Symmetry and Separability In Spatial-Temporal Processes

Transcription

1 Symmetry and Separability In Spatial-Temporal Processes Man Sik Park, Montserrat Fuentes Symmetry and Separability In Spatial-Temporal Processes 1

2 Motivation In general, environmental data have very complex spatial-temporal dependency structures. Standard assumptions are not enough to capture these dependency structures. Nonseparability, even nonstationarity may not be helpful. New solution may be needed. Here we develop new models and methods to estimate and predict air-pollution data. Symmetry and Separability In Spatial-Temporal Processes 2

3 Research Objectives 1. New Class of Asymmetric Spatial -Temporal Covariance Models Define Symmetry under the Space-Time Setting Propose New Classes of Asymmetric Covariance Models 2. New Tests for Lack of Symmetry in Spatial-Temporal Processes Introduce New Formal Tests for Lack of Symmetry 3. Simulation Study & Real Application Study the Performance of the Testing Methods Apply to Real Air-Pollution Datasets Symmetry and Separability In Spatial-Temporal Processes 3

4 Literature Review 1. Tests for Separability and Stationarity Modjeska and Rawlings (1983), Shitan and Blockwell (1995), Haas (1998), Guo and Billard (1998), Brown et al. (2000), Mitchell (2002), Mitchell, Genton and Gumpertz (2002), Chen (2004) Fuentes (2005): spectral methods for testing separability and stationarity in space Symmetry and Separability In Spatial-Temporal Processes 4

5 Literature Review 2. New class of Nonseparable Covariance Models Rodriguez-Iturbe and Mejia (1974), Myers and Journel (1990), Rouhani and Myers (1990), Jones and Zhang (1997) Cressie and Huang (1999), Gneiting (2002) Fuentes et al. (2005): a mixture of local spectrums C(s i s j ; t p t q )= M K(s i s n )K(s j s n )C n (s i s j ; t p t q ) n=1 Symmetry and Separability In Spatial-Temporal Processes 5

6 Literature Review 3. Symmetry Scaccia and Martin (2005) Introduce Symmetry and Separability in Spatial (Lattice) Process Axial Symmetry: C(h 1,h 2 )=C(h 1, h 2 ) Diagonal Symmetry: C(h 1,h 2 )=C(h 2,h 1 ) Separability: C(h 1,h 2 ) C 1 (h 1, 0) C 2 (0,h 2 ) Propose new tests for Symmetry and Separability based on Periodograms Lu, and Zimmerman (2005), Stein (2005) Symmetry and Separability In Spatial-Temporal Processes 6

7 Limitations No generalization of symmetry in spatial-temporal processes No formal tests for symmetry in spatial-temporal processes No class of asymmetric spatial-temporal covariance models Symmetry and Separability In Spatial-Temporal Processes 7

8 Definition: Stationarity For {Z(s; t) :s =(s 1,s 2 ) D R 2 ; t T R} with Cov{Z(s i ; t k ),Z(s j ; t l )} = C(s i s j ; t k t l ), Weak (Second-Order) Stationarity 1) E{Z(s i ; t k )} = µ, where µ is constant 2) for h R 2 and u R, Strong Stationarity Cov{Z(s i ; t k + u),z(s i ; t k )} = C(h; u) < 1) for any finite N and domain D T, with {S 1,, S N } D T and z 1,,z N R, Pr(Z(S 1 + S) z 1,,Z(S N + S) z N ) =Pr(Z(S 1 ) z 1,,Z(S N ) z N ) Symmetry and Separability In Spatial-Temporal Processes 8

9 Definition: Separability For {Z(s; t) :s =(s 1,s 2 ) D R 2 ; t T R} with Cov{Z(s i ; t k ),Z(s j ; t l )} = C(s i s j ; t k t l ), Separability Cov{Z(s i + h; t k + u),z(s i ; t k )} = C s (s i + h, s i ) C T (t k + u, t k ), C s ( ) is a spatial covriance and C T ( ) is a temporal covariance Separability under Stationarity Cov{Z(s i + h; t k + u),z(s i ; t k )} = C s (h) C T (u) Symmetry and Separability In Spatial-Temporal Processes 9

10 temporal lag (u) spatial lag ( h ) Figure 1: separability - there are ridges along the lines where h =0or u =0 Symmetry and Separability In Spatial-Temporal Processes 10

11 Definitions: Symmetry For {Z(s; t) :s =(s 1,s 2,,s d ) D R d ; t T R} with Cov{Z(s i ; t k ),Z(s j ; t l )} = C(s i s j ; t k t l ) C(h; u), Definition 1. A process is called axially symmetric in time if C(h; u) =C(h; u), where all possible h =(h 1,h 2,,h d ) 0 for u 0. Symmetry and Separability In Spatial-Temporal Processes 11

12 second spatial coordinate (a 1 + h 1,a 2 + h 2 ;t a + u) (c 1 + h 1,c 2 + h 2 ;t c u) (a 1,a 2 ;t a ) (c 1,c 2 ;t c ) first spatial coordinate Figure 2: Axial Symmetry in Time Symmetry and Separability In Spatial-Temporal Processes 12

13 Definition 2. A process is called axially symmetric in space if C(h; u) =C( h; u), where h =(h 1,,h k 1, h k,h k+1,,h d ) for k fixed. Symmetry and Separability In Spatial-Temporal Processes 13

14 second spatial coordinate (a 1 + h 1,a 2 + h 2 ;t a + u) (a 1,a 2 ;t a ) (c 1 h 1,c 2 + h 2 ;t c + u) first spatial coordinate (c 1,c 2 ;t c ) Figure 3: Axial Symmetry in Space Symmetry and Separability In Spatial-Temporal Processes 14

15 Definition 3. A process is called diagonally symmetric in space if C(h; u) =C(ḧ; u), where ḧ =(h 1,,h k 1,h l,h k+1,,h l 1,h k,h l+1,,h d ) for k l. Symmetry and Separability In Spatial-Temporal Processes 15

16 second spatial coordinate (a 1 + h 1,a 2 + h 2 ;t a + u) (a 1,a 2 ;t a ) (c 1 + h 2,c 2 + h 1 ;t c + u) (c 1,c 2 ;t c ) first spatial coordinate Figure 4: Diagonal Symmetry in Space Symmetry and Separability In Spatial-Temporal Processes 16

17 Symmetry (2-Dim Spatial Domain) For {Z(s; t) :s =(s 1,s 2 ) D R 2 ; t T R} with Cov{Z(s i ; t k ),Z(s j ; t l )} = C(s i s j ; t k t l ) C(h; u), Axial Symmetry in Time: C(h; u) =C(h 1,h 2 ; u) =C(h 1,h 2 ; u) =C(h; u) Axial Symmetry in Space: C(h; u) =C(h 1,h 2 ; u) =C(h 1, h 2 ; u) =C( h; u) Diagonal Symmetry in Space: C(h; u) =C(h 1,h 2 ; u) =C(h 2,h 1 ; u) =C(ḧ; u) Symmetry and Separability In Spatial-Temporal Processes 17

18 Background: Matern-type Covariance C(u) = πγ 2 ν 1 Γ(ν) (α u )ν K ν (α u ),f(ω) =γ ( α 2 + ω 2) ν 1/2 covariance nu=1/2 nu=3/2 spectral density nu=3/2 nu=1/ distance frequency - ν measures the degree of smoothness - α explains the rate of decay of correlation Symmetry and Separability In Spatial-Temporal Processes 18

19 A New Class of Asymmetric Stationary Models Propose the spatial-temporal spectral density function given by f v (ω; τ) =γ ( α 2 β 2 + β 2 ω + τv α 2 (τ + v 2ω) 2) ν = f 0 (ω + τv 1 ; τ + v 2ω), where f 0 (ω,τ)=γ(α 2 β 2 + β 2 ω 2 + α 2 τ 2 ) ν and v =(v 1, v 2) f v (ω,τ) > 0 everywhere f v (ω,τ) < for v Symmetry and Separability In Spatial-Temporal Processes 19

20 A New Class of Asymmetric Covariance Models Asymmetric Stationary Covariance Function: C v (h; u) = exp{ih ω + iuτ}f 0 (ω + τv 1 ; τ + v 2ω) dτ dω R d R ( ) 1 h uv2 = 1 v 1 v C v 1 v ; u h v v 1 v, 2 where C 0 is a stationary spatial-temporal covariance function simply transformed from f 0,and } d h = { hi = 1 d v 1j v 2j h i + v 2i v 1j h j i=1 j i j i i=1 Symmetry and Separability In Spatial-Temporal Processes 20

21 A New Class of Asymmetric Covariance Models Closed Form of Asymmetric Covariance Function: 1 C v (h; u) = 1 v 1 v 2 M ν d+1 2 γπ(d+1)/2 α 2ν+d β 2ν+1 2 ν (d+1)/2 1 Γ(ν) { } 2 { α h uv 2 β(u h v 1 ) 1 v 1 v v 1 v 2 } 2, M ν (r) r ν K ν (r) Symmetry and Separability In Spatial-Temporal Processes 21

22 A New Class of Asymmetric Covariance Models v =(v 1, v 2) controls (A)symmetry. Axial Symmetry in Time if v 1 = v 2 = 0, Axial Symmetry in Space if v 11 0orv 21 0and v 12 = v 22 =0, Diagonal Symmetry in Space if v 11 = v 12 = v 10, v 21 = v 22 = v 20, and at least one of v 10 and v 20 is nonzero, Asymmetry in Space and Time otherwise. Symmetry and Separability In Spatial-Temporal Processes 22

23 A New Class of Asymmetric Covariance Models C v (h; u) = 1 1 v 1 v 2 ( ) h uv2 C 0 1 v 1 v ; u h v v 1 v 2 Temporal component (u h v 1 ) the units of v 1 are time divided by distances reciprocals of speed Spatial component ( h uv 2 ) the units of v 2 are distances divided by time velocities Symmetry and Separability In Spatial-Temporal Processes 23

24 A New Class of Asymmetric Covariance Models Subclass of Asymmetric Covariance Models v 2 = 0 C v1 (h; u) M ν d+1 2 {β(u } 2 h v 1 ) α + h α 2 v 1 = 0 C v2 (h; u) M ν d+1 2 (βu ) 2 α + h uv 2 α 2 Symmetry and Separability In Spatial-Temporal Processes 24

25 Axial Symmetry in Time C(h 1,h 2 ; u) =C(h 1,h 2 ; u) spatial lag (h2) temporal lag (u) spatial lag (h1) (a) for any u spatial lag (h1 or h2) (b) for any h 2 (or h 1 ) - the covariance function is invariant to the change of any lags. Symmetry and Separability In Spatial-Temporal Processes 25

26 Axial Symmetry in Space C(h 1,h 2 ; u) =C(h 1, h 2 ; u) spatial lag (h2) temporal lag (u) spatial lag (h1) (c) u = 10, 0, spatial lag (h2) (d) h 1 = 200, 0, the covariance function is invariant to the change of h 2. - the centroids lie on the axis of h 2 =0. Symmetry and Separability In Spatial-Temporal Processes 26

27 Diagonal Symmetry in Space C(h 1,h 2 ; u) =C(h 2,h 1 ; u) spatial lag (h2) temporal lag (u) spatial lag (h1) (e) u = 10, 0, spatial lag (h1 or h2) (f) h 2 (h 1 )= 200, 0, the covariance function depends on the spatial lags, h 1 and h 2. - the centroids lie on the line with slope of v 12 /v 11 =1. Symmetry and Separability In Spatial-Temporal Processes 27

28 Asymmetry in Space and Time: v 11 0,v 12 0,v 11 v 12 spatial lag (h2) spatial lag (h1) (g) u = 10, 0, 10 - the covariance function depends on the spatial lags, h 1 and h 2. - the centroids lie on the line with slope of v 12 /v 11. Symmetry and Separability In Spatial-Temporal Processes 28

29 Asymmetry in Space and Time: v 11 0,v 12 0,v 11 v 12 temporal lag (u) temporal lag (u) spatial lag (h1) (h) h 2 = 400, 0, spatial lag (h1) (i) h 1 = 400, 0, the centroids lie on the axes of the spatial lags. - how much the centroids are shifted is related to v 11 and v 12. Symmetry and Separability In Spatial-Temporal Processes 29

30 Symmetry and Separability Both consider spatial-temporal dependency (interaction) Separability (Fuentes et al. (2005)) f ɛ (ω,τ)=γ ( α 2 β 2 + β 2 ω 2 + α 2 τ 2 + ɛ ω 2 τ 2) ν Symmetry (Our Model) f v (ω; τ) =γ ( α 2 β 2 + β 2 ω + τv α 2 (τ + v 2ω) 2) ν Every symmetric covariance is always nonseparable Symmetry and Separability In Spatial-Temporal Processes 30

31 latitude longitude Figure 5: PM 2.5 daily concentrations obtained from FRM network (18 stations; 08/01/03 08/31/03) Symmetry and Separability In Spatial-Temporal Processes 31

32 Table 1: Parameter Estimates based on Gaussian Asymmetric Spatial-Temporal Covariance from WLS and ML methods. Note that ( 4) Θ M.1 M.2 M.3 M.4 WLS ML WLS ML WLS ML WLS ML σ φ α β v 11 6.(-4) 1.(-4) 1.(-3) 2.(-3) v 12-6.(-4) -3.(-4) -1.(-3) 1.(-3) v v WSSE log(l) Symmetry and Separability In Spatial-Temporal Processes 32

33 Discussion Introduce new concepts of symmetry in spatial-temporal processes Propose new classe of asymmetric stationary spatial-temporal covariance models The environmental data influenced by some external meteorological conditions Symmetry and Separability In Spatial-Temporal Processes 33

34 1. Axial Symmetry in Time: C(h 1,h 2 ; u) =C(h 1,h 2 ; u) Cross-spectrum between Z(a; t) andz(b; t) as f ab (τ) (2π) 1 exp{ iuτ}c(a b; u) du = fba(τ) c Under H 0 : C(h; u) =C(h; u) R C(h; u) =C(h; u) f ab (τ) =f ba (τ) Im.f ab (τ) =0 φ ab (τ) tan 1 { Im.fab (τ) Re.f ab (τ) } =0 Symmetry and Separability In Spatial-Temporal Processes 34

35 1. Axial Symmetry in Time: C(h 1,h 2 ; u) =C(h 1,h 2 ; u) [ Let φ ab(τ) = φ ab (τ)/ R ] 1/2 ab (τ) 2 1 Asymptotic Distribution of B T T ( ( N 0,π R [ φ ab (τ) φ ab(τ)] ) ) W 2 (α) dα [1 η{2τ}], (Brillinger (2001)) Symmetry and Separability In Spatial-Temporal Processes 35

36 Fuentes (2006) Asymptotic normality of cross-spectral density function, f ab (τ) f ai b i (τ k ) is approximately independent of f aj b j (τ l ) if either C.1 τ k + τ l is sufficiently large so that W (α + τ k ) 2 W (α + τ l ) 2 dα =0, R i.e. τ k + τ l bandwidth of W (α) 2 or C.2 the distance between pairs (a i, b i )and(a j, b j ) is greater than the bandwidth of the function {g ρ (s)}. Symmetry and Separability In Spatial-Temporal Processes 36

37 1. Axial Symmetry in Time: C(h 1,h 2 ; u) =C(h 1,h 2 ; u) [ Let φ ik φ ai b i (τ k )/ R ] 1/2 ai b (τ i k) 2 1 Under H 0 : φ ai b i (τ k )=0andφ aj b j (τ l )=0 Asymptotic Distribution of Φ ( φ ik, φ jl ) BT T ( Φ Φ ) N 2 (0, A ΣA), A = Φ / θ, where θ =(φ ik,r ik,φ jl,r jl ) Σ is a variance-covariance matrix of ) B T T ( θ θ f ai b i (τ k ) ind f aj b j (τ l ) φ ai b i (τ k ) ind φ aj b j (τ l ) Symmetry and Separability In Spatial-Temporal Processes 37

38 1. Axial Symmetry in Time: C(h 1,h 2 ; u) =C(h 1,h 2 ; u) Two-way ANOVA procedure: φ ik = α i + β k + ɛ ik, ɛ ik N ( ) 0,σɛ 2, i, k Cov (ɛ ik,ɛ jl )=0, i, j, k, l satisfying C.1 or C.2 Check axial symmetry in time by studying if β k =0, k Check stationarity by studying if α i =0, i Symmetry and Separability In Spatial-Temporal Processes 38

39 2. Axial Symmetry in Space: C(h 1,h 2 ; u) =C( h 1,h 2 ; u) Cross-spectrum between Z(s, a 2 + h 2 ; t + u) andz(s, a 2 ; t) as k(ω; h 2,u)=(2π) 1 exp{ ih 1 ω}c(h 1,h 2 ; u) dh 1 R = k c ( ω; h 2,u) Under H 0 : C(h 1,h 2 ; u) =C( h 1,h 2 ; u) C(h 1,h 2 ; u) =C( h 1,h 2 ; u) k(ω; h 2,u)=k( ω; h 2,u) Im.k(ω; h 2,u)=0 { } Im.k(ω; ψ(ω; h 2,u) tan 1 h2,u) Re.k(ω; h 2,u) =0 Symmetry and Separability In Spatial-Temporal Processes 39

40 2. Axial Symmetry in Space: C(h 1,h 2 ; u) =C( h 1,h 2 ; u) Suppose Z is observed at N(N 1 N 2 ) spatial sites on regular grids with unit spacing Δ =(Δ 1, Δ 2 ) and at same measuring times T. Let ψ Δ 1 (ω; h 2,u)= ψ Δ1 (ω; h 2,u)/ [ Q Δ1 (ω; h 2,u) 2 1 ] 1/2 Asymptotic Distn of [ ] B N1 N 1 ψ Δ1 (ω; h 2,u) ψδ 1 (ω; h 2,u) ( ( N 0,π R ) ) W 2 (α) dα [1 η{2ω}], (Brillinger (2001)) Symmetry and Separability In Spatial-Temporal Processes 40

41 2. Axial Symmetry in Space: C(h 1,h 2 ; u) =C( h 1,h 2 ; u) Let ψ im ψ ( Δ ωm 1 ; a i 2 b i 2,t a i ) tb i Under H 0 : ψ im =0andψ jn =0 Asymptotic Distribution of Ψ ( ψ im, ψ jn ( Ψ BN1 N 1 Ψ ) N 2 (0, B ΩB), ) B = Ψ / θ, where θ =(ψ im,q im,ψ jn,q jn ) Ω is a variance-covariance matrix of B N1 N 1 ( θ θ ) ) ind k ( Δ1 ωm ; a i 2 b i 2,t a i tb i k Δ1 (ω n ; a j 2 bj 2,ta j tb j ψ ( Δ1 ωm ; a i 2 b i 2,t a i ) ind tb i ψ ) Δ1 (ω n ; a j 2 bj 2,ta j tb j ) Symmetry and Separability In Spatial-Temporal Processes 41

42 2. Axial Symmetry in Space: C(h 1,h 2 ; u) =C( h 1,h 2 ; u) Two-way ANOVA procedure: ψ im = γ i + δ m + e im, e im N ( 0,σe) 2, i, m Cov (e im,e jn )=0 Check axial symmetry in space by studying if δ m =0, m Check spatial-temporal interaction by studying if γ i =0, i Symmetry and Separability In Spatial-Temporal Processes 42

43 Description: Air-Polution Data Models-3/Community Multiscale Air Quality (CMAQ) model the hourly concentrations of air pollutants with the resolution of 36km 36km the Particulate Matter with diameter less than 2.5μm (PM 2.5 ) one of the critical factors for public health problem Symmetry and Separability In Spatial-Temporal Processes 43

44 latitude longitude Figure 6: PM 2.5 daily concentrations obtained from CMAQ output (61(N 1 ) 61(N 2 ) 363(T )) Symmetry and Separability In Spatial-Temporal Processes 44

45 1. Testing Lack of Axial Symmetry in Time row index column index Figure 7: The plot of the locations of 16 pairs; ENE direction (D1), NNE direction (D2), NNW direction (D3), and WNW (D4). Symmetry and Separability In Spatial-Temporal Processes 45

46 1. Testing Lack of Axial Symmetry in Time Table 2: Analysis of variance Item DF Sum of squares F value Pr(F ) Directions Locations Frequencies Residuals Apparant evidence of nonstationrity exists. Nonzero φ ijk mainly comes from the Directions. We can not reject Axial Symmetry in Time. Symmetry and Separability In Spatial-Temporal Processes 46

47 1. Testing Lack of Axial Symmetry in Time Table 3: Analysis of variance Direction Item DF SS F value Pr(F ) D1 D2 D3 D4 Locations Frequencies Residuals Locations Frequencies Residuals Locations Frequencies Residuals Locations Frequencies Residuals Axial Symmetry in Time can be rejected for some directions. Symmetry and Separability In Spatial-Temporal Processes 47

48 1. Testing Lack of Axial Symmetry in Time Sample Quantiles Sample Quantiles Theoretical Quantiles (a) for D Theoretical Quantiles (b) for D4. Figure 8: The QQplots of the residuals The normality condition for error term is reasonable. Symmetry and Separability In Spatial-Temporal Processes 48

49 2. Testing Lack of Axial Symmetry in Space time index D5 D row index Figure 9: The plot of the locations of 16 pairs; D5, D6. Symmetry and Separability In Spatial-Temporal Processes 49

50 2. Testing Lack of Axial Symmetry in Space Table 4: Analysis of variance Item DF Sum of squares F value Pr(F ) Directions Interactions Frequencies Residuals Lack of Axial Symmetry in Space exists regardless of the direction. Symmetry and Separability In Spatial-Temporal Processes 50

51 2. Testing Lack of Axial Symmetry in Space Table 5: Analysis of variance Direction Item DF SS F value Pr(F ) D5 D6 Interactions Frequencies Residuals Interactions Frequencies Residuals Lack of Axial Symmetry in Space comes from the direction D6. Symmetry and Separability In Spatial-Temporal Processes 51

52 Conclusions & Further Study + Concept of Symmetry in Spatial-Temporal Processes + Formal Tests for Lack of Symmetry + Classes of Asymmtric Spatial-Temporal Covariance Models Bayesian Estimation Approach Symmetry and Separability In Spatial-Temporal Processes 52

53 Thank you very much!! Symmetry and Separability In Spatial-Temporal Processes 53