Global processes

1
Global processes

• Problems such as global warming require modeling
  of processes that take place on the globe (an
  oriented sphere). Optimal prediction of
  quantities such as global mean temperature need
  models for global covariances.
• Note spherical covariances can take values in
  -1,1not just imbedded in R3.
• Also, stationarity and isotropy are identical
  concepts on the sphere.

2
Isotropic covariances on the sphere
Isotropic covariances on a sphere are of the
form where p and q are directions, ?pq the angle
between them, and Pi the Legendre
polynomials. Example ai(2i1)ri
3
Global temperature

• Global Historical Climatology Network 7280
  stations with at least 10 years of data. Subset
  with 839 stations with data 1950-1991 selected.

4
Isotropic correlations
5
The Fourier transform
6
Properties of Fourier transforms

• Convolution
• Scaling
• Translation

7
Parcevals theorem

• Relates space integration to frequency
  integration. Decomposes variability.

8
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.

9
Illustration of aliasing

• Aliasing applet

10
Spectral representation

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

11
Estimating the spectrum

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

12
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

13
Some common isotropic spectra

• Squared exponential
• Matérn

14
A simulated process
15
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

16
Spectrum of canopy heights
17
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).

18
Using non-gridded data

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

19
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

20
A simulated example
21
Estimated variogram
22
Evidence of anisotropy
15o red 60o green 105o blue 150o brown
23
Another view of anisotropy
24
Geometric anisotropy

• Recall that if we have an isotropic
  covariance (circular isocorrelation curves).
• If for a linear transformation A, we have
  geometric anisotropy (elliptical isocorrelation
  curves).
• General nonstationary correlation structures are
  typically locally geometrically anisotropic.

25
Lindgren Rychlik transformation