Download - Space-time processes
Space-time processes
NRCSE
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
SARMAP revisited
Spatial correlation structure depends on hour of the day:
QuickTime™ and a decompressor
are needed to see this picture.
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)
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Y(t,x) = μ(t,x)+ σ t (x)(α xZ1(t)+ Z2 (t,x))+ ε(t,x)
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σt (x )
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
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Cω(u) = e−ihTω∫ C(h,u)dh
C(h,u) = exp(iuτ)C1(h, τ)κ(τ)
f(h; τ)1 24 34 dτ∫
Spectral density
Under stationarity and separability,
If spatially nonstationary, write
Define the spatial coherency as
Under separability this is independent
of frequency τ
f(h;τ) =ϕ (h)κ(τ)
fa,b (τ) =12π
exp(−iuτ)Cov(Z(a, t+u),Z(b, t))du∫
Ra,b (τ) =fa,b (τ)
fa,a (τ)fb,b (τ)⎡⎣ ⎤⎦12
Estimation
Let
(variance stabilizing)
where R is estimated using
φa,b (τ) = tanh−1(Ra,b (τ))
f̂a,b (τ) = gρ (a−s)gρ (b−s)Ia+s ,b+s∗ (τ)ds
R2∫
Models-3 output
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
Coherence plot
a3,b3 a6,b6
A class of Matérn-type nonseparable covariances
=1: separable
=0: time is space (at a different rate)
f(,τ) =γ(α2β2 +β2 2 + α2τ2 + 2 τ2 )−ν
scale spatialdecay
temporaldecay
space-timeinteraction
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.
Z(s, t) = K (s −s i, t−ti )Zi (s, t)i=1
5
∑
Another approach
Gneiting (2001): A function f is completely monotone if (-1)nf(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
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f(t) = e−rtdF(r)0
∞∫
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f( h 2)
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C(h,u) =σ2
ψ(u 2)d/ 2ϕ
h 2
ψ(u 2 )
⎛
⎝ ⎜
⎞
⎠ ⎟
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ϕ
Some examples
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ϕ (t) = exp(−ctγ ), c > 0,0 < γ ≤ 1
ϕ(t) =cνtν/ 2
2ν−1Γ(ν)Kν (ct1/ 2 ), c > 0,ν > 0
ϕ(t) = (1+ ctγ )− ν , c,ν > 0,0 < γ ≤ 1
ψ(t) = (atα + 1)β , a > 0,0 < α ≤ 1,0 ≤ β ≤ 1
ψ(t) =ln(atα + b)
ln(b), a > 0,b > 1,0 < α ≤ 1
A particular case
QuickTime™ and a decompressor
are needed to see this picture.
α=1/2,γ=1/2 α=1/2,γ=1
α=1,γ=1/2 α=1,γ=1
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C(h,u) = (u 2α + 1)−1exp −h 2
(u 2α + 1)γ
⎛
⎝ ⎜
⎞
⎠ ⎟
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)
C(h,u) =EVCS (h−Vu)
C(h,−u) =C0 (h+ vu) ≠C0 (h−vu) =C(h,u)
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
CT (u) =(1+ a u2α )−1
Evidence of asymmetry
Time lag 1Time lag 2Time lag 3
A national US health effects study
Region 6: S Calif, all 94 sites, fitting and validation
Fitting (63)Validation (31)
Los Angeles County
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 1(si) = 0(si) for now.
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(si,t) = μ1(si ) + μ2 (si,t)
μ1(si ) = μ0 (si ) + δkV∑ ik
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2 (si,t) = ρj (si )fj (t)∑
SVD computation
Singular values of T=2912 x S=545 observation matrix
Index, 1:545
Singular value
0 100 200 300 400 500
0
200
400
600
800
EOF 1
dates87to94[1:1456]
01/01/1987 10/01/1987 07/01/1988 04/01/1989 01/01/1990 10/01/1990
dates87to94[1457:2912]
01/01/1991 10/01/1991 07/01/1992 04/01/1993 01/01/1994 10/01/1994
Annual Trend Component 1
EOF 2
dates87to94[1:1456]
01/01/1987 10/01/1987 07/01/1988 04/01/1989 01/01/1990 10/01/1990
dates87to94[1457:2912]
01/01/1991 10/01/1991 07/01/1992 04/01/1993 01/01/1994 10/01/1994
Annual Trend Component 2
EOF 3
dates87to94[1:1456]
01/01/1987 10/01/1987 07/01/1988 04/01/1989 01/01/1990 10/01/1990
dates87to94[1457:2912]
01/01/1991 10/01/1991 07/01/1992 04/01/1993 01/01/1994 10/01/1994
Annual Trend Component 3
1987-1994
sqrt(max 8hr O3)0.0
0.2
0.4
01/01/1989 01/01/1990 01/01/1991 01/01/1992 01/01/1993 01/01/1994
60370113
1987-1994
sqrt(max 8hr O3)0.0
0.2
0.4
01/01/1989 01/01/1990 01/01/1991 01/01/1992 01/01/1993 01/01/1994
61112003
1987-1994
sqrt(max 8hr O3)0.0
0.2
0.4
01/01/1989 01/01/1990 01/01/1991 01/01/1992 01/01/1993 01/01/1994
61111003
Kriging of 0
Kriging of ρ2
Quality of trend fits
Date
sq rt Ozone
0.0
0.2
0.4
01/01/1989 01/01/1990 01/01/1991 01/01/1992 01/01/1993 01/01/1994
Fitted trend (solid) vs Predicted (dashed): 060371002
Date
sq rt Ozone
0.0
0.2
0.4
01/01/1989 01/01/1990 01/01/1991 01/01/1992 01/01/1993 01/01/1994
Fitted trend (solid) vs Predicted (dashed): 060371301
Date
sq rt Ozone
0.0
0.2
0.4
01/01/1989 01/01/1990 01/01/1991 01/01/1992 01/01/1993 01/01/1994
Fitted trend (solid) vs Predicted (dashed): 060375001
Observed vs. predicted
Date
sq rt Ozone
0.0
0.2
0.4
01/01/1989 04/01/1989 07/01/1989 10/01/1989 01/01/1990 04/01/1990 07/01/1990 10/01/1990
Date
sq rt Ozone
0.0
0.2
0.4
01/01/1991 04/01/1991 07/01/1991 10/01/1991 01/01/1992 04/01/1992 07/01/1992 10/01/1992
Date
sq rt Ozone
0.0
0.2
0.4
01/01/1993 04/01/1993 07/01/1993 10/01/1993 01/01/1994 04/01/1994 07/01/1994 10/01/1994
Observed (points) vs Predicted (lines): 060371301
A model for counts
Work by Monica Chiogna, Carlo Gaetan, U. Padova
Blue grama (Bouteloua gracilis)
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).
Some views
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
Principal fields
Coefficients
Years