Joint spatio-temporal analysis of a linear and a

directional variable: space-time modeling of wave

heights and wave directions in the Adriatic Sea

Fangpo Wang*, Alan E. Gelfand* and Giovanna Jona-Lasinio†

*Department of Statistical Science, Duke University

*{fw19, alan}@stat.duke.edu

†Department of Statistical Science, University of Rome Sapienza

†giovanna.jonalasinio@uniroma1.it

February 25, 2014

Abstract

It is valuable to have a better understanding of factors that influence sea motion and to provide

more accurate forecasts. In particular, we are motivated by data on wave heights and outgoing wave

directions over a region in the Adriatic sea during the time of a storm, with the overarching goal

of understanding the association between wave directions and wave heights to enable improved

prediction of wave behavior. Our contribution is to develop a fully model-based approach to cap-

ture joint structured spatial and temporal dependence between a linear and an angular variable.

Model fitting is carried out using a suitable data augmented Markov chain Monte Carlo (MCMC)

algorithm. We illustrate with data outputs from a deterministic wave model for a region in the

1

Adriatic Sea. The proposed joint model framework enables both spatial interpolation and temporal

forecasting.

KEYWORDS Angular variable; Bayesian kriging; hierarchical model; latent variables; Markov

chain Monte Carlo; projected Gaussian process; significant wave height

1. INTRODUCTION

Better understanding and more accurate prediction of sea motion might help to reduce risks in

various marine related operations, such as navigation and safety of ships, coastal erosion and oil

spill motion. However, sea motion data are mostly available as the outputs from deterministic

models, especially if we are interested in exploring a relatively broad area of the sea. Recently,

deterministic models based on the dynamics and thermodynamics of the atmosphere have been

used to forecast the weather with increasing reliability. The outputs from such models are usu-

ally computed at several spatial and temporal resolutions. Building upon earlier work (Wang and

Gelfand, 2012) which focused solely upon wave directions, in this paper, we extend our interest

to include wave heights along with outgoing wave directions and provide a framework for joint

modeling these two measurements. More generally, we offer a modeling approach for joint spatial

and spatio-temporal analysis of an angular and a linear variable.

Wang and Gelfand (2012) have dealt with wave directions associated with spatial locations over

time. In fact, our available data include both significant wave heights and outgoing wave directions

(formally defined below) at the same spatial and temporal resolution. Figure 1 displays both the

outgoing wave directions and wave heights over a region of Adriatic sea, for a subset of available

locations, at a particular time point during a storm. The wave heights are displayed in the image

plot, which is an additional layer over the arrow plots of the directions. Perhaps not surprising,

there is strong visual evidence of spatial dependence for both heights and directions, although with

quite different patterns. Rather than separately modeling the wave heights and wave directions, it

is natural to think of building a joint model to accommodate the underlying association between

them. (The analysis in Section 2.4 and Table 2 below, supports this.)

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The association between wave directions and wave heights might seem to be similar to that say,

between wind directions and wind speeds. However, for wind data, we usually observe the N-S

and W-E components as linear data which induce a direction. We can then add wind speed as a

third linear variable. In the marine application, we observe a height and only a direction. For wind

data, this would be akin to having wind speed and only wind direction. To our knowledge, there

are no such joint models for spatial wave heights and wave directions.

More broadly, the contribution of this paper is to develop a fully model-based approach to cap-

ture joint structured spatial dependence for modeling linear data and directional data at different

spatial locations. We employ a projected Gaussian process for directions and, given direction, a

linear Gaussian process for heights. We show that Bayesian model fitting under such specification

is straightforward using a suitable data augmented Markov chain Monte Carlo (MCMC) algorithm.

This joint model framework allows natural extension to space-time data and can directly incorpo-

rate space-time covariate information, enabling, within its specification, both spatial interpolation

and temporal forecasting.

Wave height, like wind speed, is a linear variable, which can be treated in several ways (Kalnay,

2002; Wilks, 2006; Jona Lasinio et al., 2007). Work on space and space-time models of significant

wave heights can be found in Baxevani et al. (2009) and Ailliot et al. (2011). Again, wave direc-

tions are circular variables, measured in degrees relative to a fixed orientation. There is a relatively

smaller literature on modeling wave directions. Some recent work to build a space and space-time

model of wave directions can be found in Jona Lasinio et al. (2012) and Wang and Gelfand (2012).

Modeling linear and circular variables in a marine context has been considered in a likelihood

framework by Lagona and Picone (2013) and Bulla et al. (2012). These papers focus on classifica-

tion of sea regimes in the Adriatic basin. Classification is based on wind and waves directions and

wind speed and wave heights coming from a coastal meteorological monitoring station along with

the Ancona buoy. A hidden Markov model for the analysis of the time series of bivariate circular

observations (wind and wave directions) is proposed in Lagona and Picone (2013). They assume

the data are sampled from bivariate circular densities, whose parameters are driven by the evolu-

tion of a latent Markov chain. In Bulla et al. (2012), wind and wave data are clustered, again by

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Figure 1: Plot of wave heights (meters) and wave directions for 200 locations at 12:00 on April5th, 2010

pursuing a hidden Markov approach. Toroidal clusters are defined by a class of bivariate von Mises

densities while skew-elliptical clusters are defined by mixed linear models with positive random

effects. With just a single location, none of this work is spatial.

We often simultaneously observe realizations of a circular variable Θ and a linear variable

X as pairs (θ1, x1), . . . , (θn, xn). In addition to the foregoing examples, Jammalamadaka and

SenGupta (2001, chap. 8.5) suggest pairs such as the direction of departure and the distance to the

destination or wind direction and humidity. In the literature to date, the discussion usually focuses

on conditional modeling. For instance, if the angular variable Θ depends on linear variables, it

is referred to as a Linear-Circular regression model. i.e. a regression specification to use the

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linear covariates to explain a directional response. Such modeling typically adopts the von Mises

distribution and with a suitable link function which maps Rk to (−π, π), e.g., the arctan function,

where k is the dimension of covariates (Gould, 1969; Laycock, 1975; Johnson and Wehrly, 1978;

Fisher and Lee, 1992). Regression under the general projected normal was proposed in Wang and

Gelfand (2013).

Conditioning in the opposite direction, now using an angular covariate to explain a linear re-

sponse, i.e., if the linear variable X depends on independent angular variables Θ’s, we have a

regression model with a linear response and angular covariates, known as a Circular-Linear re-

gression model. Again, a link function is employed (Mardia, 1976; Mardia and Sutton, 1978;

Johnson and Wehrly, 1978). A flexible approach is to employ trigonometric polynomials (see

Jammalamadaka and SenGupta, 2001).

Measuring association between a circular and a linear variable is not as obvious as that between

two linear variables. Since the angular variable Θ has the support on the circle, the pair (Θ, X)

has the support on a cylinder. If their relationship can be written as X = a + b cos Θ + c sin Θ,

Mardia (1976) proposes a measure using the ordinary multiple correlation in the regression setting

between X and (cos Θ, sin Θ). Similarly, Johnson and Wehrly (1977) discuss the dominant canon-

ical correlation coefficient between X and (cos Θ, sin Θ). The modeling that we offer below is in

the spirit of this representation, of conditioning X on Θ. In particular, we build our joint model

through a conditional times a marginal specification where the marginal specification is a space

or space-time directional data process and then, conditionally, we specify a space or space-time

linear process. The only other spatial work involving a circular-linear regression model we are

aware of is the recent paper of Modlin et al. (2012). They model wind angle and wind speed given

wind angle. They model wind angle as a wrapped normal model introducing conditionally autore-

gressive (CAR) spatial random effects. They model wind speed on the log scale, conditional on

wind angle with a cosine link, again adding CAR spatial random effects. For our setting with wave

directions, we feel a projected Gaussian process is physically more appropriate than a wrapping

model. Furthermore, adding wave height given wave direction we find a natural link function.

The format of this paper is as follows. Section 2 develops a static joint model of spatial wave

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heights and wave directions, with a data illustration. In Section 3, we consider strategies for spatio-

temporal extension. For example, a dynamic model specification is straightforward but difficult to

fit. A specification that imagines time to be continuous is easier to work with. For the joint space-

time setting, we propose an illustrative model for the process to investigate and compare calm sea

state with stormy sea state. We illustrate using data for the Adriatic sea, off the coast of Italy.

Section 4 offers a summary and future challenges.

2. STATIC JOINT MODELS

In this section, we propose a framework for jointly modeling spatial linear variables and circular

variables. In Section 2.1, we provide the model details. Section 2.2 discusses the model fitting un-

der this specification using data obtained at a collection of spatial locations. Section 2.3 illustrates

the post-model fitting Bayesian kriging within this framework. In Section 2.4, we provide real data

examples for illustration.

2.1 Model specification

We begin with a single linear variable and a single directional variable which, illustratively, we

will refer to as the wave height and wave direction in the sequel. We build a joint parametric model

for the wave height H and the wave direction Θ by introducing a latent variable R (which will be

specified below), in the form

f(H,Θ|Ψh,Ψθ) =

∫f(H,Θ, R|Ψh,Ψθ) dR

=

∫f(H|Θ, R,Ψh)f(Θ, R|Ψθ) dR, (1)

where Ψh and Ψθ are sets of parameters associated with the conditional model for height and the

marginal model for direction, respectively, and are elaborated below.

To briefly review the projected normal model, suppose a random vector Y = (Y1, Y2)T follows

a bivariate normal distribution, with mean µ and covariance matrix Σ. The corresponding random

unit vector U = Y/‖Y‖ is said to follow a circular projected normal distribution (Small, 1996;

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Mardia and Jupp, 2000) with the same parameters, denoted as PN2(µ,Σ).

Transforming Y (equivalently U) to an angular random variable Θ through Θ = arctan∗(Y2/Y1) =

arctan∗(U2/U1), the density for Θ can be derived (Mardia, 1972, pp. 52). Here arctan∗(S/C) is

formally defined as arctan(S/C) if C > 0, S ≥ 0; π/2 if C = 0, S > 0; arctan(S/C) + π if

C < 0; arctan(S/C) + 2π if C ≥ 0, S < 0; undefined if C = 0, S = 0.

This distribution is almost intractable and suggests that we introduce a latent variable R =

||Y|| and work with the joint distribution of R and Θ, easily obtained through polar coordinate

transformation from the joint distribution of Y1 and Y2. In fact, it takes the form,

f(r, θ|µ,Σ) = (2π)−1|Σ|−12 exp

(−(ru− µ)TΣ−1(ru− µ)

2

)r, (2)

where u = (cos θ, sin θ)T. Wang and Gelfand (2013) note that the general projected normal

distribution is not fully identified; U = Y/‖Y‖ is invariant to scale transformation. In Σ = σ21 ρσ1σ2

ρσ1σ2 σ22

, Wang and Gelfand (2013) set σ1 = τ and σ2 = 1 to ensure identifiability,

resulting in a four-parameter (µ1, µ2, τ , ρ) distribution with Σ = T =

τ 2θ ρτθ

ρτθ 1

.

Next, for location s in the domain of interestD, denote the linear variable (height) by H(s) and

the angular variable (direction) by Θ(s). As the spatial model for Θ(s)|Ψθ, we propose a stationary

projected Gaussian process (Wang and Gelfand, 2012) with a constant mean µ = (µ1, µ2)T and

separable cross-covariance Cθ(s, s′) = %θ(s − s′;φθ) · T , where φθ is the decay parameter asso-

ciated with the correlation function %θ(·) and T is a 2 × 2 matrix defined as T =

τ 2θ ρτθ

ρτθ 1

.

Altogether, we let Ψθ = {µ, T, φθ}.

In particular, as in Wang and Gelfand (2012), the projected Gaussian process is induced from

an inline bivariate Gaussian process Y(s) = (Y1(s), Y2(s))T with a constant mean µ and cross-

covarianceCθ(s, s′). Essentially, we can transform back and forth between the two random variable

spaces (Θ(s), R(s)) and (Y1(s), Y2(s)) through Y1(s) = R(s) cos Θ(s) and Y2(s) = R(s) sin Θ(s),

thus defining the latent process, R(s). As above and as detailed in Wang and Gelfand (2012), this

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latent R(s) is introduced to facilitate model fitting of the projected Gaussian process. Thus, we

obtain f(Θ, R|Ψθ) under the integral in expression (1) in the same fashion as we obtain expression

(2).

At the top level of the hierarchy in (1), f(H|Θ, R,Ψh) is specified as a univariate Bayesian

spatial regression (a customary geostatistical model) for the wave height H(s)|Θ(s), R(s),Ψh,

with Θ(s) and R(s) included in the mean through a link function g(·). That is,

H(s) = g(Θ(s), R(s)) + w(s) + ε(s).

As usual, the residual is partitioned into the spatial effect term w(s) and the non-spatial error term

ε(s), where w(s) is assumed to follow a zero mean stationary Gaussian process with covariance

function Ch(s − s′) and ε(s)’s are uncorrelated pure errors. Under this conditional specification,

we are actually implementing a regression model with both circular (Θ) and linear (R) covariates.

A natural choice for the link function g(·) will revert to the linear regression on Y1(s) and Y2(s)

from the “unobserved” inline Gaussian process Y(s). Explicitly, we have

H(s) = β0 + β1R(s) cos Θ(s) + β2R(s) sin Θ(s) + w(s) + ε(s) (3)

= β0 + β1Y1(s) + β2Y2(s) + w(s) + ε(s)

= X(s)Tβ + w(s) + ε(s) (4)

where the spatial random effect w(s) follows a zero-centered GP with covariance function Ch =

σ2h%h(s− s′;φh), the error term ε(s)

iid∼ N(0, τ 2h).

In expression (4), we denote the regression coefficient vector β = (β0, β1, β2)T and the lo-

cation specific covariate vector as X(s) = (1, R(s) cos Θ(s), R(s) sin Θ(s))T. Under this model

specification, the parameters associated with the heights are Ψh = {β, φh, σ2h, τ

2h}. Altogether,

(H(s), Y1(s), Y2(s))T specifies a trivariate Gaussian process whose mean structure and cross-covariance

structure, under the above specification, is easily calculated. (See Appendix A.)

The coefficients of the spatial regression, β1 and β2, naturally provide information regarding

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the association between the circular random variable Θ and the linear variableH under the specific

specification in (3). When β1 and β2 are both 0, fitting the joint model essentially becomes fitting

the wave heights and the wave directions separately as a spatial regression model and a projected

Gaussian process model. Also, we can use the square of the multiple correlation coefficient, R2H|Y,

as a measure of the strength of the conditional dependence (explanation), as is customary in linear

regression. Notably, it is for a regression where the covariates are not observed. However, using

Appendix A, it can be obtained, as a parametric function, from the joint dependence structure and

is free of location. In particular, if ∆ = β21τ

2θ + β2

2 + 2β1β2τθρ,

R2H|Y =

∆

∆ + σ2h + τ 2

h

. (5)

The posterior distribution of R2H|Y can be obtained directly from posterior samples of the parame-

ters after model fitting.

2.2 Fitting

Suppose we have a joint spatial model for (H(s),Θ(s)), as specified in the previous subsection. We

then have the following parameters to estimate: Ψθ = {µ, τ 2θ , ρ, φθ} and Ψh = {β, φh, σ2

h, τ2h}.

We have observations (h,θ), where h = (h(s1), . . . , h(sn))T and θ = (θ(s1), . . . , θ(sn))T. We

need the likelihood, the joint distribution f(h(s1), . . . , h(sn), θ(s1), . . . , θ(sn); Ψθ,Ψh).

Since Θ(s)|Ψθ follows a stationary projected Gaussian process, according to the definition, we

have the corresponding “unobserved” latent linear variable Y1(si) = R(si) cos Θ(si) and Y2(si) =

R(si) sin Θ(si), i = 1, . . . , n. We note that Y1 = (Y (s1), . . . , Y1(sn))T and Y2 = (Y2(s1), . . . , Y2(sn))T,

the realizations of the inline Gaussian process Y = (YT1 ,Y

T2 )T follow a multivariate normal

with mean (µ111×n, µ211×n)T and covariance matrix Σ̃θ = T ⊗ Γθ(φθ), where {Γθ(φθ)}j,k =

%θ(sj − sk;φθ), j, k = 1, . . . , n. For the conditional layer, the model fitting is exactly the same as

that for a customary spatial regression. Therefore, the update of β, τ 2h and σ2

h is rather standard.

We denote Σh = σ2hΓh(φh) + τ 2

hIn, where {Γh(φh)}j,k = %h(sj − sk;φh), j, k = 1, . . . , n.

The full conditional distribution for R(si) is a product of two terms, one from f(Θ, R|Ψθ)

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and the other from f(H|Θ, R,Ψh). Block updating of the latent vector R is more difficult under

the joint modeling framework than that for just the marginal projected Gaussian process. There-

fore, we resort to conditional updating as R(si)|R(−si),Ψθ,Ψh,θ,h. As described in Wang and

Gelfand (2012), the properties of the inline GP are utilized to obtain the conditional distribution

Y(si)|Y(−si),Ψθ. The details of the posterior computation steps are shown in Appendix B.

To complete the Bayesian model, we specify the priors for the hyperparameters as follows.

For Ψθ, conjugacy for µ arises under a bivariate normal prior, e.g. µ ∼ N2(0, λµI2). For τ 2θ , we

choose the prior as an inverse Gamma IG(aτθ, bτθ) with mean bτθ/(aτθ − 1) = 1 while a uniform

prior on (−1, 1) is used for ρ. For the decay parameter φθ of the exponential covariance function,

we have worked with continuous uniform priors having support which allows small ranges up to

ranges larger than the maximum distance over the region. For the MCMC implementation, the full

conditional distributions are given in Appendix B. The parameters τ 2θ , ρ and φθ are updated using

a Metropolis-Hastings step. For Ψh, conjugacy for β arises under a multivariate Gaussian prior,

e.g., β ∼ N3(0, λβI3). For the decay parameter φh, similarly, we use continuous uniform priors

with support allowing small ranges up to ranges to the maximum distance over the region. As

discussed in Wang and Gelfand (2012), from our limited experience, a projected Gaussian process

seems to require a broader range than a linear process over the same region. As for σ2h and τ 2

h ,

inverse Gamma priors are used, IG(aσh, bσh) and IG(aτh, bτh), respectively.

2.3 Implementing kriging

For prediction at a new location s0, our goal is to obtain a joint distribution of H(s0) and Θ(s0)

given the data (θ,h), expressed as

f(θ(s0), h(s0)|θ,h)) =

∫ ∫ ∫f(θ(s0), h(s0), r(s0)|r,Ψ,θ,h)f(r,Ψ|θ,h) drdr0dΨ,

whereR(s0) is the latent random variable associated with Θ(s0) and the parameters Ψ = {Ψθ,Ψh}.

For kriging of the circular variable at the new location, Θ(s0), we simplify by starting from the

lower level, the joint distribution of Y(s0) = (Y1(s0), Y2(s0))T and Y∗ = (Y1(s1), Y2(s1), . . . , Y1(sn), Y2(sn))T

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is, Y(s0)

Y∗

∼MVN

µ(s0)

µ∗

,

1 ρT0,Y(φθ)

ρ0,Y(φθ) ΓY(φθ)

⊗ T ,

where µ∗ = (µT(s1), . . . ,µT(sn))T, {ΓY(φθ)}j,k = %θ(sj − sk;φθ) and {ρ0,Y(φθ)}j = %θ(s0 −

sj;φθ), j, k = 1, . . . , n. Thus, the conditional distribution for Y(s0)|Y∗ is a bivariate normal with

mean Es0 and covariance matrix Σs0 , where

Es0 = µ(s0) + ρT

0,Y(φθ)⊗ T · Γ−1Y (φθ)⊗ T−1(Y∗ − µ∗),

Σs0 = (1− ρT

0,Y(φθ)Γ−1Y (φθ)ρ0,Y(φθ))⊗ T.

In fact, the conditional distribution for Θ(s0)|Y∗ is a general projected normal PN2(Es0 ,Σs0).

The latent variable R(si) associated with the i-th location is updated during the model fitting. At

the g-th iteration, we gather the posterior samples of Y∗ through y1(si)(g) = r(si)

(g) cos θ(si) and

y2(si)(g) = r(si)

(g) sin θ(si), i = 1, . . . , n and g = 1, . . . , G. Finally, we are able to draw sam-

ples of Θ(s0) from the predictive distribution f(θ(s0)|θ,h), since there exists an explicit form for

Θ(s0)|Y∗, equivalently, Θ(s0)|Θ,R. As a byproduct of sampling Θ(s0), we also obtain a sample

ofR(s0) at the g-th iteration, r(s0)(g). Let r(g) be the realization ofR = (R(s1), . . . , R(sn))T at the

g-th iteration. We can evaluate f(θ(s0)|r(g),Ψ(g),θ) using a fine grid of points on [0, 2π) and take

the average of the density values on each grid. This resulting average is a usual Rao-Blackwellized

estimate of the predictive density at the kriged location s0. As a mixture of general projected

normal densities, its form is very flexible. (Theorem 1 in Wang and Gelfand (2012) shows that

mixtures of projected normals are dense in the class of all directional data distributions.)

For kriging of the linear variable at the new location, H(s0), again we simplify, now starting

from the joint distribution of H(s0) and H = (H(s1), . . . , H(sn))T,

H(s0)

H

∼MVN

X(s0)Tβ

Xβ

,

τ 2h + σ2

h τ 2hρ

T0,H(φh)

τ 2hρ0,H(φh) τ 2

hΓh(φh) + σ2hIn

,

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where X(s0) = (1, R(s0) cos Θ(s0), R(s0) sin Θ(s0))T, {Γh(φh)}j,k = %h(sj−sk;φh), {ρ0,H(φh)}j =

%h(s0−sj;φh), j, k = 1, . . . , n and X is a n×3 matrix with the i-th row as (1, R(si) cos Θ(si), R(si) sin Θ(si)),

i = 1, . . . , n. At the g-th iteration, the posterior samples, θ(s0)(g) and r(s0)(g), are included in

x(s0)(g) as the covariates. The remaining kriging steps for the linear variable H(s0) are standard,

and thus we do not provide further details here.

2.4 A data example

We study wave heights and directions in the Adriatic Sea, a semi-enclosed basin (Picone, 2013).

The orography of the Adriatic Sea plays a key role in any study of associated wave behavior as

most of the waves tend to travel from north-northwest in a south-easterly direction along the major

axis of the basin.

The wave height data we employ in this paper are significant wave heights. The significant

wave height (SWH) is essentially the average height of the highest waves in a wave group. The

traditional definition of the significant wave height, denoted as Hs, is the average of the one third

highest wave heights (trough to crest) observed in the region. Recently, a more commonly used

definition is four times the standard deviation of the vertical displacement of the sea surface (eleva-

tion), which is calculated on the basis of the wave spectrum. Corresponding to the latter definition,

a notation Hm0 is often used. With regard to directions, outgoing wave directions are often pre-

ferred to incoming and are recorded in degrees relative to a fixed orientation.

Monitored buoy data, supplying wave height and wave direction measurements would seem to

be attractive for such analysis. However, at present, buoy networks are too sparse to be used as a

data source for spatial analysis. Therefore, we employ an alternative data source for wave heights

and wave directions, outputs from deterministic models, usually climatic forecasts computed at

several spatial and temporal resolutions. These models are increasingly accurate and may be

eventually calibrated using buoy data, a data fusion problem. However, here we illustrate solely

with deterministic model output.

We employ the foregoing general modeling framework, illustrating with deterministic model

output. In particular, we use data outputs from a deterministic wave model implemented by ISPRA

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(Istituto Superiore per la Protezione e la Ricerca Ambientale) using the Hydro-Meteorological-

Marine System (‘Sistema Idro-Meteo-Mare’, SIMM) (Speranza et al., 2004, 2007). This integrated

forecasting system includes a wind forecast model (BOLAM) predicting the surface wind (wind

at 10 meters above the sea surface) over the entire Mediterranean and wave models (in deep and

shallow waters).

The deep water WAve Model (WAM) describes the evolution of sea wave spectra by solving

energy transport equations, using as input only the wind forecast produced from the BOLAM.

Wave spectra are locally modified using a source function describing the wind energy, the energy

redistribution due to nonlinear wave interactions, and energy dissipation due to wave fracture. The

WAM produces significant wave height, mean wave directions (always computed over one hour

time window), and the peak wave period. The model provides forecasts over a grid with 10 × 10

km cells every hour. The ISPRA dataset has forecasts on a total of 4941 grid points over the Italian

Mediterranean. Over the Adriatic Sea area, we have 1494 points.

According to meteorologists, the definition of “calm” in the Adriatic Sea refers to a time period

with wave height lower than 1 meter. When the wave height is between 1 and 2 meters, a pre-

storm warning is triggered and above 2 meters there is a storm warning. We plot the data, wave

directions and wave heights, for an illustrative hour under each sea state, in Figures 2(a)-(c), with

arrow plots for the wave directions and image plot (color) for the wave heights. We apply the joint

modeling framework proposed in Section 2.1 separately to each of the three datasets, respectively

representing calm, transition and storm in the Adriatic Sea, as shown in Figures 2(a)-(c). In the

absence of covariates, we employ the joint model in Section 2.1.

We model the heights directly. With concern about negative predictions, we also fitted our

model with log wave heights. The results do not change; negative predictions are infrequent, even

in the calm state. We also note that the stationarity assumption is likely unrealistic for coastal data

due to anticipated orographic influences; our model specification is an initial attempt.

As above, there are 1494 locations (grid points) in the Adriatic Sea and we randomly select

300 (approximately 20%) locations for the purpose of validation. We note that the maximum

interpoint distance over set of locations is 850 km. For the fitting, we run the MCMC algorithm for

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(c) storm

Figure 2: Plot of wave heights (meters) and wave directions under three sea states: (a) calm, (b)transition and (c) storm

25000 iterations with a burn-in of 5000 and thinning by collecting every fifth sample. The kriging

estimates are computed using 4000 posterior samples.

The prior setting for Ψθ is as follows: µ ∼ N(0, 4I), τ 2θ ∼ IG(2, 1) and ρ ∼ Unif(−1, 1).

A continuous uniform prior Unif(0.0004, 0.1) is used for φθ, corresponding to a maximum and

minimum range of 7500 km and 30 km respectively. In fact, the strong spatial dependence for the

14

directions under storm conditions suggest that we may require a range beyond the largest pairwise

distance in the dataset when there is little variation in wave directions. For the parameters associ-

ated with the heights Ψh = {β, τ 2h , σ

2h, φh}, a continuous uniform prior Unif(0.005, 0.5) is used for

φh, corresponding to a maximum and minimum range of 600 km and 60 km respectively. Since φh

is the decay parameter in the regular linear spatial regression model, we can adopt customary pri-

ors for ranges with linear variables. We employ a non-informative prior for β as β ∼ N3(0, 10I).

Inverse gamma priors are used for τ 2h and for σ2

h, again IG(2, 1). These inverse gamma priors are

weak in that they have infinite variance with mean 1. We do not find evidence of much sensitivity

to their centering.

We compare our results with those based upon independently modeling the spatial wave height

and the spatial wave direction, a nested case of our joint modeling framework obtained by setting

β1 and β2 to be zero. In the independence case, the marginal model for wave heights is a spatial

regression with constant mean β0, written as

H(s) = β0 + w(s) + ε(s), w(s) ∼ GP (0, σ2h%h(s− s′;φh)), ε(s)

iid∼ N(0, τ 2h),

where the parameters set is Ψhm = {β0, φh, σ2h, τ

2h}. The marginal model for wave directions

becomes a constant mean projected Gaussian process model.

The posterior summaries of parameters are shown in Table 1, for both the joint model and the

independence model above. The 95% credible intervals are obtained as approximate highest pos-

terior density (HPD) intervals following Chen and Shao (1999). Model comparison between the

joint model and the independence model is provided in Table 2. Regarding choice of model com-

parison criterion, for circular variables we utilize the continuous ranked probability score (CRPS)

and the predictive log scoring loss (PLSL), following the discussion in Wang and Gelfand (2013).

For the linear variables, predictive mean square error and average length of 95% credible intervals

are evaluated for each model.

During a calm day, we observe small values of wave heights (lower than 1 meter) and large

variation in wave directions over the region of interest. Conversely, at a time point during the storm,

15

Table 1: Posterior summaries of the parameters: the joint model (H,Θ) and independent modelsH , Θ

parameter mean lower upper mean lower upper

data (Calm) (H , Θ) H , Θ

β0 0.2446 0.1965 0.2950 0.2331 0.1859 0.2809β1 0.0125 -0.0209 0.0471 - - -β2 0.0653 0.0501 0.0796 - - -σ2h 0.0051 0.0046 0.0056 0.0055 0.0051 0.0060τ2h 0.0011 0.0010 0.0013 0.0011 0.0010 0.0013φh 0.0101 0.0084 0.0126 0.0126 0.0105 0.0116φθ 0.0020 0.0019 0.0022 0.0019 0.0017 0.0022µ1 0.2856 -0.4046 0.9755 0.3039 -0.4310 1.0000µ2 -0.5055 -1.7615 0.8260 -0.4141 -1.6671 0.9952τ2θ 0.2717 0.2275 0.3137 0.2747 0.2388 0.3180ρ -0.4874 -0.5713 -0.4268 -0.4790 -0.5451 -0.4078

data (Transition) (H , Θ) H , Θ

β0 0.4660 0.3244 0.6176 0.5394 0.3685 0.6892β1 0.1887 0.1281 0.2453 - - -β2 0.3062 0.2478 0.3706 - - -σ2h 0.0344 0.0306 0.0389 0.0413 0.0376 0.0482τ2h 0.0021 0.0019 0.0024 0.0021 0.0018 0.0023φh 0.0101 0.0084 0.0126 0.0084 0.0072 0.0090φθ 0.0006 0.0005 0.0007 0.0006 0.0005 0.0007µ1 -0.1066 -2.3975 2.1270 0.0617 -2.0922 2.2151µ2 -0.2935 -1.7928 1.3506 -0.2152 -1.8580 1.3850τ2θ 2.4400 2.0958 2.8055 2.2318 1.9602 2.5818ρ -0.2264 -0.3069 -0.1473 -0.2257 -0.2994 -0.1430

data (Storm) (H, Θ) H , Θ

β0 0.3795 0.1897 0.5776 0.4737 0.0734 0.8790β1 -0.3052 -0.4049 -0.2066 - - -β2 0.8794 0.7588 0.9883 - - -σ2h 0.0702 0.0546 0.0826 0.1930 0.1677 0.2283τ2h 0.0033 0.0028 0.0038 0.0034 0.0028 0.0039φh 0.0105 0.0087 0.0122 0.0070 0.0063 0.0076φθ 0.0006 0.0005 0.0007 0.0006 0.0005 0.0007µ1 0.5853 -0.8418 1.8745 0.6990 -0.5824 2.0369µ2 -0.6064 -2.0949 0.9097 -0.5152 -2.0546 0.9437τ2θ 0.7216 0.6515 0.7917 0.7312 0.6652 0.8070ρ -0.6433 -0.6827 -0.6018 -0.6631 -0.7107 -0.6105

we observe that wave directions are fairly homogeneous while there is a large variation in wave

heights over the region (ranging from 1 meter up to 6 meters). During the transition period, there is

considerable variation both in wave heights and wave directions. From Table 1, the posterior means

16

Table 2: Model comparison: the joint model (H,Θ) and independent models H , Θ

(a) calm (b) transition (c) stormFeature (H,Θ) H , Θ (H,Θ) H , Θ (H,Θ) H , Θ

Predictive Mean Square Error (height) 0.0006 0.0006 0.0031 0.0030 0.0109 0.0108Average Length of 95% Credible Interval (height) 0.1821 0.1788 0.3405 0.3586 0.5163 0.6345

mean CRPS for wave direction 0.0407 0.0408 0.0276 0.0279 0.0213 0.0223PLSL for wave direction -977 -974 -1098 -1104 -1321 -1318

of the spatial variance component for wave heights under the joint model (σ2h) are 0.0051, 0.0344

and 0.0702 for the three different datasets. Since the contribution of τ 2h is relatively negligible,

these differences across time points agree with our remark regarding increasing variation of wave

heights from calm to storm.

For the wave directions, there is no single parameter which can capture the variation across the

region. Specifically, it is difficult to glean much from the posterior summaries of the parameters of

the marginal projected normal distribution since the four parameters of a general projected normal

altogether determine the shapes in a complex fashion. However, the decay parameter φθ associated

with the spatial correlation kernel of the projected Gaussian process provide some indication of the

smoothness of the surface of directions. For the calm day, φθ is much larger than in the other two

cases, indicating less smoothness in this surface, as expected.

With prediction as an objective, we can consider the predictive densities of wave height and

wave direction at kriged locations. For illustration, the results for one selected location at the three

different sea states are shown in Figure 3. During the calm period, the joint model does not show

much improvement in predictions in terms of concentration of the predictive densities. During the

transition period, the difference between the joint model and independent models becomes a bit

more clear; both of the predictive densities in Figure 3(b) show that the joint model outperforms

the independent one. Finally, in Figure 3(c), we see considerable benefit to the joint model.

Turning to the model comparison results in Table 2, during the calm days, the model compari-

son criteria find the performance of the two models to be essentially equivalent. For the transition

case, the joint model emerges as slightly better, with shorter average length of 95% credible in-

tervals and slightly smaller average CPRS and PLSL. For the time point chosen during a storm,

17

−1.0 −0.5 0.0 0.5 1.0

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18

the biggest gain is achieved. We have a substantially narrower average length of credible intervals

and a roughly 5% smaller value of CRPS. Such model comparison results are supported by the

predictive density examination in Figure 3.

As mentioned previously, the independence model is nested in the joint model framework. The

posterior summaries of β1 and β2 in Table 1 concur with these findings. For the time point during

a calm day, β1 and β2 are nearly zero, which supports the independence of wave heights and wave

directions. As we move from calm to transition to storm, the β1 and β2 become increasingly differ-

ent from 0. The multiple correlation coefficient is computed, as in (5) and we show the posterior

densities of the multiple correlation coefficient for all three scenarios in Figure 4. The figure con-

cisely supports Table 1, revealing stronger association between height and direction as we move

from calm to storm. Finally, we note that the negative β1 and positive β2 are not unexpected. With

storms tending to travel from the north-northwest to the southeast, and with the sin increasing and

cos decreasing in these directions, this supplies an expectation of an increase in wave height as a

function of θ as the storm develops.

3. JOINT SPACE-TIME MODELS

In Section 2, we focused on joint modeling of spatial wave directions and wave heights at a static

time slice and showed the benefit from jointly modeling these two variables, especially during

a storm. It is natural to envision an underlying process for heights and directions in continuous

space and time; fitting such a model would enable both interpolation and forecasting at future time

points. Since such data are also available across time at hourly resolution using the ISPRA output,

we consider such a process model.

To see the behavior of wave heights over the region, we select 10 illustrative locations, shown

in upper panel of Figure 5. Their corresponding time series of hourly heights across the first ten

days in April, 2010 are plotted in the lower panel of Figure 5. Following the foregoing definition

of sea motion states based on significant wave heights, we observe a range of behavior, including

a transition from calm to storm and back to calm.

19

0.0 0.2 0.4 0.6 0.8 1.0

05

1015

2025

3035

the square of the multiple correlation coefficient

Den

sity

calmtransitionstorm

Figure 4: The posterior densities of the square of the multiple correlation coefficient R2H|Y

So, now, denote the wave height and wave direction measurements at location s and time t

as H(s, t) and Θ(s, t). In Section 3.1, we offer a space-time model for the pair (H(s, t),Θ(s, t))

under the continuous space-time setting. For illustration, we use two portions of the ISPRA output,

a calm window and a storm window (highlighted in Figure 5).

3.1 Model specification

We have the wave height H(s, t) and the wave direction Θ(s, t) both over the space-time domain,

s ∈ D and t ∈ (0, K). We adopt the same joint framework shown in expression (1), where Ψh and

Ψθ are sets of parameters associated with the conditional model for height and the marginal model

for direction, respectively.

Again, this joint space-time model is provided conditionally. For the marginal distribution

of the circular variables Θ(s, t)|Ψθ, we propose a stationary spatio-temporal projected Gaussian

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Figure 5: Time series of wave heights (lower panel) for 10 illustrative locations (upper panel) fromn = 200 in the Adriatic Sea region in April 2010. The two time windows (calm and stormy) usedin Section 3.3 are highlighted.

21

process with a constant mean µ = (µ1, µ2)T and separable cross-covariance form,

Cθ ((s, t), (s′, t′)) = %θ,s(s− s′;φθ,s)%θ,t(t− t′;φθ,t) · T, (6)

where %θ,s is the spatial correlation and %θ,t is the temporal correlation. The parameters associated

with the directions Ψθ = (µ, T, φθ,s, φθ,t), φθ,s and φθ,t are the decay parameters associated with

their corresponding correlation functions %θ,s and %θ,t and T is still defined as in the previous

section, T =

τ 2θ ρτθ

ρτθ 1

. This model has been considered in Wang and Gelfand (2012).

As in the static case, we only need one space-time covariance function. For simplicity, we

assume separability in space and time in expression (6). Future work will have us investigating

space-time dependent covariance functions such as those in Gneiting (2002) and Stein (2005). Un-

der this model specification, the projected Gaussian process is induced from an inline Gaussian

process Y(s, t) with a constant mean µ and cross-covariance Cθ((s, t), (s′, t′)). So, we can trans-

form back and forth between the spaces for the pairs of random variables, (Θ(s, t), R(s, t)) and

(Y1(s, t), Y2(s, t)) through Y1(s, t) = R(s, t) cos Θ(s, t) and Y2(s, t) = R(s, t) sin Θ(s, t). A latent

variable R(s, t) has again been introduced to facilitate the model fitting.

At the conditional level, we introduce

H(s, t) = X(s, t)Tβ + w(s, t) + ε(s, t)

= β0 + β1Y1(s, t) + β2Y2(s, t) + w(s, t) + ε(s, t).(7)

In fact, we simplify w(s, t) to w(s) where, again, the spatial random effect w(s) follows a

zero-centered GP with covariance function Ch = σ2h%h(s − s′;φh) and the random the error term

ε(s, t)iid∼ N(0, τ 2

h). Now the parameters associated with the heights are Ψh = {β, φh, σ2h, τ

2h}.

This model asserts that the temporal dependence between H(s, t) is fully contributed by the latent

components Y1(s, t) and Y2(s, t). Restoring w(s, t) and perhaps allowing time dependent coeffi-

cients introduces identifiability challenges and thus is deferred to future consideration. Note, from

the previous section, that the coefficient vector, β, changes with sea state. Since, for now, we are

22

not considering dynamics in β, it is only sensible to fit the model in (7) to data in either a calm

window or a storm window.

3.2 Fitting, kriging and forecasting

The model fitting for the joint space-time model is straightforward and only requires some modifi-

cation of the fitting for the static case with details in Section 2.2. We now have observations h,θ at

a collection of locations s1, . . . , sn and a collection of time points t1, . . . , tk, where h = {h(si, tj)}

and θ = {θ(si, tj)}, where i = 1, . . . , n and j = 1, . . . , k. The parameters we need to estimate

are Ψh = {β, φh, σ2h, τ

2h} and Ψθ = (µ, T, φθ,s, φθ,t). In addition, we have to update the latent

R(si, tj), where i = 1, . . . , n and j = 1, . . . , k. Again, we update R(si, tj) conditioning on all the

other latent variables R(−(si, tj)) by collecting the relevant terms in the likelihood, from both the

conditional, f(H|Θ, R,Ψh), and marginal, f(Θ, R|Ψθ), specifications.

Kriging of both measurements at a new location s0 at any of the observed time points is stan-

dard; we omit the details here. Instead, we focus on the one-step ahead prediction to the time point

tk+1 for the n locations with observations. We start from the joint distribution,

Ytk+1

Yt1:tk

∼MVN

µtk+1

µt1:tk

,Γk+1(φθ,t)⊗ Γn(φθ,s)⊗ T

,

where Ytk+1= (Y1(s1, tk+1), Y2(s1, tk+1), . . . , Y1(sn, tk+1), Y2(sn, tk+1)T, Yt1:tk = (YT

t1, . . . ,YT

tk)T,

Ytj = (Y1(s1, tj), Y2(s1, tj), . . . , Y1(sn, tj), Y2(sn, tj))T, Γk+1(φθ,t) = {%θ,t(tj − tq;φθ,t)}, j, q =

1, . . . , k + 1 and Γn(φθ,s) = {%θ,s(si − sp;φθ,s)}, i, p = 1, . . . , n. Thus, we obtain the conditional

distribution Ytk+1|Yt1:tk ,Ψθ from which the posterior samples of Ytk+1

can be easily obtained at

each iteration. Then, the posterior sampling of the wave height at a future time point H(si, tk+1)

remains the same as that in the static case.

3.3 Data examples

For illustration, we randomly select a small fraction of the locations (n = 200) from the same data

set in Section 2.4. These locations are those in both Figure 1 and Figure 5. Recalling the definition

23

of calm and storm in Section 2.4, Figure 5 also provides information regarding the time intervals

of calm and storm. Our goal here is to illustrate the prediction at a future time point tk+1 within

each interval, using our joint space-time model.

First, we select data from a calm period, April 2nd to April 3rd, 2010. The time points are

collected every two hours. Thus starting from April 2nd, 00:00 to April 3rd 22:00, we have 24

time points in total. In addition, we select data from a severe storm period, April 5th to April 6th.

Again, the time points are collected at the same temporal resolution, every two hours. Thus starting

from April 5th, 00:00 to April 6th 22:00, with 24 time points in total. In both cases, we hold out

the observation at the last time point for forecasting validation.

The posterior summaries of the regression coefficients for the conditional specification are

reported in Table 3. During the calm period, β1 and β2 are nearly zero, agreeing with the results

for a single time slice during calm period shown in Section 2.4. This suggests that approximate

independence of wave heights and wave directions sustains during a calm period. However, during

a storm period, β1 and β2 clearly depart from zero, indicating strong dependence of wave heights

and wave directions during the storm, again consistent with the results in Section 2.4. We see that,

again, σ2h dominates the variation in heights and we see more than a twenty-fold increase during

the storm period. With regard to the decay parameters, φθ,s is roughly double in calm compared

with storm, i.e., the spatial range is roughly half. On the other hand, φθ,t is roughly half in calm

compared with storm, i.e., the temporal range is roughly double.

Lastly, we are interested in the result of prediction at the future time point tk+1. Table 4 pro-

vides the summaries of prediction for both wave heights and wave directions. We do not use any

information at time tk+1 to fit the model; however, the model provides reasonable prediction. We

predict direction better in a storm (expected since there is less variability in direction during a

storm) and height dramatically better during a calm period (again, expected because heights are

lower and less variable when it is calm), in agreement with the results from Table 3.

As a last comment here, the reader might wonder why, with time observed discretely, we

didn’t consider a dynamic model specification. In fact, we have investigated such forms. The

implicit challenge lies in the transition specification. That is, the marginal distributions of the

24

Table 3: Posterior summaries of the parameters (mean and HPD)(calm) (storm)

parameter mean lower upper mean lower upperβ0 0.1612 0.0563 0.2998 0.5483 -0.0218 1.1750β1 -0.0117 -0.0388 0.0161 -0.1439 -0.2805 -0.0171β2 0.0329 0.0112 0.0563 0.2061 0.0650 0.3529σ2h 0.3561 0.2916 0.4315 8.2797 6.8463 10.0983τ2h 0.0020 0.0017 0.0024 0.0091 0.0064 0.0119φh 0.0051 0.0048 0.0056 0.0047 0.0043 0.0051µ1 0.2999 -0.2808 0.8955 0.1835 -0.6192 0.9448µ2 -0.1431 -0.9733 0.6035 -0.0868 -0.9079 0.6537τ2 0.6129 0.5665 0.6448 0.9245 0.8879 0.9972ρ -0.7095 -0.7359 -0.6878 -0.6251 -0.6656 -0.5905φθ,s 0.0043 0.0037 0.0050 0.0022 0.0020 0.0024φθ,t 0.0775 0.0750 0.0803 0.1667 0.1487 0.1782

Table 4: Summaries for one-step ahead predictioncalm storm

Predictive Mean Square Error (height) 0.0522 0.4497mean CRPS for wave direction 0.0973 0.0647

directions under the projected normal have shapes that depend jointly, in a complex way, on all four

parameters in µ and T . We would need to introduce a four dimensional joint dynamic specification

for them; dynamics in just say, µ are not enough. Then, we would have to address dynamics in

the parameters for the conditional specification for heights given directions. Altogether, we found

such second stage vector autoregressive models very difficult to specify, even more difficult to fit,

especially with shorter time series. Moreover, if these parameters are only indexed by time, then

we will still have space-time separability.

4. SUMMARY AND FUTURE WORK

In this paper, we have proposed a general framework for jointly modeling spatially indexed linear

variables and circular variables. Also, we have discussed the extension to the space-time setting.

We have focused on introducing the joint modeling, without much attention to covariates. In

particular, for wave data, wind information would be a potentially useful covariate. Also, if the

region is large enough, a trend surface might be worthwhile to consider. However, it may be

25

challenging to interpret the trend through the projection.

Much more work can be done for the joint space-time modeling of directional data and linear

data. Working within a calm period or a storm period, we can consider space-time dependence as

suggested in Section 3.1, that is space-time dependence in the bivariate Gaussian process inducing

space-time dependence in the projected Gaussian process. More attractive, and potentially more

useful, would be a model across time that would accommodate calm, storm, and transition. To

do so with regard to heights would require, at the least, introduction of a β0(t) process, i.e., a

time-dependent intercept. In fact, we would also want time-dependent slope parameters, β1(t) and

β2(t). Anticipating dependence between slopes and intercept, this leads to a three dimensional

process over time. Another aspect to consider is the fact that variability in height changes over sea

state. This suggest that we also add σ2h(t), perhaps as a log Gaussian process. (Of course, range

might be time dependent, adding a further model feature.) Then, there is the matter of dynamics

in the space-time projected Gaussian process. Here, we would want to extend the specification

in Section 3.1 to allow continuous time in µ and T . We encounter similar difficulties to those

raised at the end of the previous section. All told, there are clearly lots of challenging modeling

opportunities.

Included in the output of the deterministic models is mean wave period and peak wave period.

The mean wave period, Tm, is the mean of all wave periods in time series representing a certain

sea state, as opposed to peak wave period, Tp, which is the wave period with the highest energy.

Also, a trend surface might also be worthwhile to introduce. It would be useful to investigate these

additional variables with regard to improved interpretation and prediction.

Lastly, the reader will also appreciate that we can extend our joint modeling approach to handle

multiple linear and directional variables in space and in space and time. Following our condition-

ing approach, with say p directional variables we would specify a marginal 2p dimensional linear

Gaussian process to project to a p dimensional directional process. Then, with say r linear vari-

ables, we would need an r dimensional Gaussian process, appropriately conditioned on the 2p

dimensional process. It is easy to imagine such specifications but model fitting will become ex-

tremely demanding.

26

ACKNOWLEDGMENT

The work of the first two authors was supported in part by NSF CMG 0934595 and NSF CDI

0940671. The authors thank ISPRA (Istituto Superiore per la Protezione e la Ricerca Ambientale)

for the use of data output from the wave model of its SIMM hydro-meteo-marine forecasting

system.

APPENDIX

A: Mean and cross-covariance structures of (H(s), Y1(s), Y2(s))T

Recall from Section 2.1, (H(s), Y1(s), Y2(s))T follows a trivariate Gaussian process. The mean

structure of this Gaussian process is (β0 + β1µ1 + β2µ2, µ1, µ2)T and the cross-covariance is

CH,Y(s, s′) = %θ(s− s′;φθ)

v11 v12 v13

v21T

v31

+ %h(s− s′;φh)

σ2h + τ 2

h1s=s′ 01×2

02×1 02×2

,

where v11 = β21τ

2θ + 2β1β2ρτθ + β2

2 , v12 = v21 = β1τ2θ + β2ρτθ, v13 = v31 = β1ρτθ + β2 and

T =

τ 2θ ρτθ

ρτθ 1

.

B: Posterior Computation: The full conditionals of Ψh and Ψθ

Recall from Section 2.2, Ψh = {β, φh, σ2h, τ

2h} and Ψθ = {µ, T, φθ}, where T =

τ 2θ ρτθ

ρτθ 1

.

We define two covariance matrices Σ̃θ = T ⊗Γθ(φθ) and Σ̃∗θ = Γθ(φθ)⊗T . Their correspond-

ing precision matrices are Q̃θ = Σ̃−1θ = T−1 ⊗ Γ−1

θ (φθ) and Q̃∗θ = Σ̃∗−1

θ = Γ−1θ (φθ) ⊗ T−1. Γ−1

θ,ij

denotes the element on the i-th row and j-th column of the inverse matrix of Γθ(φθ).

Following expression (3), the top or conditional hierarchy can be written as,

H(s) = β0 +R(s)[β1 cos Θ(s) + β2 sin Θ(s)] + w(s) + ε(s),

= β0 +R(s)β4(s) + w(s) + ε(s).

27

As defined in Section 2.2, Σh = σ2hΓ(φh) + τ 2

hIn. The inverse matrix of Σh is denoted as Qh, with

Qh,ij representing its element on the i-th row and j-th column. We then denote H̃(s) = H(s)−β0,

thus H̃(s) = R(s)β4(s) +w(s) + ε(s). The conditional distribution of H̃(si) given H̃(−si) can be

easily obtained as a normal distribution, with R(si) involved in the mean.

The full conditionals for the parameters of the joint process model:

• µ ∼ N2(Eµ, Q−1µ ), where the precision matrix of this bivariate normal Qµ = ATΣ̃−1

θ A +

I2/λµ, Eµ = Q−1µ ATΣ̃−1

θ Y and A =

1n×1 0n×1

0n×1 1n×1

.

•

R(si) ∝ riI(0,∞)(ri)

exp

(−

(Γ−1θ,iiu

Ti T−1ui + β2

4(si))r2i − 2ri(Γ

−1θ,iiu

Ti T−1Esi + β4(si)Esi2)

2

),

where ui = (cos θ(si), sin θ(si))T, Esi = µ − Q−1

si

∑j 6=i Q̃

∗θ,ij(Y(sj) − µ),Qsi = Q̃∗θ,ii =

Γ−1θ,ii · T−1, Esi2 = h̃(si) +Q−1

h,ii

∑j 6=iQh,ij[h̃(sj)− rjβ4(sj)] and Q̃∗θ,ij = Γ−1

θ,ij · T−1.

• φθ ∝ |Γθ(φθ)|−1exp(−(Y − Aµ)TT−1 ⊗ Γ−1

θ (φθ)(Y − Aµ)/2).

•

τ 2θ , ρ ∝ |T |−n/2exp

(−(Y − Aµ)TT−1 ⊗ Γ−1

θ (φθ)(Y − Aµ)/2)

(τ 2θ )−aτθ−1exp(−bτθ/τ 2

θ ).

• β ∼ N3(Eβ, Q−1β ), where the precision matrix of the multivariate normal Qβ = XTΣ−1

h X,

Eβ = Q−1β XTΣ−1

h h, h = (h(s1), . . . , h(sn))T, X is a n × 3 matrix with the i-th row as

(1, r(si) cos θ(si), r(si) sin θ(si)), i = 1, . . . , n.

• φh ∝ |Σh|−1/2exp(− (h−Xβ)TΣ−1

h (h−Xβ)

2

), where Σh = τ 2

hΓ−1h (φh) + σ2

hIn.

28

•

τ 2h , σ

2h ∝ |Σh|−1/2exp

(−(h−Xβ)TΣ−1

h (h−Xβ)

2

)(τ 2h)−aτh−1

exp(−bτh/τ 2h)(σ2

h)−aσh−1exp(−bσh/τ 2

h).

REFERENCES

Ailliot, P., Baxevani, A., Cuzol, A., Monbet, V., and Raillard, N. (2011), “Space-time Models for

Moving Fields with an Application to Significant Wave Height Fields,” Environmetrics, 22, 354–

369.

Baxevani, A., Caires, S., and Rychlik, I. (2009), “Spatio-temporal Statistical Modelling of Signifi-

cant Wave Height,” Environmetrics, 20, 14–31.

Bulla, J., Lagona, F., Maruotti, A., and Picone, M. (2012), “A Multivariate Hidden Markov Model

for the Identification of Sea Regimes from Incomplete Skewed and Circular Time Series,” Jour-

nal of Agricultural, Biological, and Environmental Statistics, 17(4), 544–567.

Chen, M.-H., and Shao, Q.-M. (1999), “Monte Carlo Estimation of Bayesian Credible and HPD

Intervals,” Journal of Computational and Graphical Statistics, 8(1), 69–92.

Fisher, N. I., and Lee, A. J. (1992), “Regression Models for an Angular Response,” Biometrics,

48, 665–677.

Gneiting, T. (2002), “Nonseparable, Stationary Covariance Functions for Space-Time Data,” Jour-

nal of the American Statistical Association, 97, 590–600.

Gould, A. L. (1969), “A Regression Technique for Angular Variates,” Biometrics, 25(4), 683–700.

Jammalamadaka, S. R., and SenGupta, A. (2001), Topics in Circular Statistics World Scientific.

Johnson, R. A., and Wehrly, T. E. (1977), “Measures and Models for Angular Correlation and

Angular-Linear Correlation,” Journal of the Royal Statistical Society. Series B, 39(2), 222–229.

29

Johnson, R. A., and Wehrly, T. E. (1978), “Some Angular-Linear Distributions and related regres-

sion models,” Journal of the American Statistical Association, 73(363), 602–606.

Jona Lasinio, G., Gelfand, A. E., and Jona Lasinio, M. (2012), “Spatial Analysis of Wave Direction

Data Using Wrapped Gaussian Processes,” The Annals of Applied Statistics, 6(4), 1349–1997.

Jona Lasinio, G., Orasi, A., Divino, F., and Conti, P. L. (2007), “Statistical Contributions to the

Analysis of Environmental Risks along the Coastline,” in Societá Italiana di Statistica - rischio

e previsione, Venezia: CLEUP, pp. 255–262.

Kalnay, E. (2002), Atmospheric Modeling, Data Assimilation and Predictability Cambridge Uni-

versity Press.

Lagona, F., and Picone, M. (2013), “Maximum Likelihood Estimation of Bivariate Circular Hidden

Markov Models from Incomplete Data,” Journal of Statistical Computation and Simulation,

83, 1223–1237.

Laycock, P. (1975), “Optimal Design: Regression Models for Directions,” Biometrika, 62, 305–

311.

Mardia, K. V. (1972), Statistics of Directional Data Academic Press (London and New York).

Mardia, K. V. (1976), “Linear-Circular Correlation Coefficients and Rhythmometry,” Biometrika,

63(2), 403–405.

Mardia, K. V., and Jupp, P. E. (2000), Directional Statistics John Wiley & Sons.

Mardia, K. V., and Sutton, T. W. (1978), “A model for cylindrical variables with applications,”

Journal of the Royal Statistical Society. Series B (Methodological), 40, 229–233.

Modlin, D., Fuentes, M., and Reich, B. (2012), “Circular Conditional Autoregressive Modeling of

Vector Fields,” Environmetrics, 23, 46–53.

Picone, M. (2013), Model Based Clustering of Mixed Linear and Circular Data, PhD thesis, Roma

Tre University.

30

Small, C. G. (1996), The Statistical Theory of Shape Springer.

Speranza, A., Accadia, C., Casaioli, M., Mariani, S., Monacelli, G., Inghilesi, R., Tartaglione, N.,

Ruti, P. M., Carillo, A., Bargagli, A., Pisacane, G., Valentinotti, F., and Lavagnini, A. (2004),

“POSEIDON: An integrated system for analysis and forecast of hydrological, meteorological

and surface marine fields in the Mediterranean area,” Nuovo Cimento C, 27, 329–345.

Speranza, A., Accadia, C., Mariani, S., Casaioli, M., Tartaglione, N., G. Monacelli, P. M. R., and

Lavagnini, A. (2007), “SIMM: An integrated forecasting system for the Mediterranean area,”

Meteorol. Appl., 14(4), 337–350.

Stein, M. L. (2005), “Space-Time Covariance Functions,” Journal of the American Statistical As-

sociation, 100, 310–321.

Wang, F., and Gelfand, A. E. (2012), Modeling Space and Space-time Directional Data Using

Projected Gaussian Processes„ Technical report, Duke University.

Wang, F., and Gelfand, A. E. (2013), “Directional Data Analysis under the General Projected

Normal Distribution,” Statistical Methodology, 10(1), 113–127.

Wilks, D. S. (2006), “Comparison of Ensemble-MOS Methods in the Lorenz ’96 Setting,” Meteo-

rological Applications, 13, 243–256.

31