Nonstationary covariance structures II

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Nonstationary covariance structures II
NRCSE
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Drawbacks with deformation approach

• Oversmoothing of large areas
• Local deformations not local part of global fits
• Covariance shape does not change over space
• Limited class of nonstationary covariances

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Nonstationary spatial covariance

• Basic idea the parameters of a local variogram
  modelnugget, range, sill, and anisotropyvary
  spatially.
• For deformation, which do?
• Not nugget nor sill.

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Major approaches

• Haas, 1990, Moving window kriging
• Kim, Mallock Holmes, 2005, Piecewise Gaussian
  modeling
• Nott Dunsmuir, 2002, Biometrika, Average of
  locally stationary processes
• Fuentes, 2002, Kernel averaging of orthogonal,
  locally stationary processes.
• Pintore Holmes, 2005, Fourier and
  Karhunen-Loeve expansions
• Higdon Swall, 1998, 2000, Gaussian moving
  averages or process convolution model
• Nychka, Wikle Royle, 2002. Wavelet expansion.

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Thetford revisitedSpectrum of canopy heights
Ridge in neg quadrat 46.4 of var
Low freq ridge in pos quadrat 19.7 of var
High freq ridge in pos quadrat 6.1 of var
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Looking at several adjacent plots

• Features depend on spatial location

HF qlt0
LF qgt0
LF qlt0
HF qgt0
other
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Moving window approach

• For kriging at s, consider only sites near s
• Near defined small enough to make
  stationarity/isotropy reasonable

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The moving window approach

• In purely spatial context, pick the ns nearest
  sites (for spacetime, use a window in space and
  time and grow until get ns sites)
• Fit a model based only on the chosen sites and
  perform appropriate kriging
• Difficulty not a single model, so can be
  discontinuous
• Overall covariance may not be positive definite

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Kernel averaging

• Fuentes (2000) Introduce uncorrelated stationary
  processes Zk(s), k1,...,K, defined on subregions
  Sk and construct
• where wk(s) is a weight function related to
  dist(s,Sk). Then

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Spectral version

• so
• where
• Hence

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Estimating spectrum

• Asymptotically

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Details

• K 9 h 687 km
• Mixture of Matérn spectra

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An example
Models-3 output, 81x87 grid, 36km x 36km. Hourly
averaged nitric acid concentration week of
950711.
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Models-3 fit
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A spectral approach to nonstationary processes

• Spectral representation
• ?s slowly varying square integrable, Y
  uncorrelated increments
• Hence is the space-varying spectral
  density
• Observe at grid use FFT to estimate in nbd of s

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Testing for nonstationarity

• U(s,w) log has approximately mean
  f(s,w) log fs(w) and constant variance in s and
  w.
• Taking s1,...,sn and w1,...,wm sufficently well
  separated, we have approximately Uij U(si,wj)
  fijeij with the eij iid.
• We can test for interaction between location and
  frequency (nonstationarity) using ANOVA.

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Details

• The general model has
• The hypothesis of no interaction (?ij0)
  corresponds to additivity on log scale
• (uniformly modulated process
• Z0 stationary)
• Stationarity corresponds to
• Tests based on approx ?2-distribution (known
  variance)

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Models-3 revisited
Source df sum of squares ?2
Between spatial points 8 26.55 663.75
Between frequencies 8 366.84 9171
Interaction 64 30.54 763.5
Total 80 423.93 10598.25
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Moving averages

• A simple way of constructing stationary sequences
  is to average an iid sequence .
• A continuous time counterpart is
  , where x is a random measure which is stationary
  and assigns independent random variables to
  disjoint sets, i.e., has stationary and
  independent increments.

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Covariance

• In the squared exponential kernel case

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Lévy-Khinchine

• General representation for psii
• n is the Lévy measure, and xt is the Lévy
  process. We can construct it from a Poisson
  measure H(du,ds) on R2 with intensity
  E(H(du,ds))n(du)ds and a standard Brownian
  motion Bt as

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Examples

• Brownian motion with drift xtN(mt,s2t)
• n(du)0.
• Poisson process xtPo(lt)
• ms20, n(du)ld1(du)
• Gamma process xtG(at,b)
• ms20, n(du)ae-bu1(ugt0)du/u
• Cauchy process
• ms20, n(du)bu-2du/p

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Spatial moving averages

• We can replace R for t or s with R2 (or a general
  metric space)
• We can replace b(t-s) by bt(s) to relax
  stationarity
• We can let the intensity measure for H be an
  arbitrary measure n(ds,du)

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Gaussian moving averages

• Higdon (1998), Swall (2000)
• Let x be a Brownian motion without drift,
  and . This is a Gaussian process with
  correlation
• Account for nonstationarity by letting the kernel
  b vary with location

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Details

• yields an explicit covariance function which is
  squared exponential with parameter changing with
  spatial location.
• The prior distribution describes local
  ellipses, i.e., smoothly changing random kernels.

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Prior parametrization

• Bivariate normal covariance can be described by
  an ellipse (level curve), determined by area,
  center and one focus.
• Focus chosen independently Gaussian with
  isotropic squared exponential covariance.
• Another parameter describes the range of
  influence (scale) of a given ellipse. Prior gamma.

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Local ellipses
Focus
Ellipse determined by focus, center, and fixed
area Focus components independent Gaussian time
series
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Posterior

• Compute posterior on a grid
• Mostly Metropolis-Hastings algorithm
• Full model allows the size of the kernels to vary
  across locations

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Example

• Piazza Road Superfund cleanup. Dioxin applied to
  road to prevent dust storms seeped into
  groundwater.

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Posterior distribution