Paper Review: NONSTATIONARY COVARIANCE MODELS FOR GLOBAL DATA

Transcription

1 Paper Review: NONSTATIONARY COVARIANCE MODELS FOR GLOBAL DATA BY MIKYOUNG JUN AND MICHAEL L. STEIN Presented by Sungkyu Jung April, 2009

2 Outline 1 Introduction 2 Covariance Models 3 Application: Level 3 TOMS data

3 Introduction Global data Spatial data covering a large portion of the Earth Such data show non-stationary covariance structure on a global scale. Figure: Level 3 Total Ozone Mapping Spectrometer (TOMS) day on a particular date on regular grid. (longitude 180 to 180, latitude 50 to 50 )

4 Introduction Focus of the paper 1. Introduce a flexible class of parametric models with properties: Spatial process covers the globe. Covariance model captures nonstationary. 2. Compare different models by criterion: Maximized Likelihood Values Comparison between empirical data and fitted values.

5 Introduction Distances on Earth The distance between two locations on the Earth, d[(l 1, l 1 ), (L 2, l 2 )], where L denotes the latitude, l denotes the longitude. Chordal distance Euclidean distance between two locations. Great Circle distance Length of the shortest path between two locations. Chordal distance is used in this paper.

6 Covariance Models Stationary (or homogeneous) model Let Z 0 (L, l) be a spatial process on a sphere. Z 0 is homogeneous if its covariance function K 0 only depends on the distance. i.e. Cov(Z 0 (L 1, l 1 ), Z 0 (L 2, l 2 )) = K 0 (d[(l 1, l 1 ), (L 2, l 2 )]). Example: Matérn covariance function: K 0 (d[(l 1, l 1 ), (L 2, l 2 )]) = α(d/β) ν K ν (d/β)

7 Covariance Models Axially symmetric model Z is called axially symmetric if Z is stationary with respect to longitude, i.e. Cov(Z(L 1, l 1 ), Z(L 2, l 2 )) = K(L 1, L 2, l 1 l 2 ). Figure: Empirical standard deviations along longitude and latitude (TOMS). Standard deviations varies along latitude.

8 Covariance Models Axially symmetric model + nugget effect Let Z 0 be a homogeneous process and K 0 be its covariance function. The simplest model for axially symmetric Z is a rescaled version of Z 0 plus a nugget effect (for measurement error), i.e. Z(L, l) = P (L)Z 0 (L, l) + ψ(l, l), where ψ is a white noise process, and P (L) is a linear combination of Legendre polynomials (will be introduced in the next slide).

9 Covariance Models Legendre Polynomials A class of orthogonal polynomials suited for functions on spheres. Figure: Graphs of nth order Legendre polynomials. (Source: wikipedia.org) Define P (L; p 0,..., p m ) = m i=0 p ip i (sin L), where P i denotes the Legendre polynomials or order i.

10 Covariance Models Nonstationary cov. models through Differential Operators Rescaled Z 0, Z, has the same covariance structure for different latitudes (up to scale). Define a process Z by applying differential operators; Z(L, l) = { A(L) L + B(L) l } Z 0 (L, l), (1) where A(L), B(L) are a linear combination of Legendre polynomials, Z 0 is a homogenous process with the covariance function K 0. The covariance function of this can be written in terms of derivatives of K. (Stein (2005)) That A,B only depending on L (latitude) makes Z axially symmetric. (Jun and Stein (2007)) Z has different covariance structure along different latitudes.

11 Covariance Models General axially symmetric model for TOMS data First, homogeneous Matern model: K 0 (d[(l 1, l 1 ), (L 2, l 2 )]; α, β, ν) = α(d/β) ν K ν (d/β), Then, rescaled axially symmetric model: K 1 (L 1, L 2, l; α, β, ν, ɛ, k 0,..., k m ) =P (L 1 ; k 0,..., k m )P (L 2 ; k 0,..., k m )K 0 (d; α, β, ν) + ɛ1 (L1 L 2 =l=0) Then add a covariance function of model (1): K 2 (L 1, L 2, l; α, β, ν, ɛ, k 0,..., k m, α 1, β 1, ν 1, a 0,..., a n1, b 0,..., b n2 ) =K Z (L 1, L 2, l; α 1, β 1, ν 1, a 0,..., a n1, b 0,..., b n2 ) + K 1 (L 1, L 2, l; α, β, ν, ɛ, k 0,..., k m ) (2)

12 Application: Level 3 TOMS data Application: Level 3 TOMS data Level 3 TOMS data gives daily total column ozone levels, spatially gridded, focus on particular dates: May 14-15, 1990

13 Application: Level 3 TOMS data Modeling Mean Structure + Tapering Subtract mean structure, and taper the residual.

14 Application: Level 3 TOMS data Covariance Model Comparisons: Maximized LKHD Various non-stationary covariance models (2) fitted to the tapered residuals.

15 Application: Level 3 TOMS data Covariance Model Comparisons: Graphics a) standard deviations along different longitude b) standard deviations along different latitude c) standard deviations of lag-1 difference along different latitude

16 Application: Level 3 TOMS data Summary & Discussion Summary Axially symmetric spatial process on the Earth Compared various covariance models informally on TOMS data Discussion Need more formal criterion (AIC, BIC) to penalize the #parameters Can be extended to spatial-temporal process naturally