{"corpus_id":14663311,"paper_sha":"225cbd16bf3f529c574866c0b48a21cd8e655601","doi":"10.1214/AOS/1009210544","arxiv_id":null,"pmid":null,"pmcid":null,"mag_id":1520752838,"dblp_id":null,"acl_id":null,"title":"On the distribution of the largest eigenvalue in principal components analysis","year":2001,"publication_date":"2001-04-01","venue":"","journal":{"name":"Annals of Statistics","pages":"295-327","volume":"29"},"journal_issn":null,"journal_title":null,"publication_types":[],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics"],"reference_count":51,"citation_count":2226,"influential_citation_count":267,"is_open_access":true,"arxiv_categories":null,"arxiv_license":null,"arxiv_journal_ref":null,"mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":"https://projecteuclid.org/journals/annals-of-statistics/volume-29/issue-2/On-the-distribution-of-the-largest-eigenvalue-in-principal-components/10.1214/aos/1009210544.pdf","s2_open_access_landing_url":"https://www.semanticscholar.org/paper/225cbd16bf3f529c574866c0b48a21cd8e655601","s2_open_access_license":null,"s2_open_access_status":"BRONZE","pmc_open_access_pdf_url":null,"pmc_open_access_landing_url":null,"pmc_open_access_license":null,"pmc_open_access_status":null,"unpaywall_open_access_pdf_url":null,"unpaywall_open_access_landing_url":null,"unpaywall_open_access_license":null,"unpaywall_open_access_status":null,"abstract":"Let x(1) denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x(1) is the largest principal component variance of the covariance matrix $X'X$, or the largest eigenvalue of a p­variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with $n/p = \\gamma \\ge 1$. When centered by $\\mu_p = (\\sqrt{n-1} + \\sqrt{p})^2$ and scaled by $\\sigma_p = (\\sqrt{n-1} + \\sqrt{p})(1/\\sqrt{n-1} + 1/\\sqrt{p}^{1/3}$, the distribution of x(1) approaches the Tracey-Widom law of order 1, which is defined in terms of the Painlevé II differential equation and can be numerically evaluated and tabulated in software. Simulations show the approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts.","claims":[{"public_id":"cl_9178971c3e736341b703ae869fa4aea1","status":"active","text":"After centering by μ_p = (\\sqrt{n-1} + \\sqrt{p})^2 and scaling by σ_p, the largest eigenvalue distribution for a p-variate Wishart matrix with identity covariance converges to the Tracy-Widom law of order 1 as p and n grow with n/p = γ ≥ 1.","confidence":0.98,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_9178971c3e736341b703ae869fa4aea1"},{"public_id":"cl_e87c1677d83522e75d65c39569ca2f55","status":"active","text":"The Tracy-Widom approximation is informative in simulations for matrix dimensions as small as 5.","confidence":0.93,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous 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