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# Introduction to Time Series APPM 4540

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This 2 page Class Notes was uploaded by Dr. Filomena Hegmann on Thursday October 29, 2015. The Class Notes belongs to APPM 4540 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 32 views. For similar materials see /class/231870/appm-4540-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 General Overview Questions Not Speci c to the Last Part of the Course 1 De ne causality of an ARMApq series Xt Is the ARMA model Xt Xtil 4Xt72 4Xt73 Zt 8054Zt712 where Zt N WN0lt72 causal 2 Explain What the Durbin Levinson and Innovations algorithms do Why are they useful 3 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 fA 005A for A E 77r7 7r7 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 4 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 l 2 3 4 5 6h 0819 0770 0709 0660 0575 3402 i 0819 0301 0074 0033 0116 a What model do you propose for this series b How might you estimate the parameters in the model Don t actually do it 5 Suppose you were going to analyze a time series with a linear trend Give two methods 1 2 rPPi for removing the trend from the series Last Part of the Course Questions What does a Kalman lter do Suppose that Xt is a mean zero7 real valued7 stationary series with spectral density fA A for A E 07 7r Compute yh CovXtXth for h 2 0 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 fXA for 77r lt A 3 7r Find the spectral density for the ltered process Yt de ned by l Yt g in terms of the spectral density of Xt Xt Xtil Xt72gt7 H 0 01 O 9051 H Suppose that Xt and are uncorrelated stationary processes with spectral den sities fX and fy respectively Consider the process Wt Xt Yt Show that fWO fXO fY gt7 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 Suppose that Xt is a mean zero real valued stationary series with spectral density f 1 for A E ingrl a Compute the autocovariance function y b ldentify this process Yeahi in the interest of getting these review problems out7 l m stealing a problem from the last homework assignment It s a little messy but important in terms of the concepts covered The spectral density of a real valued process Xt is de ned on O7 7r by 100 g7001lt Ag g001 fOF O otherwise and on 77r0 by f f7 a Evaluate the autocovariance function of Xt at lags O and 1 b Find the spectral density of the process where Yt Vngt Xt 7 Xt12 c What is the variance of Y d Sketch the power transfer function of the lter V127 and use the sketch to explain the effect of the lter on sinusoids with frequencies near 0 and near 7r6

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