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The Fourier transform

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Properties of Fourier transforms Convolution Scaling Translation

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Parceval’s 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(N 3 ).

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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 J x is the grid square with center x and n x 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

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Evidence of anisotropy 15 o red 60 o green 105 o blue 150 o brown

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Another view of anisotropy

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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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Lindgren & Rychlik transformation

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