Stat 592A Spatial Statistical Methods [e-mail] [e-mail]

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Stat 592ASpatial Statistical
Methodspeter_at_stat.washington.edupds_at_stat.washin
gton.edu
NRCSE
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Course content

• 1. Kriging
• 1. Gaussian regression
• 2. Simple kriging
• 3. Ordinary and universal kriging
• 4. Effect of estimated covariance
• 2. Spatial covariance
• 1. Isotropic covariance in R2
• 2. Covariance families
• 3. Parametric estimation
• 4. Nonparametric models
• 5. Covariance on a sphere

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• 3. Nonstationary structures I deformations
• 1. Linear deformations
• 2. Thin-plate splines
• 3. Classical estimation
• 4. Bayesian estimation
• 5. Other deformations
• 4. Nonstationary structures II linear
  combinations etc.
• 1. Moving window kriging
• 2. Integrated white noise
• 3. Spectral methods
• 4. Testing for nonstationarity
• 5. Kernel methods

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• 5. Space-time models
• Separability
• A simple non-separable model
• Stationary space-time processes
• Space-time covariance models
• Testing for separability
• 6. Assessment and use of deterministic models
• The kriging approach
• Bayesian hierarchical models
• Bayesian melding
• Data assimilation
• Model approximation

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Programs

• R
• geoR
• Fields
• S-Plus
• S SpatialStats

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Course requirements

• Active participation
• Submit two lab reports
• eight homework problems
• Three homework problems can be replaced by an
  approved project
• Lab Thursdays 2-4 in
  B-027 Communications

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1. Kriging
NRCSE
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Research goals in air quality research

• Calculate air pollution fields for health effect
  studies
• Assess deterministic air quality models against
  data
• Interpret and set air quality standards
• Improved understanding of complicated systems

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The geostatistical model

• Gaussian process
• ?(s)EZ(s) Var Z(s) lt 8
• 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 locations Z(s1),...,Z(sn)
  
• estimate
• Z(s0) (the process at an unobserved place)
  
• (an average of the process)
• In the environmental context often time series of
  observations at the locations.

or
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Some history

• Regression (Galton, Bartlett)
• Mining engineers (Krige 1951, Matheron, 60s)
• Spatial models (Whittle, 1954)
• Forestry (Matérn, 1960)
• Objective analysis (Grandin, 1961)
• More recent work Cressie (1993), Stein (1999)

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A Gaussian formula

• If
• then

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Simple kriging

• Let X (Z(s1),...,Z(sn))T, Y Z(s0), so that
• ?X?1n, ?Y?,
• ?XXC(si-sj), ?YYC(0), and
• ?YXC(si-s0).
• Then
• This is the best unbiased linear predictor when ?
  and C are known (simple kriging).
• The prediction variance is

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Some variants

• Ordinary kriging (unknown ?)
• where
• Universal kriging (? (s)A(s)???for some spatial
  variable A)
• Still optimal for known C.

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The (semi)variogram

• Intrinsic stationarity
• Weaker assumption (C(0) needs not exist)
• Kriging predictions can be expressed in terms of
  the variogram instead of the covariance.

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An example

• Ozone data from NE USA (median of daily one hour
  maxima JuneAugust 1974, ppb)

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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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Effect of estimated covariance structure

• The usual geostatistical method is to consider
  the covariance known. When it is estimated
• the predictor is not linear
• nor is it optimal
• the plug-in estimate of the
  variability often has too low mean
• Let . Is
  a good estimate of m2(?) ?

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Some results

• 1. Under Gaussianity, m2??? m1(?? with equality
  iff p2(X)p(X?) a.s.
• 2. Under Gaussianity, if is sufficient, and
  if the covariance is linear in ??? then
• 3. An unbiased estimator of m2(???is
• where is an unbiased estimator of m1(?).

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Better prediction variance estimator

• (Zimmerman and Cressie, 1992)
• (Taylor expansion often approx. unbiased)
• A Bayesian prediction analysis takes account of
  all sources of variability (Le and Zidek, 1992)