Multivariate Gaussian Random Fields with SPDEs

Transcription

1 Multivariate Gaussian Random Fields with SPDEs Xiangping Hu Daniel Simpson, Finn Lindgren and Håvard Rue Department of Mathematics, University of Oslo PASI, 214

2 Outline The Matérn covariance function and SPDEs Multivariate GRFs with SPDEs Multivariate GRFs with oscillating covariance functions Sampling the multivariate GRFs Fast inference approach Results with simulated datasets and real datasets Conclusion and Discussion 2 / 44

3 Multivariate Gaussian random elds are needed in real world Why we need Multivariate random elds? Used to model the correlated datasets; Capture the spatial dependence structures; Helpful to predict the other elds; Big area and a lot of issues needed to be solved. 3 / 44

4 Multivariate Gaussian random elds are needed in real world lat lat lon lon 4 / 44

5 Motivation Why systems of SPDEs for Multivariate GRFs??? Automatically fulll the non-negative denite constrain; The precision matrix Q is sparse; The parameters in the (systems of) SPDEs are interpretable; Fast infernece can be achieved; Can be extended in various of ways. 5 / 44

6 Matérn covariance function The Matérn covariance function is isotropic and has the form Cov (x(), x(h)) = σ 2 M(h ν, κ) = σ2 2 1 ν Γ(ν) (κ h )ν K ν (κ h ), h denotes the Euclidean distance; ν > is the smoothness parameter; κ > is the scaling parameter; σ 2 is the marginal variance; K ν is the modied Bessel function of second kind. 6 / 44

7 Link to SPDE The important relationship is that a Gaussian Field x(s) with the Matérn covariance function is a stationary solution to the linear fractional SPDE (Lindgren et al, 211) (κ 2 ) α/2 x(s) = W(s), α = ν + d/2, ν > (κ 2 ) α/2 is a pseudo-dierential operator, W(s) is standard spatial Gaussian white noise is the Laplacian 7 / 44

8 The Matérn covariance function, restrictive??? At the rst glance, modelling with the Matérn covariance function seems quite restrictive, But actually it is not since it covers the most important and mostly used covariance models in spatial statistics. Stein (1999), on Page 14, recommend with Use the Matérn model". 8 / 44

9 Multivariate Gaussian random eld A multivariate GRF is a collection of continuously indexed multivariate normal random vectors x(s) MVN(, Σ(s)), where, Σ(s) is a non-negative denite matrix which depends on the points s R d. 9 / 44

10 Multivariate SPDE model Dene system of SPDEs L 11 L L 1p x 1 (s) ε 1 (s) L 21 L L 2p x 2 (s) = ε 2 (s).. L p1 L p2... L pp x p (s) ε p (s) L ij = b ij (κ 2 ij )α ij /2 are dierential operators; ε i are independent but not necessarily identically distributed noise processes; Currently, we can only take integer valued α ij. 1 / 44

11 Solve the SPDEs The idea is taken from nite element analysis. The solution of the SPDE could be represented by x(s) = n ψ i (s)ω i, i=1 ψ i (s) is some chosen basis-function; ω i is some Gaussian distributed weights; n is the number of the vertices in the triangulation. 11 / 44

12 Piece-wise linear basis function We choose the piece-wise linear basis function. 12 / 44

13 The Precision matrix Denote Q αi as the precision matrix for the Gaussian weights ω i for α i = 1, 2, 3, as a function of κ ij Q 1,κ 2 = K ij κ 2 ij Q 2,κ 2 = K T C 1 K ij κ 2 κ 2 ij ij Q α,κ 2 = K T C 1 Q ij κ 2 α 2,κ 2 C 1 K ij ij κ 2, for α = 3, 4,. ij, denote the inner product; C ij = ψ i, ψ j, G ij = ψ i, ψ j, K κ 2 ij = κ 2 ij C ij + G ij. 13 / 44

14 GMRF approximation C ij = ψ i, ψ j is replaced by Cii = ψ i, 1. Cii is a diagonal matrix; diagonal matrix Cii yields a Markov approximation; dierence is negligible. 14 / 44

15 Covariance-based model Gneinting et al. (21) "Matérn Cross-Covariance Functions for Multivariate Random Fields": C 11 (h) C 12 (h) C 1p (h) C 21 (h) C 22 (h) C 2p (h) C(h) =......, C p1 (h) C p2 (h) C pp (h) C ii (h) = σ ii M(h ν ii, a ii ) is the marginal covariance function; C ij (h) = ρ ij σ i σ j M(h ν ij, a ij ) is the cross-covariance function. 15 / 44

16 Model matching The models based on the SPDEs approach and the covariance-based approach are compared by matching the corresponding elements of the spectral matrix. Under the assumption that a 11 = a 21 = a 22 and ν 11 = ν 21 = ν 22, the models constructed by using SPDEs approach and the covariance-based approach becomes equivalent when b 22 b 21 = σ 1 ρσ 2, b 2 22 b b2 21 = σ 11 σ / 44

17 Sampling the positively correlated bivariate GRFs / 44

18 Sampling the negatively correlated bivariate GRFs / 44

19 The correlation functions / 44

20 Example and application It can be shown that the logarithm of posterior distribution of θ is log(π(θ y)) = Const + log(π(θ)) log( Q(θ) ) 1 2 log( Q c(θ) ) µ c(θ) T Q c (θ)µ c (θ). 2 / 44

21 Triangular system of SPDEs The triangular system of SPDEs is commonly used: L x 1 (s) ε 1 (s) L 21 L x 2 (s) = ε 2 (s).. L p1 L p2... L pp x p (s) ε p (s) Can be solved sequentially; Much faster and robust; Possible to model high dimensional multivariate GRFs. 21 / 44

22 Results from simulated data Table : Inference with simulated dataset 1 Parameters True value Estimated Standard deviations κ κ κ b b b / 44

23 Statistical inference with real data example Show how to use the SPDEs approach for constructing the multivariate GRFs in real application; The same meteorological dataset used by Gneiting et al. (21) was chosen and analyzed; It contains the following data: pressure errors (in Pascal), temperature errors (in Kelvin), measured against longitude and latitude; This meteorological dataset contains one observation at 157 locations in the north American Pacic Northwest; Valid on 18th, December of 23 at 16: local time. 23 / 44

24 The system of SPDEs The system of SPDEs has been used for this dataset can be written down explicitly as b 11 (κ 2 11 ) α 11/2 x1 (s) = ε 1 (s), b 22 (κ 2 22 ) α 22/2 x2 (s) + b 21 (κ 2 21 ) α 21/2 x1 (s) = ε 2 (s). The noise processes are from SPDEs (κ 2 n1 )αn 1 /2 ε 1 (s) = W 1 (s), (κ 2 n2 )αn 1 /2 ε 2 (s) = W 2 (s). 24 / 44

25 The re-construct bivariate elds from SPDEs approach / 44

26 The re-construct bivariate elds from Gneiting et al. (21) approach / 44

27 The predictive performance predicted value predicted value 2 predicted value predicted value observed value observed value observed value observed value Figure : the covariance-based approach (left) and the predictive performances of SPDEs approach (right) 27 / 44

28 The predictive performance Table : predictive errors for the SPDEs approach and the covariance-based models Models relative errors pressure eld temperature eld Covariance-based model SPDEs approach / 44

29 Now we go even further! Construct the multivariate GRFs with oscillating covariance functions. Two approaches can be considered. Re-parametrization the systems of the SPDEs, Using the noise process with oscillating covariance function. 29 / 44

30 Question: Why oscillating covariance functions Some random elds could have the oscillating covariance structure, such as the pressure on the globe; Using the SPDE approach, it is not hard to do that. Didn't nd many literatures doing this in spatial statistics. (Am I Wrong?) 3 / 44

31 Systems of SPDEs with oscillating noise processes L 11 L L 1p x 1 (s) ε 1 (s) L 21 L L 2p x 2 (s) = ε 2 (s).. L p1 L p2... L pp x p (s) ε p (s) L ij = b ij (κ 2 ij )α ij /2 are dierential operators; ε i are independent but not necessarily identically distributed noise processes; Some ε i can have oscillating covariance functions 31 / 44

32 Triangular system of SPDEs We recommend the lower triangular operator matrix L 11 x 1 (s) ε 1 (s) L 21 L 22 x 2 (s) = ε 2 (s).. L p1 L p2... L pp x p (s) ε p (s). Many advantages: less parameters, fast inference, easy interpreting, and easy locating the position of the oscillating elds. 32 / 44

33 Triangular system of SPDEs If only the noise process ε i (s) has oscillating covariance function, then All random elds x j (s), j < i will have non-oscillating covariance functions; Random elds x j (s), j = i will be sure to have oscillating covariance function; Random elds x j (s), j > i could possibly have oscillating covariance functions; 33 / 44

34 systems of SPDEs with oscillating noise processes / 44

35 The correlation function / 44

36 systems of SPDEs with oscillating noise processes / 44

37 The correlation function / 44

38 Results from simulated data 1 Table : Inference for the simulated dataset 1 Parameters True values Estimates Standard deviations b b b h h κ n ω This is with the case only the second random eld is oscillating and the rst elds is a Matérn random eld. 38 / 44

39 Results from simulated data 2 Table : Inference for the simulated dataset 2 Parameters True values Estimates Standard deviations b b b h h κ n κ n ω This is with the case both the random elds are with oscillating covariance functions. 39 / 44

40 Practical settings for inference κ 11 = κ n 1 and κ 22 = κ n 2 ; Model selection: x α ij and α n i at dierent values; Multivariate GRFs with oscillating covariance functions: Fewer noise processes with oscillating covariance functions when possible; Pre-analysis for the location the oscillating random elds. 4 / 44

41 Inference with real dataset This dataset is from the ERA 4 database, and this dataset contains the temperature and pressure data on the whole globe on 4th of September, lat lat lon lon (a) (b) Figure : Real dataset from ERA 4 database with temperature (a) and pressure (b) 41 / 44

42 Estimated bivariate random elds: 2D (a) (b) Figure : Estimated conditional mean of bivariate random elds for temperature (a) and pressure (b) 42 / 44

43 Prediction predicted value 5 predicted value observed value observed value Figure : Prediction for the bivariate random elds at another 5 data points for temperature (left) and pressure (right) 43 / 44

44 Conclusion, discussion and future work Illustrated the possibility of construction the multivariate GRFs with the system of SPDEs; the GRFs constructed by the system of SPDEs fulll the non-negative denite constrain; The precision matrix for the GMRFs are sparse and hence fast inferences are feasible even for large datasets; Demonstrate the connection between the covariance-based models and SPDE-based models; Looking for real datasets for good applications!!! 44 / 44