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 [-1,1]–not just imbedded in R3.

Also, stationarity and isotropy are identical concepts on the sphere.

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=(2i+1)i

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C(p,q) = aii= 0

∞

∑ Pi (cosγpq )

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C(p,q) =1− ρ2

1− 2ρcos γpq + ρ2 − 1

Global temperature

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

Isotropic correlations

The Fourier transform

g:Rd → R

G(ω) =F (g) = g(s)exp(iωTs)ds∫

g(s) =F −1(G) =

12π( )

d exp(-iωTs)G(ω)dω∫

Properties of Fourier transforms

Convolution

Scaling

Translation

F (f∗g) =F (f)F (g)

F (f(ag)) =

1a

F(ω / a)

F (f(g−b)) =exp(ib)F (f)

Parceval’s theorem

Relates space integration to frequency integration. Decomposes variability.

f(s)2ds∫ = F(ω) 2dω∫

Aliasing

Observe field at lattice of spacing . Since

the frequencies ω and ω’=ω+2πm/ are aliases of each other, and indistinguishable.

The highest distinguishable frequency is π, the Nyquist frequency.

Zd

exp(iωTk) = exp(i ωT+ 2π mT

⎛

⎝⎜⎞

⎠⎟k)

= exp(iωTk)exp(i2πmTk)

Illustration of aliasing

Aliasing applet

Spectral representation

Stationary processes

Spectral process Y has stationary increments

If F has a density f, it is called the spectral density.

Z(s) = exp(isTω)dY(ω)Rd∫

E dY(ω) 2 =dF(ω)

Cov(Z(s1),Z(s2 )) = e i(s1-s2 )Tωf(ω)dωR2∫

Estimating the spectrum

For process observed on nxn grid, estimate spectrum by periodogram

Equivalent to DFT of sample covariance

In,n (ω) =1

(2πn)2z(j)eiωTj

j∈J∑

2

ω =2πj

n; J = (n − 1) / 2⎢⎣ ⎥⎦,...,n − (n − 1) / 2⎢⎣ ⎥⎦{ }

2

Properties of the periodogram

Periodogram values at Fourier frequencies (j,k)π are

•uncorrelated

•asymptotically unbiased

•not consistent

To get a consistent estimate of the spectrum, smooth over nearby frequencies

Some common isotropic spectra

Squared exponential

Matérn

f(ω)=σ2

2παexp(− ω 2 / 4α)

C(r) =σ2 exp(−α r2 )

f(ω) =φ(α2 + ω 2 )−ν−1

C(r) =πφ(α r )νK ν (α r )2 ν−1Γ(ν + 1)α2 ν

A simulated process

Z(s) = gjk cos 2πjs1

m+

ks2

n⎡⎣⎢

⎤⎦⎥+Ujk

⎛⎝⎜

⎞⎠⎟k=−15

15

∑j=0

15

∑

gjk =exp(− j+ 6 −ktan(20°) )

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

Spectrum of canopy heights

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).

l (θ) = logf(ω;θ) +

IN,N(ω)f(ω;θ)

⎧⎨⎩

⎫⎬⎭ω

∑

Using non-gridded data

Consider

where

Then Y is stationary with spectral density

Viewing Y as a lattice process, it has spectral density

Y(x) =−2 h(x−s)∫ Z(s)ds

h(x) =1( xi ≤ / 2, i =1,2)

fY (ω) =12 H(ω) 2

fZ(ω)

f,Y (ω) = H(ω +2πq

)2

fZq∈Z2∑ (ω +

2πq

)

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

Yn2 (x) =

1nx

h(s i −x)Z(s i )i∈J x

∑

Ig1Yn2(ω) =

1g1

2 (x)∑g1(x)Yn2 (x)e−ixTω∑

2

g1(x) =nx / n

LY

=n2

2π( )2 logf,Y (2πj / n) +

Ig1,Yn2(2πj / n)

f,Y (2πj / n)

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪j∑

A simulated example

Estimated variogram

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Evidence of anisotropy15o red60o green105o blue150o brown

Another view of anisotropy

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σe2 = 127.1(259)

σs2 = 68.8 (255)

θ = 10.7 (45)

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σe2 = 154.6 (134)

σs2 = 141.0 (127)

θ = 29.5 (35)

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.

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C(x,y) = C( x − y )

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C(x,y) = C( Ax − Ay )

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Lindgren & Rychlik transformation

′x = (2x + y + 109.15) / 2

′y = 4(−x + 2y − 154.5) / 3