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## TIME SERIES

by: Providenci Mosciski Sr.

20

0

9

# TIME SERIES STAT 519

Providenci Mosciski Sr.
UW
GPA 3.66

Staff

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COURSE
PROF.
Staff
TYPE
Class Notes
PAGES
9
WORDS
KARMA
25 ?

## Popular in Statistics

This 9 page Class Notes was uploaded by Providenci Mosciski Sr. on Wednesday September 9, 2015. The Class Notes belongs to STAT 519 at University of Washington taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/192513/stat-519-university-of-washington in Statistics at University of Washington.

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Date Created: 09/09/15
Introduction to Spectral Analysis Don Percival Applied Physics Lab University of Washington overheads available at http www staff washington edudbptalks html What is Spectral Analysis 0 one of the most Widely used and lucrative methods in data analysis 0 can be regarded as analysis of variance of time series using sinusoids sinusoids statistics Fourier theory statistics 0 today7s lecture introduction to spectral analysis notion of a time7 series 025 introduction to time series analysis gtk basics of time domain7 analysis gtk subject of Stat 519 notion of the spectrum methods for estimating the spectrum gtk nonparametric gtk parametric concluding comments StatBE 520 has lots more details Time Series amp Time Series Analysis 0 What is a time series one damned thing after another7 R A Fisher tt1N four examples Figures 2 and 3 0 goal of time series analysis quantify Characteristics of time series 0 univariate statistics eg sample mean amp variance 1 N 1 N 7 A2 7 2 332 th and a Et N t1 N t1 7 inadequate to say how 33 and tk are related Lagged Scatter Plots 0 bivariate distribution of separated pairs 0 t1 versus 3 t 1 N 1 lag 1 scatter plot 0 four examples Figure 4 o tk versus 3 t 1 N k lag k scatter plot 0 summarize scatter plots using linear model tk 06k kt 6m not always reasonable see Figure 9 0 Pearson product moment correlation coef cient let y1yN amp 21 2N be 2 collections of ordered values let 3 amp 2 be sample means sample correlation coef cient Elyt 17gtltZt 2 Ea 22gt 2 2W measures strength of linearity 1 g 5 g 1 6 Sample Autocorrelation Sequence oletyttkt1N k andzttt1N k o for each lag k plug these into 2 13gtZt 2 m 22gt m m and fudge things a bit to get 6 2 Exams egtltast segt E 1ltt 3quot2 o m k O N 1 called sample acs 0 four examples Figures 6 and 7 Modeling of Time Series 0 assume 33 is realization of random variable Xt 0 need to specify properties of Xt ie model 3 o simplifying assumptions related to stationarity m estimates pk E GOV Xta Xtk02 E EltXt HgtltXtk Mgt027 Where gtllt u E EXt note does not depend on t 02 EXt m2 does not depend on t X s are multivariate Gaussian statistics of X s completely determined if we know u 02 and ms 0 critique of time domain7 characterization u 02 pk not easy to visualize mt from pk7s statistical properties of pfs di cult to use Frequency Domain Modeling I 0 based on idea of expressing Xt in terms of sinusoids 0 top ve rows of Figure Al show cos27rft for t1128 amp p gigg where f is frequency of sinusoid lf is period 0 bottom row shows addition of ve sinusoids highly structured and nonrandom 0 Figure A2 shows cos27rft with gb chosen ran domly one for each f rattier looking but still highly structured c Figure A3 shows additions of 64 sinusoids with fre 63 1 2 quenc1es 128 128 128 128 amp randoni phases very ratty looking with no apparent structure 0 note cos27rft gb Acoslt27rftgt Bsin27rft where A cosgz5 and B singz5 EA EB O var A var B covA B O ie uncorrelated W 6 Frequency Domain Modeling II o generalize to following simple model for Xt N2 Xt u Z Aj cos lt27Tfjtgt Bj sin 27rfjt j1 holds for t 12 N where N is even fj E jN xed frequencies cycles unit time called Fourier or standard frequencies As and Bj7s are random variables gtk EAj EBj O gtllt varAj var Bj a cov A77 Ak cov 37 Bk O for j k gtllt cov Aj7 Bk O for all j k note a now allowed to depend on j

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