Space-time processes

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Space-time processes
NRCSE
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Separability

• Separable covariance structure
• Cov(Z(x,t),Z(y,s))CS(x,y)CT(s,t)
• Nonseparable alternatives
• Temporally varying spatial
  covariances
• Fourier approach
• Completely monotone functions

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SARMAP revisited

• Spatial correlation structure depends on hour of
  the day

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Brunos 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)

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General stationary space-time covariances

• Cressie Huang (1999) By Bochners theorem, a
  continuous, bounded, symmetric integrable C(hu)
  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

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Spectral density

• Under stationarity and separability,
• If spatially nonstationary, write
• Define the spatial coherency as
• Under separability this is independent
• of frequency t

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Estimation

• Let
• (variance stabilizing)
• where R is estimated using

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Models-3 output
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ANOVA results
Item df rss P-value
Between points 1 0.129 0.68
Between freqs 5 11.14 0.0008
Residual 5 0.346
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Coherence plot
a3,b3
a6,b6
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A class of Matérn-type nonseparable covariances

• ?1 separable
• ?0 time is space (at a different rate)

spatial decay
temporal decay
scale
space-time interaction
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Chesapeake Bay wind field forecast (July 31, 2002)
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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.

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Another approach

• Gneiting (2001) A function f is completely
  monotone if (-1)nf(n)0 for all n. Bernsteins
  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

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Some examples
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A particular case
a1/2,g1/2
a1/2,g1
a1,g1/2
a1,g1
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Velocity-driven space-time covariances

• CS covariance of purely spatial field
• V (random) velocity of field
• Space-time covariance
• Frozen field model P(Vv)1 (e.g. prevailing
  wind)

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Irish wind data

• Daily average wind speed at 11 stations, 1961-70,
  transformed to velocity measures
• Spatial exponential with nugget
• Temporal
• Space-time mixture of Gneiting model and frozen
  field

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Evidence of asymmetry
Time lag 1 Time lag 2 Time lag 3
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A national US health effects study
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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.

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SVD computation
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EOF 1
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EOF 2
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EOF 3
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Kriging of m0
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Kriging of r2
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Quality of trend fits
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Observed vs. predicted
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A model for counts

• Work by Monica Chiogna, Carlo Gaetan, U. Padova
• Blue grama (Bouteloua gracilis)

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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).
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Some views
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Modelling

• 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

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Principal fields
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Coefficients
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Years