Lecture 3: Kriging and Parameter Estimation Gaussian Random Fields

Semivariograms Lecture 3: Kriging and parameter estimation In geostatistics, it is common do describe random fields in Spatial Statistics and Image Analysis • terms of semivariograms instead of covariance functions. For a random field X(s), the semivariogram is defined as • 1 γ(s, t)= V(X(s) X(t)) 2 − and the variogram is V(X(s) X(t)). David Bolin − University of Gothenburg For an isotropic random field with covariance r(h), the • semivariogram is Gothenburg γ(h)=r(0) r(h) − April 1, 2019 (exersice!) Title page David Bolin Gaussian random fields Matérn variograms ArandomfieldX(s) is Gaussian if (X(s ),...,X(s ))T has a • 1 n multivariate Gaussian distribution for each choice of s1,...,sn. X(s) is uniquely specified by • 1 The mean value function µ(s)=E(X(s)),and 2 The covariance function r(s1, s2)=C(X(s1),X(s2)). X(s) is • 1 stationary if µ(s) µ and if r(s1, s2) depends only on the ⌘ separation between the locations, h = s s . 1 − 2 2 isotropic if µ(s) µ and if r(s1, s2) only depends on the ⌘ distance between the locations, h = s s . k 1 − 2k Examples of isotropic covariance functions: • σ2 ⌫ 1 Matérn: r(h)= ⌫ 1 (h) K⌫ (h) Γ(⌫)2 − 2 Exponential: r(h)=σ2 exp( h) 2 3 h− 1 h3 3 Spherical: r(h)=σ (1 + 3 ),ifh ✓ and r(h)=0 − 2 ✓ 2 ✓ otherwise. Title page David Bolin Title page David Bolin Some terminology Image analysis applications Image reconstruction Noise reduction Title page David Bolin Title page David Bolin Statistical models including random fields Conditional distributions For a multivariate Gaussian variable We have measurements yi,...,yn taken at some spatial • X1 µ1 ⌃11 ⌃12 locations s1,...,sn. N , X2 ⇠ µ2 ⌃21 ⌃22 Given that we also have some explanatory variables ✓ ◆ ✓✓ ◆ ✓ ◆◆ • B1,...,BK , we use a model we have that K 1 1 X2 X1 N(µ2 + ⌃21⌃11− (X1 µ1), ⌃22 ⌃21⌃11− ⌃12) Yi = Bk(si)βk + X(si)+"i . | ⇠ − − Xk=1 where X(s) is a mean-zero Gaussian random field. If X2 represents a random field at some unobserved locations, and Questions: X1 the observations, the conditional mean • 1 How do we estimate the parameters of the model? 1 E(X X )=µ + ⌃ ⌃− (X µ ) 2 How can we perform prediction for an unobserved location s0? 2| 1 2 21 11 1 − 1 is often called the Kriging predictor. Title page David Bolin Title page David Bolin Kriging prediction Covariates Afirstideaistouselinearregressiontointerpolatethedata: Traditionally, one has separated between three cases • Simple kriging: µ(s)=B(s)β is known. k • 2 Ordinary kriging: µ(s)=β is unknown but constant. Y (s)= βiBi(s)+"s, where "s are iid N(0, σ ) • i=1 Universal kriging: µ(s)=B(s)β is unknown. X • Possible covariates • For ordinary and universal kriging, we have to estimate the mean-value together with the covariance parameters ✓ before Longitue Latitude Altitude computing the prediction. So we have to East coast West coast Estimate the model parameters β, σ2, ✓ . • { e } Given the parameters, compute the kriging prediction. • Title page David Bolin Example David Bolin Example: US temperatures OLS estimate Estimate the parameters using ordinary least squares: • 1 βˆ =(B>B)− B>Y, where Bij = Bi(sj) and Yi = Y (si). Calculate the prediction Xˆ(s)= k βˆ B (s). • i=1 i i P Mean summer temperatures (June-August) in the continental • US 1997 recorded at 250 weather stations. We want to estimate all US temperatures based on the data. • Example David Bolin Example David Bolin Residudals Regression parameters Update regression parameters using GLS: How do we test whether the prediction is reasonable? • If the model assumptions hold, the residuals Y (s) Xˆ(s) βˆ =(B ⌃ 1B) 1B ⌃ 1Y, • − GLS > − − > − should be independent identically distributed. Confidence interval for βi: I =(βˆ 1.96 V ) βi i ± ii 1 1 p where V =(B>⌃− B)− . OLS GLS Intercept 21.6317 20.4688 ⇤ ⇤ Longitude 1.2897 1.0022 − ⇤ Latitude 2.6959 2.6845 − ⇤ ⇤ Altitude 2.6693 4.2177 − ⇤ ⇤ East coast 0.0952 0.0096 − − West coast 1.3064 1.0139 − ⇤ ⇤ Example David Bolin Example David Bolin Variogram estimate Kriging estimation The kriging estimator is 2 1 E(X(s) Y, ✓ˆ)=ˆµ(s)+r(⌃ + σ I)− (Y Bβˆ) | e − K ˆ where ⌃ij = r(si, sj), ri = r(s, si), and µˆ = k=1 Bk(s)βk. P Example David Bolin Example David Bolin Kriging residuals There is now less spatial structure in the residuals. Example David Bolin Empirical covariances of residuals Example David Bolin.

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