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Space-time processes

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Cov(Z(x,t),Z(y,s))=CS(x,y)CT(s,t) Nonseparable alternatives ... Nonseparability generated by seasonally changing ... Blue grama (Bouteloua gracilis) The data ... – PowerPoint PPT presentation

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Title: Space-time processes


1
Space-time processes
NRCSE
2
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

3
SARMAP revisited
  • Spatial correlation structure depends on hour of
    the day

4
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)

5
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

6
Spectral density
  • Under stationarity and separability,
  • If spatially nonstationary, write
  • Define the spatial coherency as
  • Under separability this is independent
  • of frequency t

7
Estimation
  • Let
  • (variance stabilizing)
  • where R is estimated using

8
Models-3 output
9
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
10
Coherence plot
a3,b3
a6,b6
11
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
12
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13
Chesapeake Bay wind field forecast (July 31, 2002)
14
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.

15
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16
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

17
Some examples
18
A particular case
a1/2,g1/2
a1/2,g1
a1,g1/2
a1,g1
19
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)

20
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

21
Evidence of asymmetry
Time lag 1 Time lag 2 Time lag 3
22
A national US health effects study
23
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24
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.

25
SVD computation
26
EOF 1
27
EOF 2
28
EOF 3
29
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30
Kriging of m0
31
Kriging of r2
32
Quality of trend fits
33
Observed vs. predicted
34
A model for counts
  • Work by Monica Chiogna, Carlo Gaetan, U. Padova
  • Blue grama (Bouteloua gracilis)

35
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).
36
Some views
37
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

38
Principal fields
39
Coefficients
40
Years
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