Class Note for PUBHLTH 744 at UMass(1)
Class Note for PUBHLTH 744 at UMass(1)
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This 17 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 13 views.
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Date Created: 02/06/15
Linear models With applications in R PUBHLTH 744 Handout 5Pr0jecfion Opemiors Instructor Andrea S Foulkes Division of Biostatistics and Epidemiolog UMass School of Public Health and Health Sciences Fall 2007 Projection operators Definition Suppose M is a vector space and N1 and N2 are two subspaces of M where N1 l N2 M and N1 N2 0 Consider the unique decomposition of z x l y where z 6 N1 and y 6 N2 The linear transformation PNI NZz z is called the projection of 2 onto the subspace N1 along the subspace N2 See Figure HandouLS 1 Projection operators N 7 m wax 1le W2me HandouLS 2 Projection operators Definition Let A be an n x 71 matrix A is called a projection operator onto CA along NA if for any 1 E CA A U 1 Example LetM RZ N1 Slt i and 2 N2 S 1 Since N1 and N2 are linearly independent we have have N1 l N2 R2 To find PNIWT See Figure HandouLS 3 Projection operators Theorem A2 A idempotent if and only if A is a projection operator HandouLS 4 Orthogonal projection operators Definition M is an orthogonal projection operator onto CX if and only if v E CX a M U v and ii w E CXL a Mu 0 See Figure HandouLS 5 Orthogonal projection operaturs clx HandouLS 6 Orthogonal projection operators 1 2 1 2 Example Let X lt 1 gt We then have CXSlt1gt and CXL Slt 31 For M to be an orthogonal projection operator onto CX we need M U v where v alt i gt and ii Mu 0 where w lt H33 33 71 ConSIder HandouLS 7 Orthogonal projection operators ltMUMltgtlt gtltgtv lt gtMwMlti gtltz z gto HandouLS 8 Orthogonal projection operators Theorem If M is an orthogonal projection operator onto CX then CM CX Theorem M is an orthogonal projection operator onto CM if and only if M M2 idempotent and M M symmetric In order to prove M is the orthogonal projection operator onto CX we need to show 1 M M 2 M M2 and 3 CX CM HandouLS 9 Orthogonal projection operaturs Example For X defined in Example above we have M M and 12 12 12 12 M2lt12 1 2gtlt1 2 J2 7 14141414 1414 1414gt M HandouLS 10 Orthogonal projection operators Theorem Orthogonal projection operators are unique Theorem Suppose M is an n x n orthogonal projection operator for rank r g n then 1 The eigenvalues of M are 0 or 1 2 rM trM r and 3 M is positive semidefinite HandouLS 11 Orthogonal projection operators Theorem Suppose X is an n x p matrix of rank r minnp and let A 11704 where ai is n x 1 and the columns of A form an orthonormal basis for CX Then AA is the unique orthogonal projection operator onto CX HandouLS 12 Orthogonal projection operators Proof gt AATT ATTAT AAT 2 AAT2 AATAAT AATAAT AIAT AAT Note this is true since A is orthonormal and so by definition the product of each column with itself is 1 and the product of each column with another is O a Let x E CAA Then z AA t for some 25 Let 25 A t Then we have x AA t At which implies z E CX since CX z At So CAA C CX gt s Let x E CX We know z E CA since A forms an orthonormal basis for X and so we can write x At for some 256 72 But A is full rank and so any vector 25 6 RP can be written as t A z where z E R Thus z At AA z which implies z E CAA Therefore CX C CAA D HandouLS 13 v1 S snoPMEH D 39X0 m 9 1 2 X 3 2 w 9 z 1X14XXX5 2 mg 39d 93 Z 1X 3 z X0 Ileoea lt W X14XXX X14XXXX14XXX ZW 39W X14XXX XI XXX X14XXX JV yomd 39TXO 3U0B XO oluo JOlBJedO uogpefwd leuo oquo sq s W ueqi 39X14XXX W SUHSG 39d 1UBJ Jo XJ18LU d x u ue s X esoddns IUJBJOBILL smqmado louoafold 21103011110 Orthogonal projection operators Theorem 58 If M is the unique orthogonal projection operator onto CXi HandouLS 15 Least squares regression Suppose EY u X If X is full rank then X X is invertible also of full rank and the least squares estimator of B is 3 X X 1X Y The least squares estimator of u is ID X3 XX X 1X Y MY where M is the orthogonal projection operator onto CX and MY is the projection of Y onto this space HandouLS 16
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