×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

13

0

17

# Class Note for PUBHLTH 744 at UMass(1)

Marketplace > University of Massachusetts > Class Note for PUBHLTH 744 at UMass 1

No professor available

These notes were just uploaded, and will be ready to view shortly.

Either way, we'll remind you when they're ready :)

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

COURSE
PROF.
No professor available
TYPE
Class Notes
PAGES
17
WORDS
KARMA
25 ?

## Popular in Department

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.

×

## Reviews for Class Note for PUBHLTH 744 at UMass(1)

×

×

### What is Karma?

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

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

×

×

### BOOM! Enjoy Your Free Notes!

×

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

Bentley McCaw University of Florida

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Allison Fischer University of Alabama

#### "I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over \$600 per month. I LOVE StudySoup!"

Jim McGreen Ohio University

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

Parker Thompson 500 Startups

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com