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 +2πmT

⎛

⎝⎜⎞

⎠⎟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)2 z(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)πare

•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 )ν K 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) =1

2 H(ω) 2fZ(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)Y

n2 (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 )

€

C(x,y) = C( Ax − Ay )

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are needed to see this picture.

Lindgren & Rychlik transformation

′x = (2x + y + 109.15) / 2

′y = 4(−x + 2y − 154.5) / 3