What s for today. Random Fields Autocovariance Stationarity, Isotropy. c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13

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1 What s for today Random Fields Autocovariance Stationarity, Isotropy c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13

2 Stochastic Process and Random Fields A stochastic process is a family or collection of random variables and the members of it can be identified or indexed according to some metric Example: a time series X(t), t = t 1,,t n We call a spatial process, Z(s), s D, D R d, a random field Typically d = 2 but d can be greater than 2 In the same way, we can define a spatial-temporal process (or random field), Z(s,t), s D, t R c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13

3 Autocovariance The covariance function of Z(s) is defined as K(s 1,s 2 ) = Cov{Z(s 1 ),Z(s 2 )} = E[{Z(s 1 ) µ(s 1 )}{Z(s 2 ) µ(s 2 )}] if µ(s) = E{Z(s)} It is called autocovariance since it is covariance of itself (you have sample size 1!) c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13

4 Autocovariance Suppose you have the following data What is the domain of your spatial process? How do you calculate covariance between the two locations? c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13

5 Autocovariance With one sample, to estimate autocovariance, we make certain assumptions to the covariance structure. Examples: (Weak) stationarity: K(s 1,s 2 ) = K 1 (s 1 s 2 ) for a valid covariance function K 1 isotropy: K(s 1,s 2 ) = K 2 ( s 1 s 2 ) for a valid covariance function K 2 There are certain requirements that a covariance function should satisfy. We will discuss this soon. c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13

6 Why autocovariance matters? (a) (b) Both (a) and (b) give random fields on the domain [0,10] [0,10] but in (a), every point is independent and in (b), nearby points have correlations In (a), if a pixel is missing, what would be the best guess for that missing pixel? How about (b)? c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13

7 Why autocovariance matters? This lead to the concept of Kriging Kriging is another name of the Best Linear Unbiased Prediction (BLUP), and so your predicted value at a new location will be a linear combination of your observations In determining the coefficients of the linear combination, your autocovariance plays an important role. If there is no correlation among all the observations, what should be the coefficients of the linear combination? c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13

8 Covariance function of a random field For a real random field Z on D with E{Z(s) 2 } < for all s D, the covariance function K(s 1,s 2 ) = Cov{Z(s 1 ),Z(s 2 )} should satisfy n c j c k K(s j,s k ) 0 j,k=1 for all finite n, all s 1,,s n D and all real c 1,,c n. Where does this condition come from? We call such function K nonnegative definite (or positive semi-definite). Nonnegative definiteness is the basic requirement for a function to be a valid covariance function In many cases, we may want positive definite covariance functions c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13

9 Positive definite function If K 1 and K 2 are p.d., then a 1 K 1 + a 2 K 2 is p.d. for all a 1,a 2 0 If K 1,K 2, are p.d. and lim n K n (s) = K(s) for all s D, then K is p.d. If K 1 and K 2 are p.d. then K( ) = K 1 ( )K 2 ( ) is p.d. c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13

10 Stationarity We need to make certain assumptions to overcome the one sample problem A random field Z is called weakly stationary if it has finite second moments, its mean function is constant and its covariance function satisfies the following: Cov{Z(s 1 ),Z(s 2 )} = K(s 1 s 2 ) That is, stationarity means the covariance is translation invariant Sometimes we assume strict stationarity, which requires the probability distribution to be translation invariant c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13

11 Isotropy A random field Z is called weakly isotropic if it has finite second moments, its mean function is constant and its covariance function satisfies the following: Cov{Z(s 1 ),Z(s 2 )} = K( s 1 s 2 ) That is, isotropy means that the covariance is translation and rotation invariant Note that if a random field is isotropic, it is stationary We can define strict isotropy in a similar way to strict stationarity c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13

12 Anisotropy We say a random field is anisotropic if it is not isotropic. However, there is a special kind of anisotropy. If a process is isotropic after a linear transformation of coordinates, we say the random field is geometrically anisotropic. That is, for a nonsingular matrix V, if Z(Vs) is isotropic, we say Z is geometrically anisotropic. c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13

13 Property of autocovariance Suppose Z is weakly stationary on R d with autocovariance function K. Then, K should satisfy 1 K(0) 0 2 K(s) = K( s) 3 K(s) K(0) How do you derive 1-3? c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13