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Introduction Geostatistical Model Covariance structure Cokriging Geostatistical Model, Covariance structure and Cokriging Hans Wackernagel¹ ¹Equipe de Géostatistique — MINES ParisTech www.cg.ensmp.fr Statistics and Machine Learning Interface Meeting Manchester, July 2009 Statistics & Machine Learning • Manchester, July 2009 Introduction Geostatistical Model Covariance structure Cokriging Introduction Statistics & Machine Learning • Manchester, July 2009 Geostatistics and Gaussian processes Geostatistics is not limited to Gaussian processes, it usually refers to the concept of random functions, it may also build on concepts from random sets theory. Geostatistics Geostatistics: is mostly known for the kriging techniques, nowadays deals much with geostatistical simulation. Bayesian inference of geostatistical parameters has also become a topic of research. Sequential data assimilation is an extension of geostatistics using a mechanistic model to describe the time dynamics. In this talk: we will stay with linear (Gaussian) geostatistics, concentrate on kriging in a multi-scale and multi-variate context. A typical application may be: the response surface estimation problem eventually with several correlated response variables. Statistical inference of parameters will not be discussed. Statistics vs Machine Learning ? Necessity of an interface meeting Differences (subjective): geostatistics favours interpretability of the statistical model, machine learning stresses prediction performance and computational perfomance of algorithms. Ideally both should be achieved. Geostatistics: definition Geostatistics is an application of the theory of regionalized variables to the problem of predicting spatial phenomena. (G. MATHERON, 1970) Usually we consider the regionalized variable z(x) to be a realization of a random function Z(x). Stationarity For the top series: we think of a (2nd order) stationary model For the bottom series: a mean and a finite variance do not make sense, rather the realization of a non-stationary process without drift. Second-order stationary model Mean and covariance are translation invariant The mean of the random function does not depend on x: h i E Z(x) = m The covariance depends on length and orientation of the vector h linking two points x and x0 = x + h: h i cov(Z(x);Z(x0)) = C(h) = E Z(x) − m · Z(x+h) − m Non-stationary model (without drift) Variance of increments is translation invariant The mean of increments does not depend on x and is zero: h i E Z(x+h) − Z(x) = m(h) = 0 The variance of increments depends only on h: h i var Z (x+h)−Z (x) = 2g(h) This is called intrinsic stationarity. Intrinsic stationarity does not imply 2nd order stationarity. 2nd order stationarity implies stationary increments. The variogram With intrinsic stationarity: 1 h 2 i g(h) = E Z (x+h) − Z(x) 2 Properties - zero at the origin g(0) = 0 - positive values g(h) ≥ 0 - even function g(h) = g(−h) The covariance function is bounded by the variance: C(0) = s 2 ≥ jC(h)j The variogram is not bounded. A variogram can always be constructed from a given covariance function: g(h) = C(0) − C(h) The converse is not true. What is a variogram ? A covariance function is a positive definite function. What is a variogram? A variogram is a conditionnally negative definite function. In particular: any variogram matrix Γ = [g(xa −xb )] is conditionally negative semi-definite, > > [wa ] g(xa −xb ) [wa ] = w Γw ≤ 0 for any set of weights with n ∑ wa = 0: a=0 Ordinary kriging n n ? Estimator: Z (x0) = ∑ wa Z(xa ) with ∑ wa = 1 a=1 a=1 Solving: " n # ? argmin var(Z (x0) − Z(x0)) − 2m( ∑ wa − 1) w1;:::;wn;m a=1 yields the system: 8 n > w g(x −x ) + m = g(x −x ) 8a > ∑ b a b a 0 < b =1 n > w = 1 :> ∑ b b =1 n 2 and the kriging variance: sK = m + ∑ wa g(xa −x0) a=1 Kriging the mean Stationary model: Z (x) = Y (x) + m n n ? Estimator: M = ∑ wa Z(xa ) with ∑ wa = 1 a=1 a=1 Solving: " n # ? argmin var(M − m) − 2m( ∑ wa − 1) w1;:::;wn;m a=1 yields the system: 8 n > w C(x −x ) − m = 0 8a > ∑ b a b < b =1 n > w = 1 :> ∑ b b =1 2 and the kriging variance: sK = m Kriging a component S Stationary model: Z(x) = ∑ Yu(x) + m (Yu ? Yv foru 6= v) u=0 n n Estimator: Y ? (x ) = w Z(x ) with w = 0 u0 0 ∑ a a ∑ a a=1 a=1 Solving: " # n argmin var Y ? (x ) − Y (x ) − 2m w u0 0 u0 0 ∑ a w1;:::;wn;m a=1 yields the system: 8 n > w C(x −x ) − m = C (x −x ) 8a > ∑ b a b u0 a 0 < b =1 n > w = 0 :> ∑ b b =1 Mobile phone exposure of children by Liudmila KUDRYAVTSEVA http://perso.rd.francetelecom.fr/joe.wiart/ Child heads at different ages Phone position and child head Head of 12 year old child SAR exposure (simulated) Max SAR for different positions of phone The phone positions are characterized by two angles The SAR values are normalized with respect to 1 W. Variogram Anisotropic linear variogram model Max SAR kriged map Kriging standard deviations Kriging standard deviations Different sample design Introduction Geostatistical Model Covariance structure Cokriging Geostatistical filtering: Skagerrak SST Statistics & Machine Learning • Manchester, July 2009 NAR16 images on 26-27 april 2005 Sea-surface temperature (SST) NAR SST − 200504261000S NAR SST − 200504261200S 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ 59˚ 59˚59˚ 59˚ 58˚ 58˚58˚ 58˚ 57˚ 57˚57˚ 57˚ 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ 6 7 8 9 6 7 8 9 NAR SST − 200504262000S NAR SST − 200504270200S 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ 59˚ 59˚59˚ 59˚ 58˚ 58˚58˚ 58˚ 57˚ 57˚57˚ 57˚ 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ 6 7 8 9 6 7 8 9 Nested variogram model Nested scales modeled by sum of different variograms: micro-scale nugget-effect of .005 small scale spherical model (range .4 deg longitude, sill .06) large scale linear model Geostatistical filtering Small-scale variability of NAR16 images NAR SST − SHORT26apr10h NAR SST − SHORT26apr12h 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ 59˚ 59˚59˚ 59˚ 58˚ 58˚58˚ 58˚ 57˚ 57˚57˚ 57˚ 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ −1 0 1 −1 0 1 NAR SST − SHORT26apr20h NAR SST − SHORT27apr02h 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ 59˚ 59˚59˚ 59˚ 58˚ 58˚58˚ 58˚ 57˚ 57˚57˚ 57˚ 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ 6˚ 7˚ 8˚ 9˚ 10˚ 11˚ −1 0 1 −1 0 1 Introduction Geostatistical Model Covariance structure Cokriging Geostatistical Model Statistics & Machine Learning • Manchester, July 2009 Introduction Geostatistical Model Covariance structure Cokriging Linear model of coregionalization The linear model of coregionalization (LMC) combines: a linear model for different scales of the spatial variation, a linear model for components of the multivariate variation. Statistics & Machine Learning • Manchester, July 2009 Two linear models S Linear Model of Regionalization: Z(x) = ∑ Yu(x) u=0 h i E Yu(x+h) − Yu(x) = 0 h i E Yu(x+h) − Yu(x) · Yv (x+h) − Yv (x) = gu(h)duv N Linear Model of PCA: Zi = ∑ aip Yp p=1 h i E Yp = 0 cov Yp;Yq = 0 for p 6= q Linear Model of Coregionalization Spatial and multivariate representation of Zi (x) using p u uncorrelated factors Yu (x) with coefficients aip: S N u p Zi (x) = ∑ ∑ aip Yu (x) u=0 p=1 p Given u, all factors Yu (x) have the same variogram gu(h). This implies a multivariate nested variogram: S Γ(h) = ∑ Bu gu(h) u=0 Coregionalization matrices The coregionalization matrices Bu characterize the correlation between the variables Zi at different spatial scales. In practice: 1 A multivariate nested variogram model is fitted. 2 Each matrix is then decomposed using a PCA: N h u i h u u i Bu = bij = ∑ aip ajp p=1 yielding the coefficients of the LMC. LMC: intrinsic correlation When all coregionalization matrices are proportional to a matrix B: Bu = au B we have an intrinsically correlated LMC: S Γ(h) = B ∑ au gu(h) = B g(h) u=0 In practice, with intrinsic correlation, the eigenanalysis of the different Bu will yield: different sets of eigenvalues, but identical sets of eigenvectors. Regionalized Multivariate Data Analysis With intrinsic correlation: The factors are autokrigeable, i.e. the factors can be computed from a classical MDA on the variance-covariance matrix V =∼ B and are kriged subsequently. With spatial-scale dependent correlation: The factors are defined on the basis of the coregionalization matrices Bu and are cokriged subsequently. Need for a regionalized multivariate data analysis! Regionalized PCA ? Variables Zi (x) # no Intrinsic Correlation ? −! gij (h) = ∑ Bu gu(h) u # yes # PCA on B PCA on Bu ## Transform into Y Cokrige Y ? (x) u0p0 ## Krige Y ? (x) −! Map of PC p0 Introduction Geostatistical Model Covariance structure Cokriging Geostatistical filtering: Golfe du Lion SST Statistics & Machine Learning • Manchester, July 2009 Modeling of spatial variability as the sum of a small-scale and a large-scale process Variogram of SST SST on 7 june 2005 The variogram of the Nar16 im- age is fitted with a short- and a long-range structure (with geo- metrical anisotropy). The small-scale components of the NAR16 image and of corresponding MARS ocean-model output are extracted by geostatistical filtering. Geostatistical filtering Small scale (top) and large scale (bottom) features 3˚ 4˚ 5˚ 6˚ 7˚ 3˚ 4˚ 5˚ 6˚ 7˚ 43˚ 43˚ 43˚ 43˚ 3˚ 4˚ 5˚ 6˚ 7˚ 3˚ 4˚ 5˚ 6˚ 7˚ −0.5 0.0 0.5 −0.5 0.0 0.5 NAR16 image MARS model output 3˚ 4˚ 5˚ 6˚ 7˚ 3˚ 4˚ 5˚ 6˚ 7˚ 43˚ 43˚ 43˚ 43˚ 3˚ 4˚ 5˚ 6˚ 7˚ 3˚ 4˚ 5˚ 6˚ 7˚ 15 20 15 20 Zoom into NE corner 3˚ 4˚ 5˚ 6˚ 7˚ 8001000 900 700 100 600 600 200500 400 200500 43˚ 700 1100 43˚ 300 9001000 14001200 400 100 8001600 15001300300 1700 2400 200018002300 2100 1900 2200 300200 800 900 500 1000 1200 400 600700 1300 Bathymetry 1100 1500 1400 1700 1600 1800 1900 2000 3˚ 4˚ 5˚ 6˚ 7˚ 0 500
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  • Covariance Functions

    Covariance Functions

    C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, the MIT Press, 2006, ISBN 026218253X. c 2006 Massachusetts Institute of Technology. www.GaussianProcess.org/gpml Chapter 4 Covariance Functions We have seen that a covariance function is the crucial ingredient in a Gaussian process predictor, as it encodes our assumptions about the function which we wish to learn. From a slightly different viewpoint it is clear that in supervised learning the notion of similarity between data points is crucial; it is a basic similarity assumption that points with inputs x which are close are likely to have similar target values y, and thus training points that are near to a test point should be informative about the prediction at that point. Under the Gaussian process view it is the covariance function that defines nearness or similarity. An arbitrary function of input pairs x and x0 will not, in general, be a valid valid covariance covariance function.1 The purpose of this chapter is to give examples of some functions commonly-used covariance functions and to examine their properties. Section 4.1 defines a number of basic terms relating to covariance functions. Section 4.2 gives examples of stationary, dot-product, and other non-stationary covariance functions, and also gives some ways to make new ones from old. Section 4.3 introduces the important topic of eigenfunction analysis of covariance functions, and states Mercer’s theorem which allows us to express the covariance function (under certain conditions) in terms of its eigenfunctions and eigenvalues. The covariance functions given in section 4.2 are valid when the input domain X is a subset of RD.