NRCSE. Space-time processes. 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.
Separable covariance structure:
Spatial correlation structure depends on hour of the day:
Nonseparability generated by seasonally changing spatial term
(uniformly modulated at each time)
Z1 large-scale feature
Z2 separable field of local features
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 w.
Usage: Fourier transform of Cw(u)
Problem: Need to know Fourier pairs
Under stationarity and separability,
If spatially nonstationary, write
Define the spatial coherency as
Under separability this is independent
of frequency τ
where R is estimated using
=0: time is space (at a different rate)
Prior equal weight on =0 and =1.
Posterior: mass (essentially) 0 for =0 for regions 1, 2, 3, 5; mass 1 for region 4.
Gneiting (2001): A function f is completely monotone if (-1)nf(n)≥0for 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 y with completey monotone derivative into a space-time covariance
CS covariance of purely spatial field
V (random) velocity of field
Frozen field model: P(V=v)=1 (e.g. prevailing wind)
Daily average wind speed at 11 stations, 1961-70, transformed to “velocity measures”
Spatial: exponential with nugget
Space-time: mixture of Gneiting model and frozen field
Time lag 1
Time lag 2
Time lag 3
where Vik are covariates, such as population density, proximity to roads, local topography, etc.
where the fj are smoothed versions of temporal singular vectors (EOFs) of the TxN data matrix.
We will set m1(si) = m0(si) for now.
Work by Monica Chiogna, Carlo Gaetan, U. Padova
Blue grama (Bouteloua gracilis)
Yearly counts of blue grama plants in a series of 1 m2 quadrats in a mixed grass prairie (38.8N, 99.3W) in Hays, Kansas, between 1932 and1972 (41 years).
Aim: See if spatial distribution is changing with time.
Y(s,t)(s,t) ~ Po((s,t))
log((s,t)) = constant
+ fixed effect of temp & precip
+ weighted average of principal fields