Spatio-temporal Models Again point-referenced vs. areal unit data - - PowerPoint PPT Presentation

Spatio-temporal Models

Again point-referenced vs. areal unit data Continuous time vs. discretized time Association in space versus association in time! For point-referenced data, t continuous, Gaussian data,

Y (s, t) = µ(s, t) + w(s, t) + ǫ(s, t)

For non-Gaussian data, instead use appropriate likelihood with link g(E(Y (s, t))) = µ(s, t) + w(s, t) Don’t treat time as a third coordinate – scale issue! sensible: Cov(Y (s, t), Y (s′, t′)) = C(s − s′, t − t′) NOT sensible: Cov(Y (s, t), Y (s′, t′)) = C((s, t) − (s′, t′))

Spatiotemporal Modeling – p. 1

Spatio-temporal Models

Separable form:

C(s − s′, t − t′) = σ2ρ1(s − s′; φ1)ρ2(t − t′; φ2)

Limitation Nonseparable form: Sum of independent separable processes More generally, mixing of separable covariance functions Spectral domain approaches - space-time spectral density and inverse Fourier transform

Spatiotemporal Modeling – p. 2

Spatio-temporal Models

An EDA idea Now suppose time is discretized, i.e. data are

Yt(s), t = 1, . . . , T

Type of data: time series versus cross-sectional (e.g., real estate sales). Latter is problematic For time series data, exploratory analysis: Arrange into an n × T matrix Y with entries Yt(si) Center by row averages of Y yields Yrows Center by column averages of Y yields Ycols sample spatial covariance matrix: 1

T YrowsY T rows

sample autocorrelation matrix: 1

nY T colsYcols

E, residuals matrix after a regression fitting

Spatiotemporal Modeling – p. 3

Empirical Orthogonal Functions

Can understand the structure of Y, Yrows, Ycols, E using empirical orthogonal functions Say for E, use singular value decomposition

E = UDV T =

T

• j=1

djujvT

j

where U is n × n orthogonal, V is T × T orthogonal and

D is “almost diagonal"

If we arrange the dj in decreasing order then ujvT

j is

the jth empirical orthogonal function Typically, we only need a few terms in the sum to well approximate EET. With just the first term it would suggest approximating E(s, t) by u1(s)v1(t).

Spatiotemporal Modeling – p. 4

Spatio-temporal Models

If t discrete, a time series of spatial process realizations Modeling: Yt(s) = µt(s) + wt(s) + ǫt(s),

• r perhaps g(E(Yt(s)) = µt(s) + wt(s)

For ǫt(s), independent N(0, τ2

t )

For wt(s)

wt(s) = αt + w(s), αt ∼? wt(s) independent for each t wt(s) = wt−1(s) + ηt(s), independent spatial process

innovations Inference? Interpolate for new locations at an

• bserved time; predict for a current location at a new

time; new location and new time

Spatiotemporal Modeling – p. 5

Multivariate spatio-temporal models

Again, connecting space and time, extending versions above

Y(s, t) = µ(s, t) + w(s, t) + ǫ(s, t) with µ(s, t) = X(s, t)Tβ

For w(s, t), can supply an additive form in random effects Can supply a multiplicative form in random effects (from EOF idea) Perhaps, most natural is a coregionalization specification

Spatiotemporal Modeling – p. 6

cont.

Coregionalization approach In the LMC, consider w(s, t) = Av(s, t) where the components of v(s, t) are independent univariate space-time processes Can extend to At, to A(s) or even A(s, t) Can make the evolution dynamic, e.g., wt(s) = Avt(s) where vlt(s) = φvl,t−1(s) + ǫlt(s) Further variations Could also try cross-convolution using valid Cl(s, t)

Spatiotemporal Modeling – p. 7

Dynamic Space Time models

Again, t is discrete. More general two-stage

• specification. Dynamics in the mean

Stage 1: Measurement equation

Y (s, t) = µ (s, t) + ǫ (s, t) ; ǫ (s, t) ind

∼ N

• 0, σ2

ǫ

• .

µ (s, t) = xT (s, t) ˜ β (s, t) . ˜ β (s, t) = βt + β (s, t)

Stage 2: Transition equation

βt = βt−1 + ηt, ηt

ind

∼ Np

• 0, Ση
• .

β (s, t) = β (s, t − 1) + w (s, t) . w(s, t) is a multivariate space-time process

Spatiotemporal Modeling – p. 8

cont.

So, as above w (s, t) = Av (s, t), with

v (s, t) = (v1 (s, t) , ..., vp (s, t))T.

The vl (s, t) are replications of a Gaussian processes with unit variance and correlation function ρl (φl, ·) Can connect to linear Kalman filter

Spatiotemporal Modeling – p. 9

An example:

Modelling Temperature given precipitation. Sampled temperature data (maximum monthly temperature) and sampled precipitation data (maximum monthly precipitation), 50 sites across Colorado, January through December in 1997 (12 months). temp(s, t) = ˜

β0(s, t) + ˜ β1(s, t)precip(s, t) + ǫ(s, t)

Independent Gaussians for ηt Space-time varying intercept process and slope

• process. Coregionalization for w(s, t)

Spatiotemporal Modeling – p. 10

Spatial domain (with elevation)

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Spatiotemporal Modeling – p. 11

Parameter estimates

Parameters median (2.5%, 97.5%) Ση [1, 1] 0.296 (0.130, 0.621) Ση [2, 2] 0.786 (0.198, 1.952) Ση [1, 2]

• Ση [1, 1] Ση [2, 2]
• 0.562 (-0.807, -0.137)

Σw [1, 1] 0.017 (0.016, 0.019) Σw [2, 2] 0.026 (0.0065, 0.108) Σw [1, 2]

• Σw [1, 1] Σw [2, 2]
• 0.704 (-0.843, -0.545)

σ2

ǫ

0.134 (0.106, 0.185) φ1 1.09 (0.58, 2.04) φ2 0.58 (0.37, 1.97) Range for intercept process 2.75 (1.47, 5.17) Range for slope process 4.68 (1.60, 6.21)

Spatiotemporal Modeling – p. 12

Time varying parameters

0.0 1.5 0.0 1.5 0.0 1.5 Month beta_0t Jan Mar May Jul Sep Nov

Intercept

−2.0 −0.5 −2.0 −0.5 −2.0 −0.5 Month beta_1t Jan Mar May Jul Sep Nov

• Coeff. of Precipitation

Spatiotemporal Modeling – p. 13

Intercept process

Time-sliced image-contour plots displaying the posterior mean surface of the spatial residuals

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Jan

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Feb

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Mar

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Apr

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May

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Jun

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Jul

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Aug

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Sep

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Oct

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Nov

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Dec Spatiotemporal Modeling – p. 14

Slope process

Time-sliced image-contour plots displaying the posterior mean surface of the spatial residuals

−108 −104 37 38 39 40 41 Longitude Latitude −108 −104 37 38 39 40 41

Jan

−108 −104 37 38 39 40 41 Longitude Latitude −108 −104 37 38 39 40 41

Feb

−108 −104 37 38 39 40 41 Longitude Latitude −108 −104 37 38 39 40 41

Mar

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Apr

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May

−108 −104 37 38 39 40 41 Longitude Latitude −108 −104 37 38 39 40 41

Jun

−108 −104 37 38 39 40 41 Longitude Latitude −108 −104 37 38 39 40 41

Jul

−108 −104 37 38 39 40 41 Longitude Latitude −108 −104 37 38 39 40 41

Aug

−108 −104 37 38 39 40 41 Longitude Latitude −108 −104 37 38 39 40 41

Sep

−108 −104 37 38 39 40 41 Longitude Latitude −108 −104 37 38 39 40 41

Oct

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Nov

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Dec Spatiotemporal Modeling – p. 15

Multivariate Dynamic Model

For (s, t) yielding Y (s, t) = (Y1 (s, t) , ..., Yp (s, t))T, and covariate vectors (including an intercept) xl (s, t),

l = 1, 2, ..., p.

The model becomes

Y (s, t) = µ (s, t) + ǫ (s, t) ; ǫ (s, t) ind

∼ N (0, Σǫ) ; Σǫ = σ2

ǫIp

µ (s, t) = X (s, t) ˜ β (s, t) ,

where ˜

β (s, t) =

• ˜

β

T 1 (s, t) , ..., ˜

β

T m (s, t)

T is m

l=1 kl × 1,

X (s, t) = Diag

• xT

1 (s, t) , ..., xT m (s, t)

• is p × m

l=1 kl.

Spatiotemporal Modeling – p. 16

cont.

The transition equation is,

˜ β (s, t) = βt + β (s, t) βt = βt−1 + ηt, ηt

ind

∼ Np

• 0, Ση
• .

β (s, t) = β (s, t − 1) + w (s, t) ,

where Ση is a m

k=1 pk × p l=1 kl covariance matrix,

w (s, t) is a p

l=1 kl dimensional spatial process.

Hence, β (s, t) =

• βT

1 (s, t) , ..., βT m (s, t)

T is a m

k=1 pk

dimensional spatial process. To reduce computing effort, we work only with a spatio-temporally varying intercept for each component

Spatiotemporal Modeling – p. 17

Areal unit data

Yi(t), temporal process for each unit (rare!) Yit, a time series for each unit (and occasionally, Yijt), is

more common Again, Yit = µit + φit + ǫit,

• r for non-Gaussian data, g(E(Yit)) = µit + φit

Again, ǫit ∼ N(0, τ2

t )

Modeling for φit?? CAR in space and time! Again, additive form, φi + αt For space nested within time, model φ(t)

i

∼ CAR(λt),

with say λt

iid

∼ Gamma(a, b) φit|φ−(it), space, time neighbors, weight for space,

weight for time MCAR, φi = (φi1, φi2, ...φiT ), if short series

Spatiotemporal Modeling – p. 18

Neighbors in time and space

Spatiotemporal Modeling – p. 19

Large datasets

Comments on the problem of handling large n in space (possibly p-dim at each location) possibly large T in time for point referenced datasets Large matrices n × n, nT × nT, npT × npT Joint density approximation (Vecchia, Stein) Spectral methods (Whittle likelihood, Fuentes) Lattice modelling instead or lattice approximation (MRFs) to spatial process models (Rue) Create sparsity (covariance tapering) and fast matrix multiplication (Nychka et al.) Reduced rank kriging (Cressie and Johannesson) - representation for C(s, s′)

Spatiotemporal Modeling – p. 20

Dimension reduction

Idea of dimension reduction: Represent the high dimensional vector of variables in a lower dimensional space Write the vector w ( say, n × 1) in the form w = L ˜

w, ˜ w is m × 1

Choosing L and ˜

w?

A basis representation with usual basis functions A representation through kernel convolution (Higdon) Predictive process approach

Spatiotemporal Modeling – p. 21

An attractive approach

The spatial regression model:

Y (s) = XT(s)β + w(s) + ǫ(s)

Obvious multivariate version Fitting a fully Bayes model over sites S = {s1, . . . , sn} involves matrix decompositions and determinants for the n × n Σw(φ). When Φ is estimated this must be done in each MCMC iteration. Approach: Replace w(s) by a process ˜

w(s) that projects

the process into a smaller subspace.

Spatiotemporal Modeling – p. 22

The “Kriging” Equation

Consider “knots” S∗ = {s∗

1, . . . , s∗ m} with m << n.

Predict w(s) based upon S∗ at s0:

˜ w(s0) = E[w(s0)|w∗] = Cov(w(s0), w∗)V ar(w∗)−1w∗; w∗ = [w(s∗

j)]m j=1,

Cov(w(s0), w∗) = [Γw(s0, s∗

1; Φ), . . . , Γw(s0, s∗ m; Φ)],

V ar(w∗) = [Γw(s∗

j, s∗ j ′; Φ)]m j,j′=1.

Spatiotemporal Modeling – p. 23

The Predictive Process

˜ w(s) defined in this way create a new multivariate

Gaussian spatial process with

Γ ˜

w(s, s′) = Cov(w(s), w∗)V ar(w∗)−1CovT (w(s), w∗).

We can write ˜

w(s) = L(s)w∗ L(s) = Cov(w(s), w∗)V ar(w∗)−1 is an 1 × m vector.

Process realizations over S = s1, . . . , sn:

˜ w = [ ˜ w(si)]n

i=1 = Lw∗;

∼ MV N(0, LΣ−1

w∗LT)

with L = [L(si)]n

i=1 being n × m.

Spatiotemporal Modeling – p. 24

Reducing the dimension

We fit the reduced model:

Y(s) = XT(s)β + ˜ w(s) + ǫ(s) = XT(s)β + L(s)w∗ + ǫ(s).

Computations involve Σw∗ (m × m) instead of Σw (n × n). Can make a huge difference with m << n. Extension to multivariate and temporal settings; knots in space and in time Concerns: How many knots will suffice? How do we choose the knots? Impact of knot selection in practical data analysis?

Spatiotemporal Modeling – p. 25

Attractive properties

Flexibility with knots compared to other methods. Don’t require a regular grid as for spectral methods and other likelihood approximation methods. Derives from the original process. No need to specify kernels or other covariance functions. No restrictions to stationarity or isotropy Adapts seamlessly to spatiotemporal and other settings since it is a “model” rather than an approximation! Given S∗ you cannot do better in a (reverse entropy) Kullback-Leibler sense with any other process of the form L(s)w∗. When S∗ = S we recover the original model.

Spatiotemporal Modeling – p. 26