In [[statistics]], the '''projection matrix''' (<math>(\mathbf{P})</math>),<ref>{{cite book |first=Alexander |last=Basilevsky |title=Applied Matrix Algebra in the Statistical Sciences |___location= |publisher=Dover |year=2005 |isbn=0-486-44538-0 |pages=160–176 |url=https://books.google.com/books?id=ScssAwAAQBAJ&pg=PA160 }}</ref> sometimes also called the '''influence matrix'''<ref>{{cite web |title=Data Assimilation: Observation influence diagnostic of a data assimilation system |url=http://old.ecmwf.int/newsevents/training/lecture_notes/pdf_files/ASSIM/ObservationInfluence.pdf }}{{dead link|date=April 2018 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> or '''hat matrix''' (<math>(\mathbf{H})</math>), maps the vector of [[response variable|response values]] (dependent variable values) to the vector of [[fitted value]]s (or predicted values). It describes the [[influence function (statistics)|influence]] each response value has on each fitted value.<ref name="Hoaglin1977" >{{Cite journal | title = The Hat Matrix in Regression and ANOVA
| first1= David C. | last1= Hoaglin |first2= Roy E. | last2=Welsch |journal= [[The American Statistician]] | volume=32 |date=February 1978| pages=17–22 | doi = 10.2307/2683469 |issue=1| jstor = 2683469 |url=http://dspace.mit.edu/bitstream/1721.1/1920/1/SWP-0901-02752210.pdf }}</ref><ref name = "Freedman09">{{cite book |author=[[David A. Freedman]] |year=2009|title=Statistical Models: Theory and Practice |publisher=[[Cambridge University Press]]|quote= |page=}}</ref> The diagonal elements of the projection matrix are the [[leverage (statistics)|leverage]]s, which describe the influence each response value has on the fitted value for that same observation.
