7a_spatial stats-pt1

7a_spatial stats-pt1 - Lecture outline Advanced Topics in...

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11/21/2007 1 Advanced Topics in Forest Biometrics - FOR6934 Introduction to statistical models for spatial data – Part 1 Lecture outline Spatial data Stationarity Isotropy Semivariogram analysis Why spatial statistics? Spatial statistics first came about in the cartography and surveying Significant contributions have been made by researchers in the mining industry (e.g., D G Krige) Spatial statistics help us to characterize the spatial continuity or roughness of a data set While ordinary (one-dimensional) statistics may be nearly identical for different data sets, their spatial continuity may be quite different A sample size of one? Consider monitoring 100 seedlings planted in a 1-hectare contiguous area This study will give rise to 100 spatially referenced observations: Z ( s i ) ( Z ( s 1 ) , Z ( s 2 ) , … Z ( s 100 ) ) where: s is a vector representing the location of the observations Are the observations independent? Spatial data is a special case of clustered data A set of spatial data is one realization of sampling from a random field (a stochastic process in two-dimensions) We expect observations taken close together to be more related than observations taken far apart Types of spatial data Geostatistical data The domain, D , of the data is a fixed, continuous set Data can be observed at points ( x, y ), where Lattice data Data can only be observed at fixed points in D , i.e., all possible sample locations can be enumerated For example, data by county or city block, biomass harvested in contiguous 1x1 m 2 areas in 100 m 2 Spatial point patterns Data consist of locations where a particular attribute is observed For example, locations of Brazil nut trees in an area 2 ) , ( y x Questions of interest Whereas the Z process is of interest in geostatistical and lattice data, the D process is of interest with spatial point patterns Continuous surface maps are often of interest with geostatistical data Kriging is used to predict Z ( s 0 ) at some unsampled location s 0 The relative randomness or uniformity of pattern is often of interest with spatial point patterns Answering these questions often involves making two assumptions: stationarity and isotropy
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11/21/2007 2 Assumption of Stationarity Stationarity is a property of self-replication of a stochastic process Absolute coordinates are not important – only the distance between points is The stochastic properties of Z ( s ) and Z ( s + h ) are similar The random field looks similar over the domain, D There are several kinds of stationarity Strong (strict) Second-order (weak) Intrinsic Types of Stationarity Strong stationarity the spatial distribution is invariant under translation (rotating or stretching of coordinate system) Second-order (weak) stationarity The mean of the random field is constant and the covariance between observations only depends on the distance between them
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7a_spatial stats-pt1 - Lecture outline Advanced Topics in...

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