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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Thetford revisited

• Features depend on spatial location

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Kernel averaging

• Fuentes (2000) Introduce uncorrelated stationary
  processes Zk(s), k1,...,K, defined on disjoint
  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
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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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Lévy-Khinchine

• n is the Lévy measure, and xt is the Lévy
  process. W 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 with R2 (or a general
  metric space)
• We can replace R for 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
  correlogram
• 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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Local ellipses
Foci
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Prior parametrization

• Foci chosen independently Gaussian with isotropic
  squared exponential covariance
• Another parameter describes the range of
  influence of a given ellipse. Prior gamma.

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Example

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

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Posterior distribution
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Estimating nonstationary covariance using wavelets

• 2-dimensional wavelet basis obtained from two
  functions?? and??
• First generation scaled translates of all four
  subsequent generations scaled translates of the
  detail functions. Subsequent generations on finer
  grids.

detail functions
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W-transform
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Karhunen-Loeve expansion

• and
• where Ai are iid N(0,1)
• Idea use wavelet basis instead of
  eigenfunctions, allow for dependent Ai

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Covariance expansion

• For covariance matrix ? write
• Useful if D close to diagonal.
• Enforce by thresholding off-diagonal elements
  (set all zero on finest scales)

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Surface ozone model

• ROM, daily average ozone 48 x 48 grid of 26 km x
  26 km centered on Illinois and Ohio. 79 days
  summer 1987.
• 3x3 coarsest level (correlation length is about
  300 km)
• Decimate leading 12 x 12 block of D by 90,
  retain only diagonal elements for remaining
  levels.

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ROM covariance