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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.

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Separable covariance structure:


Nonseparable alternatives

  • Temporally varying spatial covariances
  • Fourier approach
  • Completely monotone functions
sarmap revisited
SARMAP revisited

Spatial correlation structure depends on hour of the day:

bruno s seasonal nonseparability
Bruno’s seasonal nonseparability

Nonseparability generated by seasonally changing spatial term

(uniformly modulated at each time)

Z1 large-scale feature

Z2 separable field of local features

(Bruno, 2004)

general stationary space time covariances
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 w.

Usage: Fourier transform of Cw(u)

Problem: Need to know Fourier pairs

spectral density
Spectral density

Under stationarity and separability,

If spatially nonstationary, write

Define the spatial coherency as

Under separability this is independent

of frequency τ



(variance stabilizing)

where R is estimated using

coherence plot
Coherence plot



a class of mat rn type nonseparable covariances
A class of Matérn-type nonseparable covariances

=1: separable

=0: time is space (at a different rate)








fuentes model
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
Another approach

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

a particular case
A particular case





velocity driven space time covariances
Velocity-driven space-time covariances

CS covariance of purely spatial field

V (random) velocity of field

Space-time covariance

Frozen field model: P(V=v)=1 (e.g. prevailing wind)

irish wind data
Irish wind data

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

evidence of asymmetry
Evidence of asymmetry

Time lag 1

Time lag 2

Time lag 3

trend model
Trend model

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.

a model for counts
A model for counts

Work by Monica Chiogna, Carlo Gaetan, U. Padova

Blue grama (Bouteloua gracilis)

the data
The data

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

+ trend

+ weighted average of principal fields