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# Statistical Analysis of Time Series STAT 635

OSU

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This 8 page Class Notes was uploaded by Alison Vandervort on Monday September 21, 2015. The Class Notes belongs to STAT 635 at Ohio State University taught by Peter Craigmile in Fall. Since its upload, it has received 47 views. For similar materials see /class/209997/stat-635-ohio-state-university in Statistics at Ohio State University.

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Date Created: 09/21/15

Stat 635 Autumn 2011 Peter F Craigmile Homework 2 Solutions This homework is graded out of a total of 40 points 1 3 points For an AR1 process we have that v is 02 2 v WW125 Thus approximate 95 CI for u is given by i100i196 vn 271i196 125100 0271i0693 7042270964 The interval contains zero so there is evidence from the data to support the hypothesis that the process mean7 u is zero E0 9 points a 2 points The mean process of Xt is uXt EXt l gt 0 l gt which depends on t The process is not stationary b 4 points For Y X 7 thl the mean is MYW EXt EXt71 i zt 7 i 2t 7 1 27 Which does not depend on t Next7 COVYt7 Yth C0VXt 7 Xt717 Xth 7 Xth71 7 covX7 XHh 7 covX 7 Xth1 7 covX17 Xth covX17 Xth1 COVWt7 VVthgt 7 COVWt7 mh7l 7 OWN1717 Wthgt COVVVt717VVth71 MVW 7 WM 7 1 7 WWW 17 which does not depend on 25 Thus is stationary c 3 points The mean is 1 q 1 q MVt EWt mingO ql mjq 1 2tij 1 2t7 by the symmetry ofj about 0 Since the white noise process is uncorrelated7 the variance is 1 q 1 q V 7 X 7 39 7 7 39 Var t 2q 12 quva t 7 2q DZ quvarUVt a 02 1 q 2 7 a 7 261 12 7 261 1 3 8 points We rst show that a trend of degree 3 is passed without distortion For coef cients k let 3 m Z ktk 160 be a polynomial of degree 3 Then 2 2 3 3 2 2 W7 Z HZmink Z k Z wink j72 739772 k0 k0 739772 We show the result if we can show this expression is equal to Mt ie7 2 4 k 7 k 7 Zat7g 7t k7071723 1 F First note that 2372 aj 71 4 3 4 719 99 1 2372337 232j3a7 0 and 3433 7 47114 03 14 4719 7 0 We now prove the result For k O7 equation 1 is equal to 2372 aj 17 which is to For k 1 we have 15272 Li 7 23723 017 t7 0 t For k 2 we have t2 237207 7 222723 017 237234207 t2 7 2 t 0 0 252 For k 3 t3 2372 17325 2372 jaj3tZ72 farzi fa t370070 t3 Thus polynomials of degree 3 pass without distortion under this ltering Next7 we show that the lter eliminates seasonal components with period 3 Let st be our seasonal component with period 3 ie7 st st3 for all t We also know that the seasonal component has mean or sum 07 ie7 ELI st 0 Thus 2 E ajStij a723t72 171871 108 a13t1 a23t2 F72 8t 38t1 38t2 9 3231 3t 0 3 7 g 1 g 7844 48t2 38 48t1 8t2l 3 1 i 7872 4871 38 48t1 8t2l use seasonality here since seasonal components sum to 0 We have shown that this lter eliminates seasonal components of period 3 4 5 points a 2 points Using the trigonometric identity7 we have Xt Acos27rft 7 Acos27rft cos gt 7 Asin27rft sin gt 7 l sin27rft g cos27rft m where l iAsin gt and g Acos gt b 3 points We solve for A and 1 Since A is nonnegative7 A f and arctani 1 27 arctani 1 2 7 71397 arctani 1 2 7r7 77r2 7r27 undefined7 2gt0 1gt07 2lt07 1 07 2lt07 1gt07 207 1lt07 207 107 20 5 15 points R question square root of wind speed a 1 point The square root transformation is used stabilize any possible mean variance relationship in the data The distribution of the data should look more symmetric under transformation 20 30 40 hours since midnight 23 Oct 2005 Figure 1 Time series plot of the square root of the wind speed b 3 points The time series plot of the square root wind speed is shown in Figure 1 The most predominant feature of the data is that the square root of the wind speed increases from 3 MPH12 around 9 MPH on the original scale at hour 0 midnight on 23 October 20057 until it reaches a peak of 75 MPH12 N 56 MPH around 357 36 hours later noon on 24 October 2005 This increase does not look linear After that time the square root Wind speeds decreases to around 4 MPH12 ie7 16 MPH Around this increasing and decreasing trend7 the variance of the errors look fairly constant This is because the square root transformation helped stabilize the variance c 4 points The output of the model from R is Call lmformula winds quot hours Ihoursquot2 Ihoursquot3 Residuals Min 1Q Median 3Q Max 11516 05169 01010 04489 17242 to U m g quot 0 E u 39 m 9 m 8 9 sf 6 3 g m Q T 39l l l l l 0 10 20 30 40 0 10 20 30 40 hours since midnight 23 Oct 2005 hours since midnight 23 Oct 2005 Figure 2 Left panel shows time series plot with an estimate of polynomial trend superimposed as the solid smooth line The right hand panel shows the residuals of the trend t Coefficients Estimate Std Error t value Prgtt Intercept 3934e00 2262e01 17390 lt 2e16 hours 2973e01 4305e02 6905 183e10 Ihoursquot2 2412e02 2195e03 10988 lt 2e16 Ihoursquot3 3911e04 3159e05 12380 lt 2e16 Signif codes 0 0001 001 005 01 1 Residual standard error 06824 on 134 degrees of freedom Multiple RSquared 07707 Adjusted Rsquared 07656 Fstatistic 1501 on 3 and 134 DF pvalue lt 22e16 Letting t be the index denoting the number of 20 minutes elapsing since midnight on 23 October 2005 our model for the square root wind speed7 Xi7 is Xt th 77 where is a polynomial trend of degree 37 and m is a mean zero noise process which we hope is stationary Using least squares7 our estimate of M is t 3934 7 02973ht 002412ht2 7 00003911710337 where here ht t3 is the number of hours since midnight on 23 October 2005 From the plots of the tted line over the original time series and the residuals Figure 27 we ACF Figure 3 Sample ACF of the residuals from the polynomial trend model can see the t seems to capture some of the changes in the mean of the square root wind series but not everything Although the residuals are centered around zero there seems to be signi cant correlations remaining in the residuals see part d 1 point The sample ACF plot of the residuals Figure 3 have a cosine shape NOTE The cosine indicates evidence of a possible periodicity in the residuals e 2 points The rst order differences of the square root wind speeds are shown in Figure 4 The rst difference series has many peaks in the series and the variability seems to be smaller for the later hours of the series not weakly stationary7 From the sample ACF right panel we see that the rst differences look close to white noise although there are three non zero lags lags 1 8 and 9 that are slightly outside the con dence bands contradictory to the white noise assumption f 2 points The second order differences of the square root wind speeds are shown in Figure 5 The second order differences are more spiky than the rst order differences Again the variance may be slightly lower at the end of the series breaking an assumption of stationarity There is evidence of negative lag one and two autocorrelation both evident in the time series plot and in the sample ACF This process is not white noise This is an example of over differencing 7 we have differenced the time series too many times and induced negative sample autocorrelations 7 something we do not tend to see in data very often g 2 points Figure 6 shows a plot of the simple MAq lter smooths for q 1 over 40 minutes q 2 80 minutes q 3 2 hours q 6 4 hours and q 12 8 hours The wind speed on the square root scale are smoother as we increase q The variance decreases with increasing q but the bias increases The q 3 averages over two hours or q 6 averages over 4 hours smooths seems the most pleasing in that they pick up the peak at around 12 15 hours as well as the peak at around 36 hours Investigating the sample ACF plots of the residuals from the smooths may aid us in deciding the value of q to choose 0 Q v 3 o D Q I 390 0 c l E 0 LL 0 6 N N cgt o 6 m N CIgt I I I I I C I I I I I 0 10 20 30 40 0 5 10 15 20 hours since midnight 23 Oct 2005 Lag Figure 4 Left panel shows a time series plot of the rst order differences of the square root wind series and the right hand panel shows the associated sample ACF 0 gm 00 w 0 E 0quot 0 LI 0 lt B Em quot0039 UI C l O 1 O 20 30 40 O 5 1 O 1 5 20 hours since midnight 23 Oct 2005 Lag Figure 5 Left panel shows a time series plot of the second order differences of the square root wind series and the right hand panel shows the associated sample ACF smoothed sqrt Wind speed 4 smoothed sqrt Wind speed 4 O 10 20 30 40 O 10 20 30 40 hours since midnight 23 Oct 2005 hours since midnight 23 Oct 2005 q 3 q 6 Q Q 3 0 3 00 E E E 0 E 0 6 6 I v I V U U 1 E l E l 8 8 g 0 10 20 30 40 g 0 10 20 30 40 hours since midnight 23 Oct 2005 hours since midnight 23 Oct 2005 U g q 12 0 0 E E 0 t 039 I V U I C I 0 10 20 30 40 hours since midnight 23 Oct 2005 Figure 6 For different values of q plots of the smooths obtained using an Mq lter applied to the square root wind series

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