reduced rank spatial covariance: a multiresolution approach · reduced rank covariance references...
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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
Reduced Rank CovarianceReferences
Summary
1 Reduced Rank CovarianceWavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
2 References
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
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.)
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
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).
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
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.
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
Outline
1 Reduced Rank CovarianceWavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
2 References
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
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!
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
The matrix W (Kwong and Tang, 1994)
0.0 0.2 0.4 0.6 0.8 1.0
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x
W[,
(1:M
M)
+ o
ffset
]
−1.
5−
1.0
−0.
50.
00.
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5
−1.
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00.
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5
W[,
(1:M
M)
+ o
ffset
]
f x f m x f
f x m m x m
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
D and H
0.0 0.2 0.4 0.6 0.8 1.0
0.0
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1.0
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0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
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1.0
D matrix
0.0 0.2 0.4 0.6 0.8 1.0
0.0
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1.0
H matrix
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
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)
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
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!!!
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
Outline
1 Reduced Rank CovarianceWavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
2 References
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
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
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1
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
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.
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
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
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
Outline
1 Reduced Rank CovarianceWavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
2 References
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
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.
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30.50 0.55 0.60
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
Application to AOT (54× 32)
7 8 9 10 11 12
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day 207, 2006
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x
y
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day 207, 2006
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
Non-stationary covariance obtained after 5 iterationsof MC simulations.
−0.02
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Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
Daily max 8 hour ozone, June 18, 1987
−92 −88 −84
3840
4244
(a)
0
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150
−92 −88 −8437
3941
43
(b)
60
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140
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
Daily NO2 in California
−120 −119 −118 −117 −116
33.0
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Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
Covariance NO2 in California
0.15
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Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
Forecasting NO2 in California
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Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
Outline
1 Reduced Rank CovarianceWavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
2 References
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
Wavelet multiresolution approachThe reduced rank covarianceApplications: artificial data, AOT and ozone dataConclusions
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)
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE
Reduced Rank CovarianceReferences
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.
Orietta Nicolis REDUCED RANK SPATIAL COVARIANCE