1.2 Consider a signal-plus-noise model of the general form xt = St + wt, where w

Question

1.2 Consider a signal-plus-noise model of the general form xt = St + wt, where wt is Gaussian white noise with = I. Simulate and plot n 200 observations from each of the following two models (a) x,-st , for, 200, where 0 10 ex t=l, . . . , 100 t-100 t-1p (-T00) } cos(2mt/4), t- 101,...,200 Hint s c(repC0,100), 10*exp(-(1:100)/20)*cos (2*pi*1:100/4)) x = s + rnorm(200) plot.ts(x) (b) X.-St + wt, for t = l, . . . . 200, where t=l, . . . , 100 t= 101, 0 St(10exp{-(,200) } cos(2nt/4), 200 (c) Compare the general appearance of the series (a) and (b) with the earthquake series and the explosion series shown in Figure 1.7. In addition, plot (or sketch) and compare the signal modulators (a) exp{-t/20j and (b) expí-t/200), for t=1.2, . . . . 100.

Explanation / Answer

1.2: answer

(a)

s1=c(rep(0,100), 10*exp(-(101:200)/20)*cos(2*pi*(101:200)/4))
x1=ts(s1+rnorm(200,0,1))
plot(x1)

Time Series:
Start = 1
End = 200
Frequency = 1
[1] -0.233026171 -1.759954265 -0.167985922 -0.516095021 -0.841062483
[6] 0.938025589 -0.054161987 0.101381693 -0.215298330 1.020757523
[11] 0.411740102 1.192287688 -0.255535330 -1.401650356 0.719805795
[16] 2.054389866 -0.296141992 0.781382853 0.258284989 -0.807717473
[21] -0.001503069 0.070768091 0.338189655 1.012135459 -0.121372263
[26] 0.988211308 -0.919651910 2.068229156 -2.781263985 1.015028846
[31] 1.429575180 -0.948139361 -0.539580826 -0.365006901 -0.821882664
[36] 0.245087413 0.237901167 0.236321590 1.321223936 0.586372236
[41] 0.292250497 -0.302622837 -1.251942699 0.880474256 -2.077045144
[46] -0.997225284 1.516565258 0.094787169 -1.618029900 0.793268343
[51] 0.636960903 -0.345965330 0.546364494 0.961178773 -0.008874651
[56] -0.230502565 -0.276616532 -0.525407749 0.368020920 1.801004266
[61] 1.832131775 -2.058314952 -0.879232802 0.399625144 0.935040390
[66] 1.733018365 0.647045468 1.125679207 1.093305405 -0.862908566
[71] -1.303222316 1.251075713 0.940317523 1.332937167 -0.421842503
[76] -0.462625758 -1.459978336 1.458174352 -0.097805710 -1.455394158
[81] 0.631236184 -0.937352510 0.707983646 0.272190473 0.393504063
[86] -0.089055978 0.606237469 -0.186913882 0.406806224 -1.728051837
[91] -0.787351393 0.184054623 0.489818675 -1.433005923 -0.158408562
[96] -0.228879568 0.644017826 0.528697244 -0.147871886 -0.707011983
[101] 1.131163962 -2.431686494 -0.641725202 -0.430762603 -1.084124963
[106] -1.133613591 -0.074799463 0.502444996 0.270918273 0.625783703
[111] 1.486480362 0.431789212 0.674750909 1.076300454 -0.088867302
[116] -1.598685221 -0.949070516 0.034957609 -0.205915272 -0.201686910
[121] -0.548976125 -0.440368145 -0.197540799 -1.117804447 0.888539526
[126] -1.872345349 -0.596786051 0.832320504 -0.556928348 0.316393624
[131] 0.679131041 0.086819058 1.277969561 -0.071968640 -1.166908534
[136] -0.277802373 0.455802302 1.258322955 -0.875863796 -0.072880043
[141] 1.399194440 1.690123329 1.796996340 0.882650377 -0.091073408
[146] -0.066955264 0.722193709 -0.254240574 1.184907422 -0.179441973
[151] 0.598868874 0.217055979 1.099154845 0.885053935 0.079114235
[156] -0.174506603 -0.150071281 -0.179965231 -0.106832302 -2.846721509
[161] -0.333521722 -1.456741294 -0.262079230 -0.517806144 -0.515495217
[166] -1.448801099 -0.308345811 -0.758227220 1.513724122 0.688212430
[171] 0.527123858 1.336667816 0.023608439 0.265772279 -0.958509542
[176] 1.045668145 1.176886249 -0.857169387 -0.608813874 1.092616433
[181] -1.007707004 0.419173259 -0.060172090 -0.401519128 1.241736999
[186] 1.793313278 -1.057039353 -0.699338931 -0.877240244 -2.703161387
[191] 1.878014097 0.504955558 0.933357551 1.181458384 1.505609677
[196] 0.340673666 -2.286607062 -0.446583170 2.421269873 -0.428687762

(b)

s2=c(rep(0,100), 10*exp(-(101:200)/200)*cos(2*pi*(101:200)/4))
x2=ts(s1+rnorm(200,0,1))
plot(x2)

Time Series:
Start = 1
End = 200
Frequency = 1
[1] -0.059844837 -0.994952997 -1.677571453 0.917001689 0.017294366
[6] 0.296268463 -0.746180904 0.440955423 0.973485088 0.112835652
[11] -0.205914173 -0.791318464 0.725025728 -0.914665801 0.219122205
[16] 1.996711861 -0.154524843 0.417007216 -1.276248039 1.733738762
[21] 0.219444310 -3.712212906 -0.637648670 -1.396445307 -1.118087260
[26] 0.060213826 -0.209168132 0.716330033 -1.241212357 -1.314329044
[31] -1.274467515 -0.277682185 -0.548191178 -0.570704021 1.518680292
[36] 0.668321156 0.257179702 1.053551685 -0.003870614 -1.113833194
[41] -1.021393350 0.626876016 1.498569631 -0.640828937 -0.287644574
[46] 0.952557139 -1.866224620 -0.710232967 0.808225402 1.274739845
[51] -1.176637692 -0.137166307 0.145615502 -0.804758155 -0.038593142
[56] 0.075599234 -0.765758073 1.821422151 1.476301104 -0.697732017
[61] -0.608100198 0.334347312 -1.039518265 0.694362492 1.038561800
[66] 0.860550114 0.484893397 -0.392216998 -0.121109357 0.068744183
[71] 0.182311086 -0.250872112 -0.491016180 -0.270950345 0.585368180
[76] 0.473196057 -0.459814906 0.706490626 -0.551943134 -0.717733902
[81] 2.165993373 0.172822887 0.168183517 -0.148331634 -1.054599726
[86] -0.727241221 0.103187292 0.915144905 -0.373737182 0.318047778
[91] -0.513946163 1.042474096 0.415230832 -0.130182908 -0.864724461
[96] -0.382323114 -0.276325372 -0.205251502 -1.054337374 0.318688888
[101] -0.442905665 -0.241275664 -1.041464281 -0.245628988 -1.721057266
[106] 1.273011584 -0.263257672 -0.295372612 0.076518828 -0.094420552
[111] 0.599061834 0.736919464 0.390357852 1.008793735 -1.651157568
[116] 0.786044631 -1.582408195 1.003609891 0.786767599 -0.103549512
[121] -0.741791227 -1.250385782 0.942616034 0.816577343 -0.501539054
[126] -1.209904972 -0.554546447 0.748177021 -0.716261989 -0.189760761
[131] -0.271305548 0.587980196 -0.148694796 -0.201147165 0.918337482
[136] -0.672052121 -0.234913081 0.687620780 1.626320287 1.119193329
[141] 1.797463379 0.040552803 0.722543068 -0.140244548 1.156961906
[146] 0.158128064 0.067759234 -1.566295620 0.494111939 1.916912351
[151] -1.584150511 1.752028046 1.837396335 -2.164385587 1.174515791
[156] -0.529095473 1.089031753 1.022567849 1.283552572 -0.165074126
[161] 0.110399481 1.265871002 0.744848288 0.038714660 0.398918098
[166] 0.002225787 -0.032647475 -0.615968252 -0.944647485 -0.329497152
[171] -1.247572598 -0.030722084 1.371232478 -1.574654841 0.948524184
[176] -1.387898916 -1.021821509 0.230575409 -1.381707465 -0.530796806
[181] 1.423704480 0.537790377 -0.075984505 -0.506667993 0.806940255
[186] -1.608009859 0.522547764 0.552379282 0.667672082 -2.175114349
[191] 0.612154443 1.381829907 -0.721060624 1.265375538 -0.465369711
[196] 0.563712962 -0.394095068 0.041252757 -0.264512465 0.943389579

(c)

Varing the series in (a) ans (b) is some time equal

(a) Case 1. exp(-t/20)

s1= exp(-(1:100)/20)
x1=ts(s1+rnorm(100,0,1))

plot(x1)

(b)

s2=exp(-(1:100)/200)
x2=ts(s1+rnorm(200,0,1))
plot(x2)

In this series the (b) series varing is more than (a).

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.