Meet the professor

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Meet the professor

• Friday, January 23 at SFU
• 430 Beer and snacks reception

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Spatial Covariance
NRCSE
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Valid covariance functions

• Bochners theorem The class of covariance
  functions is the class of positive definite
  functions C
• Why?

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Spectral representation

• By the spectral representation any isotropic
  continuous correlation on Rd is of the form
• By isotropy, the expectation depends only on the
  distribution G of .
• Let Y be uniform on the unit sphere. Then

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Isotropic correlation

• Jv(u) is a Bessel function of the first kind and
  order v.
• Hence
• and in the case d2
• (Hankel transform)

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The Bessel function J0
0.403
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The exponential correlation

• A commonly used correlation function is ?(v)
  ev/?. Corresponds to a Gaussian process with
  continuous but not differentiable sample paths.
• More generally, ?(v) c(v0) (1-c)ev/? has a
  nugget c, corresponding to measurement error and
  spatial correlation at small distances.
• All isotropic correlations are a mixture of a
  nugget and a continuous isotropic correlation.

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The squared exponential

• Using yields
• corresponding to an underlying Gaussian field
  with analytic paths.
• This is sometimes called the Gaussian covariance,
  for no really good reason.
• A generalization is the power(ed) exponential
  correlation function,

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The spherical

• Corresponding variogram

nugget
sill
range
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The Matérn class

• where is a modified Bessel function of
  the third kind and order ?. It corresponds to a
  spatial field with ?1 continuous derivatives
• ?1/2 is exponential
• ?1 is Whittles spatial correlation
• yields squared exponential.

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Some other covariance/variogram families
Name Covariance Variogram
Wave
Rational quadratic
Linear None
Power law None
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Estimation of variograms

• Recall
• Method of moments square of all pairwise
  differences, smoothed over lag bins
• Problems Not necessarily a valid variogram
• Not very robust

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A robust empirical variogram estimator

• (Z(x)-Z(y))2 is chi-squared for Gaussian data
• Fourth root is variance stabilizing
• Cressie and Hawkins

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Least squares

• Minimize
• Alternatives
• fourth root transformation
• weighting by 1/?2
• generalized least squares

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Maximum likelihood

• ZNn(?,?) ? ??(si-sj?) ? V(?)
• Maximize
• and q 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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Asymptotics

• Increasing domain asymptotics let region of
  interest grow. Station density stays the same
• Bad estimation at short distances, but
  effectively independent blocks far apart
• Infill asymptotics let station density grow,
  keeping region fixed.
• Good estimates at short distances. No effectively
  independent blocks, so technically trickier

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Steins result

• Covariance functions C0 and C1 are compatible if
  their Gaussian measures are mutually absolutely
  continuous. Sample at si, i1,...,n, predict at
  s (limit point of sampling points). Let ei(n) be
  kriging prediction error at s for Ci, and V0 the
  variance under C0 of some random variable.
• If limnV0(e0(n))0, then

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The Fourier transform
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Properties of Fourier transforms

• Convolution
• Scaling
• Translation

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Parcevals theorem

• Relates space integration to frequency
  integration. Decomposes variability.

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Aliasing

• Observe field at lattice of spacing ?.
  Since
• the frequencies ? and ??2?m/??are aliases of
  each other, and indistinguishable.
• The highest distinguishable frequency is ???, the
  Nyquist frequency.

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Illustration of aliasing

• Aliasing applet

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Spectral representation

• Stationary processes
• Spectral process Y has stationary increments
• If F has a density f, it is called the spectral
  density.

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Estimating the spectrum

• For process observed on nxn grid, estimate
  spectrum by periodogram
• Equivalent to DFT of sample covariance

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Properties of the periodogram

• Periodogram values at Fourier frequencies
  (j,k)????are
• uncorrelated
• asymptotically unbiased
• not consistent
• To get a consistent estimate of the spectrum,
  smooth over nearby frequencies

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Some common isotropic spectra

• Squared exponential
• Matérn

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A simulated process
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Thetford canopy heights

• 39-year thinned commercial plantation of Scots
  pine in Thetford Forest, UK
• Density 1000 trees/ha
• 36m x 120m area surveyed for crown height
• Focus on 32 x 32 subset

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Spectrum of canopy heights
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Whittle likelihood

• Approximation to Gaussian likelihood using
  periodogram
• where the sum is over Fourier frequencies,
  avoiding 0, and f is the spectral density
• Takes O(N logN) operations to calculate
• instead of O(N3).

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Using non-gridded data

• Consider
• where
• Then Y is stationary with spectral density
• Viewing Y as a lattice process, it has spectral
  density

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Estimation

• Let
• where Jx is the grid square with center x and nx
  is the number of sites in the square. Define the
  tapered periodogram
• where . The Whittle
  likelihood is approximately

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A simulated example
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Estimated variogram