More Related Content Similar to Regularized Estimation of Spatial Patterns (20) Regularized Estimation of Spatial Patterns14. . . . . . . . . . . . . . . . . . . . .
Outline
1 Background: Spatial Pattern Estimation
2 Proposed Method: Spatial MCA
3 Real Data Analysis
4 Summary
15. . . . . . . . . . . . . . . . . . . . .
Background
• Bivariate spatial processes:
{(η1i(s1), η2i(s2)) : s1 ∈ D1, s2 ∈ D2}; i = 1, . . . , n
• D1, D2 ⊂ Rd
16. . . . . . . . . . . . . . . . . . . . .
Background
• Bivariate spatial processes:
{(η1i(s1), η2i(s2)) : s1 ∈ D1, s2 ∈ D2}; i = 1, . . . , n
• D1, D2 ⊂ Rd
• Example:
• η1(s1): sea surface temperature; D1: Indian Ocean
• η2(s2): rainfall; D2: East Africa
17. . . . . . . . . . . . . . . . . . . . .
Background
• Bivariate spatial processes:
{(η1i(s1), η2i(s2)) : s1 ∈ D1, s2 ∈ D2}; i = 1, . . . , n
• D1, D2 ⊂ Rd
• Assumption:
• η11(s1), . . . , η1n(s1): uncorrelated and mean zero
• η21(s2), . . . , η2n(s2): uncorrelated and mean zero
• common spatial covariance function:
• C11(s1, s∗
1) = cov(η1i(s1), η1i(s∗
1))
• C22(s2, s∗
2) = cov(η2i(s2), η2i(s∗
2))
• C12(s1, s2) = cov(η1i(s1), η2i(s2))
18. . . . . . . . . . . . . . . . . . . . .
Background
• Data at locations s11, . . . , s1p1 ∈ D1 and s21, . . . , s2p2 ∈ D2
•
Y1i(s1j) = η1i(s1j) + ϵ1ij; j = 1, . . . , p1
•
Y2i(s2j) = η2i(s2j) + ϵ2ij; j = 1, . . . , p2
• ϵℓij ∼ (0, σ2
ℓ ), uncorrelated with ηℓ(·), ℓ = 1, 2
• i = 1, . . . , n
19. . . . . . . . . . . . . . . . . . . . .
Background
• Data at locations s11, . . . , s1p1 ∈ D1 and s21, . . . , s2p2 ∈ D2
•
Y1i(s1j) = η1i(s1j) + ϵ1ij; j = 1, . . . , p1
•
Y2i(s2j) = η2i(s2j) + ϵ2ij; j = 1, . . . , p2
• ϵℓij ∼ (0, σ2
ℓ ), uncorrelated with ηℓ(·), ℓ = 1, 2
• i = 1, . . . , n
• Example:
20. . . . . . . . . . . . . . . . . . . . .
Background
• Data at locations s11, . . . , s1p1 ∈ D1 and s21, . . . , s2p2 ∈ D2
•
Y1i(s1j) = η1i(s1j) + ϵ1ij; j = 1, . . . , p1
•
Y2i(s2j) = η2i(s2j) + ϵ2ij; j = 1, . . . , p2
• ϵℓij ∼ (0, σ2
ℓ ), uncorrelated with ηℓ(·), ℓ = 1, 2
• i = 1, . . . , n
• Example:
• Y1(s1): observed sea surface temperature (red dot)
• Y2(s2): observed rainfall (blue dot)
21. . . . . . . . . . . . . . . . . . . . .
Aim
• Find dominant coupled patterns between η1i(·) and η2i(·)
• to study how variations of η1i(·) affect η2i(·)
22. . . . . . . . . . . . . . . . . . . . .
Rank-K Cross-Covariance Model
• D1, D2 ⊂ Rd: continuous domain
• Azaïez and Belgacem (2015),
C12(s1, s2) = cov(η1i(s1), η2i(s2)) =
∞∑
k=1
dkuk(s1)vk(s2)
• nonnegative singular values: d1 ≥ d2 ≥ · · ·
• {uk(·)} and {vk(·)}: sets of orthonormal basis functions
• similar to the Karhunen-Loéve expansion
23. . . . . . . . . . . . . . . . . . . . .
Rank-K Cross-Covariance Model
• D1, D2 ⊂ Rd: continuous domain
• Azaïez and Belgacem (2015),
C12(s1, s2) = cov(η1i(s1), η2i(s2)) =
∞∑
k=1
dkuk(s1)vk(s2)
• nonnegative singular values: d1 ≥ d2 ≥ · · ·
• {uk(·)} and {vk(·)}: sets of orthonormal basis functions
• similar to the Karhunen-Loéve expansion
• Assume dK+1 = 0.
• (u1(·), v1(·)), . . . , (uK(·), vK(·)): K dominant coupled patterns
24. . . . . . . . . . . . . . . . . . . . .
Goal
• Find (u1(·), v1(·)), . . . , (uK(·), vK(·)) as K coupled patterns
25. . . . . . . . . . . . . . . . . . . . .
Goal
• Find (u1(·), v1(·)), . . . , (uK(·), vK(·)) as K coupled patterns
• Common approach: maximum covariance analysis (MCA)
26. . . . . . . . . . . . . . . . . . . . .
Bivariate Data Vector
•
(
Y1i
Y2i
)
=
(
η1i
η2i
)
+
(
ϵ1i
ϵ2i
)
; i = 1, . . . , n
•
(
Y1i
Y2i
)
=
(
(Y1i(s11), . . . , Y1i(s1p1 ))′
(Y2i(s21), . . . , Y2i(s2p2 ))′
)
•
(
η1i
η2i
)
=
(
(η1i(s11), . . . , η1i(s1p1 ))′
(η2i(s21), . . . , η2i(s2p2 ))′
)
•
(
ϵ1i
ϵ2i
)
=
(
(ϵ1i1, . . . , ϵ1ip1 )′
(ϵ2i1, . . . , ϵ2ip2 )′
)
• Assume p1 ≥ p2
27. . . . . . . . . . . . . . . . . . . . .
Maximum Covariance Analysis (MCA)
• Bivariate data vector
(
Y1i
Y2i
)
i.i.d.
∼
((
0
0
)
,
(
Σ11 Σ12
Σ21 Σ22
))
• Σ12 = cov(Y1i, Y2i) = cov(η1i, η2i)
• Idea: find u ∈ Rp1 and v ∈ Rp2 , with ∥u∥2 = ∥v∥2 = 1, which
maximize
d = cov(u′
Y1i, v′
Y2i) = u′
Σ12v
28. . . . . . . . . . . . . . . . . . . . .
Maximum Covariance Analysis (MCA)
• Bivariate data vector
(
Y1i
Y2i
)
i.i.d.
∼
((
0
0
)
,
(
Σ11 Σ12
Σ21 Σ22
))
• Σ12 = cov(Y1i, Y2i) = cov(η1i, η2i)
• Singular value decomposition (SVD):
Σ12 = UDV ′
• Singular values: DK×K = diag(d1, . . . , dK ); d1 ≥ · · · ≥ dK > 0
• Left singular vectors: Up1×K = {u1, . . . , uK }
• Right singular vectors:Vp2×K = {v1, . . . , vK }
29. . . . . . . . . . . . . . . . . . . . .
Maximum Covariance Analysis (MCA)
• Bivariate data vector
(
Y1i
Y2i
)
i.i.d.
∼
((
0
0
)
,
(
Σ11 Σ12
Σ21 Σ22
))
• Σ12 = cov(Y1i, Y2i) = cov(η1i, η2i)
• Singular value decomposition (SVD):
Σ12 = UDV ′
• Singular values: DK×K = diag(d1, . . . , dK ); d1 ≥ · · · ≥ dK > 0
• Left singular vectors: Up1×K = {u1, . . . , uK }
• Right singular vectors:Vp2×K = {v1, . . . , vK }
• Coupled pattern: (u1, v1), . . . , (uK, vK)
30. . . . . . . . . . . . . . . . . . . . .
Sample Maximum Covariance Analysis
• Bivariate data matrix: (Y1, Y2)
• Y1 = (Y11, . . . , Y1n)′
; Y2 = (Y21, . . . , Y2n)′
• Sample cross-covariance matrix: S12 = Y ′
1 Y2/n
• Singular value decomposition (SVD):
S12 = ˜U ˜D ˜V ′
• ˜Dp1×p1 = diag( ˜d1 . . . ˜dp2 ); ˜d1 ≥ · · · ≥ ˜dp1 ≥ 0
• ˜Up1×p2 = {˜u1, . . . , ˜up2 }
• ˜Vp1×p1 = {˜v1, . . . , ˜vp1 }
• (˜u1, ˜v1), . . . , (˜uK, ˜vK): estimates of (u1, v1), . . . , (uK, vK)
31. . . . . . . . . . . . . . . . . . . . .
Sample Maximum Covariance Analysis
• Bivariate data matrix: (Y1, Y2)
• Y1 = (Y11, . . . , Y1n)′
; Y2 = (Y21, . . . , Y2n)′
• Sample cross-covariance matrix: S12 = Y ′
1 Y2/n
• Singular value decomposition (SVD):
S12 = ˜U ˜D ˜V ′
• ˜Dp1×p1 = diag( ˜d1 . . . ˜dp2 ); ˜d1 ≥ · · · ≥ ˜dp1 ≥ 0
• ˜Up1×p2 = {˜u1, . . . , ˜up2 }
• ˜Vp1×p1 = {˜v1, . . . , ˜vp1 }
• (˜u1, ˜v1), . . . , (˜uK, ˜vK): estimates of (u1, v1), . . . , (uK, vK)
• Problem:
• high estimation variability: n small, p1 or p2 large
• noisy patterns → low interpretation
• without a spatial structure of (u, v)
32. . . . . . . . . . . . . . . . . . . . .
Example:
u~(s1) v~(s2)
MCA
−1.0 −0.5 0.0 0.5 1.0
33. . . . . . . . . . . . . . . . . . . . .
Example:
u(s1) v(s2)
TRUE
u~(s1) v~(s2)
MCA
−1.0 −0.5 0.0 0.5 1.0
34. . . . . . . . . . . . . . . . . . . . .
Motivation
To enhance the interpretability, we implement MCA by incorporating
35. . . . . . . . . . . . . . . . . . . . .
Motivation
To enhance the interpretability, we implement MCA by incorporating
1 spatial structure of (uk(·), vk(·))
36. . . . . . . . . . . . . . . . . . . . .
Motivation
To enhance the interpretability, we implement MCA by incorporating
1 spatial structure of (uk(·), vk(·))
2 spatial localized patterns of (uk(·), vk(·))
37. . . . . . . . . . . . . . . . . . . . .
Motivation
To enhance the interpretability, we implement MCA by incorporating
1 spatial structure of (uk(·), vk(·))
2 spatial localized patterns of (uk(·), vk(·))
retain that orthogonal constraints for (u, v).
38. . . . . . . . . . . . . . . . . . . . .
Outline
1 Background: Spatial Pattern Estimation
2 Proposed Method: Spatial MCA
3 Real Data Analysis
4 Summary
39. . . . . . . . . . . . . . . . . . . . .
Quick Recap
• Data: Yℓi = (Yℓi(sℓ1), . . . , Yℓi(sℓpℓ
))′, i = 1, . . . , n
•
Yℓi(sℓj) = ηℓi(sℓj) + ϵℓij; j = 1, . . . , pℓ
• ϵℓij ∼ (0, σ2
ℓ )
• ϵℓij: uncorrelated with ηℓ(·)
• ℓ = 1, 2
40. . . . . . . . . . . . . . . . . . . . .
Quick Recap
• Data: Yℓi = (Yℓi(sℓ1), . . . , Yℓi(sℓpℓ
))′, i = 1, . . . , n
•
Yℓi(sℓj) = ηℓi(sℓj) + ϵℓij; j = 1, . . . , pℓ
• ϵℓij ∼ (0, σ2
ℓ )
• ϵℓij: uncorrelated with ηℓ(·)
• ℓ = 1, 2
• Spatial cross-covariance function:
C12(s1, s2) = cov(η1i(s1), η2i(s2)) =
K∑
k=1
dkuk(s1)vk(s2)
• d1 ≥ · · · ≥ dK ≥ 0
• u1(·), . . . , uK (·): K unknown orthonormal functions
• v1(·), . . . , vK (·): K unknown orthonormal functions
41. . . . . . . . . . . . . . . . . . . . .
MCA (alternative version)
• Sample cross-covariance matrix: S12 = Y ′
1 Y2/n
• MCA: perform SVD of S12
42. . . . . . . . . . . . . . . . . . . . .
MCA (alternative version)
• Sample cross-covariance matrix: S12 = Y ′
1 Y2/n
• MCA: perform SVD of S12
• Alternative method:
( ˜U, ˜V ) = arg max
U,V
tr(U′
S12V ),
subject to U′U = V ′V = IK
• Up1×K = (u1, . . . , uK ) with ujk = uk(s1j)
• Vp2×K = (v1, . . . , vK ) with vjk = vk(s2j)
43. . . . . . . . . . . . . . . . . . . . .
Regularized MCA
• Sample cross-covariance matrix: S12 = X′Y /n
• Up1×K = (u1, . . . , uK) with ujk = uk(s1j)
• Vp2×K = (v1, . . . , vK) with vjk = vk(s2j)
• Objective function:
tr(U′
S12V )
subject to U′U = V ′V = IK
44. . . . . . . . . . . . . . . . . . . . .
Regularized MCA
• Sample cross-covariance matrix: S12 = X′Y /n
• Up1×K = (u1, . . . , uK) with ujk = uk(s1j)
• Vp2×K = (v1, . . . , vK) with vjk = vk(s2j)
• Objective function:
tr(U′
S12V ) −
K∑
k=1
{
τ1uJ(uk)+τ2u
p1∑
j=1
|uk(s1j)| + τ1vJ(vk)+τ2v
p2∑
j=1
|vk(s2j)|
}
subject to U′U = V ′V = IK
45. . . . . . . . . . . . . . . . . . . . .
Regularized MCA
• Sample cross-covariance matrix: S12 = X′Y /n
• Up1×K = (u1, . . . , uK) with ujk = uk(s1j)
• Vp2×K = (v1, . . . , vK) with vjk = vk(s2j)
• Objective function:
tr(U′
S12V ) −
K∑
k=1
{
τ1uJ(uk)+τ2u
p1∑
j=1
|uk(s1j)| + τ1vJ(vk)+τ2v
p2∑
j=1
|vk(s2j)|
}
subject to U′U = V ′V = IK
• J(uk(·)) =
∑
z1+···+zd=2
∫
Rd
(
∂2
uk(s)
∂xz1
1 . . . ∂x
zd
d
)2
ds
• s = (x1, . . . , xd)′
• τ1u, τ1v: smoothness parameter
• τ2u, τ2v: sparseness parameter
46. . . . . . . . . . . . . . . . . . . . .
Spatial MCA (SpatMCA)
• J(uk(·)) = u′
kΩ1uk, J(vk(·)) = v′
kΩ2vk
• Ω1 ≻ 0: determined only by s11, . . . , s1p1
• Ω2 ≻ 0: determined only by s21, . . . , s2p2
• Green and Silverman (1994)
47. . . . . . . . . . . . . . . . . . . . .
Spatial MCA (SpatMCA)
• J(uk(·)) = u′
kΩ1uk, J(vk(·)) = v′
kΩ2vk
• Ω1 ≻ 0: determined only by s11, . . . , s1p1
• Ω2 ≻ 0: determined only by s21, . . . , s2p2
• Green and Silverman (1994)
• SpatMCA: ( ˆU, ˆV ) maximizes:
tr(U′S12V )−
K∑
k=1
{
τ1uu′
kΩ1uk+τ2u
p1∑
j=1
|ujk| + τ1vv′
kΩ2vk+τ2v
p2∑
j=1
|vjk|
}
subject to U′U = V ′V = IK
48. . . . . . . . . . . . . . . . . . . . .
Spatial MCA (SpatMCA)
• J(uk(·)) = u′
kΩ1uk, J(vk(·)) = v′
kΩ2vk
• Ω1 ≻ 0: determined only by s11, . . . , s1p1
• Ω2 ≻ 0: determined only by s21, . . . , s2p2
• Green and Silverman (1994)
• SpatMCA: ( ˆU, ˆV ) maximizes:
tr(U′S12V )−
K∑
k=1
{
τ1uu′
kΩ1uk+τ2u
p1∑
j=1
|ujk| + τ1vv′
kΩ2vk+τ2v
p2∑
j=1
|vjk|
}
subject to U′U = V ′V = IK
• As τ1u = τ1v = τ2u = τ2v = 0, ( ˆU, ˆV ) = ( ˜U, ˜V ) (sample MCA )
49. . . . . . . . . . . . . . . . . . . . .
SpatMCA: (ˆu1(·), ˆv1(·)), . . . , (ˆuK(·), ˆvK(·))
• (ˆu1(·), ˆv1(·)), . . . , (ˆuK(·), ˆvK(·)) maximize
tr(U′
S12V ) −
K∑
k=1
{
τ1uJ(uk)+τ2u
p1∑
j=1
|uk(s1j)| + τ1vJ(vk)+τ2v
p2∑
j=1
|vk(s2j)|
}
,
subject to U′
U = V ′
V = IK
50. . . . . . . . . . . . . . . . . . . . .
SpatMCA: (ˆu1(·), ˆv1(·)), . . . , (ˆuK(·), ˆvK(·))
• (ˆu1(·), ˆv1(·)), . . . , (ˆuK(·), ˆvK(·)): smoothing spline
51. . . . . . . . . . . . . . . . . . . . .
SpatMCA: (ˆu1(·), ˆv1(·)), . . . , (ˆuK(·), ˆvK(·))
• (ˆu1(·), ˆv1(·)), . . . , (ˆuK(·), ˆvK(·)): smoothing spline
•
ˆuk(s1) =
p1∑
i=1
a1ig(∥s1 − s1i∥) + b10 +
d∑
j=1
b1jx1j
ˆvk(s2) =
p2∑
i=1
a2ig(∥s2 − s2i∥) + b20 +
d∑
j=1
b2jx2j
• s1 = (x11, . . . , x1d)′
; s2 = (x21, . . . , x2d)′
• g(r) =
1
16π
r2
log r; if d = 2,
Γ(d/2 − 2)
16πd/2
r4−d
; if d = 1, 3,
• a1 = (a11, . . . , a1p1 )′
and b1 = (b10, b11, . . . , b1d)′
based on ˆuk
• a2 = (a21, . . . , a2p2 )′
and b2 = (b20, b21, . . . , b2d)′
based on ˆvk
52. . . . . . . . . . . . . . . . . . . . .
Why roughness and Lasso penalties?
53. . . . . . . . . . . . . . . . . . . . .
2D Example
u(s1) v(s2)
TRUE
u~(s1) v~(s2)
MCA
−1.0 −0.5 0.0 0.5 1.0
54. . . . . . . . . . . . . . . . . . . . .
Case 1: Fix τ2u = τ2v = 0 (only smoothness)
u(s1) v(s2)
TRUE
−1.0 −0.5 0.0 0.5 1.0
u^(s1) v^(s2)
= 0τ1 u = τ1 v =
55. . . . . . . . . . . . . . . . . . . . .
Case 1:Fix τ2u = τ2v = 0 (only smoothness)
u(s1) v(s2)
TRUE
−1.0 −0.5 0.0 0.5 1.0
u^(s1) v^(s2)
= 0τ1 u = τ1 v =
56. . . . . . . . . . . . . . . . . . . . .
Case 2: Fix τ1u = τ1v = 0 (only sparseness)
u(s1) v(s2)
TRUE
−1.0 −0.5 0.0 0.5 1.0
u^(s1) v^(s2)
= 0τ2 u = τ2 v =
57. . . . . . . . . . . . . . . . . . . . .
Case 2: Fix τ1u = τ1v = 0 (only sparseness)
u(s1) v(s2)
TRUE
−1.0 −0.5 0.0 0.5 1.0
u^(s1) v^(s2)
= 0τ2 u = τ2 v =
58. . . . . . . . . . . . . . . . . . . . .
Case 3: Fix τ1u = τ1v = τ2u = τ2v
u(s1) v(s2)
TRUE
−1.0 −0.5 0.0 0.5 1.0
u^(s1) v^(s2)
= = 0τ1 u = τ1 v = τ2 u = τ2 v =
59. . . . . . . . . . . . . . . . . . . . .
Case 3: Fix τ1u = τ1v = τ2u = τ2v
u(s1) v(s2)
TRUE
−1.0 −0.5 0.0 0.5 1.0
u^(s1) v^(s2)
= = 0τ1 u = τ1 v = τ2 u = τ2 v =
60. . . . . . . . . . . . . . . . . . . . .
Estimation of D
Given the SpatMCA estimate ( ˆU, ˆV ),
ˆD = arg min
d1,...,dK ≥0
∥S12 − ˆUD ˆV ′
∥2
F = diag( ˆd1, . . . , ˆdK)
• ˆdk = min(ˆu′
kS12ˆvk, 0)
61. . . . . . . . . . . . . . . . . . . . .
Selection of (τ1u, τ2u, τ1v, τ2v)
• The proposed CV criterion is
CV(τ1u, τ2u, τ1v, τ2v)
=
1
M
M∑
m=1
∥S
(m)
12 − ˆU(−m)
τ1u,τ2u
ˆD(−m)
τ1u,τ2u,τ1v,τ2v
( ˆV (−m)
τ1v,τ2v
)′
∥2
F
• Partition {(Y11, Y21), . . . , (Y1n, Y2n)} into M parts with equal size nM
• S
(m)
12 = (Y
(m)
1 )′
Y
(m)
2 /nM based on the m-th part data (Y
(m)
1 , Y
(m)
2 )
• ˆU
(−m)
τ1,τ2 , ˆV
(−m)
τ1,τ2 , ˆD
(−m)
τ1u,τ2u,τ1v,τ2v : based on (Y
(−m)
1 , Y
(−m)
2 )
• (Y
(−m)
1 , Y
(−m)
2 ): remaining data, i.e., Y1, Y2 excluding (Y
(m)
1 , Y
(m)
2 )
62. . . . . . . . . . . . . . . . . . . . .
Selection of (τ1u, τ2u, τ1v, τ2v)
• The proposed CV criterion is
CV(τ1u, τ2u, τ1v, τ2v)
=
1
M
M∑
m=1
∥S
(m)
12 − ˆU(−m)
τ1u,τ2u
ˆD(−m)
τ1u,τ2u,τ1v,τ2v
( ˆV (−m)
τ1v,τ2v
)′
∥2
F
• Partition {(Y11, Y21), . . . , (Y1n, Y2n)} into M parts with equal size nM
• S
(m)
12 = (Y
(m)
1 )′
Y
(m)
2 /nM based on the m-th part data (Y
(m)
1 , Y
(m)
2 )
• ˆU
(−m)
τ1,τ2 , ˆV
(−m)
τ1,τ2 , ˆD
(−m)
τ1u,τ2u,τ1v,τ2v : based on (Y
(−m)
1 , Y
(−m)
2 )
• (Y
(−m)
1 , Y
(−m)
2 ): remaining data, i.e., Y1, Y2 excluding (Y
(m)
1 , Y
(m)
2 )
• (τ1u, τ2u, τ1v, τ2v): selected by minimizing CV(τ1u, τ2u, τ1v, τ2v)
63. . . . . . . . . . . . . . . . . . . . .
Selection of (τ1u, τ2u, τ1v, τ2v)
• High computation cost to select {τ1u, τ2u, τ1v, τ2v} simultaneously
64. . . . . . . . . . . . . . . . . . . . .
Selection of (τ1u, τ2u, τ1v, τ2v)
• High computation cost to select {τ1u, τ2u, τ1v, τ2v} simultaneously
• Two-step procedure:
1 Select smoothness parameters τ1u and τ1v:
(ˆτ1u, ˆτ1v) = arg min
{τ1u,τ1v}⊂[0,∞)2
CV(τ1u, 0, τ1v, 0),
2 Select sparseness parameters τ2u and τ2v:
(ˆτ2u, ˆτ2v) = arg min
{τ2u,τ2v}⊂[0,∞)2
CV(ˆτ1u, τ2u, ˆτ1v, τ2v).
65. . . . . . . . . . . . . . . . . . . . .
Example (1D): 5-fold CV
u(·)
−0.40.00.4
MCA SpatMCA Smooth (only) Sparse (only)
v(·)
−0.40.00.4
MCA SpatMCA Smooth (only) Sparse (only)
66. . . . . . . . . . . . . . . . . . . . .
Outline
1 Background: Spatial Pattern Estimation
2 Proposed Method: Spatial MCA
3 Real Data Analysis
4 Summary
67. . . . . . . . . . . . . . . . . . . . .
Sea Surface Temperature vs. Rainfall
• Bivariate data:
• Sea surface temperature (SST):
• Region: Indian Ocean
• Number of grids: p1 = 3, 591 (resolution: 1◦
× 1◦
)
– Rainfall:
• Region: East African
• Number of grids: p2 = 255 (resolution: 5◦
× 5◦
)
• Time period (monthly average): Jan. 2011- Dec. 2015 → n = 60
• Remove monthly mean for each grid
68. . . . . . . . . . . . . . . . . . . . .
Sea Surface Temperature vs. Rainfall
• Bivariate data:
• Sea surface temperature (SST):
• Region: Indian Ocean
• Number of grids: p1 = 3, 591 (resolution: 1◦
× 1◦
)
– Rainfall:
• Region: East African
• Number of grids: p2 = 255 (resolution: 5◦
× 5◦
)
• Time period (monthly average): Jan. 2011- Dec. 2015 → n = 60
• Remove monthly mean for each grid
• Goal: find coupled patterns of the SST and rainfall data
• Reference: Omondi et al. (2013)
69. . . . . . . . . . . . . . . . . . . . .
Real Data Analysis
• Randomly decompose the data into two parts with 30 time points
• Training data
• Validation data
• SpatMCA: based on 5-fold CV
70. . . . . . . . . . . . . . . . . . . . .
Result: CV vs. K
2 4 6 8 10
107108109110111112
K
CV
MCA SpatMCA
• ˆK = 1 for MCA and SpatMCA
71. . . . . . . . . . . . . . . . . . . . .
Result: 1st Coupled Pattern of SpatMCA
72. . . . . . . . . . . . . . . . . . . . .
Result: 1st Coupled Pattern of SpatMCA
73. . . . . . . . . . . . . . . . . . . . .
Result: 1st Coupled Pattern of SpatMCA
74. . . . . . . . . . . . . . . . . . . . .
Result: 1st Coupled Pattern of MCA
75. . . . . . . . . . . . . . . . . . . . .
Result: 1st Maximum Covariance Variables
• 1st maximum covariance variables: {ˆu′
1Y1i} ; {ˆv′
1Y2i}
76. . . . . . . . . . . . . . . . . . . . .
Result: 1st Maximum Covariance Variables
• 1st maximum covariance variables: {ˆu′
1Y1i} ; {ˆv′
1Y2i}
MCA SpatMCA
• Pearson’s correlation: 0.6 for MCA and SpatMCA
77. . . . . . . . . . . . . . . . . . . . .
Result: Average Squared Error (ASE)
• ASE = 1
p1p2
∥Sv
12 − ˆUK
ˆDK
ˆV ′
K∥2
F
• Sv
12: sample cross-covariance matrix based on validation data
• Result:
2 4 6 8 10
0.00220.00260.00300.0034
K
ASE
MCA SpatMCA
78. . . . . . . . . . . . . . . . . . . . .
Outline
1 Background: Spatial Pattern Estimation
2 Proposed Method: Spatial MCA
3 Real Data Analysis
4 Summary
79. . . . . . . . . . . . . . . . . . . . .
Summary
SpatMCA:
• high-dimensional structure → low-dimensional structure
• with the roughness and Lasso penalties
• enhance physical interpretation, e.g., spatial localized patterns
80. . . . . . . . . . . . . . . . . . . . .
Summary
SpatMCA:
• high-dimensional structure → low-dimensional structure
• with the roughness and Lasso penalties
• enhance physical interpretation, e.g., spatial localized patterns
• can cope with irregular spaced locations
81. . . . . . . . . . . . . . . . . . . . .
GitHub: egpivo/SpatMCA
82. . . . . . . . . . . . . . . . . . . . .
Thanks for your attention!