c173c273_lec1_w11[1]

c173c273_lec1_w11[1] - University of California Los Angeles...

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University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou Introduction What is geostatistics? Geostatistics is concerned with estimation and prediction for spatially continuous phe- nomena, using data obtained at a limited number of spatial locations. Here, with phenomena we mean the distribution in a two- or three-dimensional space of one or more random variables called regionalized variables . The phenomenon for which the regionalized variables are referred to it is called regionalization . For example, the dis- tribution of mineral ore grades in the three-dimensional space. Or the distribution of ozone, etc. History: The term geostatistics was coined by Georges Matheron (1962). Matheron and his colleagues (at Fontainebleau, France) used this term in prediction for problems in the mining industry. The prefix “geo” concerns data related to earth. Today, geostatistical methods are applied in many areas beyond mining such as soil science, epidemiology, ecology, forestry, meteorology, astronomy, corps science, envi- ronmental sciences, and in general where data are collected at geographical locations (spatial locations). The spatial locations throughout the course will be denoted with s 1 , s 2 , · · · , s n and the spatial data collected at these locations will be denoted with z ( s 1 ) , z ( s 2 ) , · · · , z ( s n ). Spatial locations are determined by their coordinates ( x, y ). We will mainly focus in two-dimensional space data. Very important in the analysis of spatial data is the distance between the data points. We will use mostly Euclidean distances. Suppose data point s i has coordinates ( x i , y i ) and data point s j has coordinates ( x j , y j ). The Euclidean distance between points s i and s j is given by: d ij = q ( x i - x j ) 2 + ( y i - y j ) 2 Other forms of distances can be used (great-circle distance, azimuth distance, travel distance from point to point, time needed to get from point to point, etc.). 1
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The problem: - Present and explain the distribution of the random function Z ( s ) : s D - Predict the value of the function Z ( s ) at spatial location s 0 (in other words the value z ( s 0 )) using the observed data vector z ( s 1 ) , z ( s 2 ) , · · · , z ( s n ) (see figure be- low). 62 64 66 68 70 72 74 128 130 132 134 136 138 140 x coordinate y coordinate s0 s1 s2 s3 s4 s5 s6 s7 - Environmental protection agencies set maximum thresholds for harmful substances in the soil, atmosphere, and water. Therefore given the data we should also like to know the probabilities that the true values exceed these thresholds at unsampled locations. A random function Z ( s ) can be seen as a set of random variables Z ( s i ) defined at each point s i of the random field D : Z ( s ) = Z ( s i ) , s i D . These random variables are correlated and this correlation depends on the vector h that separates two points s and s + h , the direction (south-north, east-west, etc.), but also on the nature of the variables considered. The data can be thought as a realization of the function Z ( s ) with s
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