Statistics for Spatial Functional Data

Transcription

1 Statistics for Spatial Functional Data Marcela Alfaro Córdoba North Carolina State University February 11, 2016

2 INTRODUCTION Based on Statistics for spatial functional data: some recent contributions from Delicado, Giraldo, Comas and Mateu (2010). Disclaimer: Not so recent contributions. Functional data analysis (FDA): Random functions as statistical atoms. (Ramsay and Silverman, 2005) Spatial statistics (SS): geostatistical data, market point pattern, areal data. (Cressie, 1993) Spatial functional process (SFP): {χ s : s D R d } Location s, in the d-dimensional Euclidean space, χs are functional random variables.

3 Definition: when D is a fixed subset of R d with positive volume and n points s 1,..., s n in D are chosen to observe the random functions χ si, i = 1,..., n. We assume: E[χ s (t)] = m(t) and Var[χ s (t)] = σ 2 (t) for all t [a, b]. Cov[χ s (t), χ s+h (u)] = C(h; t, u) for all t, u [a, b] There exists a distance h γ(h; t, u) -Variogram-.

4 Goal: To predict curves at unvisited sites.

5 Approach I: A. Goulard and Voltz (1993) proposed three approaches: Multivariate approach 1: Cokrige first, Fit later Predictor Multivariate approach 2: Fit first, Cokrige later Predictor A Curve Kriging Predictor

6 Cokrige first, Fit later Predictor (CFP) (χ si (t 1 ),..., χ si (t M )) is a M-dimensional r.v. at site s i Cokriging to get (ˆχ s0 (t 1 ),..., ˆχ s0 (t M )) Fit a parametric model χ( ; θ) to reconstruct the whole function at s 0 : χ( ; ˆθ s0 ). Fit first, Cokrige later Predictor (FCP) Fit a parametric model χ( ; θ) to the curves: χ( ; ˆθ si ) p-dimensional (ˆθ si,..., ˆθ sn ) is a multivariate random field Cokriging to get ˆθ s 0 and evaluate in χ( ; ˆθ s 0 )

7 Curve Kriging Predictor (CKP) BLUP for χ s0 is n ˆχ s0 (t) = λ i χ s0 (t), t [a, b], λ 1,..., λ n R i=1 Optimization problem is defined as: min λ 1,...,λ n b a Var(ˆχ s0 (t) χ s0 (t))dt, n s.t. λ i = 1 i=1 Use a parametric model to approximate function χ si ˆχ s0 = n λ i χ( ; ˆθ si ) i=1

8 Approach II: B. Giraldo et al (2008, 2009a, 2009b): Ordinary kriging for functional-valued data Point-wise functional kriging Functional kriging (total model) Cokriging predictor based on functional data

9 Ordinary kriging for functional-valued data: Non-parametric version of the curve kriging predictor (Goulard and Voltz, 1993) Non-parametrically fit the observed functions using FCV to choose the smoothing parameter: SSE FCV = n SSE FCV (i) = i=1 n i=1 M j=1 (ˆχ (i) s i (t j ) χ si (t j )) 2 Use the fitted model to approximate function χ si ˆχ s0 = n λ i ˆχ si i=1

10 Point-wise functional kriging: Coefficients λ i ( ) are functions now ˆχ s0 (t) = n λ i (t)ˆχ si (t) i=1 Optimization problem is defined as: b min λ 1 ( ),...,λ n( ) a Var(ˆχ s0 (t) χ s0 (t))dt, n s.t. λ i (t) = 1 t [a, b] i=1 Solved using K basis functions, B 1 (t),..., B K (t) for both χ si (t) and λ i (t). Choice of K makes this option computationally expensive.

11 Functional kriging (total model): Allow coefficients λ i to be defined in [a, b] [a, b] ˆχ s0 (t) = n b i=1 a λ i (t, v)ˆχ si (v)dv t [a, b], i = 1,..., n Coherent with the functional linear model for functional responses (Ramsay and Silverman, 2005). In a similar way as the previous method: λ i (t, v) = K K c i jl B j(t)b l (v) j=1 l=1

12 Goal: To predict curves at unvisited sites.

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14 Geostatistics for funcional data or space-time geostatistics? One person s deterministic mean structure may be another person s correlated error structure (Cressie, 1993) Author s recommendations: Use space-time geostatistics when the number M of observed values of functional data χ s ( ) is small or when the interest is predicting a specific value χ s0 (t 0 ) for an unvisited site s 0 and/or unobserved time t 0. In other cases, geostatistics for functional data should be the default approach.

15 Other references Dabo-Niang and Yao (2007) propose non-parametric kernel regression with scalar response Y s and functional predictors. The objective is to non-parametrically estimate E(Y s χ s ) taking into account the spatial dependence. Yamanishi and Tanaka (2003) develop a regression model where both response and predictors are functional data, and the relation among variables may change over the space.

16 Other references Baladandayuthapani et al. (2008) show an alternative for analyzing an experimental design with a spatially correlated functional response. They use Bayesian hierarchical models allowing the inclusion of spatial dependence among curves into standard FDA techniques, such as functional multiple regression and functional analysis of variance. Rodriguez et al. (2008, 2009) and Petrone et al. (2008) propose hierarchical models that are extensions of the Dirichlet process mixture of Gaussians.

17 POINT PROCESSES Definition: When a complete function χ si is observed at each point s 1,..., s n in D generated by a standard point process. Goal: Is there spatial dependence in the functional marks?

18 POINT PROCESSES Comas et al. (2008): Functional mark-correlation function. Null hypothesis: no spatial dependence between functional marks Statistic as a function of r = s 1 s 2 : ĝ f (r) = 1 2πrˆλ 2 p W h(χ s1, χ s2 )K( s 1 s 2 r) s 1,s 2 Ψ Ê[h(χ s1, χ s2 )]e(s 1, s 1 s 2 ) h(χ s1, χ s2 ) = b a (χ s1 (t) χ(t))(χ s2 (t) χ(t))dt where χ(t) is the average function over the observed functions.

19 POINT PROCESSES Statistic as a function of r = s 1 s 2 : ĝ f (r) = 1 2πrˆλ 2 p W h(χ s1, χ s2 )K( s 1 s 2 r) s 1,s 2 Ψ Ê[h(χ s1, χ s2 )]e(s 1, s 1 s 2 ) Ψ is the observed point pattern, ˆλ p is an estimator of the point density, K( ) is a kernel function that is non-negative and symmetric wrt the origin, and e( ) is a factor to correct for the edge-effects. ĝ f (r) > 1 pairs of functions at distances r are more similar than the average (+ correlation), ĝ f (r) < 1 corresponds to the opposite and ĝ f (r) = 1 implies spatial independence between functions.

20 POINT PROCESSES

21 AREAL DATA Definition: When D is a fixed and countable set χ si is a summary function of an event happened at area s i. Goals: Detection of spatial dependence, identification of spatial clusters, and modeling the spatial dependence.

22 AREAL DATA Distance-based version of LISA: (Delicado and Broner (2008)) 5 step algorithm to identify spatial clusters based on local tests. For their example (population pyramids) they use Kullback-Leibler divergence as distance: d KL (f i, f j ) = b a log ( ) fi (x) f i (x)dx + f j (x) b a ( ) fj (x) log f j (x)dx f i (x)

23 AREAL DATA

24 SUMMARY (FDA + SS) is a very fertile area of research. Contributions on geostatistics, point processes and areal data with functional observations have been described. As of 2009, geostatistics for functional data was the most developed topic.

25 SOME (RECENT) REFERENCES Kokoszka: Dependent Functional Data (2012) Ramsey et al: Spatial Functional Data Analysis (2011) Staicu et al: Fast Methods for Spatially Correlated Multilevel Functional Data (2010) This is not an exhaustive list!