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