STAT 592A(UW) 526 (UBC-V) 890-4(SFU) Spatial Statistical Methods [e-mail] www.stat.washington.edu/peter/592

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STAT 592A(UW) 526 (UBC-V) 890-4(SFU)
Spatial Statistical Methodspeter_at_stat.washingto
n.eduwww.stat.washington.edu/peter/592
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
• 5. Bayesian kriging
• 2. Spatial covariance
• 1. Isotropic covariance in R2
• 2. Covariance families
• 3. Parametric estimation
• 4. Nonparametric models
• 5. Fourier analysis
• 6. Covariance on a sphere

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• 3. Nonstationary structures I deformations
• Linear deformations
• Thin-plate splines
• Classical estimation
• Bayesian estimation
• Other deformations
• 4. Markov random fields
• The Markov property
• Hammersley-Clifford
• Ising model
• Gaussian MRF
• Conditional autoregression
• The non-lattice case

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• 5. Nonstationary structures II linear
  combinations etc.
• Moving window kriging
• Integrated white noise
• Spectral methods
• Testing for nonstationarity
• Wavelet methods
• 6. Space-time models
• Mean surface
• Separability
• A simple non-separable model
• Stationary space-time processes
• Space-time covariance models
• Testing for separability
• The Le-Zidek approach

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• 7. Statistics, data and deterministic models
• The kriging approach
• Bayesian hierarchical models
• Bayesian melding
• Data assimilation
• Model approximation
• 8. Statistics of compositions
• An algebra for compositions
• The logistic normal distribution
• Source apportionment
• Analysis of variance
• A space-time model

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• 9. Wavelet tools
• Basic wavelet theory
• Multiscale analysis
• Longterm memory models
• Wavelet analysis of trends
• 10. Setting air quality standards
• Bayesian model averaging
• Standards as hypothesis tests
• Potential network bias
• Maxima of spatial processes
• Operational evaluation of air quality standards

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Programs

• R
• geoR
• fields
• spBayes
• RandomFields
• GMRFLib a C-library for fast and exact
  simulation of Gaussian Markov random fields

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Course requirements

• Submit at least 8 homework problems
• (3 can be replaced by an approved project)
• Submit at least three lab reports
• Every other Thursday will be a lab day.
• Virtual lab machine (get Remote Desktop
  Connection)

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Office hours

• MTh 10-11 B213 Padelford
• Skype name guttorp
• Homework solutions and lab reports must be
  submitted electronically

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Kriging
NRCSE
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Research goals in environmental research

• Calculate pollution fields for health effect
  studies
• Assess deterministic models against data
• Interpret and set environmental standards
• Improve 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 location)
  
• (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 (Bravais, Galton, Bartlett)
• Mining engineers (Krige 1951, Matheron, 60s)
• Spatial models (Whittle, 1954)
• Forestry (Matérn, 1960)
• Objective analysis (Gandin, 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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Universal kriging variance
simple kriging variance
variability due to estimating ?
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Some other kriging variants

• Indicator kriging
• Block kriging
• Co-kriging
• Using a covariate to improve kriging
• Disjunctive kriging
• A nonlinear version of kriging expand the
• field into CONS and co-krige these

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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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Ordinary kriging

• where
• and kriging variance

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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)