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Publication Metadata only A comparison of analysis of covariate-adjusted residuals and analysis of covariance(Taylor & Francis Inc, 2009) Goad, Carla L.; Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of Sciences; N/AVarious methods to control the influence of a covariate on a response variable are compared. These methods are 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). Covariate-adjusted residuals are obtained from the overall regression line fit to the entire data set ignoring the treatment levels or factors. It is demonstrated that the methods on covariate-adjusted residuals are only appropriate when the regression lines are parallel and covariate means are equal for all treatments. 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. The parametric methods on residuals and ANCOVA exhibited similar size and power when error terms have symmetric distributions with variances having the same functional form for each treatment, and covariates have uniform distributions within the same interval for each treatment. In such cases, parametric tests have higher power compared to the K-W test on residuals. When error terms have asymmetric distributions or have variances that are heterogeneous with different functional forms for each treatment, the tests are liberal with K-W test having higher power than others. The methods on covariate-adjusted residuals are severely affected by the clustering of the covariates relative to the treatment factors when covariate means are very different for treatments. For data clusters, ANCOVA method exhibits the appropriate level. However, such a clustering might suggest dependence between the covariates and the treatment factors, so makes ANCOVA less reliable as well.Publication Metadata only A new family of random graphs for testing spatial segregation(Wiley, 2007) Priebe, Carey E.; Marchette, David J.; Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of Sciences; N/AThe authors discuss a graph-based approach for testing spatial point patterns. This approach falls under the category of data-random graphs, which have been introduced and used for statistical pattern recognition in recent years. The authors address specifically the problem of testing. complete spatial randomness against spatial patterns of segregation or association between two or more classes of points on the plane. To this end, they use a particular type of parameterized random digraph called a proximity catch digraph (PCD) which is based on relative positions of the data points from various classes. The statistic employed is the relative density of the PCD, which is a U-statistic when scaled properly. The authors derive the limiting distribution of the relative, density, using the standard asymptotic theory of U-statistics. They evaluate the finite-sample performance of their test statistic by Monte Carlo simulations and assess its asymptotic performance via Pitman's asymptotic efficiency, thereby yielding the optimal parameters for testing. They further stress that their methodology remains valid for data in higher dimensions.Publication Metadata only A solution method for linear and geometrically nonlinear MDOF systems with random properties subject to random excitation(Elsevier, 1998) Micaletti, RC; Çakmak, Ahmet Ş.; Nielsen, Søren R.K.; Department of Mathematics; Köylüoğlu, Hasan Uğur; Teaching Faculty; Department of Mathematics; College of Sciences; N/AA method for computing the lower-order moments of response of randomly excited multi-degree-of-freedom (MDOF) systems with random structural properties is proposed. The method is grounded in the techniques of stochastic calculus, utilizing a Markov diffusion process to model the structural system with random structural properties. The resulting state-space formulation is a system of ordinary stochastic differential equations with random coefficients and deterministic initial conditions which are subsequently transformed into ordinary stochastic differential equations with deterministic coefficients and random initial conditions, This transformation facilitates the derivation of differential equations which govern the evolution of the unconditional statistical moments of response. Primary consideration is given to linear systems and systems with odd polynomial nonlinearities, for in these cases there is a significant reduction in the number of equations to be solved. The method is illustrated for a five-story shear-frame structure with nonlinear interstory restoring forces and random damping and stiffness properties. The results of the proposed method are compared to those estimated by extensive Monte-Carlo simulation.Publication Metadata only An investigation of new graph invariants related to the domination number of random proximity catch digraphs(Springer, 2012) Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of Sciences; N/AProximity catch digraphs (PCDs) are a special type of proximity graphs based on proximity maps which yield proximity regions. PCDs are defined using the relative allocation of points from two or more classes in a region of interest and have applications in various fields. We introduce some auxiliary tools for PCDs and graph invariants related to the domination number of the PCDs and investigate their probabilistic properties. 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.Publication Metadata only Balanced and strongly balanced 4-kite designs(Utilitas Mathematica Publishing, 2013) Gionfriddo, Mario; Milazzo, Lorenzo; Department of Mathematics; Küçükçifçi, Selda; Faculty Member; Department of Mathematics; College of Sciences; 105252A G-design is called balanced if the degree of each vertex x is a constant. A G-design is called strongly balanced if for every i = 1, 2, ⋯, h, there exists a constant Ci such that dAi(x)= Ci for every vertex x, where AiS are the orbits of the automorphism group of G on its vertex-set and dAi(x) of a vertex is the number of blocks of containing x as an element of Ai. We say that a G-design is simply balanced if it is balanced, but not strongly balanced. In this paper we determine the spectrum of simply balanced and strongly balanced 4-kite designs.Publication Metadata only 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/ASpatial 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.Publication Metadata only Class-specific tests of spatial segregation based on nearest neighbor contingency tables(Wiley, 2009) Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of Sciences; N/AThe spatial interaction between two or more classes might cause multivariate clustering patterns such as segregation or association, which can be tested using a nearest neighbor contingency table (NNCT). The null hypothesis is randomness in the nearest neighbor structure, which may result from random labeling (RL) or complete spatial randomness of points from two or more classes (which is henceforth called CSR independence). We consider Dixon's class-specific segregation test and introduce a new class-specific test, which is a new decomposition of Dixon's overall chi-squared segregation statistic. We analyze the distributional properties and compare the empirical significant levels and power estimates of the tests using extensive Monte Carlo simulations. We demonstrate that the new class-specific tests have comparable performance with the currently available tests based on NNCTs. For illustrative purposes, we use three example data sets and provide guidelines for using these tests.Publication Metadata only 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/AWe 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.Publication Metadata only Correlated coalescing Brownian flows on R and the circle(IMPA - Instituto de Matemática Pura e Aplicada, 2018) Hajri, Hatem; Department of Mathematics; N/A; Çağlar, Mine; Karakuş, Abdullah Harun; Faculty Member; Master Student; Department of Mathematics; College of Sciences; Graduate School of Sciences and Engineering; 105131; N/AWe consider a stochastic differential equation on the real line which is driven by two correlated Brownian motions B+ and B- respectively on the positive half line and the negative half line. We assume |d〈B+,B-〉t| ≤ ρ dt with ρ ∈ [0, 1). We prove it has a unique flow solution. Then, we generalize this flow to a flow on the circle, which represents an oriented graph with two edges and two vertices. We prove that both flows are coalescing. Coalescence leads to the study of a correlated reflected Brownian motion on the quadrant. Moreover, we find the distribution of the hitting time to the origin of a reflected Brownian motion. This has implications for the effect of the correlation coefficient ρ on the coalescence time of our flows.Publication Metadata only Defining set spectra for designs can have arbitrarily large gaps(Util Math Publ Inc, 2008) Havas, George; Lawrence, Julie L.; Ramsay, Colin; Street, Anne Penfold; Department of Mathematics; Yazıcı, Emine Şule; Faculty Member; Department of Mathematics; College of Sciences; 27432A set of blocks which is a subset of a unique t-(v, k, lambda) design D is a defining set of D. A defining set is minimal if it does not properly contain a defining set. Define the spectrum of minimal defining sets of D by spec(D) = {vertical bar M vertical bar : M is a minimal defining set of D}. Call h a hole in spec(D) if h is not an element of spec(D), but there are minimal defining sets of D with cardinalities both larger and smaller than h. If spec(D) does not contain a hole, then it is said to be continuous. Previously, the spectra of only a limited number of designs were known and all of these were continuous. The question "whether the spectrum is continuous for all designs" was raised by B. Gray et al. (Discrete Mathematics 261 (2003), 277-284). We describe a new algorithm which finds all minimal defining sets of t-(v, k, lambda) designs. Using this algorithm we investigated the spectra for a variety of small designs, and found several examples of non-continuous spectra. We also derive some theoretical results which enable us to construct an infinite family of designs with arbitrarily large sequences of consecutive holes in their spectra.