[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-160102-en":3,"doc-seo-160102-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},160102,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","Principal Component Analysis - Introductory concepts and multivariate exploratory analysis","Principal Component Analysis (PCA) is introduced as one of three core exploratory multivariate tools used to summarize many variables simultaneously. The material explains how vectors represent variables geometrically, how correlations correspond to angles between vectors, and how analyses arise by projecting these vectors onto low-dimensional planes. It also describes variables as dimensions, illustrating how cases form clusters whose shapes reflect the relative spread of observations across variables, supporting visual discovery of inter-relationships.","ADVANCED BIOSTATICS ABSTAT18  \nPrincipal Component Analysis  \nCarina Silva  \n([carina.silva@estesl.ipl.pt](carina.silva@estesl.ipl.pt)) Higher School of Technologies and Health of Lisbon & Center of Statistics and Applications, University of Lisbon j CEAUL  \nIGC, April 3rd-6th, 2018  \nAdvanced Biostatistics ~~ ~~Introduction  \nIntroduction  \nI The tools used for exploratory analyses are mainly multivariate methods, i.e. statistical methods dealing with many variables at the same time.  \nAdvanced Biostatistics ~~ ~~Introduction  \nIntroduction  \nI The tools used for exploratory analyses are mainly multivariate methods, i.e. statistical methods dealing with many variables at the same time.  \nI There are three basic tools of exploratory analysis (= exploratory statistics): Principal Component Analysis (PCA); Correspondence Analysis (CA or COA) and Multi-Dimensional Scaling (MDS) .  \nAdvanced Biostatistics ~~ ~~Introduction  \nIntroduction  \nI The tools used for exploratory analyses are mainly multivariate methods, i.e. statistical methods dealing with many variables at the same time.  \nI There are three basic tools of exploratory analysis (= exploratory statistics): Principal Component Analysis (PCA); Correspondence Analysis (CA or COA) and Multi-Dimensional Scaling (MDS) .  \nI These three methods are based on the same fundamental principles. Their role is to visually summarize all analyzed variables, and to reveal their inter-relationships.  \nAdvanced Biostatistics ~~ ~~Introduction  \nPrincipal Component Analysis  \nThe principles are:  \nI variables are expressed geometrically as vectors,  \nAdvanced Biostatistics ~~ ~~Introduction  \nPrincipal Component Analysis  \nThe principles are:  \nI variables are expressed geometrically as vectors, I correlations as angles between vectors,  \nAdvanced Biostatistics ~~ ~~Introduction  \nPrincipal Component Analysis  \nThe principles are:  \nI variables are expressed geometrically as vectors,  \nI correlations as angles between vectors,  \nI analyses are seen as projections of these vectors onto planes.  \nAdvanced Biostatistics ~~ ~~Introduction  \nVariables as dimensions  \nA variable is represented by a straight line (i.e. a vector), having an origin (point 0), and a direction (positive values or negative values) .  \nI If we consider 3 variables, they correspond to 3 distinct lines, hence de􀀌ning 3 dimensions, i.e. a space:  \n4 / 43  \nAdvanced Biostatistics ~~ ~~Introduction  \nI Let's consider water quality sampling in a river. In one given location, temperature, oxygen rate and nitrates concentration are measured; this corresponds to one data point (i.e. one case, on site S at time t) in which 3 variables are expressed. This point is located somewhere in the space created by the 3 variables, according to the values taken for each variable:  \nAdvanced Biostatistics ~~ ~~Introduction  \nVariables as dimensions  \nI If we have for example 15 locations (cases) to study 3 variables, then the 15 points constitute a cluster in a space of dimension 3 .  \nI This cluster can have a relatively spherical shape if data points are more or less distributed in the same way for each variable I On the contrary the cluster will have an elongated shape if the points are more scattered on one variable than on the others.","cbCaip8wYvU0Jnc2","https://ap.wps.com/l/cbCaip8wYvU0Jnc2","pdf",649858,1,65,"English","en",105,"# Introduction\n## Exploratory multivariate tools\n## Geometric principles of PCA\n## Variables as dimensions and clusters","[{\"question\":\"What is the role of PCA in exploratory analysis?\",\"answer\":\"PCA is used to visually summarize analyzed variables and reveal relationships among them using multivariate exploratory statistics.\"},{\"question\":\"How are variables and correlations represented in PCA?\",\"answer\":\"In PCA, variables are expressed as vectors; correlations are represented as angles between those vectors.\"},{\"question\":\"What does a cluster shape indicate when using variables as dimensions?\",\"answer\":\"With multiple cases, the points form a cluster in a space; a more spherical cluster suggests similar spread across variables, while an elongated cluster indicates greater scattering on one variable than the others.\"}]","Principal Component Analysis - Introductory concepts and multivariate exploratory analysis | PDF",1788050438,164,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"principal-component-analysis-introductory-concepts-and-multivariate-exploratory-analysis","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/principal-component-analysis-introductory-concepts-and-multivariate-exploratory-analysis/160102/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-30",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the role of PCA in exploratory analysis?","Question",{"text":75,"@type":76},"PCA is used to visually summarize analyzed variables and reveal relationships among them using multivariate exploratory statistics.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are variables and correlations represented in PCA?",{"text":80,"@type":76},"In PCA, variables are expressed as vectors; correlations are represented as angles between those vectors.",{"name":82,"@type":73,"acceptedAnswer":83},"What does a cluster shape indicate when using variables as dimensions?",{"text":84,"@type":76},"With multiple cases, the points form a cluster in a space; a more spherical cluster suggests similar spread across variables, while an elongated cluster indicates greater scattering on one variable than the others.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]