Mean square continuity

Transcription

1 Mean square continuity Suppose Z is a random field on R d We say Z is mean square continuous at s if lim E{Z(x) x s Z(s)}2 = 0 If Z is stationary, Z is mean square continuous at s if and only if K is continuous at the origin It can be shown that a stationary random field is either mean square continuous everywhere or nowhere (HW2 for only statistics students, due September 17) Mikyoung Jun (Texas A&M) stat647 lecture 4 September 10, / 14

2 Mean square differentiability A process Z on R with finite second moments is mean square differentiable at x if {Z(x + h n ) Z(x)}/h n converges in L 2 for all sequences {h n } converging to 0 as n If Z is stationary, we can show that Z is mean square differentiable if and only if K (0) exists and is finite For a stationary process Z, Z is m-times mean square differentiable if and only if K (2m) (0) exists and is finite and the autocovariance function of Z (m) is ( 1) m K (2m) Mikyoung Jun (Texas A&M) stat647 lecture 4 September 10, / 14

3 Mean square differentiability If Z is stationary, it is either mean square differentiable everywhere or nowhere Mean square differentiability and the smoothness of its realizations have no relationship! Mikyoung Jun (Texas A&M) stat647 lecture 4 September 10, / 14

4 Bochner s Theorem A complex valued function K on R d is the autocovariance function for a weakly stationary mean square continuous complex-valued random field on R d if and only if it can be represented as K(x) = exp (iw T x)f (dw) R d where F is a positive finite measure. Mikyoung Jun (Texas A&M) stat647 lecture 4 September 10, / 14

5 Implication of Bochner s Theorem If Z is a real valued stationary mean square continuous random field on R d, its covariance function K should have a positive spectral density f The spectral density f can be obtained by f (w) = 1 (2π) d exp ( iw T x)k(x)dx R d On the other hand, if we have a positive spectral density f, we obtain the covariance function from K(x) = exp (iw T x)f (w)dw R d Mikyoung Jun (Texas A&M) stat647 lecture 4 September 10, / 14

6 Examples of spectral densities on R For φ, α > 0, if f (w) = φ(α 2 + w 2 ) 1, then K(x) = πφα 1 e αx, x > 0 f (w) = φ(α 2 + w 2 ) 2 gives K(x) = 1 2 πφα 3 e αx (1 αx) A Gaussian model, K(x) = ce αx2 has the spectral density, f (w) = 1 2 c(πα) 1/2 e w 2 /(4α) The triangular model (K(x) = c(a x) + ) has the spectral density of the form f (w) = cπ 1 {1 cos (aw)}/w 2 A spectral density, K(x) = ( 4ν ρ 2 σ 2 ν 1 Γ(ν) ( 2ν1/2 x ρ where c(ν, ρ) = Γ(ν+ d 2 )(4ν)ν π d/2 Γ(ν)ρ 2ν σc(ν, ρ), gives a Matérn class, + u 2 ) ν+d/2 ) ν K ν ( 2ν1/2 x ρ ) (another parameterization!) Mikyoung Jun (Texas A&M) stat647 lecture 4 September 10, / 14

7 Point process A point process {Z(s : s D R 2 } consists of a pattern of points in the random set D Bernoulli and Binomial process: If a single event s is distributed in D such that P(s A) = ν(a)/ν(d) for all sets A D, where ν(a) gives the area of the set A, then we call the process a Bernoulli process If n Bernoulli processes are superposed to form a process of n events in D, we call the resulting process a Binomial point process If N(A) denotes the number of events in the set A D, then for a Binomial process, N(A) is a Binomial random variable with sample size N(D) and success probability π(a) = ν(a)/ν(d) Mikyoung Jun (Texas A&M) stat647 lecture 4 September 10, / 14

8 Point process The intensity λ(s) is the average number of events per unit area We define λ(s) = E{N(ds)} lim ν(ds) 0 ν(ds) If the intensity does not change with spatial location, we say the process is homogeneous Binomial process is a homogeneous process Mikyoung Jun (Texas A&M) stat647 lecture 4 September 10, / 14

9 Poisson process If N(A) Poisson(λν(A)) (0 < λ < ) and if N(A 1 ) and N(A 2 ) are independent for disjoint subregions of D, A 1 and A 2, then we say the process is a homogeneous Poisson process If the intensity λ varies over space, then we say the process is inhomogeneous Poisson process: N(A) Poisson(λ(A)) where λ(a) = A λ(s)ds Mikyoung Jun (Texas A&M) stat647 lecture 4 September 10, / 14

10 Examples of point process Poisson proc (lambda=29) Inhomogemeous Poisson proc redwood cells Mikyoung Jun (Texas A&M) stat647 lecture 4 September 10, / 14

11 Second-order properties of point patterns Second-order intensity function is defined as λ 2 (s i, s j ) = E{N(ds i )N(ds j )} lim ds i 0, ds j 0 ds i ds j A point process is stationary if λ 2 (s i, s j ) = λ 2 (s i s j ) Isotropy should be defined in the obvious way Mikyoung Jun (Texas A&M) stat647 lecture 4 September 10, / 14

12 Estimation of the intensity function Suppose k is a kernel function Kernel function is of a similar shape to covariance functions Kernel function is usually nonnegative and has largest mass in the center (origin) Examples of kernel functions: Gaussian function k(x) = 1 ( x h) k(x) = 0.75(1 x 2 )1 ( x 1) We use a kernel to estimate the intensity function in R: In R 2, we may do: ˆλ(s 0 ) = 1 ν(a)h n i=1 k( s i s 0 ) h ˆλ(s 0 ) = 1 ν(a)h x h y n i=1 k( x i x 0 h x )k( y i y 0 h y ) Mikyoung Jun (Texas A&M) stat647 lecture 4 September 10, / 14

13 Examples of estimated intensity functions Use R package spatstat cells redwood Mikyoung Jun (Texas A&M) stat647 lecture 4 September 10, / 14

14 Some more data for team projects NHDPlus Case Study: precipitation, temperature etc data over Sabine River Soil Pollution data near Factory in Spelter, West Virginia Mikyoung Jun (Texas A&M) stat647 lecture 4 September 10, / 14