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Unformatted text preview: needs to be any obvious particular problem the consideration. • page_30there adapted to the discontinuities inunderdata? If so, what does this mean? Are Page you • Do 30 make sense context? Have the of theariables been ’v • Does itunderstand theto transform any ‘rightand variables?of measured? problem being tackled. combine statistical theory with sound common sense knowledge the particular • • Isave all the time seriesso, what should be done about it? trend present? following checklist of <Hprevious pageIf been plotted? possible actions, while noting that the list is not exhaustive and page_30 next page > We close by any missing values? If so, what should be done about them? • Are there giving the • eeds30 be adapted to the If so, whatwhat should beabout them? it? n Is seasonality present? If so,problem under consideration. to particular should be done done about Page there any outliers? • Do you understand the Are • Exercisesany obviouscontext? Have the ‘right’ data? If so, whatof the particular problem being tackled. combine statistical theory with sound commonthe variables been measured? mean? • Have all the time series been plotted? in sense and knowledge does this discontinuities • Are there giving the We close make sensefollowing checklist of possible actions, company that successive 4week and 2.1oes itbyany missing to transform whatof the sales of about them? X in the list is not exhaustive periods over 1995–1998. • AreThe following data show the coded variables?while noting • D to be values? If so, any should be done needs there adapted to the particular problem under consideration. • Ao you understand If context? should be done ariables them? VI • I re there any outliers? If what Have the right’IV about been measured? VII II III V VIII IX X XI XII • Ds trend present?Itheso, so, what should ‘be doneabout it? v • Hs seasonality present? If so, whatin the data? done what does this mean? • I re there any obvious been plotted? • Aave all the time seriesdiscontinuities should be If so, about it? Exercises 1995 153 189 215 223 201 173 121 106 86 87 • Does it make missing transform any of should be done about them? • Are there any sense tovalues? If so,221 the variables? 302 what • I re he any outliers? show the coded sales of it? 2.1 Ttherepresent? data what should be done aboutcompany X • As trendfollowing If so,If so, what should be done about them? in successive 4week periods over 1995–1998. 1996 any present? If(Due day: thebe 228IfVso, 283 does255 mean? 133 177 III should done 238 164 X128XI 108 XIII 87 74 I re there what this • As seasonality obvious discontinuities inSept 29 (Monday schedule), in class) I IIso, what 241 IVdata? about it? VI VII VIII IX XII Exercises • Does it make sense to transform any of the variables? 1997 145 189 200 coded done 201 220 233 121 172 1995–1998. 87 81 65 76 2.1 trend present? If so, what should besales about it? 292 153 302 223 201 173 86 108 •1995he following data show the221 187215 of company X in successive 4week periods over106119 Is T • Is seasonality present?IIIf so, what should be done about it? VII VIII I III IV228 178 V 283 VI248 IX X XI 120 XII87 XIII 1998 111 177 170 241 243 202 163 128 139 108 96 95 53 1996 133 255 238 164 74 95 Exercises 1995 173 121 106 2.1 The the153 successive 1995 1998. (a) Plotfollowing data show the 187 215201of company X220 data. 189 221coded sales 302 292223 in 201 233 4week periods over8681 –8765 108 1997 145 200 172 119 76 74 1996 133 177 241 238 VIII 164 128 108 XI 87 XII 74 XIII 95 I II IV V VII IX X (b) Assess the trend andIII243 228178 283 248255 202 seasonal effects. VI 1998 111 170 163 139 120 96 95 53 94 1997 200 187 201 292 220 233 172 121 119 81 65 76 108 1.2, 0.5, 0.9, 1.1, 74 2.2Plot the data. 189 observations302 a stationary time series are106 follows:87 S successive 221 215 on 223 201 as 86 1.6, 0.8, 1995 ixteen145 153 173 (a) 119960.6, 1.5, trend170 seasonal228 283 1.2 .1, 0.9, 241 1998 111 178 163 139 128 120 108 96 87 95 74 53 94 133 164 95 (b) Assess the 0.8,177 1.2, 0.5, 1.3, 248 255 and 243 effects.0.8, 202 238 (a) Plot the data. 200 observations on a292 220 time series are as119 observations. (a) Sixteen successive 2.2 Plot the145 stationary follows:81 65 1.2, 0.5, 74 1.1, 1.6, 0.8, 76 0.9, 1997 187 201 233 172 (b) 0.6, the trend and seasonal 1.3, in (b) Assess1.5, at the graph0.5, effects. (a), guess an approximate value for the autocorrelation coefficient at la 1.1,Looking 0.8, 0.9, 1.2, plotted0.8, 1.2 1998 111 170 178 248 202 163 139 120 96 95 53 2.2 (a) Sixteen observations. 243 1. Plot thesuccessive observations on a stationary time series are as follows: 1.6, 0.8, 1.2, 0.5, 0.9, 1.1, 94 1.1, Looking at the graph plotted 0.8, 1.2 0.6, 0.8, (a) Plot the data. 0.9, 1.2, 0.5, 1.3, in (a), guess an approximate value for the autocorrelation coefficient at lag (b) Plot 1.5,against xt+1, and again try to guess the value of r1. (c) Assess the trend and seasonal effects. xt (a) (b) Plot the observations. 1. Sixteen at the graph plotted in on guess an approximate are as follows: 1.6, 0.8, 1.2, 0.5, 0.9, 1.1, (d) Looking successive observations(a), a stationary time series value for the autocorrelation coefficient at lag 2.2 Calculate r1. (b) (c) Plot xt against xt+1, 0.5, 1.3, 0.8, 1.2 guess the value of r1. and again try to 1.1, 0.6, 1.5, 0.8, 0.9, 1.2, . (d) If xt gainst where to guess the value of r1. is a constant in (0, !), show that 2.3Plot theaobservations.and again try a is a constant and (a) Calculate r1. xt+1, (c) < previous page page_30 next page > AMS316 2009 HW1 (b) Looking atr1. graph plotted in (a),aguess an approximatea constant inautocorrelation that (d) If where a is . constant and is value for the (0, !), show coefficient at lag 2.3Calculate the 1. If wto use. the trigonometrical onstant inlisted in Section 7.2. Using Equation (7.2) it can be here a is constant and a is a c results (0, !), show that 2.3 (Hint: You will xt+1, and again try to guess the value of r1. (c) Plot xt against need . (d) Calculate r1. need to use the trigonometrical results listed in Section 7.2. Using Equation (7.2) it can be (Hint: You will (Hint: page_31 shown that need to use theatrigonometrical results listed in Section 7.2. show that a is a " so that is a constant in (0, !), Using Equation (7.2) it can be . Now use the re wheres N!constant and 2.3 If You will spage_31 that hown as N!" so that . Now use the result that that shown as N!" so that . Now use the result . that < previous page page_31 next be page > (Hint: You will need to use that < previous page page_31 Section 7.2. Using next page < previous page the trigonometrical results listed in page_30 Equation (7.2) it can next>page > page revious page next < previous page p31 page_30 Page < page_30 next page shown that as N!" so that . Now use the result> Page 31 that 2cos A cos B=cos(A+B)+cos(A!B) together with the result that for a suitably chosen 2cos the result that for a suitably chosen N.) previous page < A cos B=cos(A+B)+cos(A!B) together with page_30 next page > N.) 2.4 A computer generates a series of 400 observations that are supposed to be random. The first 10 sample 2.4 A computer coefficientsa series series are r1=0.02, r2=0.05, supposed to be random. The firstr6= sample utocorrelation generates the of 400 observations that are r3= !0.02, 10 0.00, a7=0.12, r8=0.06, r9=0.02,of10=!0.08.are there anyr2=0.05, r3=!!0.09, r4=0.08, r5=0.02, r6= 0.00, a 0.09, r4=0.08, r5= r utocorrelation coefficients ofr the series Is r1=0.02, evidence of nonrandomness? ! r7=0.12, r8=0.06, r9=0.02, r10= series of monthly evidence of {Xt}, for which the 2.5 Suppose we have a seasonal!0.08. Is there anyobservationsnonrandomness? seasonal factor at time t 2.5 Suppose {St}. Further suppose that monthly observations { constant through time so factor at !12 is denoted bywe have a seasonal series of the seasonal pattern is Xt}, for which the seasonalthat St=Sttime tfor is t. Denote St}. Further suppose that the seasonal by {"t} alldenoted bya{stationary series of random deviations pattern .is constant through time so that St=St!12 for all Consider the model t=a+ of St+"t h deviations by linear (a)t. Denote a stationaryXseriesbt+random aving a global {"t}. trend and additive seasonality. Show that the (a) Consider the model Xt=a+bt+St+"t having a global linear trend and additive seasonality. Show that the seasonal difference operator acts on Xt to produce a stationary series. seasonal difference operator a+bt)acts"on aving produce a stationary series. (b) Consider the model Xtt=( +bt)St+"t thh Xt to aaglobal linear trend and multiplicative seasonality. Does the Consider the model X =(a St+ aving global linear trend and multiplicative seasonality. Does the (b) operator ransform X o stationarity? If not, find differencing operator that does. operator ttransform Xtt ttostationarity? If not, find a a differencing operator that does. (Note: As stationarity is not formally defined until Chapter 3, you should use heuristic arguments. A A (Note: As stationarity is not formally defined until Chapter 3, you should use heuristic arguments. stationary process may involve aaconstant mean value (that could be zero) plus any linear combination of of the stationary process may involve constant mean value (that could be zero) plus any linear combination the stationary series {"t}, but should not include terms such as trend and seasonality.) stationary series {"t}, but should not include terms such as trend and seasonality.) < previous page < previous page page_31 page_31 next page next page > > file:///C:/Documents and Settings/Yang/
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This note was uploaded on 08/08/2010 for the course AMS 316 taught by Professor Xing during the Fall '09 term at SUNY Stony Brook.
 Fall '09

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