Lecture 18. Models for areal data. Colin Rundel 03/22/2017

Transcription

1 Lecture 18 Models for areal data Colin Rundel 03/22/2017 1

2 areal / lattice data 2

3 Example - NC SIDS SID79 3

4 EDA - Moran s I If we have observations at n spatial locations s 1,... s n ) I = n i=1 n n j=1 w ij n i=1 n j=1 w ij ysi ) ȳ ) ys j ) ȳ ) n i=1 ysi ) ȳ ) where w is a spatial weights matrix. 4

5 EDA - Moran s I If we have observations at n spatial locations s 1,... s n ) I = n i=1 n n j=1 w ij n i=1 n j=1 w ij ysi ) ȳ ) ys j ) ȳ ) n i=1 ysi ) ȳ ) where w is a spatial weights matrix. Some properties of Moran s I when there is no spatial autocorrelation): EI) = 1/n 1) VarI) = EI 2 ) EI) 2 = Something ugly but closed form Asymptotically, I EI) N 0, 1) VarI) 4

6 NC SIDS & Moran s I Lets start by using an adjacency matrix for w shared county borders). morans_i = functiony, w) { n = lengthy) y_bar = meany) num = sumw * y-y_bar) %*% ty-y_bar)) denom = sum y-y_bar)^2 ) n/sumw)) * num/denom) } morans_iy = nc$sid74, w = 1*st_touchesnc, sparse=false)) ## [1] libraryape) Moran.Inc$SID74, weight = 1*st_touchesnc, sparse=false)) %>% str) ## List of 4 ## $ observed: num ## $ expected: num ## $ sd : num ## $ p.value : num

7 EDA - Geary s C Like Moran s I, if we have observations at n spatial locations s 1,... s n ) C = n 1 2 n i=1 n j=1 w ij n i=1 n j=1 w ij ysi ) ys j ) ) 2 n i=1 ysi ) ȳ ) where w is a spatial weights matrix. 6

8 EDA - Geary s C Like Moran s I, if we have observations at n spatial locations s 1,... s n ) C = n 1 2 n i=1 n j=1 w ij n i=1 n j=1 w ij ysi ) ys j ) ) 2 n i=1 ysi ) ȳ ) where w is a spatial weights matrix. Some properties of Geary s C: 0 < C < 2 If C 1 then no spatial autocorrelation If C > 1 then negative spatial autocorrelation If C < 1 then positive spatial autocorrelation Geary s C is inversely related to Moran s I 6

9 NC SIDS & Geary s C Again using an adjacency matrix for w shared county borders). gearys_c = functiony, w) { n = lengthy) y_bar = meany) y_i = y %*% trep1,n)) y_j = ty_i) num = sumw * y_i-y_j)^2) denom = sum y-y_bar)^2 ) n-1)/2*sumw))) * num/denom) } gearys_cy = nc$sid74, w = 1*st_touchesnc, sparse=false)) ## [1]

10 Spatial Correlogram d = nc %>% st_centroid) %>% st_distance) %>% strip_class) breaks = seq0, maxd), length.out = 21) d_cut = cutd, breaks) adj_mats = map levelsd_cut), functionl) { d_cut == l) %>% matrixncol=100) %>% diag<- 0) } ) d = data_frame dist = breaks[-1], morans = map_dbladj_mats, morans_i, y = nc$sid74), gearys = map_dbladj_mats, gearys_c, y = nc$sid74) ) 8

11 0.6 morans gearys value 0.2 var morans gearys e+05 4e+05 6e+05 2e+05 4e+05 6e+05 dist 9

12 1.0 gearys morans 10

13 Autoregressive Models 11

14 AR Models - Time Lets just focus on the simplest case, an AR1) process y t = δ + ϕ y t 1 + w t where w t N 0, σ 2 ) and ϕ < 1, then Ey t ) = Vary t ) = δ 1 ϕ σ2 1 ϕ 12

15 AR Models - Time - Joint Distribution Previously we saw that an AR1) model can be represented using a multivariate normal distribution y 1 1 y 2 N 1 δ 1 ϕ.,. y n 1 σ2 1 ϕ 1 ϕ ϕ n 1 ϕ 1 ϕ n ϕ n 1 ϕ n

16 AR Models - Time - Joint Distribution Previously we saw that an AR1) model can be represented using a multivariate normal distribution y 1 1 y 2 N 1 δ 1 ϕ.,. y n 1 σ2 1 ϕ 1 ϕ ϕ n 1 ϕ 1 ϕ n ϕ n 1 ϕ n 2 1 In writing down the likelihood we also saw that an AR1) is 1st order Markovian, fy 1,..., y n ) = fy 1 ) fy 2 y 1 ) fy 3 y 2, y 1 ) fy n y n 1, y n 2,..., y 1 ) = fy 1 ) fy 2 y 1 ) fy 3 y 2 ) fy n y n 1 ) 13

17 Competing Definitions for y t y t = δ + ϕ y t 1 + w t vs. y t y t 1 N δ + ϕ y t 1, σ 2 ) 14

18 Competing Definitions for y t y t = δ + ϕ y t 1 + w t vs. y t y t 1 N δ + ϕ y t 1, σ 2 ) In the case of time, both of these definitions result in the same multivariate distribution for y. 14

19 AR in Space s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 15

20 AR in Space s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 Even in the simplest spatial case there is no clear / unique ordering, f ys 1),..., ) ys 10) = ) ) f ys 1) f ys 2) ys 1) f ys 10 ys 9), ys 8),..., ) ys 1) = ) ) f ys 10) f ys 9) ys 10) f ys 1 ys 2), ys 3),..., ) ys 10) =? 15

21 AR in Space s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 Even in the simplest spatial case there is no clear / unique ordering, f ys 1),..., ) ys 10) = ) ) f ys 1) f ys 2) ys 1) f ys 10 ys 9), ys 8),..., ) ys 1) = ) ) f ys 10) f ys 9) ys 10) f ys 1 ys 2), ys 3),..., ) ys 10) =? Instead we need to think about things in terms of their neighbors / neighborhoods. We will define Ns i ) to be the set of neighbors of location s i. If we define the neighborhood based on touching then Ns 3 ) = {s 2, s 4 } If we use distance within 2 units then Ns 3 ) = {s 1, s 2, s 3, s 4 } etc. 15

22 Defining the Spatial AR model Here we will consider a simple average of neighboring observations, just like with the temporal AR model we have two options in terms of defining the autoregressive process, Simultaneous Autogressve SAR) ys) = δ + ϕ 1 Ns) s Ns) ys ) + N 0, σ 2 ) Conditional Autoregressive CAR) ys) y s N δ + ϕ 1 Ns) s Ns) ys ), σ 2 16

23 Simultaneous Autogressve SAR) Using ys) = δ + ϕ 1 Ns) we want to find the distribution of y = s Ns) ys ) + N 0, σ 2 ) ys 1 ), ys 2 ),..., ys n )) t. 17

24 Simultaneous Autogressve SAR) Using ys) = δ + ϕ 1 Ns) we want to find the distribution of y = s Ns) ys ) + N 0, σ 2 ) ys 1 ), ys 2 ),..., ys n )) t. First we need to define a weight matrix W where {W} ij = { 1/ Ns i ) if j Ns i ) 0 otherwise 17

25 Simultaneous Autogressve SAR) Using ys) = δ + ϕ 1 Ns) we want to find the distribution of y = s Ns) ys ) + N 0, σ 2 ) ys 1 ), ys 2 ),..., ys n )) t. First we need to define a weight matrix W where {W} ij = { 1/ Ns i ) if j Ns i ) 0 otherwise then we can write y as follows, y = δ + ϕ W y + ϵ where ϵ N 0, σ 2 I) 17

26 A toy example 0 1/2 1/2 W = 1/2 0 1/2 1/2 1/2 0 18

27 Back to SAR y = δ + ϕ W y + ϵ 19

28 Conditional Autogressve CAR) This is a bit trickier, in the case of the temporal AR process we actually went from joint distribution conditional distributions which we were then able to simplify). Since we don t have a natural ordering we can t get away with this at least not easily). Going the other way, conditional distributions joint distribution is difficult because it is possible to specify conditional distributions that lead to an improper joint distribution. 20

29 Brook s Lemma For sets of observations x and y where px) > 0 x x and py) > 0 y y then py) n px) = py i y 1,..., y i 1, x i+1,..., x n ) px i=1 i x 1,..., x i 1, y i+1,..., y n ) n py i x 1,..., x i 1, y i+1,..., y n ) = px i y 1,..., y i 1, x i+1,..., x n ) i=1 21

30 A simplified example Let y = y 1, y 2 ) and x = x 1, x 2 ) then we can derive Brook s Lemma for this case, py 1, y 2 ) = py 1 y 2 )py 2 ) = py 1 y 2 ) py 2 x 1 ) px 1 ) px 1 y 2 ) = py 1 y 2 ) px 1 y 2 ) py 2 x 1 ) px 1 ) = py 1 y 2 ) py 2 x 1 ) px 1 y 2 ) px 2 x 1 ) px 1, x 2 ) = py 1 y 2 ) px 1 y 2 ) py 2 x 1 ) px 1 ) ) px2 x 1 ) px 2 x 1 ) py 1, y 2 ) px 1, x 2 ) = py 1 y 2 ) py 2 x 1 ) px 1 y 2 ) px 2 x 1 ) 22

31 Utility? Lets repeat that last example but consider the case where y = y 1, y 2 ) but now we let x = y 1 = 0, y 2 = 0) py 1, y 2 ) px 1, x 2 ) = py 1, y 2 ) py 1 = 0, y 2 = 0) py 1, y 2 ) = py 1 y 2 ) py 2 y 1 = 0) py 1 = 0 y 2 ) py 2 = 0 y 1 = 0) py 1 = 0, y 2 = 0) py 1, y 2 ) py 1 y 2 ) py 2 y 1 = 0) py 1 = 0 y 2 ) py 2 y 1 ) py 1 y 2 = 0) py 2 = 0 y 1 ) 23

32 As applied to a simple CAR s1 s2 ys 1) ys 2) N ϕw 12 ys 2), σ 2 ) ys 2) ys 1) N ϕw 21 ys 1), σ 2 ) 24

33 As applied to a simple CAR s1 s2 ys 1) ys 2) N ϕw 12 ys 2), σ 2 ) ys 2) ys 1) N ϕw 21 ys 1), σ 2 ) ) ) ) p ys 1), ys 2) p ys 1) ys 2) p ys 2) ys 1) = 0 ys ) 1) = 0 ys 2) p exp 1 2σ 2 ys 1) ϕ 2) W 12 ys 2)) exp exp 1 2σ 2 0 2) ϕw 12ys 2)) exp exp exp 1 1 2σ 2 ys 2) ϕ W 21 0) 2) ys1) ϕ W 12 ys 2)) 2 + ys 2) 2 2)) ϕw 12ys 2)) 2σ 2 1 ys 1) 2 2ϕ W 12 ys 1) ys 2) + 2)) ys 2) 2σ 2 ) ) 1 2σ y 0) 1 ϕw12 y 0) t 2 ϕw

34 Implicatiomns for y µ = 0 ) Σ 1 = 1 1 ϕw 12 σ 2 ϕw 12 1 = 1 I ϕ W) σ2 Σ = σ 2 I ϕ W) 1 25

35 Implicatiomns for y µ = 0 ) Σ 1 = 1 1 ϕw 12 σ 2 ϕw 12 1 = 1 I ϕ W) σ2 Σ = σ 2 I ϕ W) 1 we can then conclude that for y = ys 1 ), ys 2 )) t, y N 0, σ 2 I ϕ W) 1) which generalizes for all mean 0 CAR models. 25

36 General Proof Let y = ys 1 ),..., ys n )) and 0 = ys 1 ) = 0,..., ys n ) = 0) then by Brook s lemma, py) p0) = = n i=1 n py i y 1,..., y i 1, 0 i+1,..., 0 n) p0 i y 1,..., y i 1, 0 i+1,..., 0 n) exp i=1 exp = exp = exp 1 2σ 2 1 2σ 2 1 2σ 2 1 2σ 2 y i ϕ j<i Wij yj ϕ j>i 0j ) 2 ) ) ) 0 i ϕ j<i Wij yj ϕ 2 j>i 0j ) 2 ) 2 ) n n y i ϕ W ij y j + 1 ϕ W ij y j 2σ 2 i=1 j<i i=1 j<i ) n y 2 i 2ϕy i W ij y j i=1 j<i n n = exp 1 y 2 2σ 2 i ϕ i=1 i=1 j=1 ) = exp 1 2σ 2 yt I ϕw)y ) n ) y i W ij y j if W ij = W ji 26

37 CAR vs SAR Simultaneous Autogressve SAR) ys) = ϕ s W s s W s ys ) + ϵ y N 0, σ 2 I ϕw) 1 )I ϕw) 1 ) t ) Conditional Autoregressive CAR) ) W s s ys) y s N ys ), σ 2 W s s y N 0, σ 2 I ϕw) 1 ) 27

38 Generalization Adopting different weight matrices, W Between SAR and CAR model we move to a generic weight matrix definition beyond average of nearest neighbors) In time we varied p in the ARp) model, in space we adjust the weight matrix. In general having a symmetric W is helpful, but not required More complex Variance beyond σ 2 I) σ 2 can be a vector differences between areal locations) E.g. since areal data tends to be aggregated - adjust variance based on sample size 28