c173_c273_lec2_w11[1]

c173_c273_lec2_w11[1] - University of California, Los...

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University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou Spatial statistics Why spatial statistics? Noel Cressie (“Statistics for Spatial Data”) writes “why, how, when” are not enough. We need to add “where”. Today, spatial statistics models appear in areas such as mining, geology, hydrology, ecology, environmental science, medicine, image processing, crop science, epidemiology, forestry, atmospheric science, etc. Need to develop models that deal with data collected from different spatial locations. The basic components are the spatial locations { s 1 ,s 2 , ··· n } and the data observed at these locations denoted as { Z ( s 1 ) ,Z ( s 2 ) , ( s n ) } . The distance between the observations is important in analyzing spatial data. With distance we mostly mean “Euclidean distance”. However there are other forms of distances (e.g. road miles, travel time, etc.). The latter is modeled through multidi- mensional scaling. Here we will consider mostly (if not always) Euclidean distances. Consider the following example taught in all introductory statistics courses: Let the spatial data Z ( s 1 ) ( s 2 ) , ( s n ) be an i.i.d. sample from N ( μ,σ 0 ). The MVUE of μ is ¯ Z = 1 n n X i =1 Z ( s i ) We know that ¯ Z N ( μ, σ 0 n ), and therefore we can easily construct a 95% confidence interval for μ as follows: ¯ Z ± 1 . 96 σ 0 n The previous example assumes an i.i.d. sample. This can be too simplistic for spatial data. A more realistic assumption is that the data exhibit some spatial correlation. Suppose this spatial correlation is represented through the covariance function cov ( Z ( s i ) ( s j )) = σ 2 0 ρ | i - j | In the i.i.d. case cov ( Z ( s i ) ( s j )) = 0 (independent therefore the covariance is zero). 1
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What is the confidence interval for the non-i.i.d. case? Find the variance of ¯ Z first: var ( ¯ Z ) = var 1 n n X i =1 Z ( s i ) ! = 1 n 2 n X i =1 n X j =1 cov ( Z ( s i ) ,Z ( s j )) Or after some simplification ... var ( ¯ Z ) = σ 2 0 n 1 + 2 ρ 1 - ρ ! ± 1 - 1 n ² - 2 ρ 1 - ρ ! 2 1 - ρ n - 1 n ! Since Z ( s i ) is Gaussian the distribution of ¯ Z is also normal with mean μ and standard deviation the square root of the above expression. Suppose n = 10 and ρ = 0 . 26. For this example we get: var ( ¯ Z ) = σ 2 0 10 (1 . 608) ¯ Z N μ, σ 0 1 . 608 10 ! and a two-sided 95% confidence interval for μ is ¯ Z ± 1 . 96 σ 0 1 . 608 10 or ¯ Z ± 2 . 485 σ 0 10 Conclusion: If we do not realize the presence of spatial correlation in our data and we use ¯ Z ± 1 . 96 σ 0 10 , we obtain a confidence interval that is too narrow . The actual coverage is 87 . 8%, not 95%. Why? Linear models with spatially dependent error term: The classical regression model in matrix form when the error terms are i.i.d. random variables is given by Z = + ±, with var ( ± ) = σ 2 I and the estimation of β is obtained through OLS ˆ β ols = ( X 0 X ) - 1 X 0 Z When the error terms are spatially correlated the above model can be written as Z = + δ, with var ( δ ) = Σ where Σ is the n × n variance-covariance matrix. The form of Σ usually is not known.
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This note was uploaded on 02/11/2012 for the course STATS c173/c273 taught by Professor Nicolaschristou during the Spring '11 term at UCLA.

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c173_c273_lec2_w11[1] - University of California, Los...

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