What s for today Continue to discuss about nonstationary models Moving windows Convolution model Weighted stationary model c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 1 / 23
Nonstationary covariance models If you suspect the process is nonstationary, you need to have certain ideas about the types of nonstationarity For example, as in the example from the previous lecture, you may see if the covariance has dependence on latitudes (or certain coordinates) or whether the fact that the domain is land or sea Even if you know the types of nonstationarity, it is often hard to come up with valid classes of nonstationary covariance functions This is why I started from a process in the previous example, not covariance functions directly c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 2 / 23
An example of nonstationary covariance function An interesting application problem is the study of point source problem for air pollution The model in Hughes-Oliver et al. (1998a), Statistics & Probability Letters: Cor(Z(s i,s j )) = exp( θ 1 s i s j exp (θ 2 c i c j + θ 3 min[c i,c j ])) where c i = s i c and c j = s j c are the distances between sites and the point source c Note that if θ 2 = θ 3 = 0, then it reduces to an isotropic model What is the range of this model in general? c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 3 / 23
An example of nonstationary covariance function It is difficult to show that this is a valid model What should you do then? c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 4 / 23
Another example of nonstationary covariance model Let s consider another simple nonstationary model that was considered in the previous lecture For example, suppose Z 0 (s) is an isotropic process and we consider Z(s) = a(s)z 0 (s) which has nonstationary covariance function What is the condition on a required to have a valid nonstationary covariance model? Note that the covariance function of Z is nonstationary but the correlation function of Z is not a(s 1 )a(s 2 )Cov(Z 0 (s 1 ), Z 0 (s 2 )) Cor(Z(s 1 ), Z(s 2 )) = a(s 1 )a(s 2 ) Var(Z 0 (s 1 ))Var(Z 0 (s 1 )) It is in general harder to develop nonstationary correlation model, since nonstationary covariance does not necessarily mean nonstationary correlation c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 5 / 23
A note on nonstationarity You may have nonstationary process purely due to the mean part if the mean changes over space In this class, we focus on nonstationarity in the covariance structure since dealing with nonstationarity in the mean is fairly easier However, you need to have good model for the mean so that you can focus on correctly modeling the covariance (practically with the residuals) c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 6 / 23
Local stationarity Even if the process is nonstationary in general, in some situations it may be reasonable to assume that 1 the process is stationary in smaller subregions 2 the process can be decomposed into simpler, stationary processes There are generally three approaches for local stationarity: 1 moving windows 2 convolutions 3 weighted stationary processes c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 7 / 23
Moving windows The basic idea is to split the spatial region into smaller subregions and fit (isotropic) covariance models in the subregions separately Then you have sort of nonstationary model since your parameter values are different in different subregions The number of the windows as well as the size and shape of the windows should be determined This approach can save some computational time in the sense that you only fit small number of data c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 8 / 23
Moving windows Suppose our model is Z(s) = µ1 + e(s) and e (0,Σ(θ)) The ordinary kriging predictor at a new location s 0 given the observations at the locations s i, i = 1,,n is given by where Z = (Z(s 1 ),,Z(s n )) T Ẑ(s 0 ) = ˆµ + c T Σ 1 (Z µ1) It may be the case that Z at some location s j has little impact on the kriging predictor at the location s 0 if s j is far away from s 0 Using the moving windows idea, you can get the kriging predictor on the local neighborhood c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 9 / 23
Convolution method We start from a mean zero white noise X(s) such that E(X(s)) = 0, Var(X(s)) = σ 2, and Cov(X(s 1 ),X(s 2 )) = 0 if i j Then we can construct a stationary random field Z(s) by convolving X(s) with a kernel function K s (u) centered at s such that Z(s) = K s (u)x(u)du all u What are the mean and the covariance of Z? Another way of looking at it in terms of summation: c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 10 / 23
Convolution method Note that the covariance function depends on the choice of the kernel Here are some ways of choosing the kernel function Isotropic convolution: K s (u) = (2πσ 2 ) 1 exp ( 0.5((u x s x ) 2 + (u y s y ) 2 )) Note that the result of the convolution is a stationary, isotropic random field with Gaussian covariance structure Anisotropic convolution: K s (u) = A 1/2 (2π) 1 exp ( 0.5(u s) T A 1 (u s)) where A is a positive definite (but not identity) matrix This creates a geometrically anisotropic Gaussian covariance structure Nonstationary convolution: K s (u) = A(s) 1/2 (2π) 1 exp ( 0.5(u s) T A(s) 1 (u s)) This creates nonstationary covariance function. However, you have so many parameters to estimate c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 11 / 23
More on convolution method This idea mainly started from the paper, Higdon, Swall, and Kern (1999), Bayesian statistics There is also a nice paper that introduces the method and has an R code (Higdon, 2002, Quantitative methods for current environmental issues c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 12 / 23
More on convolution method As we ve seen, the convolution method also applies for isotropic or stationary model For stationary model, we can write Z(s) = K(u s)x(u)du all u Then the resulting covariance function is given by Cov(Z(s),Z(t)) = K(u (s t))k(u)du c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 13 / 23
Kernels and resulting covariance function (picture taken from Higdon (2002)) c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 14 / 23
Nonstationary kernels (picture taken from Chris Paciorek s website) c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 15 / 23
Higdon (1998), Environmental and Ecological Statistics The paper is on A process-convolution approach to modelling temperatures in the North Atlantic Ocean (note this paper deals with space-time data) First define a grid process x = (x 1,,x m ) on (w 1,τ 1 ),,(w m,τ m ) For x j independent N(µ x,σx 2 ), we let m Z(s,t) = K s (s w j,t τ j ) x j j=1 Here K s ( s, t) = C S ( s) R( t) For geometrically anisotropic Gaussian kernels on (w 1,τ 1 ),,(w m,τ m ), C lk, m we let C s ( s) = w k (s)c lk ( S) and w k (s) exp ( 1 2 s l k 2 ) k=1 c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 16 / 23
Data, Higdon (1998) c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 17 / 23
Grid locations, Higdon (1998) c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 18 / 23
Estimated mean, Higdon (1998) c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 19 / 23
Weighted stationary processes We write a nonstationary process Z(s) as the weighted sum of stationary processes Z 1 (s),,z k (s): Z(s) = k w i (s)z i (s) i=1 We assume Z i s are uncorrelated In practice, we assume Z i s have same covariance function with different parameter values You can check the covariance of Z(s) is a weighed sum of the covariances of Z i c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 20 / 23
Fuentes (2001), Environmetrics c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 21 / 23
Fuentes (2001), Environmetrics c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 22 / 23