{"corpus_id":76652424,"paper_sha":"e79728c349b00b052f6eebeb87d2e737ee3e10c2","doi":"10.1007/s41468-020-00048-w","arxiv_id":"1804.01618","pmid":null,"pmcid":null,"mag_id":3014603530,"dblp_id":"journals/jact/BerryCCF20","acl_id":null,"title":"Functional summaries of persistence diagrams","year":2018,"publication_date":"2018-04-04","venue":"Journal of Applied and Computational Topology","journal":{"name":"Journal of Applied and Computational Topology","pages":"211 - 262","volume":"4"},"journal_issn":null,"journal_title":null,"publication_types":["JournalArticle","Review"],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics","Computer Science"],"reference_count":91,"citation_count":94,"influential_citation_count":5,"is_open_access":false,"arxiv_categories":["stat.ME"],"arxiv_license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","arxiv_journal_ref":null,"mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":null,"s2_open_access_landing_url":null,"s2_open_access_license":null,"s2_open_access_status":null,"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":"One of the primary areas of interest in applied algebraic topology is persistent homology, and, more specifically, the persistence diagram. Persistence diagrams have also become objects of interest in topological data analysis. However, persistence diagrams do not naturally lend themselves to statistical goals, such as inferring certain population characteristics, because their complicated structure makes common algebraic operations—such as addition, division, and multiplication—challenging (e.g., the mean might not be unique). To bypass these issues, several functional summaries of persistence diagrams have been proposed in the literature (e.g. landscape and silhouette functions). The problem of analyzing a set of persistence diagrams then becomes the problem of analyzing a set of functions, which is a topic that has been studied for decades in statistics. First, we review the various functional summaries in the literature and propose a unified framework for the functional summaries. Then, we generalize the definition of persistence landscape functions, establish several theoretical properties of the persistence functional summaries, and demonstrate and discuss their performance in the context of classification using simulated prostate cancer histology data, and two-sample hypothesis tests comparing human and monkey fibrin images, after developing a simulation study using a new data generator we call the Pickup Sticks Simulator (STIX).","claims":[{"public_id":"cl_f499af6ab4eb4c1c72d0a4b3debca5cd","status":"active","text":"A unified framework for the functional summaries of persistence diagrams is proposed.","confidence":0.97,"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/claims/cl_f499af6ab4eb4c1c72d0a4b3debca5cd"},{"public_id":"cl_0f4f5fcb0c5cd7f8a41ab72a83dea30c","status":"active","text":"Performance is demonstrated and discussed through classification on simulated prostate cancer histology data and two-sample hypothesis tests comparing human and monkey fibrin images.","confidence":0.9,"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/claims/cl_0f4f5fcb0c5cd7f8a41ab72a83dea30c"},{"public_id":"cl_735fdcf0f9a222294b6dc23e57190014","status":"active","text":"Persistence landscape functions are generalized within the proposed framework.","confidence":0.95,"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/claims/cl_735fdcf0f9a222294b6dc23e57190014"},{"public_id":"cl_acaac6d3c4ee8643fa2f8b0bcc42985a","status":"active","text":"Pickup Sticks Simulator (STIX) is developed as a new data generator for the simulation study.","confidence":0.93,"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/claims/cl_acaac6d3c4ee8643fa2f8b0bcc42985a"},{"public_id":"cl_50b920fc5c984bcb3b00e82bf649e7be","status":"active","text":"Several theoretical properties of the persistence functional summaries are established.","confidence":0.91,"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/claims/cl_50b920fc5c984bcb3b00e82bf649e7be"}],"concepts":[{"public_id":"co_24acd46f4132ff8072741b932265d267","status":"active","name":"silhouette functions","description":"A weighted function-valued summary of a persistence diagram.","types":["functional summaries"],"aliases":[],"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/concepts/co_24acd46f4132ff8072741b932265d267"},{"public_id":"co_308b372143dbdab150849c1c7f4136e3","status":"active","name":"classification","description":"A supervised learning task used to assess how well the summaries separate labels.","types":["evaluation tasks"],"aliases":[],"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/concepts/co_308b372143dbdab150849c1c7f4136e3"},{"public_id":"co_40518448dba0a498b1cfcf64b34e4f0b","status":"active","name":"two-sample hypothesis tests","description":"Statistical tests used to compare whether two samples come from the same distribution.","types":["statistical tests"],"aliases":["two-sample tests"],"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/concepts/co_40518448dba0a498b1cfcf64b34e4f0b"},{"public_id":"co_41c060c0ecea4c9581c5c8fee34cab85","status":"active","name":"Pickup Sticks Simulator (STIX)","description":"A new data generator used to simulate persistence-diagram-related data for the study.","types":["simulators","data generators"],"aliases":["STIX","Pickup Sticks Simulator"],"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/concepts/co_41c060c0ecea4c9581c5c8fee34cab85"},{"public_id":"co_864a0198753fcfeeecc48b92f0fabe36","status":"active","name":"simulated prostate cancer histology data","description":"Synthetic prostate cancer histology data used for the classification evaluation.","types":["datasets"],"aliases":["prostate cancer histology data"],"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/concepts/co_864a0198753fcfeeecc48b92f0fabe36"},{"public_id":"co_8a4f9df13c0dff83c3302ffb91840d38","status":"active","name":"functional summaries of persistence diagrams","description":"Functions derived from persistence diagrams so they can be analyzed with standard statistical tools.","types":["statistical methods"],"aliases":["functional summaries"],"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/concepts/co_8a4f9df13c0dff83c3302ffb91840d38"},{"public_id":"co_bcf0dad9787d3815ff70db45714431bc","status":"active","name":"human and monkey fibrin images","description":"Image data from human and monkey fibrin used in the two-sample comparison.","types":["datasets","image data"],"aliases":["fibrin images"],"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/concepts/co_bcf0dad9787d3815ff70db45714431bc"},{"public_id":"co_c02fcd33458cfd5dee19136863cadc9a","status":"active","name":"unified framework","description":"A common framework that organizes multiple functional summaries of persistence diagrams.","types":["frameworks"],"aliases":["framework"],"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/concepts/co_c02fcd33458cfd5dee19136863cadc9a"},{"public_id":"co_d00446912c7faae50b9350517b5b6b6e","status":"active","name":"persistence landscape functions","description":"A family of function-valued summaries derived from a persistence diagram.","types":["functional summaries"],"aliases":["landscape functions"],"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/concepts/co_d00446912c7faae50b9350517b5b6b6e"},{"public_id":"co_dc2b913100c36a30b16a7b4a42c54097","status":"active","name":"persistence functional summaries","description":"The family of function-based summaries of persistence diagrams considered in the paper.","types":["summaries"],"aliases":["functional summaries"],"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/concepts/co_dc2b913100c36a30b16a7b4a42c54097"},{"public_id":"co_e2c10e6e7dce8a4ca1696fbeddd7e9a3","status":"active","name":"persistence diagrams","description":"A multiset representation of topological feature birth and death pairs from persistent homology.","types":["topological representations"],"aliases":[],"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/concepts/co_e2c10e6e7dce8a4ca1696fbeddd7e9a3"},{"public_id":"co_ed77646486aea372c358fc00121a81d7","status":"active","name":"persistent homology","description":"A topological data analysis framework that studies how homological features persist across scales.","types":["topological methods"],"aliases":["PH"],"contributors":[{"id":391,"public_id":"x53qfq3ny9","public_label":"kafkapple (x53qfq3ny9)","roles":["extraction"],"url":"https://sah.borca.ai/u/x53qfq3ny9"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["review"],"url":"https://sah.borca.ai/u/322360f1c1"}],"url":"https://sah.borca.ai/concepts/co_ed77646486aea372c358fc00121a81d7"}],"external_ids":{"DOI":"10.1007/s41468-020-00048-w","ArXiv":"1804.01618","PubMed":null,"PubMedCentral":null,"MAG":3014603530,"DBLP":"journals/jact/BerryCCF20","ACL":null},"open_access":{"is_open_access":true,"pdf_url":"https://arxiv.org/pdf/1804.01618","landing_url":"https://arxiv.org/abs/1804.01618","source":"arxiv","pdf_url_source":"derived_arxiv","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","reason":null},"reference_availability":{"status":"available","references_indexed":true,"full_text_available":true,"full_text_source":"arxiv","count_basis":"semantic_scholar_metadata","extraction_status":"not_applicable","reason":null},"source":{"provider":"episteme2","base_corpus":"semantic_scholar_dump","freshness_mode":"unknown","basis":["semantic_scholar_metadata","postgres_metadata"],"limits":["paper metadata is based on indexed upstream scholarly datasets","claims and concepts are available only for extracted papers","absence of claims or concepts means no extracted graph data is available in this response"],"status":"available","degraded":false,"degraded_reasons":[],"diagnostics":{"status":"available","degraded":false,"degraded_reasons":[],"metadata_status":"available","graph_status":"available","abstract_status":"available"},"source_flags":1},"paper_id":630991,"paper_uid":"8810b339-ea1a-4938-aa2a-90f8952eac43","canonical_identity":{"paper_id":630991,"paper_uid":"8810b339-ea1a-4938-aa2a-90f8952eac43","identity_status":"available","lookup_basis":"semantic_scholar_external_id","compatibility_path":"corpus_id"},"url":"https://sah.borca.ai/papers/76652424"}