Random simplicial complexes and Stein’s method
<p>The work presented in this thesis is largely motivated by the pressing need for statistical foundations in topological data analysis (TDA). By studying random simplicial complexes, which serve as null models in TDA, the thesis falls within the framework of stochastic topology. Despite vario...
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| Format: | Thesis |
| Language: | English |
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2023
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| _version_ | 1811139832892620800 |
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| author | Temčinas, T |
| author2 | Reinert, G |
| author_facet | Reinert, G Temčinas, T |
| author_sort | Temčinas, T |
| collection | OXFORD |
| description | <p>The work presented in this thesis is largely motivated by the pressing need for statistical foundations in topological data analysis (TDA). By studying random simplicial complexes, which serve as null models in TDA, the thesis falls within the framework of stochastic topology. Despite various success stories involving applications of algebraic topology, there is a lack of statistical tools that allow to rigorously evaluate topological data under uncertainty. Here we use Stein’s method for multivariate normal distributions to prove distributional approximation results for random variables that arise in the study of random simplicial complexes. We also apply the results to statistical inference and hypothesis testing using simplicial complex data. The main body of the work consists of two articles: one accepted journal publication, and one preprint, which has been submitted for publication.</p>
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<p>The first main result of the thesis is an abstract multivariate central limit theorem with explicit bounds, established via Stein’s method, for sums of locally dependent random variables, which pro- vides the greatest generality for this type of dependence structure so far. The result is then applied to prove multivariate central limit theorems for variables that arise in the study of the clique complexes of Erd ̋os–R ́enyi random graphs. The abstract central limit theorem applies to multivariate sums of random variables which contain different order of summands in each component, allowing it to be applied in the context of the random clique complexes.</p>
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<p>The subsequent chapter extends and generalises the probabilistic results of the previous chapter and applies them to statistical inference and hypothesis testing. Here we work with a more general random simplicial complex model: the multi-parameter random complex <strong>X</strong>(<em>n</em>, <strong>p</strong>). The statistical applications include properties of the maximum likelihood estimator of <strong>X</strong>(<em>n</em>, <em>p</em>) given a single observed simplicial complex as well as using the central limit theorems to devise goodness-of-fit tests.</p>
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<p>This thesis not only contributes to the Stein’s method literature but also offers better under- standing of multivariate statistics of random simplicial complexes from a probabilistic point of view as well as shows how such probabilistic understanding leads to useful statistical tools in practice.</p> |
| first_indexed | 2024-09-25T04:12:22Z |
| format | Thesis |
| id | oxford-uuid:8b5384ac-8f3c-404a-998b-bd06b414446c |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:12:22Z |
| publishDate | 2023 |
| record_format | dspace |
| spelling | oxford-uuid:8b5384ac-8f3c-404a-998b-bd06b414446c2024-07-05T14:58:15ZRandom simplicial complexes and Stein’s methodThesishttp://purl.org/coar/resource_type/c_db06uuid:8b5384ac-8f3c-404a-998b-bd06b414446cStatisticsLimit theorems (probability theory)Topological data analysisEnglishHyrax Deposit2023Temčinas, TReinert, GNanda, V<p>The work presented in this thesis is largely motivated by the pressing need for statistical foundations in topological data analysis (TDA). By studying random simplicial complexes, which serve as null models in TDA, the thesis falls within the framework of stochastic topology. Despite various success stories involving applications of algebraic topology, there is a lack of statistical tools that allow to rigorously evaluate topological data under uncertainty. Here we use Stein’s method for multivariate normal distributions to prove distributional approximation results for random variables that arise in the study of random simplicial complexes. We also apply the results to statistical inference and hypothesis testing using simplicial complex data. The main body of the work consists of two articles: one accepted journal publication, and one preprint, which has been submitted for publication.</p> <br> <p>The first main result of the thesis is an abstract multivariate central limit theorem with explicit bounds, established via Stein’s method, for sums of locally dependent random variables, which pro- vides the greatest generality for this type of dependence structure so far. The result is then applied to prove multivariate central limit theorems for variables that arise in the study of the clique complexes of Erd ̋os–R ́enyi random graphs. The abstract central limit theorem applies to multivariate sums of random variables which contain different order of summands in each component, allowing it to be applied in the context of the random clique complexes.</p> <br> <p>The subsequent chapter extends and generalises the probabilistic results of the previous chapter and applies them to statistical inference and hypothesis testing. Here we work with a more general random simplicial complex model: the multi-parameter random complex <strong>X</strong>(<em>n</em>, <strong>p</strong>). The statistical applications include properties of the maximum likelihood estimator of <strong>X</strong>(<em>n</em>, <em>p</em>) given a single observed simplicial complex as well as using the central limit theorems to devise goodness-of-fit tests.</p> <br> <p>This thesis not only contributes to the Stein’s method literature but also offers better under- standing of multivariate statistics of random simplicial complexes from a probabilistic point of view as well as shows how such probabilistic understanding leads to useful statistical tools in practice.</p> |
| spellingShingle | Statistics Limit theorems (probability theory) Topological data analysis Temčinas, T Random simplicial complexes and Stein’s method |
| title | Random simplicial complexes and Stein’s method |
| title_full | Random simplicial complexes and Stein’s method |
| title_fullStr | Random simplicial complexes and Stein’s method |
| title_full_unstemmed | Random simplicial complexes and Stein’s method |
| title_short | Random simplicial complexes and Stein’s method |
| title_sort | random simplicial complexes and stein s method |
| topic | Statistics Limit theorems (probability theory) Topological data analysis |
| work_keys_str_mv | AT temcinast randomsimplicialcomplexesandsteinsmethod |