arXiv subjects

Elvan Ceyhan

Publications and source records attributed to Elvan Ceyhan.

At least 37 records · Page 2Linked to original sources

arXiv2013

New Cell-Specific and Overall Tests of Spatial Interaction Based on Nearest Neighbor Contingency Tables

Spatial interaction patterns such as segregation and association can be tested using nearest neighbor contingency tables (NNCTs). We introduce new cell-specific (or pairwise) and overall segregation tests and determine their asymptotic distributions. In particular, we demonstrate that cell-specific tests enjoy asymptotic normality, while overall tests have chi-square distributions asymptotically. 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. In addition to the cell-specific tests as post-hoc tests for overall tests, we discuss one-class-versus-rest type of NNCT-tests after an overall test yields significant interaction. We also introduce the concepts of total, strong, and partial segregation/association to label levels of these patterns. We compare these new tests with the existing NNCT-tests in literature with simulations as well and illustrate the NNCT-tests on an ecological data set.

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arXiv2013

Segregation Indices for Disease Clustering

Spatial clustering has important implications in various fields. In particular, disease clustering is of major public concern in epidemiology. In this article, we propose the use of two distance-based segregation indices to test the significance of disease clustering among subjects whose locations are from a homogeneous or an inhomogeneous population. We derive their asymptotic distributions and compare them with other distance-based disease clustering tests in terms of empirical size and power by extensive Monte Carlo simulations. The null pattern we consider is the random labeling (RL) of cases and controls to the given locations. Along this line, we investigate the sensitivity of the size of these tests to the underlying background pattern (e.g., clustered or homogenous) on which the RL is applied, the level of clustering and number of clusters, or differences in relative abundances of the classes. We demonstrate that differences in relative abundance has the highest impact on the empirical sizes of the tests. We also propose various non-RL patterns as alternatives to the RL pattern and assess the empirical power performance of the tests under these alternatives. We illustrate the methods on two real-life examples from epidemiology.

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arXiv2013

Optimal obstacle placement with disambiguations

We introduce the optimal obstacle placement with disambiguations problem wherein the goal is to place true obstacles in an environment cluttered with false obstacles so as to maximize the total traversal length of a navigating agent (NAVA). Prior to the traversal, the NAVA is given location information and probabilistic estimates of each disk-shaped hindrance (hereinafter referred to as disk) being a true obstacle. The NAVA can disambiguate a disk's status only when situated on its boundary. There exists an obstacle placing agent (OPA) that locates obstacles prior to the NAVA's traversal. The goal of the OPA is to place true obstacles in between the clutter in such a way that the NAVA's traversal length is maximized in a game-theoretic sense. We assume the OPA knows the clutter spatial distribution type, but not the exact locations of clutter disks. We analyze the traversal length using repeated measures analysis of variance for various obstacle number, obstacle placing scheme and clutter spatial distribution type combinations in order to identify the optimal combination. Our results indicate that as the clutter becomes more regular (clustered), the NAVA's traversal length gets longer (shorter). On the other hand, the traversal length tends to follow a concave-down trend as the number of obstacles increases. We also provide a case study on a real-world maritime minefield data set.

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arXiv2012

Exploratory and Inferential Methods for Spatio-Temporal Analysis of Residential Fire Clustering in Urban Areas

The spatio-temporal analysis of residential fires could allow decision makers to plan effective resource allocations in fire management according to fire clustering levels in space and time. In this study, we provide guidelines for the use of various methods in detecting the differences in clustering patterns of fire and non-fire (i.e., background residential) locations and how these patterns change over time. As a preliminary analysis step, various exploratory data analysis methods, such as, intensity plots (i.e., kernel density estimates) are used. Moreover, the use of Diggle's D-function (a second order analysis technique) is proposed for detecting the clustering of residential fire locations (if any) and whether there is additional clustering (or regularity) in the locations of the fires compared to background residential pattern. A test for trend over time (in years, months, and weeks) of the fire location patterns are provided with a space-time interaction analysis by spatio-temporal K-function. Residential fire data from Çankaya Municipality of Ankara, Turkey is used as an illustrative example. The presented methodology is also applicable to residential fire data from similar urban settings.

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arXiv2011

Distribution of the Relative Density of Central Similarity Proximity Catch Digraphs Based on One Dimensional Uniform Data

We consider the distribution of a graph invariant of central similarity proximity catch digraphs (PCDs) based on one dimensional data. The central similarity PCDs are also a special type of parameterized random digraph family defined with two parameters, a centrality parameter and an expansion parameter, and for one dimensional data, central similarity PCDs can also be viewed as a type of interval catch digraphs. The graph invariant we consider is the relative density of central similarity PCDs. We prove that relative density of central similarity PCDs is a U-statistic and obtain the asymptotic normality under mild regularity conditions using the central limit theory of U-statistics. For one dimensional uniform data, we provide the asymptotic distribution of the relative density of the central similarity PCDs for the entire ranges of centrality and expansion parameters. Consequently, we determine the optimal parameter values at which the rate of convergence (to normality) is fastest. We also provide the connection with class cover catch digraphs and the extension of central similarity PCDs to higher dimensions.

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arXiv2010

A Comparison of Two Proximity Catch Digraph Families in Testing Spatial Clustering

We consider two parametrized random digraph families, namely, proportional-edge and central similarity proximity catch digraphs (PCDs) and compare the performance of these two PCD families in testing spatial point patterns. These PCD families are based on relative positions of data points from two classes and the relative density of the PCDs is used as a statistic for testing segregation and association against complete spatial randomness. When scaled properly, the relative density of a PCD is a U-statistic. We extend the distribution of the relative density of central similarity PCDs for expansion parameter being larger than one. We compare the asymptotic distribution of the statistic for the two PCD families, using the standard central limit theory of U-statistics. We compare finite sample performance of the tests by Monte Carlo simulations and prove the consistency of the tests under the alternatives. The asymptotic performance of the tests under the alternatives is assessed by Pitman's asymptotic efficiency. We find the optimal expansion parameters of the PCDs for testing each of the segregation and association alternatives in finite samples and in the limit. We demonstrate that in terms of empirical power (i.e., for finite samples) relative density of central similarity PCD has better performance (which occurs for expansion parameter values larger than one) under segregation alternative, while relative density of proportional-edge PCD has better performance under association alternative. The methods are illustrated in a real-life example from plant ecology.

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arXiv2010

The Distribution of the Domination Number of a Family of Random Interval Catch Digraphs

We study a new kind of proximity graphs called proportional-edge proximity catch digraphs (PCDs)in a randomized setting. PCDs are a special kind of random catch digraphs that have been developed recently and have applications in statistical pattern classification and spatial point pattern analysis. PCDs are also a special type of intersection digraphs; and for one-dimensional data, the proportional-edge PCD family is also a family of random interval catch digraphs. We present the exact (and asymptotic) distribution of the domination number of this PCD family for uniform (and non-uniform) data in one dimension. We also provide several extensions of this random catch digraph by relaxing the expansion and centrality parameters, thereby determine the parameters for which the asymptotic distribution is non-degenerate. We observe sudden jumps (from degeneracy to non-degeneracy or from a non-degenerate distribution to another) in the asymptotic distribution of the domination number at certain parameter values.

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arXiv2009

Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family

We use the domination number of a parametrized random digraph family called proportional-edge proximity catch digraphs (PCDs) for testing multivariate spatial point patterns. This digraph family is based on relative positions of data points from various classes. We extend the results on the distribution of the domination number of proportional-edge PCDs, and use the domination number as a statistic for testing segregation and association against complete spatial randomness. We demonstrate that the domination number of the PCD has binomial distribution when size of one class is fixed while the size of the other (whose points constitute the vertices of the digraph) tends to infinity and asymptotic normality when sizes of both classes tend to infinity. We evaluate the finite sample performance of the test by Monte Carlo simulations, prove the consistency of the test under the alternatives, and suggest corrections for the support restriction on the class of points of interest and for small samples. We find the optimal parameters for testing each of the segregation and association alternatives. Furthermore, the methodology discussed in this article is valid for data in higher dimensions also.

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arXiv2009

New Tests of Spatial Segregation Based on Nearest Neighbor Contingency Tables

The spatial clustering of points from two or more classes (or species) has important implications in many fields and may cause the spatial patterns of segregation and association, which are two major types of spatial interaction between the classes. The null patterns we consider are random labeling (RL) and complete spatial randomness (CSR) of points from two or more classes, which is called CSR independence. The segregation and association patterns can be studied using a nearest neighbor contingency table (NNCT) which is constructed using the frequencies of nearest neighbor (NN) types in a contingency table. Among NNCT-tests Pielou's test is liberal the null pattern but Dixon's test has the desired significance level under the RL pattern. We propose three new multivariate clustering tests based on NNCTs. We compare the finite sample performance of these new tests with Pielou's and Dixon's tests and Cuzick & Edward's k-NN tests in terms of empirical size under the null cases and empirical power under various segregation and association alternatives and provide guidelines for using the tests in practice. We demonstrate that the newly proposed NNCT-tests perform relatively well compared to their competitors and illustrate the tests using three example data sets. Furthermore, we compare the NNCT-tests with the second-order methods using these examples.

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arXiv2009

A Comparison of Analysis of Covariate-Adjusted Residuals and Analysis of Covariance

Various methods to control the influence of a covariate on a response variable are compared. In particular, ANOVA with or without homogeneity of variances (HOV) of errors and Kruskal-Wallis (K-W) tests on covariate-adjusted residuals and analysis of covariance (ANCOVA) are compared. Covariate-adjusted residuals are obtained from the overall regression line fit to the entire data set ignoring the treatment levels. It is demonstrated that the methods on covariate-adjusted residuals are only appropriate when the regression lines are parallel and means are equal for treatment factors. Empirical size and power performance of the methods are compared by extensive Monte Carlo simulations. We manipulated the conditions such as assumptions of normality and HOV, sample size, and clustering of the covariates. Guidelines on which method to use for various cases are also provided.

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arXiv2009

Relative Density of the Random r-Factor Proximity Catch Digraph for Testing Spatial Patterns of Segregation and Association (Technical Report)

Statistical pattern classification methods based on data-random graphs were introduced recently. In this approach, a random directed graph is constructed from the data using the relative positions of the data points from various classes. Different random graphs result from different definitions of the proximity region associated with each data point and different graph statistics can be employed for data reduction. The approach used in this article is based on a parameterized family of proximity maps determining an associated family of data-random digraphs. The relative arc density of the digraph is used as the summary statistic, providing an alternative to the domination number employed previously. An important advantage of the relative arc density is that, properly re-scaled, it is a U-statistic, facilitating analytic study of its asymptotic distribution using standard U-statistic central limit theory. The approach is illustrated with an application to the testing of spatial patterns of segregation and association. Knowledge of the asymptotic distribution allows evaluation of the Pitman and Hodges-Lehmann asymptotic efficacy, and selection of the proximity map parameter to optimize efficacy. Notice that the approach presented here also has the advantage of validity for data in any dimension.

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arXiv2009

Relative Edge Density of the Underlying Graphs Based on Proportional-Edge Proximity Catch Digraphs for Testing Bivariate Spatial Patterns (Technical Report)

The use of data-random graphs in statistical testing of spatial patterns is introduced recently. In this approach, a random directed graph is constructed from the data using the relative positions of the points from various classes. Different random graphs result from different definitions of the proximity region associated with each data point and different graph statistics can be employed for pattern testing. The approach used in this article is based on underlying graphs of a family of data-random digraphs which is determined by a family of parameterized proximity maps. The relative edge density of the AND- and OR-underlying graphs is used as the summary statistic, providing an alternative to the relative arc density and domination number of the digraph employed previously. Properly scaled, relative edge density of the underlying graphs is a U-statistic, facilitating analytic study of its asymptotic distribution using standard U-statistic central limit theory. The approach is illustrated with an application to the testing of bivariate spatial clustering patterns of segregation and association. Knowledge of the asymptotic distribution allows evaluation of the Pitman asymptotic efficiency, hence selection of the proximity map parameter to optimize efficiency. Asymptotic efficiency and Monte Carlo simulation analysis indicate that the AND-underlying version is better (in terms of power and efficiency) for the segregation alternative, while the OR-underlying version is better for the association alternative. The approach presented here is also valid for data in higher dimensions.

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arXiv2009

A Probabilistic Characterization of Random Proximity Catch Digraphs and the Associated Tools

Proximity catch digraphs (PCDs) are based on proximity maps which yield proximity regions and are special types of proximity graphs. PCDs are based on the relative allocation of points from two or more classes in a region of interest and have applications in various fields. In this article, we provide auxiliary tools for and various characterizations of PCDs based on their probabilistic behavior. We consider the cases in which the vertices of the PCDs come from uniform and non-uniform distributions in the region of interest. We also provide some of the newly defined proximity maps as illustrative examples.

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arXiv2009

Extension of One-Dimensional Proximity Regions to Higher Dimensions

Proximity maps and regions are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in various fields. The simplest of such maps is the spherical proximity map which maps a point from the class of interest to a disk centered at the same point with radius being the distance to the closest point from the other class in the region. The spherical proximity map gave rise to class cover catch digraph (CCCD) which was applied to pattern classification. Furthermore for uniform data on the real line, the exact and asymptotic distribution of the domination number of CCCDs were analytically available. In this article, we determine some appealing properties of the spherical proximity map in compact intervals on the real line and use these properties as a guideline for defining new proximity maps in higher dimensions. Delaunay triangulation is used to partition the region of interest in higher dimensions. Furthermore, we introduce the auxiliary tools used for the construction of the new proximity maps, as well as some related concepts that will be used in the investigation and comparison of them and the resulting graphs. We characterize the geometry invariance of PCDs for uniform data. We also provide some newly defined proximity maps in higher dimensions as illustrative examples.

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arXiv2009

Directional Clustering Tests Based on Nearest Neighbor Contingency Tables

Spatial interaction between two or more classes or species has important implications in various fields and causes multivariate patterns such as segregation or association. Segregation occurs when members of a class or species are more likely to be found near members of the same class or conspecifics; while association occurs when members of a class or species are more likely to be found near members of another class or species. The null patterns considered are random labeling (RL) and complete spatial randomness (CSR) of points from two or more classes, which is called \emph{CSR independence}, henceforth. The clustering tests based on nearest neighbor contingency tables (NNCTs) that are in use in literature are two-sided tests. In this article, we consider the directional (i.e., one-sided) versions of the cell-specific NNCT-tests and introduce new directional NNCT-tests for the two-class case. We analyze the distributional properties; compare the empirical significant levels and empirical power estimates of the tests using extensive Monte Carlo simulations. We demonstrate that the new directional tests have comparable performance with the currently available NNCT-tests in terms of empirical size and power. We use four example data sets for illustrative purposes and provide guidelines for using these NNCT-tests.

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arXiv2008

The Distribution of the Domination Number of Class Cover Catch Digraphs for Non-uniform One-dimensional Data

For two or more classes of points in $\R^d$ with $d \ge 1$, the class cover catch digraphs (CCCDs) can be constructed using the relative positions of the points from one class with respect to the points from the other class. The CCCDs were introduced by (Priebe, DeVinney, and Mar-chette, (2001). On the distribution of the domination number of random class catch cover di-graphs. Statistics and Probability Letters, 55:239-246) who investigated the case of two classes, $\X$ and $\Y$. They calculated the exact (finite sample) distribution of the domination number of the CCCDs based on $\X$ points relative to $\Y$ points both of which were uniformly distri-buted on a bounded interval. We investigate the distribution of the domination number of the CCCDs based on data from non-uniform $\X$ points on an interval with end points from $\Y$. Then we extend these calculations for multiple $\Y$ points on bounded intervals.

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arXiv2008

Class-Specific Tests of Spatial Segregation Based on Nearest Neighbor Contingency Tables

The spatial interaction between two or more classes (or species) has important consequences in many fields and might cause multivariate clustering patterns such as segregation or association. The spatial pattern of segregation occurs when members of a class tend to be found near members of the same class (i.e., conspecifics), while association occurs when members of a class tend to be found near members of the other class or classes. These patterns can be tested using a nearest neighbor contingency table (NNCT). The null hypothesis is randomness in the nearest neighbor (NN) structure, which may result from -- among other patterns -- random labeling (RL) or complete spatial randomness (CSR) of points from two or more classes (which is called the CSR independence, henceforth). In this article, we consider Dixon's class-specific tests of segregation and introduce a new class-specific test, which is a new decomposition of Dixon's overall chi-square segregation statistic. We demonstrate that the tests we consider provide information on different aspects of the spatial interaction between the classes and they are conditional under the CSR independence pattern, but not under the RL pattern. We analyze the distributional properties and prove the consistency of these tests; compare the empirical significant levels (Type I error rates) and empirical power estimates of the tests using Monte Carlo simulations. We demonstrate that the new class-specific tests also have comparable performance with the currently available tests based on NNCTs in terms of Type I error and power estimates. For illustrative purposes, we use three example data sets. We also provide guidelines for using these tests.

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arXiv2008

Analysis of Metric Distances and Volumes of Hippocampi Indicates Different Morphometric Changes over Time in Dementia of Alzheimer Type and Nondemented Subjects

In this article, we analyze the morphometry of hippocampus in subjects with very mild dementia of Alzheimer's type (DAT) and nondemented controls and how it changes over a two-year period. Morphometric differences with respect to a template hippocampus were measured by the metric distance obtained from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) algorithm which was previously used to calculate dense one-to-one correspondence vector fields between the shapes. LDDMM assigns metric distances on the space of anatomical images thereby allowing for the direct comparison and quantization of morphometric changes. We use various statistical methods to compare the metric distances in a cross-sectional and longitudinal manner. At baseline, the metric distances for demented subjects are found not to be significantly different from those for nondemented subjects. At follow-up, the metric distances for demented subjects were significantly larger compared to nondemented subjects. The metric distances for demented subjects increased significantly from baseline to follow-up but not for nondemented subjects. We also use the metric distances in logistic regression for diagnostic discrimination of subjects. We compare metric distances with the volumes and obtain similar results. In classification, the model that uses volume, metric distance, and volume loss over time together performs better in detecting DAT. Thus, metric distances with respect to a template computed via LDDMM can be a powerful tool in detecting differences in shape in cross-sectional as well as longitudinal studies.

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