Get started today!

Good to have you back!
If you've signed in to StudyBlue with Facebook in the past, please do that again.

- StudyBlue
- Washington
- University of Washington - Seattle Campus
- Statistics
- Statistics 520
- Percival
- hw 4 exer 5

Maziar R.

Advertisement

Exercise 5 of Assignment 4 (due 2/2/09) Generate a realization Y 1 , Y 2 , ..., Y 64 of the zero mean AR(2) process of Equation (45) using the procedure outlined in ?Recipe for Simulating Au- toregressive Processes? on the next page. Compute the periodogram for the Y t ?s at three adjacent Fourier frequencies, namely, f 6 = 6/64, f 7 = 7/64 and f 8 = 8/64 (we are assuming ?t = 1), and call these values ? S (p) 1 (f 6 ), ? S (p) 1 (f 7 ) and ? S (p) 1 (f 8 ). Repeat the above a ?large? number N r of times (us- ing a di?erent realization of the Gaussian white noise process each time) to obtain the sequences { ? S (p) j (f 6 ) : j = 1,...,N r }, { ? S (p) j (f 7 ) : j = 1,...,N r } and { ? S (p) j (f 8 ) : j = 1,...,N r } (here ?large? means between 100 and 10000 depending on your computer?s tolerance for repetitive tasks). Compute the sample mean and sample variance for the three sequences, and compute the sample correlation coe?cient between 1. { ? S (p) j (f 6 )} and { ? S (p) j (f 7 )}, 2. { ? S (p) j (f 6 )} and { ? S (p) j (f 8 )} and 3. { ? S (p) j (f 7 )} and { ? S (p) j (f 8 )} (cf. the equation for ?? displayed in the middle of page 5 of the text). Compare these sample values with the corresponding large sample values suggested by Equation (168b), the equation in the middle of page 199, Equation (222b) and Equation (222c). Recipe for Simulating Autoregressive Processes Let Y t = p X j=1 ? j,p Y t?j + ? t describe a stationary AR(p) process, where {? t } is a white noise process with zero mean and variance ? 2 p , and {? j,p : j = 1,...,p} is a sequence of AR coe?cients. Givene? 1 , e? 2 , ..., e? N which are taken to be uncorrelated Gaussian deviates with zero mean and unit variance (obtained on a computer from a Gaussian random number generator), we desire to generate a realization of Y 1 , Y 2 , ..., Y N . To do so, we carry out the following steps. 1. We first calculate the p ? 1 sequences {? j,p?1 : j = 1,...,p ? 1}, {? j,p?2 : j = 1,...,p?2}, ..., {? j,2 : j = 1,2} and {? 1,1 } by computing the following for k = p, p?1, ..., 2: ? j,k?1 = ? j,k + ? k,k ? k?j,k 1?? 2 k,k , 1 ? j ? k ?1. 2. Second, we calculate ? 2 k?1 = ? 2 k 1?? 2 k,k , k = p,p?1,...,1. 3. Third, we generate Y 1 , Y 2 , ..., Y p via Y 1 = ? 0 e? 1 Y 2 = ? 1,1 Y 1 + ? 1 e? 2 Y 3 = ? 1,2 Y 2 + ? 2,2 Y 1 + ? 2 e? 3 . . . Y p = ? 1,p?1 Y p?1 + ? 2,p?1 Y p?2 + ··· + ? p?1,p?1 Y 1 + ? p?1 e? p . 4. Finally, the remaining N ?p values are generated using Y t = p X j=1 ? j,p Y t?j + ? p e? t , t = p + 1,...,N. Let us consider two concrete examples, namely, the AR(2) and AR(4) processes given by Equations (45) and (46a) of Percival and Walden (1993). The AR(2) process has coe?cients ? 1,2 = 3 4 and ? 2,2 = ? 1 2 and has ? 2 2 = 1. Application of step 1 yields ? 1,1 = ? 1,2 + ? 2,2 ? 1,2 1?? 2 2,2 = 3 4 ? 1 2 · 3 4 1? 1 4 = 1 2 , while step 2 yields ? 2 1 = ? 2 2 1?? 2 2,2 = 1 1? 1 4 = 4 3 and ? 2 0 = ? 2 1 1?? 2 1,1 = 4 3 1? 1 4 = 16 9 . We thus would generate the AR(2) process using Y 1 = 4 3 e? 1 Y 2 = 1 2 Y 1 + 2 ? 3 e? 2 Y 3 = 3 4 Y 2 ? 1 2 Y 1 +e? 3 . . . Y N = 3 4 Y N?1 ? 1 2 Y N?2 +e? N . For the AR(4) process, we have ? 1,4 = 2.7607, ? 2,4 = ?3.8106, ? 3,4 = 2.6535 and ? 4,4 = ?0.9238, with ? 2 4 = 1. Application of step 1 yields ? 1,3 = ? 1,4 + ? 4,4 ? 3,4 1?? 2 4,4 . = 2.1105749802379 ? 2,3 = ? 2,4 + ? 4,4 ? 2,4 1?? 2 4,4 . = ?1.98076723152095 ? 3,3 = ? 3,4 + ? 4,4 ? 1,4 1?? 2 4,4 . = 0.70375083325625 ? 1,2 = ? 1,3 + ? 3,3 ? 2,3 1?? 2 3,3 . = 1.41977220651098 ? 2,2 = ? 2,3 + ? 3,3 ? 1,3 1?? 2 3,3 . = ?0.98160135815478 ? 1,1 = ? 1,2 + ? 2,2 ? 1,2 1?? 2 2,2 . = 0.71647720701657, while step 2 yields ? 2 3 = ? 2 4 1?? 2 4,4 . = 6.8215820667702 ? 2 2 = ? 2 3 1?? 2 3,3 . = 13.515181723107 ? 2 1 = ? 2 2 1?? 2 2,2 . = 370.69765006 ? 2 0 = ? 2 1 1?? 2 1,1 . = 761.717290031. ied ied hw-4-exer-5-09

Advertisement

"StudyBlue is great for studying. I love the study guides, flashcards and quizzes. So extremely helpful for all of my classes!"

Alice , Arizona State University"I'm a student using StudyBlue, and I can 100% say that it helps me so much. Study materials for almost every subject in school are available in StudyBlue. It is so helpful for my education!"

Tim , University of Florida"StudyBlue provides way more features than other studying apps, and thus allows me to learn very quickly!Â I actually feel much more comfortable taking my exams after I study with this app. It's amazing!"

Jennifer , Rutgers University"I love flashcards but carrying around physical flashcards is cumbersome and simply outdated. StudyBlue is exactly what I was looking for!"

Justin , LSU
StudyBlue is not sponsored or endorsed by any college, university, or instructor.

© 2015 StudyBlue Inc. All rights reserved.

© 2015 StudyBlue Inc. All rights reserved.