2.1 Spatial covariances

NRCSE

Valid covariance functionsBochner’s theorem: The class ofcovariance functions is the class ofpositive definite functions C:

Why?

aij!

i! ajC(si,sj ) " 0

aij

!i

! ajC(si,sj ) = Var( ai! Z(si ))

Spectral representation

By the spectral representation any isotropiccontinuous correlation on Rd is of the form

By isotropy, the expectation depends only onthe distribution G of . Let Y be uniform on theunit sphere. Then

�

!(v) = E eiuTX"

#

$

% , v = u ,X &Rd

�

X

�

!(v) = Eeiv X Y

= E"Y (v X )

Isotropic correlation

Jv(u) is a Bessel function of the firstkind and order v.Hence

and in the case d=2

(Hankel transform)

�

!(v) = "Y (sv)0

#

$ dG(s)

�

!Y(u) =2

u

"

#

$

%

d

2&1

'd

2

"

#

$

% Jd2&1(u)

!(v) = J0 (sv)dG(s)0

"

#

The Bessel function J0

0 200 400 600 800 1000 1200 1400 1600-0.5

0

0.5

1

–0.403

The exponential correlation

A commonly used correlation functionis ρ(v) = e–v/φ. Corresponds to aGaussian process with continuous butnot differentiable sample paths.More generally, ρ(v) = c(v=0) + (1-c)e–v/φ

has a nugget c, corresponding tomeasurement error and spatialcorrelation at small distances.All isotropic correlations are a mixtureof a nugget and a continuous isotropiccorrelation.

The squared exponential

Using yields

corresponding to an underlyingGaussian field with analytic paths.This is sometimes called the Gaussiancovariance, for no really good reason.A generalization is the power(ed)exponential correlation function,

G'(x) =2x

!2e"4x2 /!2

!(v) = e" v

#( )2

!(v) = exp " v#$% &'

(( ), 0 < ( ) 2

The spherical

Corresponding variogram

!(v) =1" 1.5v + 0.5 v

#( )3

; h < #

0, otherwise

$%&

'&

!2+"2

23 t

# + (t# )3( ); 0 $ t $ #

!2+ "

2; t > #

nugget

sill range

The Matérn class

where is a modified Bessel function ofthe third kind and order κ. It corresponds toa spatial field with κ–1 continuousderivativesκ=1/2 is exponential;κ=1 is Whittle’s spatial correlation; yields squared exponential.

G'(x) =2!

"2!

x

(x2 + "#2 )1+!

!(v) =1

2"#1$(")v

%&'(

)*+

"

K"

v

%&'(

)*+

K!

! " #

Some othercovariance/variogram

families

VariogramCovarianceName

NonePower law

NoneLinear

Rationalquadratic

Wave!2 sin("t)

"t

!2 (1"

t2

1+ #t2)

!2+ "

2 (1#sin($t)

$t)

!2+

"2t2

1+ #t2

!2+ "

2t

!2+ "

2t#

RecallMethod of moments: square of all pairwisedifferences, smoothed over lag bins

Problems: Not necessarily a valid variogramNot very robust

Estimation ofvariograms

! (h) =1

N(h)(Z(si ) " Z(sj ))

2

i,j#N(h)

$

N(h) = (i, j) : h!"h

2# si ! sj # h+

"h

2

$%&

'()

! (v) = "2 (1# $(v))

A robust empiricalvariogram estimator

(Z(x)-Z(y))2 is chi-squared for GaussiandataFourth root is variance stabilizingCressie and Hawkins:

!! (h) =

1

N(h)Z(si ) " Z(sj )

12#

$%&

'()

4

0.457 +0.494

N(h)

Least squares

Minimize

Alternatives:•fourth root transformation•weighting by 1/γ2

•generalized least squares

�

!! ([ Z(si ) " Z(sj)]2" # ( si " sj ;!)( )

j

$i

$2

Maximum likelihood

Z~Nn(µ,Σ) Σ= α[ρ(si-sj;θ)] = α V(θ)Maximize

and θ maximizes the profile likelihood

�

!(µ ,! ,") = #n

2log(2$! ) #

1

2logdetV(")

+1

2!(Z# µ )TV(")#1(Z# µ )

�

ˆ µ = 1TZ / n ˆ ! = G(ˆ " ) / n G(") = (Z# ˆ µ )TV(")#1(Z# ˆ µ )

�

! * (!) = "n

2log

G2(!)

n"1

2logdetV(!)

Parana data

ml

ls

A peculiar ml fit

Some more fits

All together now...

Bayesian kriging

Instead of estimating the parameters,we put a prior distribution on them, andupdate the distribution using the data.Model:Prior:

Posterior:

(Z !) ~ N(",#2 C($) + %2I)

f(!) = f(")f(#2 )f($)f(%2 )

f(! Z = z) " f(!) f(z #)f($2 )f(%)f(&2 )d''' $2d%d&2

geoR

Prior is assigned to φ and . Thelatter assumed zero unless specified.The distributions are discretized.Default prior on mean β is flat (if notspecified, assumed constant).(Lots of different assignments arepossible)

! "

Prior/posterior of φ

Variogram estimates

meanmedianWLS

Bayes vs universal kriging

Bayes predictive mean Universal kriging

Spectral representationStationary processes

Spectral process Y has stationaryincrements

If F has a density f, it is called thespectral density.

Z(s) = exp(isT!)dY(!)Rd

"

E dY(!)2= dF(!)

Cov(Z(s1),Z(s2 )) = ei(s1-s2 )T!f(!)d!R2

"

Estimating the spectrum

For process observed on nxn grid,estimate spectrum by periodogram

Equivalent to DFT of sample covariance

In,n (!) =1

(2"n)2z(j)ei!

Tj

j#J

$

2

! =2"j

n; J = (n# 1) / 2$% &' ,...,n# (n# 1) / 2$% &'{ }

2

Properties of theperiodogram

Periodogram values at Fourierfrequencies (j,k)π/Δ are•uncorrelated•asymptotically unbiased•not consistentTo get a consistent estimate of thespectrum, smooth over nearbyfrequencies

Some commonisotropic spectra

Squared exponential

Matérn

f(!)="2

2#$exp(% !

2/ 4$)

C(r) = "2 exp(%$ r

2)

f(!) = "(#2+ !

2)$%$1

C(r) =&"(# r )%K

%(# r )

2%$1'(% + 1)#2%

A simulated process

Z(s) = gjk cos 2!js1m

+ks2n

"#$

%&'+Ujk

()*

+,-k=.15

15

/j=0

15

/

gjk = exp(! j+ 6 ! k tan(20°) )

Thetford canopy heights

39-year thinned commercialplantation of Scots pine inThetford Forest, UKDensity 1000 trees/ha36m x 120m area surveyed forcrown heightFocus on 32 x 32 subset

Spectrum of canopy heights

Whittle likelihood

Approximation to Gaussian likelihoodusing periodogram:

where the sum is over Fourier frequencies,avoiding 0, and f is the spectral densityTakes O(N logN) operations to calculateinstead of O(N3).

!(!) = logf(";!) +IN,N (")

f(";!)

#$%

&'("

)

Using non-gridded data

Consider

where

Then Y is stationary with spectraldensity

Viewing Y as a lattice process, it hasspectral density

Y(x) = !"2 h(x " s)# Z(s)ds

h(x) = 1( xi ! " / 2, i = 1,2)

fY (!) =1

"2H(!)

2fZ (!)

f! ,Y (") = H(" +

2#q

!)

2

fZq$Z2% (" +

2#q

!)

Estimation

Let where Jx is the grid square with centerx and nx is the number of sites in thesquare. Define the tapered periodogram

where . The Whittlelikelihood is approximately

Yn2(x) =

1

nxh(si ! x)Z(si )

i"Jx

#

Ig1Yn2(!) =

1

g12 (x)"

g1(x)Yn2 (x)e# ixT!

"2

g1(x) = nx / n

!Y

=n2

2!( )2

logf" ,Y (2!j / n) +Ig1,Yn2

(2!j / n)

f" ,Y (2!j / n)

#$%

&%

'(%

)%j

*

A simulated example

Estimated variogram

trueexact mleapprox mle

Thetford revisited

Features depend on spatial location

Some references

Bertil Matern; Spatial Variation. Medd. StatensSkogsforskningsinst.49: 5, Ch 2-3.(Reprinted in Springer Lecture Notes In Statistics, vol. 36)

Cressie: ch. 2.3.1, 2.4, 2.6.

P. Guttorp, M. Fuentes and P. D. Sampson (2006): Usingtransforms to analyze space-time processes.http://www.nrcse.washington.edu/pdf/trs80.pdf (toappear,SemStat 2004 proceedings, Chapman & Hall)

Banerjee, S., Carlin, B. P. and Gelfand, A. E.: HierarchicalModeling and Analysis for Spatial Data, Chapman andHall, 2004. pp. 129-135.