Modelling non-stationarity in space and time for air quality
data Peter Guttorp University of Washington
[email protected] NRCSE
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Outline Lecture 1: Geostatistical tools Gaussian predictions
Kriging and its neighbours The need for refinement Lecture 2:
Nonstationary covariance estimation The deformation approach Other
nonstationary models Extensions to space-time Lecture 3: Putting it
all together Estimating trends Prediction of air quality surfaces
Model assessment
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Research goals in air quality modeling Create exposure fields
for health effects modeling Assess deterministic air quality models
Interpret environmental standards Enhance understanding of complex
systems
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The geostatistical setup Gaussian process (s)=EZ(s) Var Z(s)
< Z is strictly stationary if Z is weakly stationary if Z is
isotropic if weakly stationary and
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The problem Given observations at n sites Z(s 1 ),...,Z(s n )
estimate Z(s 0 ) (the process at an unobserved site) or (a weighted
average of the process)
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A Gaussian formula If then
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Simple kriging Let X = (Z(s 1 ),...,Z(S n )) T, Y = Z(s 0 ), so
X = 1 n, Y = , XX =[C(s i -s j )], YY =C(0), and YX =[C(s i -s 0
)]. Thus This is the best linear unbiased predictor for known and C
(simple kriging). Variants: ordinary kriging (unknown ) universal
kriging ( =A for some covariate A) Still optimal for known C.
Prediction error is given by
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The (semi)variogram Intrinsic stationarity Weaker assumption
(C(0) need not exist) Kriging can be expressed in terms of
variogram
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Method of moments: square of all pairwise differences, smoothed
over lag bins Problem: Not necessarily a valid variogram Estimation
of covariance functions
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Least squares Minimize Alternatives: fourth root transformation
weighting by 1/ 2 generalized least squares
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Fitted variogram
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Kriging surface
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Kriging standard error
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A better combination
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Maximum likelihood Z~N n ( , ) = [ (s i -s j ; )] = V( )
Maximize and maximizes the profile likelihood
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A peculiar ml fit
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Some more fits
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All together now...
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Effect of estimating covariance structure Standard
geostatistical practice is to take the covariance as known. When it
is estimated, optimality criteria are no longer valid, and plug-in
estimates of variability are biased downwards. (Zimmerman and
Cressie, 1992) A Bayesian prediction analysis takes proper account
of all sources of uncertainty (Le and Zidek, 1992)
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Violation of isotropy
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General setup Z(x,t) = (x,t) + (x) 1/2 E(x,t) + (x,t) trend +
smooth + error We shall assume that is known or constant t =
1,...,T indexes temporal replications E is L 2 -continuous, mean 0,
variance 1, independent of the error C(x,y) = Cor(E(x,t),E(y,t))
D(x,y) = Var(E(x,t)-E(y,t)) (dispersion)
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Geometric anisotropy Recall that if we have an isotropic
covariance (circular isocorrelation curves). If for a linear
transformation A, we have geometric anisotropy (elliptical
isocorrelation curves). General nonstationary correlation
structures are typically locally geometrically anisotropic.
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The deformation idea In the geometric anisotropic case, write
where f(x) = Ax. This suggests using a general nonlinear
transformation. Usually d=2 or 3. G-plane D-space We do not want f
to fold.
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Implementation Consider observations at sites x 1,...,x n. Let
be the empirical covariance between sites x i and x j. Minimize
where J(f) is a penalty for non-smooth transformations, such as the
bending energy
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SARMAP An ozone monitoring exercise in California, summer of
1990, collected data on some 130 sites.
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Transformation This is for hr. 16 in the afternoon
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Thin-plate splines Linear part
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A Bayesian implementation Likelihood: Prior: Linear part: fix
two points in the G-D mapping put a (proper) prior on the remaining
two parameters Posterior computed using Metropolis-Hastings
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California ozone
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Posterior samples
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Other applications Point process deformation (Jensen &
Nielsen, Bernoulli, 2000) Deformation of brain images (Worseley et
al., 1999)
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Isotropic covariances on the sphere Isotropic covariances on a
sphere are of the form where p and q are directions, pq the angle
between them, and P i the Legendre polynomials. Example: a i
=(2i+1) i
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A class of global transformations Iteration between simple
parametric deformation of latitude (with parameters changing with
longitude) and similar deformations of longitude (changing smoothly
with latitude). (Das, 2000)
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Three iterations
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Global temperature Global Historical Climatology Network 7280
stations with at least 10 years of data. Subset with 839 stations
with data 1950-1991 selected.
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Isotropic correlations
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Deformation
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Assessing uncertainty
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Gaussian moving averages Higdon (1998), Swall (2000): Let be a
Brownian motion without drift, and. This is a Gaussian process with
correlogram Account for nonstationarity by letting the kernel b
vary with location:
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Kernel averaging Fuentes (2000): Introduce orthogonal local
stationary processes Z k (s), k=1,...,K, defined on disjoint
subregions S k and construct where w k (s) is a weight function
related to dist(s,S k ). Then A continuous version has
SARMAP revisited Spatial correlation structure depends on hour
of the day (non-separable):
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Brunos seasonal nonseparability Nonseparability generated by
seasonally changing spatial term Z 1 large-scale feature Z 2
separable field of local features (Bruno, 2004)
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A non-separable class of stationary space-time covariance
functions Cressie & Huang (1999): Fourier domain Gneiting
(2001): f is completely monotone if (-1) n f (n) 0 for all n.
Bernsteins theorem : for some non- decreasing F. Combine a
completely monotone function and a function with completely
monotone derivative into a space-time covariance
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A particular case =1/2, =1/2 =1/2, =1 =1, =1/2 =1, =1
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Uses for surface estimation Compliance exposure assessment
measurement Trend Model assessment comparing (deterministic) model
to data approximating model output Health effects modeling
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Health effects Personal exposure (ambient and non- ambient)
Ambient exposure outdoor time infiltration Outdoor concentration
model for individual i at time t
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2 years, 26 10-day sessions A total of 167 subjects: 56 COPD
subjects 40 CHD subjects 38 healthy subjects (over 65 years old,
non-smokers) 33 asthmatic kids A total of 108 residences: 55
private homes 23 private apartments 30 group homes Seattle health
effects study
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pDR PUF HPEM Ogawa sampler
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HI Ogawa sampler T/RH logger Nephelometer Quiet Pump Box CO 2
monitor CAT
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PM 2.5 measurements
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Where do the subjects spend their time? Asthmatic kids: 66% at
home 21% indoors away from home 4% in transit 6% outdoors Healthy
(CHD, COPD) adults: 83% (86,88) at home 8% (7,6) indoors away from
home 4% (4,3) in transit 3% (2,2) outdoors
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Panel results Asthmatic children not on anti- inflammatory
medication: decrease in lung function related to indoor and to
outdoor PM 2.5, not to personal exposure Adults with CV or COPD:
increase in blood pressure and heart rate related to indoor and
personal PM 2.5
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Trend model where V ik are covariates, such as population
density, proximity to roads, local topography, etc. where the f j
are smoothed versions of temporal singular vectors (EOFs) of the
TxN data matrix. We will set 1 (s i ) = 0 (s i ) 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 0
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Kriging of 2
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Quality of trend fits
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Observed vs. predicted
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Observed vs. predicted, cont.
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Conclusions Good prediction of day-to-day variability seasonal
shape of mean Not so good prediction of long-term mean Need to try
to estimate
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Other difficulties Missing data Multivariate data Heterogenous
(in space and time) geostatistical tools Different sampling
intervals (particularly a PM problem)
