivariateMaternmodelformultivariatespatialdatasets,usingVecchia'sapproximationandaFisherscoringoptimizationalgo...

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