arXiv subjects

Elvan Ceyhan

Publications and source records attributed to Elvan Ceyhan.

At least 19 recordsLinked to original sources

arXiv2026

Prototype Selection Using Topological Data Analysis

Prototype selection methods compress a training set, but the existing taxonomy of condensation, edition, hybrid, competence-based, optimization-based, and clustering-based families does not include methods that operate on the multi-scale topological structure of the data. This paper introduces two different persistence-based prototype selector variants, Topological Prototype Selector (TPS) and Boundary-Conscious Topological Prototype Selector (BoundaryTPS). TPS uses two sequential Rips filtrations to retain boundary-relevant and interior-typical points. BoundaryTPS is a single-stage variant whose vertex-weighted filtration concentrates retention near the decision boundary. We evaluate both methods against seven classical baselines on fifteen real datasets and find that the topological methods occupy a different operating point in the prototype-selection design space than existing methods. BoundaryTPS achieves the lowest mean Friedman rank on $H_1$ persistence-diagram preservation and is significantly better than five of the seven baselines (Nemenyi, $α= 0.05$). TPS ranks third on the same endpoint. Both methods are more stable under fold perturbation than any chained-decision selector tested, and both inherit the source set's class proportions without label-aware machinery. On aggregate G-Mean both methods are competitive but not leading, with rank-1 frequencies of $11.3\%$ (TPS) and $9.9\%$ (BoundaryTPS) across fold combinations. Empirically, both methods scale sub-quadratically in sample size.

stat.ML↗

arXiv2026

Clustering with Uniformity- and Neighbor-Based Random Geometric Graphs

We propose a graph-based clustering method based on Cluster Catch Digraphs (CCDs) that extends their applicability to moderate-dimensional data settings. Existing CCD variants, such as RK-CCDs, rely on spatial randomness tests based on Ripley's K function, which exhibit performance degradation as dimensionality increases. To address this limitation, we introduce a nearest-neighbor-distance (NND) based Monte Carlo spatial randomness test (MC-SRT) for determining covering radii, resulting in the proposed Uniformity- and Neighbor-based CCDs (UN-CCDs). The proposed method is designed for datasets of moderate size and dimension, particularly in settings with complex cluster geometry and uniformly distributed background noise. Through Monte Carlo simulations and experiments on benchmark datasets, we show that UN-CCDs provide stable and competitive performance relative to several established clustering methods within the evaluated regimes, while remaining largely parameter-free. We also discuss computational trade-offs and identify the practical regimes in which the method is most effective. -- Keywords: Graph-based clustering; Cluster catch digraphs; Moderate-dimensional data; the nearest neighbor distance; Spatial randomness test.

cs.LG↗

arXiv2026

Adversarial Graph Traversal

Suppose a Bayesian agent seeks to traverse a graph. Each time she crosses an edge, she pays a price. The first time she reaches a node, there is a payoff. She has an opponent who can reduce the payoffs. This paper uses adversarial risk analysis to find a solution to her route selection problem. It shows how the traveler is advantaged by having an accurate subjective distribution over the costs/payoffs and by having a Bayesian prior for her opponent's strategic choices. The results are relevant to military convoy routing, corporate competition, and certain games.

math.OC↗

arXiv2025

Outlyingness Scores with Cluster Catch Digraphs

This paper introduces two novel, outlyingness scores (OSs) based on Cluster Catch Digraphs (CCDs): Outbound Outlyingness Score (OOS) and Inbound Outlyingness Score (IOS). These scores enhance the interpretability of outlier detection results. Both OSs employ graph-, density-, and distribution-based techniques, tailored to high-dimensional data with varying cluster shapes and intensities. OOS evaluates the outlyingness of a point relative to its nearest neighbors, while IOS assesses the total ``influence" a point receives from others within its cluster. Both OSs effectively identify global and local outliers, invariant to data collinearity. Moreover, IOS is robust to the masking problems. With extensive Monte Carlo simulations, we compare the performance of both OSs with CCD-based, traditional, and state-of-the-art outlier detection methods. Both OSs exhibit substantial overall improvements over the CCD-based methods in both artificial and real-world data sets, particularly with IOS, which delivers the best overall performance among all the methods, especially in high-dimensional settings. Keywords: Outlier detection, Outlyingness score, Graph-based clustering, Cluster catch digraphs, High-dimensional data.

stat.ML↗

arXiv2025

Stochastic Path Planning in Correlated Obstacle Fields

We introduce the Stochastic Correlated Obstacle Scene (SCOS) problem, a navigation setting with spatially correlated obstacles of uncertain blockage status, realistically constrained sensors that provide noisy readings and costly disambiguation. Modeling the spatial correlation with Gaussian Random Field (GRF), we develop Bayesian belief updates that refine blockage probabilities, and use the posteriors to reduce search space for efficiency. To find the optimal traversal policy, we propose a novel two-stage learning framework. An offline phase learns a robust base policy via optimistic policy iteration augmented with information bonus to encourage exploration in informative regions, followed by an online rollout policy with periodic base updates via a Bayesian mechanism for information adaptation. This framework supports both Monte Carlo point estimation and distributional reinforcement learning (RL) to learn full cost distributions, leading to stronger uncertainty quantification. We establish theoretical benefits of correlation-aware updating and convergence property under posterior sampling. Comprehensive empirical evaluations across varying obstacle densities, sensor capabilities demonstrate consistent performance gains over baselines. This framework addresses navigation challenges in environments with adversarial interruptions or clustered natural hazards.

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arXiv2025

Adversarial Obstacle Placement with Spatial Point Processes for Optimal Path Disruption

We investigate the Optimal Obstacle Placement (OOP) problem under uncertainty, framed as the dual of the Optimal Traversal Path problem in the Stochastic Obstacle Scene paradigm. We consider both continuous domains, discretized for analysis, and already discrete spatial grids that form weighted geospatial networks using 8-adjacency lattices. Our unified framework integrates OOP with stochastic geometry, modeling obstacle placement via Strauss (regular) and Matérn (clustered) processes, and evaluates traversal using the Reset Disambiguation algorithm. Through extensive Monte Carlo experiments, we show that traversal cost increases by up to 40% under strongly regular placements, while clustered configurations can decrease traversal costs by as much as 25% by leaving navigable corridors compared to uniform random layouts. In mixed (with both true and false obstacles) scenarios, increasing the proportion of true obstacles from 30% to 70% nearly doubles the traversal cost. These findings are further supported by statistical analysis and stochastic ordering, providing rigorous insights into how spatial patterns and obstacle compositions influence navigation under uncertainty.

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arXiv2025

Solving the Constrained Random Disambiguation Path Problem via Lagrangian Relaxation and Graph Reduction

We study a resource-constrained variant of the Random Disambiguation Path (RDP) problem, a generalization of the Stochastic Obstacle Scene (SOS) problem, in which a navigating agent must reach a target in a spatial environment populated with uncertain obstacles. Each ambiguous obstacle may be disambiguated at a (possibly) heterogeneous resource cost, subject to a global disambiguation budget. We formulate this constrained planning problem as a Weight-Constrained Shortest Path Problem (WCSPP) with risk-adjusted edge costs that incorporate probabilistic blockage and traversal penalties. To solve it, we propose a novel algorithmic framework-COLOGR-combining Lagrangian relaxation with a two-phase vertex elimination (TPVE) procedure. The method prunes infeasible and suboptimal paths while provably preserving the optimal solution, and leverages dual bounds to guide efficient search. We establish correctness, feasibility guarantees, and surrogate optimality under mild assumptions. Our analysis also demonstrates that COLOGR frequently achieves zero duality gap and offers improved computational complexity over prior constrained path-planning methods. Extensive simulation experiments validate the algorithm's robustness across varying obstacle densities, sensor accuracies, and risk models, consistently outperforming greedy baselines and approaching offline-optimal benchmarks. The proposed framework is broadly applicable to stochastic network design, mobility planning, and constrained decision-making under uncertainty.

cs.RO↗

arXiv2025

Modeling Home Range and Intra-Specific Spatial Interaction in Wild Animal Populations

Interactions among individuals from the same-species of wild animals are an important component of population dynamics. An interaction can be either static (based on overlap of space use) or dynamic (based on movement). The goal of this work is to determine the level of static interactions between individuals from the same-species of wild animals using 95\% and 50\% home ranges, as well as to model their movement interactions, which could include attraction, avoidance (or repulsion), or lack of interaction, in order to gain new insights and improve our understanding of ecological processes. Home range estimation methods (minimum convex polygon, kernel density estimator, and autocorrelated kernel density estimator), inhomogeneous multitype (or cross-type) summary statistics, and envelope testing methods (pointwise and global envelope tests) were proposed to study the nature of the same-species wild-animal spatial interactions. This study provides comprehensive, self-contained methodological details for investigating spatial interactions between individuals of the same species in wildlife populations. Using GPS collar data, we applied the methods to quantify both static and dynamic interactions between black bears in southern Alabama, USA. In general, our findings suggest that the black bears in our dataset showed no significant preference to live together or apart, i.e., there was no significant deviation from independence toward association or avoidance (i.e., segregation) between the bears. This can be loosely interpreted to mean that a black bear is generally indifferent to the presence of other black bears living or wandering nearby.

stat.AP↗

arXiv2024

Outlier Detection with Cluster Catch Digraphs

This paper introduces a novel family of outlier detection algorithms based on Cluster Catch Digraphs (CCDs), specifically tailored to address the challenges of high dimensionality and varying cluster shapes, which deteriorate the performance of most traditional outlier detection methods. We propose the Uniformity-Based CCD with Mutual Catch Graph (U-MCCD), the Uniformity- and Neighbor-Based CCD with Mutual Catch Graph (UN-MCCD), and their shape-adaptive variants (SU-MCCD and SUN-MCCD), which are designed to detect outliers in data sets with arbitrary cluster shapes and high dimensions. We present the advantages and shortcomings of these algorithms and provide the motivation or need to define each particular algorithm. Through comprehensive Monte Carlo simulations, we assess their performance and demonstrate the robustness and effectiveness of our algorithms across various settings and contamination levels. We also illustrate the use of our algorithms on various real-life data sets. The U-MCCD algorithm efficiently identifies outliers while maintaining high true negative rates, and the SU-MCCD algorithm shows substantial improvement in handling non-uniform clusters. Additionally, the UN-MCCD and SUN-MCCD algorithms address the limitations of existing methods in high-dimensional spaces by utilizing Nearest Neighbor Distances (NND) for clustering and outlier detection. Our results indicate that these novel algorithms offer substantial advancements in the accuracy and adaptability of outlier detection, providing a valuable tool for various real-world applications. Keyword: Outlier detection, Graph-based clustering, Cluster catch digraphs, $k$-nearest-neighborhood, Mutual catch graphs, Nearest neighbor distance.

stat.ML↗

arXiv2020

Domination Number of an Interval Catch Digraph Family and its use for Testing Uniformity

We consider a special type of interval catch digraph (ICD) family for one-dimensional data in a randomized setting and propose its use for testing uniformity. These ICDs are defined with an expansion and a centrality parameter, hence we will refer to this ICD as parameterized ICD (PICD). We derive the exact (and asymptotic) distribution of the domination number of this PICD family when its vertices are from a uniform (and non-uniform) distribution in one dimension for the entire range of the parameters; thereby determine the parameters for which the asymptotic distribution is non-degenerate. We observe 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 combinations. We use the domination number for testing uniformity of data in real line, prove its consistency against certain alternatives, and compare it with two commonly used tests and three recently proposed tests in literature and also arc density of this ICD and of another ICD family in terms of size and power. Based on our extensive Monte Carlo simulations, we demonstrate that domination number of our PICD has higher power for certain types of deviations from uniformity compared to other tests.

stat.ME↗

arXiv2019

Parameter Free Clustering with Cluster Catch Digraphs (Technical Report)

We propose clustering algorithms based on a recently developed geometric digraph family called cluster catch digraphs (CCDs). These digraphs are used to devise clustering methods that are hybrids of density-based and graph-based clustering methods. CCDs are appealing digraphs for clustering, since they estimate the number of clusters; however, CCDs (and density-based methods in general) require some information on a parameter representing the \emph{intensity} of assumed clusters in the data set. We propose algorithms that are parameter free versions of the CCD algorithm and does not require a specification of the intensity parameter whose choice is often critical in finding an optimal partitioning of the data set. We estimate the number of convex clusters by borrowing a tool from spatial data analysis, namely Ripley's $K$ function. We call our new digraphs utilizing the $K$ function as RK-CCDs. We show that the minimum dominating sets of RK-CCDs estimate and distinguish the clusters from noise clusters in a data set, and hence allow the estimation of the correct number of clusters. Our robust clustering algorithms are comprised of methods that estimate both the number of clusters and the intensity parameter, making them completely parameter free. We conduct Monte Carlo simulations and use real life data sets to compare RK-CCDs with some commonly used density-based and prototype-based clustering methods.

stat.ML↗

arXiv2019

Classification of Imbalanced Data with a Geometric Digraph Family

We use a geometric digraph family called class cover catch digraphs (CCCDs) to tackle the class imbalance problem in statistical classification. CCCDs provide graph theoretic solutions to the class cover problem and have been employed in classification. We assess the classification performance of CCCD classifiers by extensive Monte Carlo simulations, comparing them with other classifiers commonly used in the literature. In particular, we show that CCCD classifiers perform relatively well when one class is more frequent than the other in a two-class setting, an example of the class imbalance problem. We also point out the relationship between class imbalance and class overlapping problems, and their influence on the performance of CCCD classifiers and other classification methods as well as some state-of-the-art algorithms which are robust to class imbalance by construction. Experiments on both simulated and real data sets indicate that CCCD classifiers are robust to the class imbalance problem. CCCDs substantially undersample from the majority class while preserving the information on the discarded points during the undersampling process. Many state-of-the-art methods, however, keep this information by means of ensemble classifiers, but CCCDs yield only a single classifier with the same property, making it both appealing and fast.

stat.ML↗

arXiv2017

Classification Using Proximity Catch Digraphs (Technical Report)

We employ random geometric digraphs to construct semi-parametric classifiers. These data-random digraphs are from parametrized random digraph families called proximity catch digraphs (PCDs). A related geometric digraph family, class cover catch digraph (CCCD), has been used to solve the class cover problem by using its approximate minimum dominating set. CCCDs showed relatively good performance in the classification of imbalanced data sets, and although CCCDs have a convenient construction in $\mathbb{R}^d$, finding minimum dominating sets is NP-hard and its probabilistic behaviour is not mathematically tractable except for $d=1$. On the other hand, a particular family of PCDs, called \emph{proportional-edge} PCDs (PE-PCDs), has mathematical tractable minimum dominating sets in $\mathbb{R}^d$; however their construction in higher dimensions may be computationally demanding. More specifically, we show that the classifiers based on PE-PCDs are prototype-based classifiers such that the exact minimum number of prototypes (equivalent to minimum dominating sets) are found in polynomial time on the number of observations. We construct two types of classifiers based on PE-PCDs. One is a family of hybrid classifiers depend on the location of the points of the training data set, and another type is a family of classifiers solely based on class covers. We assess the classification performance of our PE-PCD based classifiers by extensive Monte Carlo simulations, and compare them with that of other commonly used classifiers. We also show that, similar to CCCD classifiers, our classifiers are relatively better in classification in the presence of class imbalance.

cs.LG↗

arXiv2016

On the Distribution of the Number of Copies of Weakly Connected Digraphs in Random $k$NN Digraphs

In a digraph with $n$ vertices, a minuscule construct is a subdigraph with $m<<n$ vertices. We study the number of copies of a minuscule constructs in $k$ nearest neighbor ($k$NN) digraph of the data from a random point process in $\mathbb{R}^d$. Based on the asymptotic theory for functionals of point sets under homogeneous Poisson process and binomial point process, we provide a general result for the asymptotic behavior of the number of minuscule constructs and as corollaries, we obtain asymptotic results for the number of vertices with fixed indegree, the number of shared $k$NN pairs and the number of reflexive $k$NN's in a $k$NN digraph.

math.PR↗

arXiv2016

A Classification of Isomorphism-Invariant Random Digraphs

We classify isomorphism-invariant random digraphs according to where randomness resides, namely, arcs, vertices, and vertices and arcs together which in turn yield arc random digraphs (ARD), vertex random digraphs (VRD) and vertex-arc random digraphs (VARD), respectively. This digraph classification can be viewed as an extension of the classification of isomorphism-invariant random graphs. We introduce randomness in the direction of the edges of a given graph and obtain direction random digraphs (DRD) as well. We classify DRDs according to which component is random in addition to the direction and study the relations of DRDs with VARDs, VRDs and ARDs. We also consider random nearest neighbor digraphs and determine their membership with respect to these digraph families.

math.CO↗

arXiv2016

On the Number of Reflexive and Shared Nearest Neighbor Pairs in One-Dimensional Uniform Data

For a random sample of points in $\mathbb{R}$, we consider the number of pairs whose members are nearest neighbors (NN) to each other and the number of pairs sharing a common NN. The first type of pairs are called reflexive NNs whereas latter type of pairs are called shared NNs. In this article, we consider the case where the random sample of size $n$ is from the uniform distribution on an interval. We denote the number of reflexive NN pairs and the number of shared NN pairs in the sample as $R_n$ and $Q_n$, respectively. We derive the exact forms of the expected value and the variance for both $R_n$ and $Q_n$, and derive a recurrence relation for $R_n$ which may also be used to compute the exact probability mass function of $R_n$. Our approach is a novel method for finding the pmf of $R_n$ and agrees with the results in literature. We also present SLLN and CLT results for both $R_n$ and $Q_n$ as $n$ goes to infinity.

math.PR↗

arXiv2014

Distribution of Relative Edge Density of the Graphs Based on a Random Digraph Family

The vertex-random graphs called proximity catch digraphs (PCDs) have been introduced recently and have applications in pattern recognition and spatial pattern analysis. A PCD is a random directed graph (i.e., digraph) which is constructed from data using the relative positions of the points from various classes. Different PCDs result from different definitions of the proximity region associated with each data point. We consider the underlying and reflexivity graphs based on a family of PCDs which is determined by a family of parameterized proximity maps called proportional-edge (PE) proximity map. The graph invariant we investigate is the relative edge density of the underlying and reflexivity graphs. We demonstrate that, properly scaled, relative edge density of these graphs is a $U$-statistic, and hence obtain the asymptotic normality of the relative edge density for data from any distribution that satisfies mild regulatory conditions. By detailed probabilistic and geometric calculations, we compute the explicit form of the asymptotic normal distribution for uniform data on a bounded region in the usual Euclidean plane. We also compare the relative edge densities of the two types of the graphs and the relative arc density of the PE-PCDs. The approach presented here is also valid for data in higher dimensions.

math.PR↗

arXiv2014

Nearest Neighbor Methods for Testing Reflexivity and Species-Correspondence

Nearest neighbor (NN) methods are employed for drawing inferences about spatial patterns of points from two or more classes. We consider Pielou's test of niche specificity which is defined using a contingency table based on the NN relationships between the data points. We demonstrate that Pielou's contingency table for niche specificity is actually more appropriate for testing reflexivity in NN structure, hence we call this table as NN reflexivity contingency table (NN-RCT) henceforth. We also derive an asymptotic approximation for the distribution of the entries of the NN-RCT and consider variants of Fisher's exact test on it. Moreover, we introduce a new test of class- or species-correspondence inspired by spatial niche/habitat specificity and the associated contingency table called species-correspondence contingency table (SCCT). We also determine the appropriate null hypotheses and the underlying conditions appropriate for these tests. We investigate the finite sample performance of the tests in terms of empirical size and power by extensive Monte Carlo simulations and the methods are illustrated on a real-life ecological data set.

stat.ME↗