{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:TBHAQPFKBJWPM3BP4HKNZCYS42","short_pith_number":"pith:TBHAQPFK","canonical_record":{"source":{"id":"1404.4272","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2014-04-16T14:44:53Z","cross_cats_sorted":[],"title_canon_sha256":"fed7e469bb82ddebda6ca63763a826f38fe3360e07e4d3a70245305d83145538","abstract_canon_sha256":"fdc24ee40ee5eb8cd42e01667d500680634220675cee6ef8a320cf3c1af7c312"},"schema_version":"1.0"},"canonical_sha256":"984e083caa0a6cf66c2fe1d4dc8b12e69fd080e985ddfe622cbad1f876dd1dce","source":{"kind":"arxiv","id":"1404.4272","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1404.4272","created_at":"2026-05-18T02:54:05Z"},{"alias_kind":"arxiv_version","alias_value":"1404.4272v1","created_at":"2026-05-18T02:54:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.4272","created_at":"2026-05-18T02:54:05Z"},{"alias_kind":"pith_short_12","alias_value":"TBHAQPFKBJWP","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"TBHAQPFKBJWPM3BP","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"TBHAQPFK","created_at":"2026-05-18T12:28:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:TBHAQPFKBJWPM3BP4HKNZCYS42","target":"record","payload":{"canonical_record":{"source":{"id":"1404.4272","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2014-04-16T14:44:53Z","cross_cats_sorted":[],"title_canon_sha256":"fed7e469bb82ddebda6ca63763a826f38fe3360e07e4d3a70245305d83145538","abstract_canon_sha256":"fdc24ee40ee5eb8cd42e01667d500680634220675cee6ef8a320cf3c1af7c312"},"schema_version":"1.0"},"canonical_sha256":"984e083caa0a6cf66c2fe1d4dc8b12e69fd080e985ddfe622cbad1f876dd1dce","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:05.800433Z","signature_b64":"ZseRWNpT7DXmUjOBUxqundm1ZIi1U1VxzKfC7iPIVwCih9iV2l/Nz6YS9B4rXPmsFJaf1NmfIv+qK+uBDHtmAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"984e083caa0a6cf66c2fe1d4dc8b12e69fd080e985ddfe622cbad1f876dd1dce","last_reissued_at":"2026-05-18T02:54:05.799752Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:05.799752Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1404.4272","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:54:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5rmWItNF+4ZIGKx+pNSifORpi8uyvspANXG9Yg6FG9WFCv1ohQFsSY1ccqBglg8sS8SUuQgrB6nulES6uAhRCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T05:49:22.658249Z"},"content_sha256":"bb0e089a46a9eca0d6176754071a04ad78bc49d974495ac5dfa4a491334a1704","schema_version":"1.0","event_id":"sha256:bb0e089a46a9eca0d6176754071a04ad78bc49d974495ac5dfa4a491334a1704"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:TBHAQPFKBJWPM3BP4HKNZCYS42","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fast and exact implementation of 3-dimensional Tukey depth regions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"stat.CO","authors_text":"Xiaohui Liu","submitted_at":"2014-04-16T14:44:53Z","abstract_excerpt":"Tukey depth regions are important notions in nonparametric multivariate data analysis. A $\\tau$-th Tukey depth region $\\mathcal{D}_{\\tau}$ is the set of all points that have at least depth $\\tau$. While the Tukey depth regions are easily defined and interpreted as $p$-variate quantiles, their practical applications is impeded by the lack of efficient computational procedures in dimensions with $p > 2$. Feasible algorithms are available, but practically very slow. In this paper we present a new exact algorithm for 3-dimensional data. An efficient implementation is also provided. Data examples i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.4272","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:54:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZcEws6GyRd9dDx5Ij7Gi4hAb6SBHCJvA0WEbn+IxNdV6Xlft2/gWiJdj2LBCaveY8XJreIdtxsipyXcs9veqAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T05:49:22.658972Z"},"content_sha256":"fc409d9a4df572a0a3f932bf234e80594abb82258b7ed610f9a8331a51113530","schema_version":"1.0","event_id":"sha256:fc409d9a4df572a0a3f932bf234e80594abb82258b7ed610f9a8331a51113530"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TBHAQPFKBJWPM3BP4HKNZCYS42/bundle.json","state_url":"https://pith.science/pith/TBHAQPFKBJWPM3BP4HKNZCYS42/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TBHAQPFKBJWPM3BP4HKNZCYS42/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-10T05:49:22Z","links":{"resolver":"https://pith.science/pith/TBHAQPFKBJWPM3BP4HKNZCYS42","bundle":"https://pith.science/pith/TBHAQPFKBJWPM3BP4HKNZCYS42/bundle.json","state":"https://pith.science/pith/TBHAQPFKBJWPM3BP4HKNZCYS42/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TBHAQPFKBJWPM3BP4HKNZCYS42/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:TBHAQPFKBJWPM3BP4HKNZCYS42","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"fdc24ee40ee5eb8cd42e01667d500680634220675cee6ef8a320cf3c1af7c312","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2014-04-16T14:44:53Z","title_canon_sha256":"fed7e469bb82ddebda6ca63763a826f38fe3360e07e4d3a70245305d83145538"},"schema_version":"1.0","source":{"id":"1404.4272","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1404.4272","created_at":"2026-05-18T02:54:05Z"},{"alias_kind":"arxiv_version","alias_value":"1404.4272v1","created_at":"2026-05-18T02:54:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.4272","created_at":"2026-05-18T02:54:05Z"},{"alias_kind":"pith_short_12","alias_value":"TBHAQPFKBJWP","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"TBHAQPFKBJWPM3BP","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"TBHAQPFK","created_at":"2026-05-18T12:28:49Z"}],"graph_snapshots":[{"event_id":"sha256:fc409d9a4df572a0a3f932bf234e80594abb82258b7ed610f9a8331a51113530","target":"graph","created_at":"2026-05-18T02:54:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Tukey depth regions are important notions in nonparametric multivariate data analysis. A $\\tau$-th Tukey depth region $\\mathcal{D}_{\\tau}$ is the set of all points that have at least depth $\\tau$. While the Tukey depth regions are easily defined and interpreted as $p$-variate quantiles, their practical applications is impeded by the lack of efficient computational procedures in dimensions with $p > 2$. Feasible algorithms are available, but practically very slow. In this paper we present a new exact algorithm for 3-dimensional data. An efficient implementation is also provided. Data examples i","authors_text":"Xiaohui Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2014-04-16T14:44:53Z","title":"Fast and exact implementation of 3-dimensional Tukey depth regions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.4272","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:bb0e089a46a9eca0d6176754071a04ad78bc49d974495ac5dfa4a491334a1704","target":"record","created_at":"2026-05-18T02:54:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"fdc24ee40ee5eb8cd42e01667d500680634220675cee6ef8a320cf3c1af7c312","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2014-04-16T14:44:53Z","title_canon_sha256":"fed7e469bb82ddebda6ca63763a826f38fe3360e07e4d3a70245305d83145538"},"schema_version":"1.0","source":{"id":"1404.4272","kind":"arxiv","version":1}},"canonical_sha256":"984e083caa0a6cf66c2fe1d4dc8b12e69fd080e985ddfe622cbad1f876dd1dce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"984e083caa0a6cf66c2fe1d4dc8b12e69fd080e985ddfe622cbad1f876dd1dce","first_computed_at":"2026-05-18T02:54:05.799752Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:54:05.799752Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZseRWNpT7DXmUjOBUxqundm1ZIi1U1VxzKfC7iPIVwCih9iV2l/Nz6YS9B4rXPmsFJaf1NmfIv+qK+uBDHtmAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:54:05.800433Z","signed_message":"canonical_sha256_bytes"},"source_id":"1404.4272","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bb0e089a46a9eca0d6176754071a04ad78bc49d974495ac5dfa4a491334a1704","sha256:fc409d9a4df572a0a3f932bf234e80594abb82258b7ed610f9a8331a51113530"],"state_sha256":"d424c3d44eef4c47933b3a40df1bad92f230daa6a716d34956d2b17206163dc5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r4Xod5yGaUG0FN1tVWkafESOUCxcmmzxamrdmeAPltujOgNiqiEwmkz+SmJIuwtIXMQC14VkYLZp3wrsA/XyDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T05:49:22.662181Z","bundle_sha256":"fa47c6396d7f43412688c073634094d0cfcf78a1fed63aa47467c2682dcb9fba"}}