Space-time processes NRCSE

Separability Separable covariance structure: Cov(Z(x,t),Z(y,s))=C S (x,y)C T (s,t) Nonseparable alternatives Temporally varying spatial covariances Fourier approach Completely monotone functions

SARMAP revisited Spatial correlation structure depends on hour of the day:

Bruno’s seasonal nonseparability Nonseparability generated by seasonally changing spatial term Z 1 large-scale feature Z 2 separable field of local features (Bruno, 2004)

General stationary space-time covariances Cressie & Huang (1999): By Bochner’s theorem, a continuous, bounded, symmetric integrable C(h;u) is a space- time covariance function iff is a covariance function for all . Usage: Fourier transform of C  (u) Problem: Need to know Fourier pairs

Spectral density Under stationarity and separability, If spatially nonstationary, write Define the spatial coherency as Under separability this is independent of frequency τ

Estimation Let where R is estimated using

Models-3 output

ANOVA results ItemdfrssP-value Between points Between freqs Residual50.346

Coherence plot a 3,b 3 a 6,b 6

A class of Matérn-type nonseparable covariances  =1: separable  =0: time is space (at a different rate) scale spatial decay temporal decay space-time interaction

Chesapeake Bay wind field forecast (July 31, 2002)

Fuentes model Prior equal weight on  =0 and  =1. Posterior: mass (essentially) 0 for  =0 for regions 1, 2, 3, 5; mass 1 for region 4.

Another approach Gneiting (2001): A function f is completely monotone if (-1) n f (n) ≥0 for all n. Bernstein’s theorem shows that for some non- decreasing F. In particular,is a spatial covariance function for all dimensions iff f is completely monotone. The idea is now to combine a completely monotone function and a function  with completey monotone derivative into a space-time covariance

Some examples

A particular case  =1/2,  =1/2  =1/2,  =1  =1,  =1/2  =1,  =1

Issues to be developed Nonstationarity Deformation approach: 3-d spline fit; penalty for roughness in time and space Antisymmetry Gneiting’s approach has C(h,u)=C(-h,u)=C(h,-u)=C(-h,-u) Unrealistic when covariance caused by meteorology Covariates Need to take explicit account of covariates driving the covariance Multivariate fields Mardia & Goodall (1992) use Kronecker product structure (like separability)

Temporally changing spatial deformations where

Velocity-driven space-time covariances C S covariance of purely spatial field V (random) velocity of field Space-time covariance Frozen field model: P(V=v)=1 (e.g. prevailing wind)

Taylor’s hypothesis C(0,u) = C(vu,0) for some v Relates spatial to temporal covariance Examples: Frozen field model Separable models

Irish wind data Daily average wind speed at 11 stations, , transformed to “velocity measures” Spatial: exponential with nugget Temporal: Space-time: mixture of Gneiting model and frozen field

Evidence of asymmetry

Time lag 0 Time lag 1 Time lag 2 Time lag 3