Multivariate Spatial Statistics
Whitney Huang Introduction to Multivariate Geostatistics Motivation
Multivariate– Covariance Whitney Huang Functions Models for multivariate– Department of Statistics covariance Purdue University function Summary September 23, 2014 References Multivariate Outline Spatial Statistics
Whitney Huang
Motivation
Multivariate– Covariance Motivation Functions Models for multivariate– covariance Multivariate–Covariance Functions function Summary
References Models for multivariate–covariance function
Summary Multivariate Motivation Spatial Statistics
Whitney Huang We often encounter the situations that multiple variables observed at spatial locations (e.g., air pollutants level at Motivation Multivariate– environmental monitoring stations, climate variables at Covariance weather stations). Functions Models for I there is a spatial dependence structure for each variable multivariate– covariance function I the variables are often correlated with each other Summary The cross correlation may help improve spatial prediction References Multivariate Motivation cont. Spatial Statistics
Whitney Huang
Motivation
If the objective is spatial prediction for a primary variable Multivariate– Covariance using other p − 1 variables. The best linear unbiased Functions
prediction is refer to Co–Kriging. In principle, it yields smaller Models for multivariate– prediction error than Kriging. covariance function
Illustration: Summary
Let Y1, Y2 be two random vectors and Z be a random References variable. Suppose (Z, Y2, Y2) is Gaussian. Then
h 2i h 2i E (Z − E (Z|Y1, Y2)) = E (Z − E (Z|Y1)) h 2i − E (E (Z|Y1, Y2) − E (Z|Y1)) Multivariate Outline Spatial Statistics
Whitney Huang
Motivation
Multivariate– Covariance Motivation Functions Models for multivariate– covariance Multivariate–Covariance Functions function Summary
References Models for multivariate–covariance function
Summary Multivariate Multivariate–Covariance Function: definitions Spatial Statistics
Whitney Huang
The multivariate process Motivation T d Multivariate– Y(s) = (Y1(s), ··· , Yp(s)) , s ∈ S ⊂ R is said to be Covariance second order stationary if for any i, j = 1, ··· , p and s, h ∈ S Functions Models for such that multivariate– covariance function E [Yi (s)] = µi Summary Cov (Yi (s + h), Yj (s)) = Cij (h) References
We have
I direct–covariance: Cii (h), i = 1, ··· , p
I cross–covariance: Cij (h) for i 6= j
The matrix-valued function C(h) = (Cij (h)) is the Multivariate covariance function. Multivariate Multivariate–Covariance Function: properties Spatial Statistics
Whitney Huang
Motivation
Multivariate– I In general, C(h) is not symmetric, i.e., Covariance Functions
Models for Cij (h) 6= Cji (h) multivariate– covariance function
when h 6= 0 Summary I By definition, References Cij (h) = Cji (−h)
I 2 |Cij (h)| ≤ Cii (0)Cjj (0)
however, |Cij (h)| ≤ Cij (0) is not necessarily true. Multivariate Outline Spatial Statistics
Whitney Huang
Motivation
Multivariate– Covariance Motivation Functions Models for multivariate– covariance Multivariate–Covariance Functions function Summary
References Models for multivariate–covariance function
Summary Multivariate Proportional covariance model Mardia & Goodall, 1993 Spatial Statistics
Whitney Huang
Motivation
Multivariate– Covariance Functions Separable forms: Models for Cij (h) = V ρ(h) multivariate– covariance function where Summary p I V = (vij )i,j=1 is a positive definite matrix References I ρ(·) is a valid correlation function
Issue: same form of marginal correlations Cii (h)and cross corrections Cij (h). Multivariate Linear model of co–regionalization (LMC) Spatial Statistics Wackernagel, 2003 Whitney Huang
Motivation
Linear combination of r ≤ p independent univariate Multivariate– Covariance stochastic processes Functions
Models for r multivariate– X covariance Cij (h) = Vk ρk (h) function
k=1 Summary where References
I Vk are are p × p positive semi–definite matrices
I ρk are valid correlation functions Issue: 1 the number of parameters increases quickly with r 2 the smoothness of any component is restricted to that of the roughest underlying univariate process Multivariate Kernel convolution method Ver Hoef & Barry, 1998 Spatial Statistics
Whitney Huang
Motivation
Multivariate– Covariance Z Z Functions Cij (h) = Ki (V1) Kj (V2) ρ (V1 − V2 − h) dV1 dV2 Models for Rd Rd multivariate– covariance where function Summary
I Ki (·) are square integrable kernel functions References I ρ(·) is a valid correlation function Issue: 1 assumes that all the spatial processes are generated from the same underlying process 2 requires Monte Carlo integration Multivariate Covariance convolution method Gaspari & Cohn, 1999 Spatial Statistics
Whitney Huang
Motivation
Multivariate– Covariance Functions
Models for multivariate– Z covariance Cij (h) = Ci (h − k) Cj (k) dk function d R Summary where Ci are square integrable functions References Issue: Although some closed–form expressions exist, this method usually requires Monte Carlo integration Multivariate Latent dimension approach Apanasovich & Genton, 2010 Spatial Statistics
Whitney Huang
Motivation
Idea: each component Yi (s) of multivariate process is Multivariate– T k Covariance represented as a point ξi = (ξi1, ··· , ξik ) ∈ R for Functions 1 ≤ k ≤ p such that Models for multivariate– covariance function Cij (s1, s2) = C˜ {(s1, ξi ) , (s2, ξj )} Summary d+k References where C˜(·) is a valid univariate covariance function on R Example: ( ) σi σj −αkhk 21 1 Cij (h) = exp β +τ (i = j) (h = 0) kξi − ξj k + 1 (kξi − ξj k + 1) 2 Multivariate Matérn covariance functions Spatial Statistics
Whitney Huang
The Matérn correlation function is isotropic and has the Motivation Multivariate– parametric form Covariance Functions ν (αkhk) Models for C (Y (s + h) , Y (s)) = M(h|ν, α) = K (αkhk) multivariate– ν−1 ν covariance 2 Γ(ν) function where Summary References I khk denotes the euclidean distance
I ν > 0 is the smoothness parameter 1 I α > 0 is the scaling parameter ( α is the range parameter)
I Kν is the modified Bessel function of the second kind of order ν Multivariate Multivariate Matérn cross-covariance functions Spatial Statistics Gneiting, Kleiber & Schlather, 2010 Whitney Huang
Motivation
Multivariate– Covariance Functions
Models for multivariate– covariance function νij σij (αij khk) Summary Cij (h) = σij Mij (h|νij , αij ) = ν −1 Kνij (αij khk) References 2 ij Γ(νij )
Remark: Some restrictive conditions necessary for νij , σij , αij to get a valid cross-covariance function Multivariate Parsimonious and full bivariate Matérn Spatial Statistics
Whitney Huang
Motivation
Multivariate– Parsimonious Matérn Covariance Functions I α = α ij Models for νii +νjj multivariate– I νij = 2 covariance function
I See Gneiting et al. JASA 2010 for condition on σij Summary It allows each process to have distinct marginal smoothness References behavior. Full bivariate Matérn In addition to the flexibility of smoothness parameter as in parsimonious Matérn, it allows for distinct scale parameters for two processes (i.e. p = 2) Multivariate Flexible Matérn Apanasovich, Genton, & Sun, 2012 Spatial Statistics
Whitney Huang
Motivation
Multivariate– Covariance Functions
Models for multivariate– covariance I a generalization of Parsimonious/full bivariate Matérn function Summary I it works for p > 2 References I it allows distinct smoothness and scale parameters Multivariate Multivariate Matérn cross-covariance functions : Spatial Statistics SPDEs approach Hu et al, 2013 Whitney Huang
Motivation
Multivariate– Covariance Functions
Gaussian process Y (s) with Matérn covariance function is a Models for stationary solution to the linear fractional stochastic partial multivariate– covariance differential equation (SPDE) Lindgren et al, 2011: function Summary κ d References α2 − ∆ 2 Y (s) = W(s), κ = ν + , ν > 0 2 where
I W(s) is a spatial Gaussian white noise
I ∆ is the Laplacian operator Multivariate Multivariate SPDE model Spatial Statistics
Whitney Huang
Motivation
Multivariate– Define system of SPDEs Covariance Functions L11 L12 ···L1p Y1(s) W1(s) Models for multivariate– L21 L22 ···L2p Y2(s) W2(s) covariance = function . . .. . .. .. . . . . . . Summary Lp1 Lp2 ···Lpp Yp(s) Wp(s) References
where κ 2 ij I Lij = bij (αij − ∆) 2 are differential operators I Wi are independent but not necessarily identically distributed white noise Multivariate Outline Spatial Statistics
Whitney Huang
Motivation
Multivariate– Covariance Motivation Functions Models for multivariate– covariance Multivariate–Covariance Functions function Summary
References Models for multivariate–covariance function
Summary Multivariate Summary Spatial Statistics
Whitney Huang
Motivation
Multivariate– Covariance Functions I Multivariate covariance function is the key component of Models for multivariate– multivariate geostatistical analysis covariance function
I The positive definiteness requirement imposes Summary
constraints on parameter space of cross-covariance References functions
I Parameter estimation and prediction become difficult when p increase Multivariate Spatial Statistics
Apanasovich, T. V., & Genton, M. G. Whitney Huang Cross-covariance functions for multivariate random fields Motivation based on latent dimensions Multivariate– Biometrika, 15–30, 2010. Covariance Functions Apanasovich, T. V., Genton, M. G., & Sun, Y. Models for multivariate– A valid Matern class of cross-covariance functions for covariance function multivariate random fields with any number of Summary components References Journal of the American Statistical Association, 180–193, 2012. Gaspari, G., & Cohn, S. E. Construction of correlation functions in two and three dimensions Quarterly Journal of the Royal Meteorological Society, 723–757, 1999. Multivariate Spatial Statistics Gneiting, T., Kleiber, W., & Schlather, M. Whitney Huang Matern cross-covariance functions for multivariate random fields Motivation Multivariate– Journal of the American Statistical Association, Covariance 1167–1177, 2010. Functions Models for Hu, X., Steinsland, I., Simpson, D., Martino, S., & Rue, multivariate– covariance H. function Spatial modelling of temperature and humidity using Summary systems of stochastic partial differential equations References arXiv, 1307.1402v1., 2013. Mardia, K. V., & Goodall, C. R. Spatial-temporal analysis of multivariate environmental monitoring data Multivariate Environmental Statistics, North-Holland Ser. Statist. Probab., 6, North-Holland, Amsterdam, 347–386, 1993. Multivariate Spatial Statistics
Whitney Huang
Motivation
Multivariate– Ver Hoef, J. M., & Barry, R. P. Covariance Constructing and fitting models for cokriging and Functions Models for multivariable spatial prediction multivariate– covariance Journal of Statistical Planning and Inference, 275–294, function
1998. Summary Wackernagel, H. References Multivariate Geostatistics: An Introduction with Applications Springer, Berlin, 3rd edition, 2003.