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Graphical Models: Mathematical Foundation and Statistical Analysis
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0000-0002-3331-4976
2024 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Working on different problems related to graph theory combined with statistics is the main purpose of this thesis. 

In paper I, we worked on the distance between trees and defined another definition for the interleaving distance that was already introduced for determining the distance between merge trees. The new definition was based on only one map from one of the trees to another one. Therefore, we could gain fixed-parameter tractable algorithms for finding the interleaving distance between merge trees with some conditions. 

In paper II, we worked on the clustering coefficient of the networks. The clustering coefficient indicates the tendency of the vertices of the network to form a triangle. We introduced another clustering coefficient, which we called it Relative clustering coefficient. Finally, the importance of the relative clustering coefficient. 

In paper III, we worked on the financial relationship between companies in Sweden, and we used two methods (the Pearson correlation coefficient (PCC) and the generalized variance decomposition (GVD). We then ap- plied these methods to the financial data consisting of the daily returns on the 28 stocks included in the computation of the OMX index (the index of the Swedish capital market). 

Gaussian Graphical Model is the main subject of paper IV. In this paper, we consider three types of precision matrices, and corresponding to each type of precision matrix, we develop the exact test theory. Finally, the new approaches are compared to the benchmark method via an extensive simulation study. 

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2024. , p. 36
National Category
Engineering and Technology
Research subject
Computational Mathematics
Identifiers
URN: urn:nbn:se:su:diva-229153ISBN: 978-91-8014-833-7 (print)ISBN: 978-91-8014-834-4 (electronic)OAI: oai:DiVA.org:su-229153DiVA, id: diva2:1857843
Public defence
2024-08-22, Cramérrummet (mötesrum 12), hus 1, Albano, Albanovägen 28, Stockholm, 13:00 (English)
Opponent
Supervisors
Available from: 2024-05-29 Created: 2024-05-14 Last updated: 2024-05-27Bibliographically approved
List of papers
1. FPT-Algorithms for computing Gromov-Hausdorff and interleaving distances between trees
Open this publication in new window or tab >>FPT-Algorithms for computing Gromov-Hausdorff and interleaving distances between trees
2022 (English)In: Journal of Computational Geometry, E-ISSN 1920-180X, Vol. 13, no 1, p. 89-124Article in journal (Refereed) Published
Abstract [en]

The Gromov-Hausdorff distance is a natural way to measure the distortion between two metric spaces. However, there has been only limited algorithmic development to compute or approximate this distance. We focus on computing the Gromov-Hausdorff distance between two metric trees. Roughly speaking, a metric tree is a metric space that can be realized by the shortest path metric on a tree. Any finite tree with positive edge weight can be viewed as a metric tree where the weight is treated as edge length and the metric is the induced shortest path metric in the tree. Previously, Agarwal et al. showed that even for trees with unit edge length, it is NP-hard to approximate the Gromov-Hausdorff distance between them within a factor of 3. In this paper, we present a fixed-parameter tractable (FPT) algorithm that can approximatethe Gromov-Hausdorff distance between two general metric trees within a multiplicative factor of 14.

Interestingly, the development of our algorithm is made possible by a connection betweenthe Gromov-Hausdorff distance for metric trees and the interleaving distance for the so-called merge trees. The merge trees arise in practice naturally as a simple yet meaningful topological summary (it is a variant of the Reeb graphs and contour trees), and are of independent interest. It turns out that an exact or approximation algorithm for the interleaving distance leads to an approximation algorithm for the Gromov-Hausdorff distance. One of the key contributions of our work is that we redefine the interleaving distance in a way that makes it easier to develop dynamic programming approaches to compute it. We then present a fixed-parameter tractable algorithm to compute the interleaving distance between two merge trees exactly, which ultimately leads to an FPT-algorithm to approximate the Gromov-Hausdorff distance between two metric trees. This exact FPT-algorithm to compute the interleaving distance between merge trees is of interest itself, as it is known that it is NP-hard to approximate it within a factor of 3, and previously the best known algorithm has an approximation factor of O(√n) even for trees with unit edge length.

National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-210621 (URN)10.20382/jocg.v13i1a4 (DOI)000862654800003 ()
Available from: 2022-10-26 Created: 2022-10-26 Last updated: 2024-05-17Bibliographically approved
2. Relative clustering coefficient
Open this publication in new window or tab >>Relative clustering coefficient
2022 (English)In: Journal of Algorithms and Computation, p. 99-108Article in journal, Meeting abstract (Refereed) Published
Abstract [en]

In this paper, we relatively extend the definition of the global clustering coefficient to another clustering, which we call it relative clustering coefficient. The idea of this definition is to ignore the edges in the network that the probability of having an edge is 0. Here, we also consider a model as an example that using the relative clustering coefficient is better than the global clustering coefficient for comparing networks and also checking the properties of the networks. 

National Category
Engineering and Technology
Identifiers
urn:nbn:se:su:diva-229150 (URN)
Available from: 2024-05-14 Created: 2024-05-14 Last updated: 2024-05-17
3. Monitoring the Dynamic Networks of Stock Returns with an Application to the Swedish Stock Market
Open this publication in new window or tab >>Monitoring the Dynamic Networks of Stock Returns with an Application to the Swedish Stock Market
2024 (English)In: Computational Economics, ISSN 0927-7099, E-ISSN 1572-9974Article in journal (Refereed) Published
Abstract [en]

In this paper, two approaches for measuring the distance between stock returns and the network connectedness are presented that are based on the Pearson correlation coefficient dissimilarity and the generalized variance decomposition dissimilarity. Using these two procedures, the center of the network is determined. Also, hierarchical clustering methods are used to divide the dense networks into sparse trees, which provide us with information about how the companies of a financial market are related to each other. We implement the derived theoretical results to study the dynamic connectedness between the companies in the Swedish capital market by considering 28 companies included in the determination of the market index OMX30. The network structure of the market is constructed using different methods to determine the distance between the companies. We use hierarchical clustering methods to find the relation among the companies in each window. Next, we obtain a one-dimensional time series of the distances between the clustering trees that reflect the changes in the relationship between the companies in the market over time. The method from statistical process control, namely the Shewhart control chart, is applied to those time series to detect abnormal changes in the financial market.

National Category
Engineering and Technology
Identifiers
urn:nbn:se:su:diva-229151 (URN)10.1007/s10614-024-10616-2 (DOI)001216054900001 ()2-s2.0-85192711903 (Scopus ID)
Available from: 2024-05-14 Created: 2024-05-14 Last updated: 2024-05-21
4. Exact test theory in Gaussian graphical models
Open this publication in new window or tab >>Exact test theory in Gaussian graphical models
2023 (English)In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 196, article id 105185Article in journal (Refereed) Published
Abstract [en]

In this paper, we derive several statistical tests on the precision matrix with application to the determination of the structure of an undirected Gaussian graph. The exact distributions of the test statistics are obtained under the null hypotheses, while the exact distributions of the random matrices, which are used in the construction of the test statistics, are deduced under the alternative hypothesis. Moreover, we present the high-dimensional asymptotic distributions of the test statistics under the null hypothesis. The testing problems that an undirected Gaussian graph possesses a structure that corresponds to the precision matrix of an AR(1) process, to the block-diagonal precision matrix and to the precision of a factor model are discussed in detail. The performance of the proposed statistical tests is further investigated via an extensive simulation study and compared to the benchmark approach.

Keywords
Gaussian graphical model, Precision matrix, Test theory
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:su:diva-229152 (URN)10.1016/j.jmva.2023.105185 (DOI)000976976000001 ()2-s2.0-85151516814 (Scopus ID)
Available from: 2024-05-14 Created: 2024-05-14 Last updated: 2024-06-13Bibliographically approved

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Farahbakhsh Touli, Elena

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