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by: Dr. Filomena Hegmann


Dr. Filomena Hegmann

GPA 3.76


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This 2 page Study Guide was uploaded by Dr. Filomena Hegmann on Thursday October 29, 2015. The Study Guide belongs to APPM 5440 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 58 views. For similar materials see /class/231875/appm-5440-university-of-colorado-at-boulder in Applied Math at University of Colorado at Boulder.




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Date Created: 10/29/15
APPM 45540 Review Problems for the Final 1 De ne causality of an ARMApq series Xt Is the ARMA model Xt Xtil 4Xt72 4Xt73 Zt 8054Zt712 where Zt N WN0lt72 causal 10 Explain What the Durbin Levinson and Innovations algorithms do Why are they useful 9quot Short Answer a True or False All strictly stationary series are stationary b True or False Every stationary mean zero series has a spectral density C Is the function de ned by yh 112 for h 7 0 and y0 2 the covariance function of some stationary series d Is the function f 005k for A E 77r 7r a legitimate spectral density e Suppose a stationary series Xt with covariance function yh CovXtXth satis es yh 0 for h gt 2 What can be said about Xt r gt A time series of length n 100 Visually appears to be mean zero and stationary The sample autocorrelations and partial autocorrelations were computed out to lag 5 and are listed below The sample variance of the series is 40 3933 h 1 2 3 4 5 6h 0819 0770 0709 0660 0575 3402 0819 0301 0074 0033 0116 a What model do you propose for this series b Estimate in any matter you see t the parameters in the model 01 Suppose that Xt is a causal AR2 series with autocorrelation function ph CorrXtXth lf p1 12 and p2 0 What are the AR model coe icients 9 Suppose that Xt is a stationary series with covariance function yh CavXtXth Derive an expression for Vadyn where Y 1 L1 Xi H T What does a Kalman lter do 90 Suppose that Xt is a mean zero real valued stationary series with spectral density f A for A E 0 7r Compute yh CovXtXth for h 2 0 3 Suppose you were going to analyze a time series with a linear trend Give two methods for removing the trend from the series 0 Suppose you are studying a causal AR1 series Xt with mean 5 when you suddenly realize you are 10 minutes late for your time series class Running out the door you leave behind some of your papers This problem is due today so you d better gure something out The only information you have when you get to class is that X15 45 X17 4875 and that the lag one autocorrelation is negative Predict X15 and give the MSE for your prediction HH 3H H 03 H q HHH 40301 Show that the spectral density for a stationary time series is an even function Let Xt be a stationary mean zero real process with spectral density fX for 77r lt A 3 7r Find the spectral density for the ltered process Yt de ned by 1 Yt gltXt Xt71 Xt727 in terms of the spectral density of Xt Suppose that Xt and are uncorrelated stationary processes with spectral den sities fX and fy respectively Consider the process Wt Xt Yt Show that fWOx fXOx fYgt7 where fW is the spectral density for Suppose you were trying to model some time series data whose variance appears to be increasing over time Name one variance stabilizing transformation you could apply to your data What is the AlC Akaike s Information Criterion used for State Herglotz s Theorem Consider the following two graphs of spectral densitites on O7 7r Graphs should be here7 but I am running against a clock right now Imagine a rst picture looking somewhat like f e A and a second picture looking somewhat like f A gt3 Which spectral density corresponds to a time series with gradual long range uctua tion Which corresponds to a time series with rapid oscillation


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