In climate and atmospheric research, many phenomena involve more than one meteorological spatial processes covarying in space. To understand how one process is affected by another, maximum covariance analysis (MCA) is commonly applied. However, the patterns obtained from MCA may sometimes be difficult to interpret. In this paper, we propose a regularization approach to promote spatial features in dominant coupled patterns by introducing smoothness and sparseness penalties while accounting for their orthogonalities. We develop an efficient algorithm to solve the resulting optimization problem by using the alternating direction method of multipliers. The effectiveness of the proposed method is illustrated by several numerical examples, including an application to study how precipitations in east Africa are affected by sea surface temperatures in the Indian Ocean.

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Regularized Estimation of Spatial Patterns
Wen-Ting Wang
June 23, 2017
Joint work with Hsin-Cheng Huang @ Academia Sinica
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Outline
1 Background: Spatial Pattern Estimation
2 Proposed Method: Spatial MCA
3 Real Data Analysis
4 Summary
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Climate Change
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Climate Change
increases the odds of extreme weather events occurring,
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Flood
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Drought
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Climate Change
affects human health and quality of life
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Drought in the East Africa
2011
10,000
People have been killed
by the worst drought in 60 years.
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Climate Change
are associated with atmospheric dynamics
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Atmospheric dynamics
can be studied through spatial patterns
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Rainfall
in East Africa
⇕?
Sea Surface Temperature
in the Indian Ocean
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Coupled Spatial Patterns
12

14. . . . . . . . . . . . . . . . . . . . .
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!