36
Nonstationary covariance structure I: Deformations

Nonstationary covariance structure I: Deformations

  • View
    238

  • Download
    0

Embed Size (px)

Citation preview

Nonstationary covariance structure I: Deformations

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Evidence of anisotropy15o red60o green105o blue150o brown

Another view of anisotropy

σe2 = 127.1(259)

σs2 = 68.8 (255)

θ = 10.7 (45)

σe2 = 154.6 (134)

σs2 = 141.0 (127)

θ = 29.5 (35)

General setup

Z(x,t) = (x,t) + (x)1/2E(x,t) + (x,t)

trend + smooth + error

We shall assume that is known or constant

t = 1,...,T indexes temporal replications

E is L2-continuous, mean 0, variance 1, independent of the error C(x,y) = Cor(E(x,t),E(y,t))

D(x,y) = Var(E(x,t)-E(y,t)) (dispersion)

Cov(Z(x,t),Z(y,t)) =ν(x)ν(y)C(x,y) x ≠ y

ν(x) + σε2 x = y

⎧ ⎨ ⎩

Geometric anisotropy

Recall that if we have an isotropic covariance (circular isocorrelation curves).

If for a linear transformation A, we have geometric anisotropy (elliptical isocorrelation curves).

General nonstationary correlation structures are typically locally geometrically anisotropic.

C(x,y) = C( x − y )

C(x,y) = C( Ax − Ay )

The deformation idea

In the geometric anisotropic case, write

where f(x) = Ax. This suggests using a general nonlinear transformation

. Usually d=2 or 3. G-plane D-space

We do not want f to fold.

C(x,y) = C( f (x) − f(y) )

f:R2 → Rd

Implementation

Consider observations at sites x1, ...,xn. Let be the empirical covariance between sites xi and xj. Minimize

where J(f) is a penalty for non-smooth transformations, such as the bending energy

ˆ C ij

(θ, f) a wij Cij −C(f(xi ), f(xj ); θ)( )

i,j∑

2+ λTP(f)

P(f)i =∂2fi∂x2

⎝⎜⎞

⎠⎟

2

+ 2∂2fi∂x∂y

⎝⎜⎞

⎠⎟

2

+∂2fi∂y2

⎝⎜⎞

⎠⎟

2⎡

⎣⎢⎢

⎦⎥⎥dxdy∫∫

SARMAP

An ozone monitoring exercise in California, summer of 1990, collected data on some 130 sites.

Transformation

This is for hr. 16 in the afternoon

Identifiability

Perrin and Meiring (1999): Let

If (1) f and f-1 are differentiable in Rn

(2) (u) is differentiable for u>0

then (f,) is unique up to a scaling for and a homothetic transformation for f (scaling, rotation, reflection)

D(x,y) =( f(x) −f(y) ), x,y ∈Rn

RichnessPerrin & Senoussi (2000): Let f and f-1 be differentiable, and let r(x,y) be continuously differentiable. Then

(stationarity) iff

Let f(0)=0,ci ith column of . Then

(isotropy) iff and

r(x,y) =ρ(f(x) −f(y))

(∗)Dxr(x,y)J f−1(x) +Dyr(x,y)J f

−1(y), x ≠y

Jf−1(0)

r(x,y) =ρ( f(x) −f(y) ) (∗)

fi (y)Dxr(0,y)c j =fj (y)Dxr(0,y)ci

The Brownian sheet

Let andThen

So the Brownian sheet can be thought of as a stationary deformation. It is however not an istropic deformation.

r(x,y) =x + y − y−x

2 x y

f1(x) =ln( x ) f2 (x) =arctan(x1 / x2 )

(s) = 12 {exp(s1 / 2 + exp(−s1 / 2)

− exp(s1 / 2 + exp(−s1 / 2) − 2cos(s2 )}

Estimating variability

Resample time slices with replacement from the original data (to maintain spatial structure).

Re-estimate deformation based on each bootstrap sample.

Kriging estimates can be made based on each of the bootstrap estimates, to get a better sense of the variability.

French rainfall data

Altitude-adjusted 10-day aggregated rainfall data Nov-Dec 1975-1992 for 39 sites from Languedoc-Rousillon region of France.

Estimated deformation

(a)

9

19

22

25

33

41

4553

(b)

9

19

22

25

33

41

45

53

G-plane equicorrelation contours

D-plane Equicorrelation Contours

Uncertainty in deformationModel refitted using 24 of the 36 sites.

The smoothing parameter λ

Cross-validation: Leave out sampling station i, estimate(θ,f) from remaining n-1 stations. Minimize the prediction error for site i, summed over i.

Together with bootstrap estimate of variability, very computer intensive.

Thin-plate splines

f(s) =c + As

linear part1 2 3 + WT %σ(s)

%σ(s) = σ(s − x1),...,σ(s − xn )( )T

σ(h) = h 2 log( h )

1T W =0 XTW=0

P(f) =tr(WT %SW)

A Bayesian implementationLikelihood:

Prior:

Linear part: fix two points in the G-D mapping put a (proper) prior on the remaining two parameters

L(S | Σ) = (2πΣ )−(T−1)/ 2 exp −T2

trΣ−1S ⎧ ⎨ ⎩

⎫ ⎬ ⎭

p(W) ∝ exp −

12τ

WiT %SWi

i=1

2

∑⎛⎝⎜

⎞⎠⎟

smoothing parameter

Computation

Metropolis-Hastings algorithm for sampling from highly multidimensional posterior.

Given estimates of D-plane locations, f(xi), the transformation is extrapolated to the whole domain using thin plate splines. Predictive distributions for

(a) temporal variance at unobserved sites,

(b) the spatial covariance for pairs of observed and/or unobserved sites,

(c) the observation process at unobserved sites.

California ozone

12

3

4

5

6

789

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

38

39

40

4142

43

444546

47

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

17

18

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 00:12:36 PST 2003

Posterior samples

12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

N=63, S. Calif: 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 22:18:29 PST 2003

Other applications

Point process deformation (Jensen & Nielsen, Bernoulli, 2000)

Deformation of brain images (Worseley et al., 1999)

Global processes

Problems such as global warming require modeling of processes that take place on the globe (an oriented sphere). Optimal prediction of quantities such as global mean temperature need models for global covariances.

Note: spherical covariances can take values in [-1,1]–not just imbedded in R3.

Also, stationarity and isotropy are identical concepts on the sphere.

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions, pq the angle between them, and Pi the Legendre polynomials.

Example: ai=(2i+1)ρi

C(p,q) = aii= 0

∑ Pi (cosγpq )

C(p,q) =1− ρ2

1− 2ρcos γpq + ρ2 − 1

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data. Subset with 839 stations with data 1950-1991 selected.

Isotropic correlations

Spherical deformation

Need isotropic covariance model on transformation of sphere/globe

Covariance structure on convex manifolds

Simple option: deform globe into another globe

Alternative: MRF approach

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude).

(Das, 2000)

Three iterations

Resulting isocovariance curves

Comparison

Isotropic Anisotropic

Assessing uncertainty

Another current climate problem

General circulation models require accurate historical ocean surface temperature records

Data from buoys, ships, satellites