Title: | Applications of spatial autocorrelation |

Author: | Prematilake, Chalani C. |

Abstract: | As we use time series analysis to study data with respect to their time of occurrence , we can also use spatial statistics to study data with respect to their locations of occurrence in space . Even though the history of spatial probabilistic analysis goes way back to the 18th century (Buffon's needle problem ) , a serious attempt to first study spatial statistics was first made at the beginning of the 20th century (Student , 1907 ) . This study examines measures of overall spatial autocorrelation or association . The word ``autocorrelation" means the correlation of a variable with itself (over time or over space , or both ) . According to Griffith (1987 ) , the quality and quantity of information contained in spatial data is reflected on spatial autocorrelation . For example , in the case of a numerical variable of interest , if most pairs of neighbouring localities have values of the variable of interest both above the average or both below the average , then spatial autocorrelation tends to be ``large" in some way (above certain number ) , while if , on the other hand , for most pairs of neighbouring localities , one locality has a value of the variable above the average and the other one has a value below the average , then the autocorrelation measure tends to be ``small" (below a certain number ) . The study of spatial statistics takes different forms according to the kind of data used . For example , when the data are nominal categorical we can use join counts as measures of spatial association . For example , we can find the number of neighbouring localities that are of the same ``type" (category of the nominal variable ) and the number of neighbouring localities of different ``types ." Moran is one of the first authors who studied join count statistics . He also calculated the moments of join counts in 1948 . Similar studies had been carried out by P . ~V . ~Krishna Iyer in 1949 and 1950 and by Florence Nightingale David in 1971 . They both came out with similar results but in different experimental environments . In the second chapter of the thesis , we review some of the work of Moran on join counts and their moments . Spatial autocorrelation of numerical data is usually carried out using Moran's $I $ coefficient and Geary's ratio $c $ (introduced in 1950 and 1954 , respectively ) . In the third chapter of this thesis , we review some of the probabilistic properties of these spatial autocorrelation coefficients that show how a variable is correlated with itself over space . We use the statistical packages R and SAS to calculate and apply the above statistics to some examples with spatial data . In addition , we show the connection of the join count statistics with Moran's $I $ coefficient and Geary's ratio $c $ , which is probably one of the new contributions in this thesis . Throughout most of the thesis , we show (using modern notation ) the randomization properties of some of the above spatial statistics , that is , we review Moran's and Geary's calculations on the probabilistic behaviour of these statistics by conditioning on the observed data (values of the variable of interest in different localities ) , but not in the order they appear . In other words , we show how statisticians in spatial statistics derive the distribution of some autocorrelation statistics under the non -free sampling scenario . |

URI: | http : / /hdl .handle .net /2346 /ETD -TTU -2011 -08 -1712 |

Date: | 2011-08 |

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