{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,9]],"date-time":"2024-09-09T11:22:52Z","timestamp":1725880972109},"reference-count":25,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2024,1,11]],"date-time":"2024-01-11T00:00:00Z","timestamp":1704931200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,6,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Classical jittered sampling partitions <jats:inline-formula id=\"j_mcma-2023-2025_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">[<\/m:mo>\n                                 <m:mn>0<\/m:mn>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mn>1<\/m:mn>\n                                 <m:mo stretchy=\"false\">]<\/m:mo>\n                              <\/m:mrow>\n                              <m:mi>d<\/m:mi>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_mcma-2023-2025_eq_0245.png\"\/>\n                        <jats:tex-math>{[0,1]^{d}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> into <jats:inline-formula id=\"j_mcma-2023-2025_ineq_9998\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>m<\/m:mi>\n                              <m:mi>d<\/m:mi>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_mcma-2023-2025_eq_0331.png\"\/>\n                        <jats:tex-math>{m^{d}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> cubes for a positive integer <jats:italic>m<\/jats:italic> and randomly places a point inside each of them, providing a point set of size <jats:inline-formula id=\"j_mcma-2023-2025_ineq_9997\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>N<\/m:mi>\n                              <m:mo>=<\/m:mo>\n                              <m:msup>\n                                 <m:mi>m<\/m:mi>\n                                 <m:mi>d<\/m:mi>\n                              <\/m:msup>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_mcma-2023-2025_eq_0215.png\"\/>\n                        <jats:tex-math>{N=m^{d}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> with small discrepancy.\nThe aim of this note is to provide a construction of partitions that works for arbitrary <jats:italic>N<\/jats:italic> and improves straight-forward constructions. We show how to construct equivolume partitions of the <jats:italic>d<\/jats:italic>-dimensional unit cube with hyperplanes that are orthogonal to the main diagonal of the cube.\nWe investigate the discrepancy of such point sets and optimise the expected discrepancy numerically by relaxing the equivolume constraint using different black-box optimisation techniques.<\/jats:p>","DOI":"10.1515\/mcma-2023-2025","type":"journal-article","created":{"date-parts":[[2024,1,10]],"date-time":"2024-01-10T17:02:25Z","timestamp":1704906145000},"page":"163-181","source":"Crossref","is-referenced-by-count":1,"title":["Partitions for stratified sampling"],"prefix":"10.1515","volume":"30","author":[{"given":"Fran\u00e7ois","family":"Cl\u00e9ment","sequence":"first","affiliation":[{"name":"Sorbonne Universit\u00e9 , CNRS, LIP6 , Paris , France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nathan","family":"Kirk","sequence":"additional","affiliation":[{"name":"Queen\u2019s University Belfast , Belfast , United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Florian","family":"Pausinger","sequence":"additional","affiliation":[{"name":"Queen\u2019s University Belfast , Belfast , United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2024,1,11]]},"reference":[{"key":"2024052809495399441_j_mcma-2023-2025_ref_001","doi-asserted-by":"crossref","unstructured":"C.  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Monthly 86 (1979), no. 1, 50\u201351.","DOI":"10.1080\/00029890.1979.11994730"},{"key":"2024052809495399441_j_mcma-2023-2025_ref_005","doi-asserted-by":"crossref","unstructured":"A. C.  Berry,\nThe accuracy of the Gaussian approximation to the sum of independent variates,\nTrans. Amer. Math. Soc. 49 (1941), 122\u2013136.","DOI":"10.1090\/S0002-9947-1941-0003498-3"},{"key":"2024052809495399441_j_mcma-2023-2025_ref_006","doi-asserted-by":"crossref","unstructured":"Y.  Cho and S.  Kim,\nVolume of hypercubes clipped by hyperplanes and combinatorial identities,\nElectron. J. Linear Algebra 36 (2020), 228\u2013255.","DOI":"10.13001\/ela.2020.5085"},{"key":"2024052809495399441_j_mcma-2023-2025_ref_007","unstructured":"J.  de Nobel, F.  Ye, D.  Vermetten, H.  Wang, C.  Doerr and T.  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