reduced rank spatial covariance: a multiresolution approach · reduced rank covariance references...

Reduced Rank CovarianceReferences

REDUCED RANK SPATIAL COVARIANCE: AMULTIRESOLUTION APPROACH

Orietta Nicolis

Instituto de Estadistica, Universidad de Valparaiso, Chile

June 18, 2014

Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE


Valparaiso



Summary

1 Reduced Rank CovarianceWavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions

2 References



Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions

The problem of large data sets...

... we propose a wavelet based method for reducing rank ofcovariance matrices of non stationary and incomplete data sets.

Examples of large data sets:data spatially distributed on a regular grid with many missingdata (satellite data: aerosols, radar, ndvi, ecc.);irregularly distributed observations (gauge data: ozone, rainfall,etc.)




Basis function representation

The general concept behind this work is the expansion of a randomfunction in terms of basis functions

f = Wγ

where W is a matrix of basis and γ is the vector of coefficients.

Fourier basis representation (Panciorek, 2006 )W is the matrix of orthogonal spectral basis functions, andγk = ak + bk, k = 1, ..., is a vector of complex-valued coefficients;Karhunen-Loève decompositionW is a matrix of orthogonal basis and the coefficients γ areindependent Gaussian random variables, γ ∼ N(0,Λ), where andΛ = diag(λ1, ..., λn).




Multiresolution representation (Matsuo et al., 2008) In thiswork

1 W is a multi-resolution or wavelet basis2 the coefficients may not be uncorrelated, γ ∼ N(0,D), where D can

be not orthogonal3 Because of the localized support of wavelet basis functions, the

expansion results in a small number of coefficients with significantcorrelations.




Outline


2 References




Spatial modelLet y be the field values on a large (regular) 2−D (N × N)-grid(stacked as a vector) with covariance function

Σ = COV(y).

Eigen decomposition of the covariance

Σ = WDWT = WHHTWT

where D = H2 and H = (W−1ΣWT)1/2.Representation of the process

y = WHγ

where γ is a vector of independent standard normal variables

W –> need not be orthogonal!

D–> need not be diagonal!




The matrix W (Kwong and Tang, 1994)

0.0 0.2 0.4 0.6 0.8 1.0

−1.

5−

1.0

−0.

50.

00.

51.

01.

5

x

0.0 0.2 0.4 0.6 0.8 1.0

−1.

5−

1.0

−0.

50.

00.

51.

01.

5

x

W[,

(1:M

M)

+ o

ffset

]

−1.

5−

1.0

−0.

50.

00.

51.

01.

5

−1.

5−

1.0

−0.

50.

00.

51.

01.

5

W[,

(1:M

M)

+ o

ffset

]

f x f m x f

f x m m x m




D and H

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

D matrix

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

H matrix




Observational data

Suppose that the observations y are samples of a centered Gaussianrandom field and are composed of two components: the observationsat irregularly distributed locations, yo, and the missing observations,ym,

y =

(yo

ym

)(1)

The observational model can be written as

yo = Ky + ε

whereyo observationsy values on a the gridK is a incidence matrixε ∼ MN(0, σ2I)




The conditional distribution of ym given yo is a multivariate normal withmean

Σo,m(Σo,o)−1yo (2)

and varianceΣm,m − Σm,o(Σo,o)−1Σo,m (3)

whereΣo,m = WoHHTWT

m is the cross-covariance between observed andmissing data,Σo,o = WoHHTWT

o + σ2I is covariance of observed data andΣm,m = WmHHTWT

m is the covariance of missing data.The matrices Wo and Wm are wavelet basis evaluated at theobserved and missing data, respectively.

ProblemΣo,m and Σo,o very big!!!




Outline


2 References




The reduced rank covariance for Σ (Nicolis andNychka, 2012).

The idea is to estimate Hon a small sub-grid G ofsize (g× g) starting from aMatérn model and using aMR approach.Monte Carlo conditionalsimulation provide anefficient estimator for theconditional mean andvariance. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1




Conditional simulation

1 Find Kriging prediction on the grid G:

yg = Σo,g(Σo,o)−1yo,

where Hg = (W−1g Σg,gWT

g )1/2 and Σg,g is stationary covariancemodel (es. Matern).

2 Generate synthetic "data": ys from ys = WHga with a ∼ N(0, 1).3 Simulated Kriging error:

u∗ = yg − ysg,

where yg = WgHga and ysg = Σo,g(Σo,o)−1ys

o.4 Find conditional field ym|yo:

yu = yg + u∗.

5 Compute the conditional covariance on T replications,Σu = COV(yu), using the new Hg in the step 1 of each iteration.




By choosing properly the filter W basis and the levels of resolutions Lwe obtain the estimation of the conditional mean and variance

Σo,m(Σo,o)−1yo (4)

and varianceΣm,m − Σm,o(Σo,o)−1Σo,m (5)

whereΣo,m = WoHgHT

g WTm, Σo,o = WoHgHT

g WTo + σ2I and

Σm,m = WmHgHTg WT

m




Outline


2 References




Non-stationary random field simulated on a 40 × 40 grid with Matèrn covariance

(θ = 0.1 and ν = 0.5)) and 50% of missing values.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

−2

−1

0

1

2

30.50 0.55 0.60




Application to AOT (54× 32)

7 8 9 10 11 12

44.0

44.5

45.0

45.5

46.0

46.5

−0.4

−0.2

0.0

0.2

0.4

0.6

day 207, 2006

7 8 9 10 11 12 13

44.0

44.5

45.0

45.5

46.0

46.5

x

y

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

day 207, 2006




Non-stationary covariance obtained after 5 iterationsof MC simulations.

−0.02

0.00

0.02

0.04

0.06

0.08

0.10

o−0.02

0.00

0.02

0.04

0.06

0.08

0.10

o

0.00

0.05

0.10

0.15

0.20

o

0.00

0.05

0.10o




Daily max 8 hour ozone, June 18, 1987

−92 −88 −84

3840

4244

(a)

0

50

100

150

−92 −88 −8437

3941

43

(b)

60

80

100

120

140




Daily NO2 in California

−120 −119 −118 −117 −116

33.0

33.5

34.0

34.5

35.0

35.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0




Covariance NO2 in California

0.15

0.20

0.25

0.30

0.35

0.40

●

0.20

0.25

0.30

0.35

0.40

0.45

0.50

●

0.2

0.3

0.4

0.5

0.6

0.7

●

0.2

0.3

0.4

0.5

0.6

●




Forecasting NO2 in California

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

o

o

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

latit

ude

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

−2 −1.5

−1 −1

−0.5

0

0

0

0.5

0.5

1

● ●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●




Outline


2 References




Further work

Find a parametrization for the matrix Hg, depending on theparameters of the Matern covariance, Hg(ν, θ), and find themaximum likelihood estimates.Consider other basis functions (frames, radial basis etc. ).Include the multiresolution covariances in spatio-temporalmodels.Extension to multivariate case (for example calibration of aerosoldata)



References

Gneiting, T. (2002) Compactly Supported Correlation Functions.J. of Multivariate Analysis, 83, 493–508

Matsuo, T., D Nychka and D. Paul (2008) NonstationaryCovariance Modeling for Incomplete Data: Monte Carlo EMapproach. In review.

Nychka, D., Wikle, C. and Royle, K. A. (2003). Multiresolutionmodels for nonstationary spatial covariance functions. Stat.Model., 2, 315–332.

Nicolis, O., Nychka. D. (2012). Reduced Rank Covariances forthe Analysis of Environmental Data. In Advanced StatisticalMethods for the Analysis of Large Data-Sets (Di Ciacco, Coli,and Angulo Ibanez, eds.), Series: Studies in Theoretical andApllied Statistics, Springer, ISBN 978-3-642-21036-5.


reduced rank spatial covariance: a multiresolution approach · reduced rank covariance references...

Documents