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    Simulation and characterization of multi-class spatial patterns from stochastic point processes of randomness, clustering and regularity
    (Springer, 2014) Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of Sciences; N/A
    Spatial pattern analysis of data from multiple classes (i.e., multi-class data) has important implications. We investigate the resulting patterns when classes are generated from various spatial point processes. Our null pattern is that the nearest neighbor probabilities being proportional to class frequencies in the multi-class setting. In the two-class case, the deviations are mainly in two opposite directions, namely, segregation and association of the classes. But for three or more classes, the classes might exhibit mixed patterns, in which one pair exhibiting segregation, while another pair exhibiting association or complete spatial randomness independence. To detect deviations from the null case, we employ tests based on nearest neighbor contingency tables (NNCTs), as NNCT methods can provide an omnibus test and post-hoc tests after a significant omnibus test in a multi-class setting. In particular, for analyzing these multi-class patterns (mixed or not), we use an omnibus overall test based on NNCTs. After the overall test, the pairwise interactions are analyzed by the post-hoc cell-specific tests based on NNCTs. We propose various parameterizations of the segregation and association alternatives, list some appealing properties of these patterns, and propose three processes for the two-class association pattern. We also consider various clustering and regularity patterns to determine which one(s) cause segregation from or association with a class from a homogeneous Poisson process and from other processes as well. We perform an extensive Monte Carlo simulation study to investigate the newly proposed association patterns and to understand which stochastic processes might result in segregation or association. The methodology is illustrated on two real life data sets from plant ecology.
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    Cell-specific and post-hoc spatial clustering tests based on nearest neighbor contingency tables
    (Korean Statistical Soc, 2017) Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of Sciences; N/A
    Spatial clustering patterns in a multi-class setting such as segregation and association between classes have important implications in various fields, e.g., in ecology, and can be tested using nearest neighbor contingency tables (NNCTs). a NNCT is constructed based on the types of the nearest neighbor (NN) pairs and their frequencies. We survey the cell-specific (or pairwise) and overall segregation tests based on NNCTs in literature and introduce new ones and determine their asymptotic distributions. We demonstrate that cell-specific tests enjoy asymptotic normality, while overall tests have chi-square distributions asymptotically. Some of the overall tests are confounded by the unstable generalized inverse of the rank-deficient covariance matrix. To overcome this problem, we propose rank-based corrections for the overall tests to stabilize their behavior. We also perform an extensive' Monte Carlo simulation study to compare the finite sample performance of the tests in terms of empirical size and power based on the asymptotic and Monte Carlo critical values and determine the tests that have the best size and power performance and are robust to differences in relative abundances (of the classes). in addition to the cell-specific tests, we discuss one(-class)-versus-rest type of tests as post-hoc,tests after a significant overall test. We also introduce the concepts of total, strong, and partial segregatioN/Association to differentiate different levels of these patterns. We compare the new tests with the existing NNCT-tests in literature with simulations and illustrate the tests on an ecological data set. (C) 2016 the Korean Statistical Society. Published by Elsevier B.V. all rights reserved.
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    Comparison of relative density of two random geometric digraph families in testing spatial clustering
    (Springer, 2014) Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of Sciences; N/A
    We compare the performance of relative densities of two parameterized random geometric digraph families called proximity catch digraphs (PCDs) in testing bivariate spatial patterns. These PCD families are proportional edge (PE) and central similarity (CS) PCDs and are defined with proximity regions based on relative positions of data points from two classes. The relative densities of these PCDs were previously used as statistics for testing segregation and association patterns against complete spatial randomness. The relative density of a digraph, D, with n vertices (i.e., with order n) represents the ratio of the number of arcs in D to the number of arcs in the complete symmetric digraph of the same order. When scaled properly, the relative density of a PCD is a U-statistic; hence, it has asymptotic normality by the standard central limit theory of U-statistics. The PE- and CS-PCDs are defined with an expansion parameter that determines the size or measure of the associated proximity regions. In this article, we extend the distribution of the relative density of CS-PCDs for expansion parameter being larger than one, and compare finite sample performance of the tests by Monte Carlo simulations and asymptotic performance by Pitman asymptotic efficiency. We find the optimal expansion parameters of the PCDs for testing each alternative in finite samples and in the limit as the sample size tending to infinity. As a result of our comparisons, we demonstrate that in terms of empirical power (i.e., for finite samples) relative density of CS-PCD has better performance (which occurs for expansion parameter values larger than one) for the segregation alternative, while relative density of PE-PCD has better performance for the association alternative. The methods are also illustrated in a real-life data set from plant ecology.