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{"title"=>"Analysis of population structure: A unifying framework and novel methods based on sparse factor analysis", "type"=>"journal", "authors"=>[{"first_name"=>"Barbara E.", "last_name"=>"Engelhardt", "scopus_author_id"=>"25635224600"}, {"first_name"=>"Matthew", "last_name"=>"Stephens", "scopus_author_id"=>"7201574987"}], "year"=>2010, "source"=>"PLoS Genetics", "identifiers"=>{"scopus"=>"2-s2.0-78049415423", "issn"=>"15537390", "pui"=>"359882467", "sgr"=>"78049415423", "pmid"=>"20862358", "doi"=>"10.1371/journal.pgen.1001117"}, "id"=>"d6fa34d3-95de-3fa6-9b2a-655f342ad4d5", "abstract"=>"We consider the statistical analysis of population structure using genetic data. We show how the two most widely used approaches to modeling population structure, admixture-based models and principal components analysis (PCA), can be viewed within a single unifying framework of matrix factorization. Specifically, they can both be interpreted as approximating an observed genotype matrix by a product of two lower-rank matrices, but with different constraints or prior distributions on these lower-rank matrices. This opens the door to a large range of possible approaches to analyzing population structure, by considering other constraints or priors. In this paper, we introduce one such novel approach, based on sparse factor analysis (SFA). We investigate the effects of the different types of constraint in several real and simulated data sets. We find that SFA produces similar results to admixture-based models when the samples are descended from a few well-differentiated ancestral populations and can recapitulate the results of PCA when the population structure is more \"continuous,\" as in isolation-by-distance models.", "link"=>"http://www.mendeley.com/research/analysis-population-structure-unifying-framework-novel-methods-based-sparse-factor-analysis", "reader_count"=>229, "reader_count_by_academic_status"=>{"Unspecified"=>1, "Professor > Associate Professor"=>16, "Librarian"=>1, "Researcher"=>63, "Student > Doctoral Student"=>2, "Student > Ph. D. Student"=>92, "Student > Postgraduate"=>10, "Student > Master"=>15, "Other"=>3, "Student > Bachelor"=>13, "Lecturer"=>4, "Lecturer > Senior Lecturer"=>1, "Professor"=>8}, "reader_count_by_user_role"=>{"Unspecified"=>1, "Professor > Associate Professor"=>16, "Librarian"=>1, "Researcher"=>63, "Student > Doctoral Student"=>2, "Student > Ph. D. Student"=>92, "Student > Postgraduate"=>10, "Student > Master"=>15, "Other"=>3, "Student > Bachelor"=>13, "Lecturer"=>4, "Lecturer > Senior Lecturer"=>1, "Professor"=>8}, "reader_count_by_subject_area"=>{"Unspecified"=>8, "Agricultural and Biological Sciences"=>158, "Veterinary Science and Veterinary Medicine"=>1, "Business, Management and Accounting"=>1, "Chemistry"=>1, "Computer Science"=>26, "Decision Sciences"=>1, "Economics, Econometrics and Finance"=>2, "Engineering"=>2, "Environmental Science"=>3, "Biochemistry, Genetics and Molecular Biology"=>11, "Mathematics"=>10, "Medicine and Dentistry"=>3, "Psychology"=>1, "Social Sciences"=>1}, "reader_count_by_subdiscipline"=>{"Medicine and Dentistry"=>{"Medicine and Dentistry"=>3}, "Social Sciences"=>{"Social Sciences"=>1}, "Decision Sciences"=>{"Decision Sciences"=>1}, "Psychology"=>{"Psychology"=>1}, "Mathematics"=>{"Mathematics"=>10}, "Unspecified"=>{"Unspecified"=>8}, "Environmental Science"=>{"Environmental Science"=>3}, "Engineering"=>{"Engineering"=>2}, "Chemistry"=>{"Chemistry"=>1}, "Economics, Econometrics and Finance"=>{"Economics, Econometrics and Finance"=>2}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>158}, "Computer Science"=>{"Computer Science"=>26}, "Business, Management and Accounting"=>{"Business, Management and Accounting"=>1}, "Biochemistry, Genetics and Molecular Biology"=>{"Biochemistry, Genetics and Molecular Biology"=>11}, "Veterinary Science and Veterinary Medicine"=>{"Veterinary Science and Veterinary Medicine"=>1}}, "reader_count_by_country"=>{"Netherlands"=>2, "Sweden"=>1, "Austria"=>1, "Belgium"=>1, "United States"=>17, "United Kingdom"=>1, "Zimbabwe"=>1, "Italy"=>1, "France"=>1, "Switzerland"=>1, "Germany"=>3, "Indonesia"=>1}, "group_count"=>5}

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• {"id"=>"458765408003457024", "text"=>"Unlike PCA, Sparse Factor Analysis / Dictionary Learning is robust to sampling scheme, actually discovers structure. http://t.co/KWqKb7DIoq", "created_at"=>"2014-04-23T00:32:51Z", "user"=>"gusl", "user_name"=>"Gustavo Lacerda", "user_profile_image"=>"http://pbs.twimg.com/profile_images/378800000272682670/cc60414e9b9902cd9faaae68bdf32b5b_normal.jpeg"}
• {"id"=>"458772659900514305", "text"=>"RT @gusl: Unlike PCA, Sparse Factor Analysis / Dictionary Learning is robust to sampling scheme, actually discovers structure. http://t.co/…", "created_at"=>"2014-04-23T01:01:40Z", "user"=>"nsaphra", "user_name"=>"Naomi Saphra", "user_profile_image"=>"http://pbs.twimg.com/profile_images/2419258103/bm0h1v99z51lynto3t1o_normal.gif"}
• {"id"=>"458775978584465408", "text"=>"RT @gusl: Unlike PCA, Sparse Factor Analysis / Dictionary Learning is robust to sampling scheme, actually discovers structure. http://t.co/…", "created_at"=>"2014-04-23T01:14:51Z", "user"=>"bobpoekert", "user_name"=>"Bob", "user_profile_image"=>"http://pbs.twimg.com/profile_images/421513627900735488/24nZosLR_normal.jpeg"}
• {"id"=>"472273614625320960", "text"=>"RT @gusl: Unlike PCA, Sparse Factor Analysis / Dictionary Learning is robust to sampling scheme, actually discovers structure. http://t.co/…", "created_at"=>"2014-05-30T07:09:38Z", "user"=>"IgorCarron", "user_name"=>"igorcarron", "user_profile_image"=>"http://pbs.twimg.com/profile_images/52435623/igor_normal.JPG"}
• {"id"=>"472290685068906496", "text"=>"RT @gusl: Unlike PCA, Sparse Factor Analysis / Dictionary Learning is robust to sampling scheme, actually discovers structure. http://t.co/…", "created_at"=>"2014-05-30T08:17:28Z", "user"=>"amodjaiga", "user_name"=>"Amod", "user_profile_image"=>"http://pbs.twimg.com/profile_images/417874665785143296/bAUz6Lye_normal.jpeg"}
• {"id"=>"488171328453611521", "text"=>"@SamWangPhD @ianholmes @joe_pickrell I like @BeEngelhardt paper 4 view of connections btwn approaches http://t.co/EFNvb3YNU4", "created_at"=>"2014-07-13T04:01:29Z", "user"=>"Graham_Coop", "user_name"=>"Graham Coop", "user_profile_image"=>"http://pbs.twimg.com/profile_images/2427468315/Portugal_graham_small_normal.jpg"}
• {"id"=>"488178184878190592", "text"=>"RT @Graham_Coop: @SamWangPhD @ianholmes @joe_pickrell I like @BeEngelhardt paper 4 view of connections btwn approaches http://t.co/EFNvb3Y…", "created_at"=>"2014-07-13T04:28:43Z", "user"=>"ianholmes", "user_name"=>"Ian Holmes", "user_profile_image"=>"http://pbs.twimg.com/profile_images/2228244796/Ian_Holmes_portrait_by_Janet_Barron_normal.jpg"}
• {"id"=>"488193802939547649", "text"=>"RT @Graham_Coop: @SamWangPhD @ianholmes @joe_pickrell I like @BeEngelhardt paper 4 view of connections btwn approaches http://t.co/EFNvb3Y…", "created_at"=>"2014-07-13T05:30:47Z", "user"=>"socalsysbio", "user_name"=>"Tau-Mu Yi", "user_profile_image"=>"http://pbs.twimg.com/profile_images/1168387482/yi_normal.jpg"}
• {"id"=>"514967044983255040", "text"=>"@bbuliksullivan http://t.co/BmZEAQmZw7", "created_at"=>"2014-09-25T02:38:05Z", "user"=>"Graham_Coop", "user_name"=>"Graham Coop", "user_profile_image"=>"http://pbs.twimg.com/profile_images/2427468315/Portugal_graham_small_normal.jpg"}
• {"id"=>"519121885611577344", "text"=>"Inviting figs. of sparse factor analysis for population structure are saddeningly low rez. http://t.co/e3o6sBNCQ0 http://t.co/T4VXdujKJT", "created_at"=>"2014-10-06T13:47:57Z", "user"=>"mja", "user_name"=>"Mark James Adams", "user_profile_image"=>"http://pbs.twimg.com/profile_images/514757536956702721/zR_MfMj4_normal.jpeg"}
• {"id"=>"759076787552935936", "text"=>"@jgschraiber another great paper for understanding structure-like analyses is https://t.co/9HUYkBvmbq by @BeEngelhardt and @mstephens999", "created_at"=>"2016-07-29T17:23:03Z", "user"=>"pontus_skoglund", "user_name"=>"Pontus Skoglund", "user_profile_image"=>"http://pbs.twimg.com/profile_images/711249110918144001/Un0orGY3_normal.jpg"}
• {"id"=>"759078669415358464", "text"=>"RT @pontus_skoglund: @jgschraiber another great paper for understanding structure-like analyses is https://t.co/9HUYkBvmbq by @BeEngelhardt…", "created_at"=>"2016-07-29T17:30:32Z", "user"=>"AmyAGoldberg", "user_name"=>"Amy Goldberg", "user_profile_image"=>"http://pbs.twimg.com/profile_images/720512702406336513/en8hQkaC_normal.jpg"}
• {"id"=>"759081501661229056", "text"=>"RT @pontus_skoglund: @jgschraiber another great paper for understanding structure-like analyses is https://t.co/9HUYkBvmbq by @BeEngelhardt…", "created_at"=>"2016-07-29T17:41:47Z", "user"=>"jgschraiber", "user_name"=>"Joshua G. Schraiber", "user_profile_image"=>"http://pbs.twimg.com/profile_images/378800000563181239/626fcb69854eedfbe60f66af22cc2d14_normal.jpeg"}
• {"id"=>"759095623702372352", "text"=>"RT @pontus_skoglund: @jgschraiber another great paper for understanding structure-like analyses is https://t.co/9HUYkBvmbq by @BeEngelhardt…", "created_at"=>"2016-07-29T18:37:54Z", "user"=>"pop_gen_JED", "user_name"=>"Jared Decker", "user_profile_image"=>"http://pbs.twimg.com/profile_images/725516876504125440/FDt-recG_normal.jpg"}
• {"id"=>"759124068041887745", "text"=>"RT @pontus_skoglund: @jgschraiber another great paper for understanding structure-like analyses is https://t.co/9HUYkBvmbq by @BeEngelhardt…", "created_at"=>"2016-07-29T20:30:56Z", "user"=>"LudovicLorlando", "user_name"=>"Ludovic Orlando", "user_profile_image"=>"http://pbs.twimg.com/profile_images/731817134624411648/V04CCMZx_normal.jpg"}
• {"id"=>"759258705418391553", "text"=>"RT @pontus_skoglund: @jgschraiber another great paper for understanding structure-like analyses is https://t.co/9HUYkBvmbq by @BeEngelhardt…", "created_at"=>"2016-07-30T05:25:56Z", "user"=>"paikusuma", "user_name"=>"Pradiptajati Kusuma", "user_profile_image"=>"http://pbs.twimg.com/profile_images/648154676622458880/mk5qAksh_normal.jpg"}
• {"id"=>"915038480622813185", "text"=>"@GreeneScientist Structure and admixture are basically also matrix factorization. https://t.co/KdF6UjYYSU", "created_at"=>"2017-10-03T02:19:09Z", "user"=>"mstephens999", "user_name"=>"Matthew Stephens", "user_profile_image"=>"http://pbs.twimg.com/profile_images/487371132345917442/Ta0v2Lvl_normal.jpeg"}
• {"id"=>"915041724245905408", "text"=>"RT @mstephens999: @GreeneScientist Structure and admixture are basically also matrix factorization. https://t.co/KdF6UjYYSU", "created_at"=>"2017-10-03T02:32:02Z", "user"=>"AmandaLea14", "user_name"=>"Amanda Lea", "user_profile_image"=>"http://pbs.twimg.com/profile_images/542754210471628800/_2UHFFo5_normal.jpeg"}
• {"id"=>"915048410184523776", "text"=>"RT @mstephens999: @GreeneScientist Structure and admixture are basically also matrix factorization. https://t.co/KdF6UjYYSU", "created_at"=>"2017-10-03T02:58:36Z", "user"=>"anshulkundaje", "user_name"=>"ANSHUL KUNDAJE", "user_profile_image"=>"http://pbs.twimg.com/profile_images/3001862173/4dc86f565ba23454ec1c9760228b3d60_normal.jpeg"}
• {"id"=>"915284537336360960", "text"=>"RT @mstephens999: @GreeneScientist Structure and admixture are basically also matrix factorization. https://t.co/KdF6UjYYSU", "created_at"=>"2017-10-03T18:36:53Z", "user"=>"hheydarii", "user_name"=>"Hamed Heydari", "user_profile_image"=>"http://pbs.twimg.com/profile_images/881627012804345856/zzxnEAcj_normal.jpg"}
• {"id"=>"915397324406276096", "text"=>"RT @mstephens999: @GreeneScientist Structure and admixture are basically also matrix factorization. https://t.co/KdF6UjYYSU", "created_at"=>"2017-10-04T02:05:04Z", "user"=>"shihuwenbo", "user_name"=>"Huwenbo Shi", "user_profile_image"=>"http://pbs.twimg.com/profile_images/850205575933603840/vSBpc0o6_normal.jpg"}
• {"id"=>"915709284788391938", "text"=>"RT @mstephens999: @GreeneScientist Structure and admixture are basically also matrix factorization. https://t.co/KdF6UjYYSU", "created_at"=>"2017-10-04T22:44:41Z", "user"=>"jwbelmon", "user_name"=>"John Belmont", "user_profile_image"=>"http://pbs.twimg.com/profile_images/906374002808983552/86P8zKYd_normal.jpg"}

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Figshare

• {"files"=>["https://ndownloader.figshare.com/files/413210", "https://ndownloader.figshare.com/files/413234", "https://ndownloader.figshare.com/files/413252"], "description"=>"<div><p>We consider the statistical analysis of population structure using genetic data. We show how the two most widely used approaches to modeling population structure, admixture-based models and principal components analysis (PCA), can be viewed within a single unifying framework of matrix factorization. Specifically, they can both be interpreted as approximating an observed genotype matrix by a product of two lower-rank matrices, but with different constraints or prior distributions on these lower-rank matrices. This opens the door to a large range of possible approaches to analyzing population structure, by considering other constraints or priors. In this paper, we introduce one such novel approach, based on sparse factor analysis (SFA). We investigate the effects of the different types of constraint in several real and simulated data sets. We find that SFA produces similar results to admixture-based models when the samples are descended from a few well-differentiated ancestral populations and can recapitulate the results of PCA when the population structure is more “continuous,” as in isolation-by-distance models.</p></div>", "links"=>[], "tags"=>["unifying", "methods", "based", "sparse"], "article_id"=>141632, "categories"=>["Mathematics", "Cancer", "Genetics"], "users"=>["Barbara E. Engelhardt", "Matthew Stephens"], "doi"=>["https://dx.doi.org/10.1371/journal.pgen.1001117.s001", "https://dx.doi.org/10.1371/journal.pgen.1001117.s002", "https://dx.doi.org/10.1371/journal.pgen.1001117.s003"], "stats"=>{"downloads"=>8, "page_views"=>11, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/Analysis_of_Population_Structure_A_Unifying_Framework_and_Novel_Methods_Based_on_Sparse_Factor_Analysis/141632", "title"=>"Analysis of Population Structure: A Unifying Framework and Novel Methods Based on Sparse Factor Analysis", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2010-09-16 00:27:12"}
• {"files"=>["https://ndownloader.figshare.com/files/830397"], "description"=>"<p>In Panel A, each dot represents a population colored according to habitat and location. Colors in Panels B and C indicate locations in Panel A. Panel B shows how SFA captures the structure with a six factor model. Loadings on the first three factors (first row of Panel B) correspond to location in the first habitat; individuals in the second habitat have essentially zero loading on these factors. Similarly, loadings on the other three factors (second row of Panel B) correspond to location in the second habitat. Panel C shows estimated loadings from PCA for the same data. Each plot shows one loading plotted against another. Although the PCA results clearly reflect the underlying structure one might struggle to infer the structure from visual inspection of these plots if the colors were unknown.</p>", "links"=>[], "tags"=>["simulated", "genotype", "2-d"], "article_id"=>500769, "categories"=>["Mathematics", "Infectious Diseases", "Genetics"], "users"=>["Barbara E. Engelhardt", "Matthew Stephens"], "doi"=>"https://dx.doi.org/10.1371/journal.pgen.1001117.g007", "stats"=>{"downloads"=>1, "page_views"=>4, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Results_on_simulated_genotype_data_from_a_two_independent_2_D_habitats_/500769", "title"=>"Results on simulated genotype data from a two independent 2-D habitats.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2010-09-16 00:12:49"}
• {"files"=>["https://ndownloader.figshare.com/files/830596"], "description"=>"<p>This plot shows good correlation between the relative admixture proportions from SFA and the estimated admixture proportions from previous work. The colors coding the groups are described in the India map.</p>", "links"=>[], "tags"=>["admixture", "proportions", "indian", "sfa"], "article_id"=>500964, "categories"=>["Mathematics", "Infectious Diseases", "Genetics"], "users"=>["Barbara E. Engelhardt", "Matthew Stephens"], "doi"=>"https://dx.doi.org/10.1371/journal.pgen.1001117.g010", "stats"=>{"downloads"=>8, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Plot_of_estimated_admixture_proportions_of_each_Indian_group_versus_the_relative_admixture_proportions_from_SFA_on_the_Indian_data_set_/500964", "title"=>"Plot of estimated admixture proportions of each Indian group versus the relative admixture proportions from SFA on the Indian data set.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2010-09-16 00:16:04"}
• {"files"=>["https://ndownloader.figshare.com/files/830285"], "description"=>"<p>In each plot the factors (-axis) are plotted against the population allele frequencies for the closest-matching population. The SFA factors were truncated to have a minimum of zero and scaled to have a maximum of one. The dashed diagonal line shows .</p>", "links"=>[], "tags"=>["scaled", "factors", "sfa", "1-d", "isolation-by-distance", "simulation", "generating", "allele"], "article_id"=>500652, "categories"=>["Mathematics", "Infectious Diseases", "Genetics"], "users"=>["Barbara E. Engelhardt", "Matthew Stephens"], "doi"=>"https://dx.doi.org/10.1371/journal.pgen.1001117.g005", "stats"=>{"downloads"=>0, "page_views"=>1, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Estimated_scaled_factors_from_SFA_and_admixture_on_the_1_D_isolation_by_distance_simulation_against_the_generating_allele_frequencies_/500652", "title"=>"Estimated scaled factors from SFA and admixture on the 1-D isolation-by-distance simulation against the generating allele frequencies.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2010-09-16 00:10:52"}
• {"files"=>["https://ndownloader.figshare.com/files/830444"], "description"=>"<p>All plots show the individuals on the -axis (colored and ordered by location with respect to the 1-D clustered isolation-by-distance model) plotted against the estimated loadings.</p>", "links"=>[], "tags"=>["pca", "clustered", "1-d"], "article_id"=>500814, "categories"=>["Mathematics", "Infectious Diseases", "Genetics"], "users"=>["Barbara E. Engelhardt", "Matthew Stephens"], "doi"=>"https://dx.doi.org/10.1371/journal.pgen.1001117.g008", "stats"=>{"downloads"=>0, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Results_from_SFA_admixture_and_PCA_for_the_clustered_1_D_simulation_/500814", "title"=>"Results from SFA, admixture, and PCA for the clustered 1-D simulation.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2010-09-16 00:13:34"}
• {"files"=>["https://ndownloader.figshare.com/files/830536"], "description"=>"<p>These results were rotated (but not rescaled) to make the correspondence to the map of Europe more immediately obvious. The results from SFAm are very similar to the results from PCA for these data, effectively recapitulating the geography of Europe.</p>", "links"=>[], "tags"=>["popres", "european"], "article_id"=>500912, "categories"=>["Mathematics", "Infectious Diseases", "Genetics"], "users"=>["Barbara E. Engelhardt", "Matthew Stephens"], "doi"=>"https://dx.doi.org/10.1371/journal.pgen.1001117.g009", "stats"=>{"downloads"=>5, "page_views"=>136, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Results_from_PCA_SFAm_and_admixture_for_the_POPRES_European_data_/500912", "title"=>"Results from PCA, SFAm, and admixture for the POPRES European data.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2010-09-16 00:15:12"}
• {"files"=>["https://ndownloader.figshare.com/files/830350"], "description"=>"<p>In Panel A, each dot represents a population colored according to location. In Panel B, each plot is of the loadings across individuals against each other, where the colors correspond to their locations in Panel A. The first row shows the three SFA loadings against each other from a three factor model. The second row shows the second two PCA loadings, the SFAm loadings, and the mapped admixture loadings (see text for details). All of the methods recapitulate, to a greater or lesser extent, the geographical structure of the habitats (up to rotation).</p>", "links"=>[], "tags"=>["applied", "simulated", "genotype", "2-d"], "article_id"=>500721, "categories"=>["Mathematics", "Infectious Diseases", "Genetics"], "users"=>["Barbara E. Engelhardt", "Matthew Stephens"], "doi"=>"https://dx.doi.org/10.1371/journal.pgen.1001117.g006", "stats"=>{"downloads"=>1, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Results_of_SFA_PCA_SFAm_and_admixture_applied_to_simulated_genotype_data_from_a_single_2_D_habitat_/500721", "title"=>"Results of SFA, PCA, SFAm, and admixture applied to simulated genotype data from a single 2-D habitat.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2010-09-16 00:12:01"}
• {"files"=>["https://ndownloader.figshare.com/files/830673"], "description"=>"<p>Relationship of terms in PCA, SFA, and admixture-based models.</p>", "links"=>[], "tags"=>["admixture-based"], "article_id"=>501044, "categories"=>["Mathematics", "Infectious Diseases", "Genetics"], "users"=>["Barbara E. Engelhardt", "Matthew Stephens"], "doi"=>"https://dx.doi.org/10.1371/journal.pgen.1001117.t001", "stats"=>{"downloads"=>11, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Relationship_of_terms_in_PCA_SFA_and_admixture_based_models_/501044", "title"=>"Relationship of terms in PCA, SFA, and admixture-based models.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2010-09-16 00:17:24"}
• {"files"=>["https://ndownloader.figshare.com/files/830026"], "description"=>"<p>Each plot shows the estimated loadings (-axis) across individuals (-axis). SFA loadings are in the first row, PCA loadings in the second, and admixture loadings in the third. European individuals are denoted with blue ‘x’s, African individuals are denoted with red triangles, and Asian individuals are denoted with green ‘+’s. A dashed horizontal line is at zero on the -axis.</p>", "links"=>[], "tags"=>["applying", "hapmap", "genotype"], "article_id"=>500405, "categories"=>["Mathematics", "Infectious Diseases", "Genetics"], "users"=>["Barbara E. Engelhardt", "Matthew Stephens"], "doi"=>"https://dx.doi.org/10.1371/journal.pgen.1001117.g003", "stats"=>{"downloads"=>3, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Results_of_applying_SFA_PCA_and_admixture_to_the_HapMap_genotype_data_/500405", "title"=>"Results of applying SFA, PCA, and admixture to the HapMap genotype data.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2010-09-16 00:06:45"}
• {"files"=>["https://ndownloader.figshare.com/files/829940"], "description"=>"<p>Each matrix in Equation 1 is illustrated by a blue rectangle and labeled. As in Equation 2, a single element of genotype matrix , is shown in red, and is computed from the product of the appropriate factor loading and factor vectors plus the corresponding random error term (all highlighted in red).</p>", "links"=>[], "tags"=>["matrix", "factorization"], "article_id"=>500311, "categories"=>["Mathematics", "Infectious Diseases", "Genetics"], "users"=>["Barbara E. Engelhardt", "Matthew Stephens"], "doi"=>"https://dx.doi.org/10.1371/journal.pgen.1001117.g001", "stats"=>{"downloads"=>2, "page_views"=>60, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Low_dimensional_matrix_factorization_via_factor_analysis_/500311", "title"=>"Low-dimensional matrix factorization via factor analysis.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2010-09-16 00:05:11"}
• {"files"=>["https://ndownloader.figshare.com/files/829980"], "description"=>"<p>In the first (sparse) representation in the first row, the factors (shown in red) each represent the mean allele frequencies for either the African population () or the European population (); this lends to sparse loadings (shown in blue) for each individual, since the African individuals are only loaded on the factor representing the African population, and likewise for the European individuals. In the second (non-sparse) representation in the second row, each factor is a combination of and , and each individual is loaded onto both factors. Note that the representations are equivalent by the equations under the table. Whereas SFA and admixture-based models tend to choose the first representation because of the sparse priors and implicit regularization, PCA tends towards the second representation (although the actual factors depend on other features of the data such as sample sizes of both groups).</p>", "links"=>[], "tags"=>["ways", "african", "european", "individuals"], "article_id"=>500351, "categories"=>["Mathematics", "Infectious Diseases", "Genetics"], "users"=>["Barbara E. Engelhardt", "Matthew Stephens"], "doi"=>"https://dx.doi.org/10.1371/journal.pgen.1001117.g002", "stats"=>{"downloads"=>4, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Illustration_of_two_different_ways_that_African_and_European_individuals_could_be_represented_/500351", "title"=>"Illustration of two different ways that African and European individuals could be represented.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2010-09-16 00:05:51"}
• {"files"=>["https://ndownloader.figshare.com/files/830075"], "description"=>"<p>In each plot the individuals are colored and ordered along the -axis by location in the 1-D habitat.</p>", "links"=>[], "tags"=>["loadings", "1-d", "isolation-by-distance"], "article_id"=>500446, "categories"=>["Mathematics", "Infectious Diseases", "Genetics"], "users"=>["Barbara E. Engelhardt", "Matthew Stephens"], "doi"=>"https://dx.doi.org/10.1371/journal.pgen.1001117.g004", "stats"=>{"downloads"=>3, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Estimated_factor_loadings_from_PCA_SFAm_SFA_and_admixture_for_the_1_D_isolation_by_distance_simulation_/500446", "title"=>"Estimated factor loadings from PCA, SFAm, SFA, and admixture for the 1-D isolation-by-distance simulation.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2010-09-16 00:07:26"}
• {"files"=>["https://ndownloader.figshare.com/files/830637"], "description"=>"<p>The columns are the four different types of matrix factorizations we considered, and the rows are the different data sets we applied each method to that show easily interpretable results. “NR” indicates that we did not run the method on those data, and a ‘–’ indicates that the results were not straightforward to describe (see <a href=\"http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1001117#s2\" target=\"_blank\">Results</a> for details). <i>Mean</i> indicates that the factor is the mean allele frequencies for the complete set of individuals; <i>contrast</i> indicates a difference in the allele frequencies along a geographical gradient.</p>", "links"=>[], "tags"=>["admixture-based"], "article_id"=>501013, "categories"=>["Mathematics", "Infectious Diseases", "Genetics"], "users"=>["Barbara E. Engelhardt", "Matthew Stephens"], "doi"=>"https://dx.doi.org/10.1371/journal.pgen.1001117.t002", "stats"=>{"downloads"=>5, "page_views"=>13, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Summary_of_results_across_PCA_SFA_and_admixture_based_models_/501013", "title"=>"Summary of results across PCA, SFA, and admixture-based models.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2010-09-16 00:16:53"}

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F1000Prime