Bo Li. National Center for Atmospheric Research. Based on joint work with:

Transcription

1 Nonparametric Assessment of Properties of Space-time Covariance Functions and its Application in Paleoclimate Reconstruction Bo Li National Center for Atmospheric Research Based on joint work with: Marc Genton: University of Geneva & TAMU Mike Sherman: TAMU Doug Nychka and Caspar Ammann: NCAR Supported by the National Science Foundation

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3 Outline of this talk Testing properties of space-time covariance function 1. Introduction 2. Development of testing methodology 3. Data analysis and simulations Reconstructing paleoclimate Apply the testing procedures to solve the problem 1. Significance and data 2. Why use BHM framework? 3. Challenges and recipe

4 Introduction Spatio-temporal data sets are common in geophysical and environmental sciences strictly stationary space-time random field {Z(s, t) : s R d, t R} C(h, u) = cov{z(s, t), Z(s + h, t + u)} Modeling such data often relies on parametric covariance models (Cressie and Huang, 1999; Gneiting, 2002; Stein, 2005) Make assumptions to simplify the covariance function in large and complex space-time data sets

5 Ways of simplifying covariance function Full symmetry: C(h, u) = C(h, u), or if C(h, u) = C( h, u) t 2 wind direction t 1 z(s 1, t 2 ) z(s 2, t 2 ) z(s 1, t 1 ) h z(s 2, t 1 ) Z: CO 2 concentration h = s 2 s 1, u = t 2 t 1 Cov(Z(s 1, t 1 ), Z(s 2, t 2 )) = C(h, u) = C( h, u) Cov(Z(s 2, t 1 ), Z(s 1, t 2 )) = C(h, u) = C( h, u)? C(h, u) = C(h, u)

6 Ways of simplifying covariance function Separability: C(h, u) = C(h, 0)C(0, u) C(0, 0) C(h, 0): pure spatial covariance C(0, u): pure temporal covariance Separability = Full symmetry Taylor Hypothesis: There exists a velocity vector v such that C(vu, 0) = C(0, u) for all u...

7 Ways of simplifying covariance function Important because they simplify the structure of the model and its inference, and ease possibly extensive computations Random field: Z(s i, t j ), i = 1,..., 100, j = 1,..., 100 Size of space-time covariance Σ: 10,000 10,000 If Σ is separable: Σ = Σ 1 Σ 2 Σ 1 = Σ 1 1 Σ 1 2 size of spatial covariance Σ 1 : size of temporal covariance Σ 2 :

8 Ways of simplifying covariance function Not always appropriate: prevailing winds, water flows, and atmosphere circulation in geoscience, meteorology and ecology Pacific Ocean wind data (Cressie and Huang,1999) Irish wind speed data (Gneiting et al., 2007) Sulfate concentration levels (Jun and Stein, 2007) Their graphic-based diagnostics are useful but difficult to assess and open to interpretation Need a formal and unified approach...

9 Existing approaches to test specific properties Developed the test in a particular situation or with the assumption of distribution Shitan and Brockwell (1995), Guo and Billard (1998), Genton and Koul (2007): test for separability in SAR/QAR Lu and Zimmerman (2001) and Guan, Sherman and Calvin (2004): nonparametric tests for spatial isotropy Scaccia and Martin (2002, 2005): spectral method to test the sym. and sep. for spatial lattice processes Fuentes (2006): nonparametric test for sep. of a spatio-temporal process based on a spectral method Mitchell, Genton and Gumpertz (2005, 2006): LRT for sep. in the context of multivariate repeated measures assuming multinormality, with extension to space-time

10 Outline of this talk Testing properties of space-time covariance function 1. Introduction 2. Development of testing methodology 3. Data analysis and simulations Reconstructing paleoclimate Apply the testing procedures to solve the problem 1. Significance and data 2. Why use BHM framework? 3. Challenges and recipe

11 Hypotheses Λ: a set of space-time lags (h, u) Ĉ(h, u): an estimator of C(h, u) G = {C(h, u), (h, u) Λ}, Ĝ = {Ĉ(h, u), (h, u) Λ} H 0 : Af(G) = 0 where A: contrast matrix of row rank q, and f = (f 1,..., f r ) T : real-valued functions, differentiable at G (Lu and Zimmerman, 2001) Full symmetry: H 1 0 : C(h, u) C(h, u) = 0, (h, u) Λ Separability: H 2 0: C(h, u)/c(h, 0) C(0, u)/c(0, 0) = 0, (h, u) Λ

12 Examples A 1 G = 0 and A 2 f(g) = 0 Λ = {(0, 0), (h 1, u 1 ), (h 1, u 1 ), (h 2, u 2 ), (h 2, u 2 ), (h 1, 0), (0, u 1 ), (h 2, 0), (0, u 2 )} G = (C(0, 0), C(h 1, u 1 ), C(h 1, u 1 ), C(h 2, u 2 ), C(h 2, u 2 ), f(g) = C(h 1, 0), C(0, u 1 ), C(h 2, 0), C(0, u 2 )) T A 1 = [ ( ) T C(h 1,u 1 ) C(h, C(h 1, u 1 ) 1,0) C(h, C(h 2,u 2 ) 1,0) C(h, C(h 2, u 2 ) 2,0) C(h, C(0,u 1), C(0,u 2) 2,0) C(0,0) C(0,0) A 2 = ]

13 Asymptotics of sample space-time covariances In many situations, the observations are taken from a fixed space S R d at regularly spaced times T n = {1,..., n} Increasing index sets D n = S T n Ĉn(h, u): sample based estimator of C(h, u) based on obs. in a sequence of increasing index sets D n Ĝn = {Ĉn(h, u) : (h, u) Λ}: estimator of G computed over D n. We make no assumptions on the marginal or the joint distribution of observations other than mild moment and mixing conditions on the random field

14 Asymptotics of sample space-time covariances Under zero mean assumption for Z: Ĉ n (h, u) = 1 n S(h) n u S(h) t=1 Z(s, t)z(s + h, t + u) Strong mixing condition: α(u) = O(u ɛ ) for some ɛ > 0 Moment condition for Ĉn(h, u): sup n E{ n{ĉn(h, u) C(h, u)} 2+δ } < for some δ > 0

15 Asymptotics of sample space-time covariances Theorem: Let {Z(s, t), s R d, t Z} be a strictly stationary spatio-temporal random field observed in D n = S T n, where S R d and T n = {1,..., n}. Assume t Z cov{z(0, 0)Z(h 1, u 1 ), Z(s, t)z(s + h 2, t + u 2 )} < for all h 1, h 2 S, s S(h 2 ) and all finite u 1, u 2, then Σ = lim n ncov(ĝn, Ĝn) exists. Further with mixing and moment conditions we have n(ĝn G) d N m (0, Σ) as n.

16 Test statistics Li, Genton and Sherman (2008): a n (Ĝn G) d N m (0, Σ) under mild moment and mixing conditions, and for a wide variety of data structures d a n {f(ĝn) f(g)} N r (0, B T ΣB) where B ij i = 1,..., m, j = 1,..., r = f j / G i Test Statistic for H 0 : Af(G) = 0: TS = a 2 n{af(ĝn)} T (AB T ΣBA T ) 1 {Af(Ĝn)} Examples: full symmetry and separability d χ 2 q Estimate the matrix B empirically (replacing G with Ĝ n ) and estimate Σ using subsampling techniques

17 Estimate Σ using Subsampling Data observed over fixed spatial domain S and increasing time domain T n Form overlapping S l(n) subblocks using moving subblock window along time Choice of optimal block length, l(n), in the sense of minimizing the MSE of estimators in a variety of contexts (e.g., Lahiri, 2003) For simplicity we follow Carlstein (1986): l(n) = ( 2γ 1 γ 2 ) 2/3 ( 3n 2 )1/3, where we estimate γ by γ n = Ĉ n (0, 1)/Ĉn(0, 0) but often works well in practice, see Lahiri (2003)

18 Choice of contrast matrices Choice of A is not unique For example: pick only the first two rows of A to form a new test statistic whose asymptotic distribution follows χ 2 2 Although these tests will have approximately the same size, the power depends on the specific choice Generally: preferable to use lags combining small spatial and temporal lags Take features of physical process into consideration while choosing testing lags

19 Outline of this talk Testing properties of space-time covariance function 1. Introduction 2. Development of testing methodology 3. Data analysis and simulations Reconstructing paleoclimate Apply the testing procedures to solve the problem 1. Significance and data 2. Why use BHM framework? 3. Challenges and recipe

20 Irish wind speed data Time series of daily average wind speed at 11 synoptic meteorological stations in Ireland during the period Prevailing westerly wind Gneiting et al.(2007) fit spatio-temporal model C(h, u) = [ { } (1 ν)(1 λ) c h exp 1 + a u 2α (1 + a u 2α ) β/2 + ν 1 ν δ h=0 ] +λ ( 1 1 2ν h 1 vu symmetry parameter λ, space-time interaction parameter β both take values in [0,1] λ = 0: fully symmetric; λ = 0, β = 0: separable Estimates from training data: ˆλ = and ˆβ = ) +

21 Testing full symmetry and separability 5 pairs of stations among the 55 pairs and time lags u = 1 and 2 days 1) prevailing westerly wind suggests choosing the 5 pairs of stations with the smallest ratio of h 2 /h 1 (h 1 WE; h 2 NS) 2) the 5 pairs of stations with the largest ratio of h 2 /h 1 3) the 5 pairs with the shortest spatial distance h among the 55 pairs

22 Malin Head smallest h 2 h 1 largest h 2 h 1 shortest h Belmullet Clones Claremorris Mullingar Birr Dublin Irish wind stations and the five pairs selected for the test Shannon Kilkenny Valentia Roche s Point

23 Testing full symmetry and separability TS1: test statistic for full symmetry TS2: test statistic for separability 1) TS1 = 262.7, p-value = 0; TS2 = 445.2, p-value = 0 reject assumptions of full symmetry and separability 2) TS1 = 20.2, TS2 = ) TS1 = 132.9, TS2 = Results in 1) detect further departure from full symmetry and separability

24 Assessing size and power Empirical sizes and powers for testing full symmetry and separability in the Irish wind speed data λ β Full Symmetry Separability 0 0 size=0.074 [0.067] (0.083) size=0.084 [0.073] (0.091) 0 1 size=0.070 [0.081] (0.092) pow=0.945 [1.000] (0.503) pow=1.000 [0.141] (1.000) pow=1.000 [0.510] (1.000) pow=1.000 [0.309] (1.000) pow=1.000 [0.959] (1.000) pow=0.900[0.095](0.529) pow=0.985 [0.125] (0.445) size=0.095 [0.075] (0.078) pow=0.717 [0.929] (0.306) pow=0.876[0.083] (0.478) pow=1.000[0.941] (0.867) Nominal level is Sizes/powers in [ ] are obtained by using the five station pairs with the largest ratio of h 2 /h 1, and sizes/powers in ( ) are obtained by using the five station pairs with the shortest h.

25 Features of the testing methodology Unified framework for testing a variety of assumptions for covariance functions Can be applied to a wide range of data sets Easy to implement Detection of asymmetric and nonseparable features in Irish wind speed data Simulation experiments show reliability and accuracy

26 Extensions for climate data Test Taylor s hypothesis in precipitation Extend to multivariate random fields: e.g., temperature, precipitation and geopotential heights in climatology Test properties of cross-covariances C ij (k), i, j: index of variates, k: lag vector C ij (k) behaves differently than C(k)

27 Extensions for climate data New definitions of simplications: Multivariate symmetry: C ij (k) = C ji (k) or C ij (k) = C ij ( k) Multivariate Separability: C ij (k) = ρ(k)t for a spatial correlation ρ(k) and covariance matrix T between variates Linear model of coregionalization (LMC): C(k) = r g=1 ρ g (k)t g

28 Outline of this talk Testing properties of space-time covariance function 1. Introduction 2. Development of testing methodology 3. Data analysis and simulations Reconstructing paleoclimate Apply the testing procedures to solve the problem 1. Significance and data 2. Why use BHM framework? 3. Challenges and recipe

29 Our paleoclimate team Caspar Ammann, Doug Nychka: NCAR Gene Wahl: Alfred University Jack Williams: University of Wisconsin-Madison Malcolm Hughes, Michael Evans: University of Arizona Mark Berliner: Ohio State University Robert Harris: Oregon State University

30 Climate change

31 Climate change

32 Climate change

33 The problem and the data Important to our life, understand climate change New research problem for statisticians Observed: Meterological variables (temperature, precipitation, 500hPa geopotential height fields) 1850-present Proxies: Tree rings, Pollens, Ice core, Borehole... past millennium Unobserved: Meteorological variables before 1849

34 Tree rings

35 Ice core

36 Pollen

37 Borehole

38 Proxies Each proxy may be skillful at different time scales Tree ring: decadal Pollen: bi-decadal to semi-centennial Borehole: centennial and onward Goal: Reconstruct the past NH temperature, and NA temperature, precipitation and 500 hpa heights field. Exploit all three sources of proxies Incorporate energy balance and external drivers

39 Bayesian Hierarchical Model (BHM) A natural framework: Three hierarchies: Data Stage: [Proxies Geophysical Process, Parameters] Likelihood of Proxies given underlying process Process Stage: [Geophysical Process Parameters] Physical model of Geophysical Process Parameter Stage: [Parameters] Specify the prior of parameters

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41 Illustration of using BHM One type of proxy, say, 14 tree rings Construct past temperature only Target is northern hemisphere mean temperature, i.e. single time series of temperature No spatial locations of proxy and temperature are considered Data Model: P t T t = β 0 + β 1 T t + e t, e t AR(2)

42 Process Model Radiative forcings: Explosive volcanism, Solar Activity Changes and Anthropogenic forcings Force

43 Process Model µ: Radiative forcings T t µ t = a + bµ t + e t, e t AR(2) Process model is essentially an AR(2) time series as the target is NH mean temperature Bring the knowledge of radiative forcings into the reconstruction

44 Prior β 0(i) N(0, σ 2 0), β 1(i) N(0, σ 2 1) a N(0, σ 2 a), b N(0, σ 2 b ) φ 2ɛ unif( 1, 1), φ 1ɛ φ 2ɛ (1 φ 2ɛ ) unif( 1, 1) φ 2 unif( 1, 1), φ 1 φ 2 (1 φ 2 ) unif( 1, 1) σ 2 P IG(q P, r P ), σ 2 T IG(q T, r T )

45 An example: simulated numerical data Output from global coupled climate model Degrees C pseudo-proxy series sampled from 14 individual grid boxes in the climate model

46 An example of BHM: simulated numerical data Input: 14 pseudo-proxies and full model temperature Reconstruct the past temperature ( ) Degree C

47 Conclusion from the example The posterior mean matches the trend of the numerical data The numerical data is within the 95% prediction band The posterior mean of parameters are close to those directly estimated from numerical data BHM works well in reconstructing the past temperature

48 Apply this simple BHM to 14 real proxies Input: 14 proxies (mostly are tree rings) and the instrumental temperature Reconstruct the past temperature ( ) Degree C

49 Reconstruct real world NH temperature Uncertainty of decadal average and decadal maximum Degree C

50 Current work Data Stage: [Proxies Temperature field, Parameters] Some issues: Proxies {}}{ TreeRing, Pollen, Borehole 1. dating errors of pollen and borehole 2. forward algorithm to convert the surface temperature to pollen assemblage and borehole depth temperature Challenge in process stage: spatio-temporal model of temperature random field

51 Current work Take advantage from long numerical simulations of climate model Apply the unified testing procedures to determine an appropriate space-time covariance function Spatially isotropic? Space-time symmetric? Space-time separable?... Hope all the answers are Yes!

52 Future work Data Stage: [Proxies Geophysical process, Parameters] Geophysical process {}}{ Temperature, Precipitation, 500hPa heights Need in process stage: A multivariate space-time geophysical process model Guide to choose this model: Multivariate symmetric? Multivariate separable? If unfortunately LMC is needed, how many components are necessary?

53 Summary Unified framework for testing a variety of assumptions for covariance functions of stationary spatiotemporal random fields Bayesian hierarchical models (BHM) to reconstruct the past Northern Hemisphere temperatures Building BHM for reconstruction of multivariate spacetime climate random field based on testing procedures

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55 References for this talk Li, B., Genton, M. G. and Sherman, M., A nonparametric assesment of properties of space-time covariance functions. (2007) JASA, Vol. 102, Li, B., Nychka, W. D. and Ammann, C. M., The Hockey Stick and the 1990s: A statistical perspective on reconstructing hemispheric temperatures. (2007), Tellus, Vol. 59, Li, B., Genton, M. G. and Sherman, M., On the asymptotic joint distribution of sample space-time covariance estimators. (2008) Bernoulli, in press Li, B., Genton, M. G. and Sherman, M., Testing the covariance structure of multivariate random fields, under review by Biometrika Li, B., Murthi, A., Bowman, K. P., North, G. R., Genton, M. G.,and Sherman, M., Statistical tests of Taylor s hypothesis: An application to precipitation fields. (2007) under review by Journal of Hydrometeorology