homework
liukejia
2018年12月8日
Chapter 7 - Exponential Smoothing
7.8.1
library(fpp2)## Warning: package 'fpp2' was built under R version 3.5.1## Loading required package: ggplot2## Warning: package 'ggplot2' was built under R version 3.5.1## Loading required package: forecast## Warning: package 'forecast' was built under R version 3.5.1## Loading required package: fma## Warning: package 'fma' was built under R version 3.5.1## Loading required package: expsmooth## Warning: package 'expsmooth' was built under R version 3.5.1ses_pigs_fit<-ses(pigs,h = 4)
ses_pigs_fit_sum <- summary(ses_pigs_fit)##
## Forecast method: Simple exponential smoothing
##
## Model Information:
## Simple exponential smoothing
##
## Call:
## ses(y = pigs, h = 4)
##
## Smoothing parameters:
## alpha = 0.2971
##
## Initial states:
## l = 77260.0561
##
## sigma: 10308.58
##
## AIC AICc BIC
## 4462.955 4463.086 4472.665
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 385.8721 10253.6 7961.383 -0.922652 9.274016 0.7966249
## ACF1
## Training set 0.01282239
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Sep 1995 98816.41 85605.43 112027.4 78611.97 119020.8
## Oct 1995 98816.41 85034.52 112598.3 77738.83 119894.0
## Nov 1995 98816.41 84486.34 113146.5 76900.46 120732.4
## Dec 1995 98816.41 83958.37 113674.4 76092.99 121539.87.8.12
a
The intervals produced in R expand as they get farther from the observed data whereas the manually produced intervals are static.
fets <- function(y, h) {
forecast(ets(y), h = h)
}
data("qcement")
tsCV_fes<-tsCV(qcement,fets,h=4)
tsCV_sna<-tsCV(qcement,snaive,h=4)
tsCV_fes## h=1 h=2 h=3 h=4
## 1956 Q1 6.700000e-02 9.600000e-02 0.1050000000 0.0640000000
## 1956 Q2 9.265000e-02 1.016500e-01 0.0606500000 0.1356500000
## 1956 Q3 -2.442844e-02 -9.704957e-02 -0.0536707064 -0.0862918406
## 1956 Q4 -7.198045e-02 -1.609598e-02 -0.0362115113 -0.0763270436
## 1957 Q1 7.500000e-02 7.400000e-02 0.0530000000 0.0250000000
## 1957 Q2 9.175809e-03 -1.182419e-02 -0.0398241914 0.0261758086
## 1957 Q3 -1.891133e-02 -4.691133e-02 0.0190886666 0.0450886666
## 1957 Q4 -3.235806e-02 3.364194e-02 0.0596419449 0.0506419449
## 1958 Q1 6.058900e-02 8.658900e-02 0.0775890010 0.0135890010
## 1958 Q2 4.919003e-02 4.019003e-02 -0.0238099701 0.0761900299
## 1958 Q3 -1.878542e-03 -6.587854e-02 0.0341214583 0.0511214583
## 1958 Q4 -6.423717e-02 3.576283e-02 0.0527628262 0.0437628262
## 1959 Q1 7.523263e-02 9.223263e-02 0.0832326297 0.0232326297
## 1959 Q2 1.086467e-02 2.125615e-02 0.0230157898 0.0249458925
## 1959 Q3 1.571592e-02 1.998057e-02 0.0236002993 0.0630252612
## 1959 Q4 -3.976007e-03 5.055201e-03 0.0538539981 0.0470000001
## 1960 Q1 1.082088e-02 5.919310e-02 0.0514999360 0.0669999978
## 1960 Q2 3.148850e-02 1.209897e-02 0.0247425902 -0.0041217142
## 1960 Q3 6.303604e-03 2.539049e-02 -0.0140123265 -0.0389318591
## 1960 Q4 2.540389e-02 -1.177932e-02 -0.0362470094 -0.0713684467
## 1961 Q1 -1.569300e-02 -4.010196e-02 -0.0734932670 -0.0682934328
## 1961 Q2 -3.216928e-02 -7.262350e-02 -0.0771603390 -0.0334272739
## 1961 Q3 -6.815775e-02 -7.731330e-02 -0.0286007344 -0.0243543250
## 1961 Q4 -4.571899e-03 4.900948e-02 0.0607687751 0.0649906612
## 1962 Q1 4.543476e-02 6.025901e-02 0.0633838796 0.0369413804
## 1962 Q2 1.206082e-02 1.578595e-02 -0.0011511826 0.0169999980
## 1962 Q3 4.557214e-03 -8.119221e-03 0.0087655374 0.0520405695
## 1962 Q4 -2.279211e-02 5.760824e-03 0.0566547289 0.0810000000
## 1963 Q1 1.909600e-02 6.251290e-02 0.0928862447 0.1229999829
## 1963 Q2 4.028200e-02 6.971538e-02 0.0953374294 0.1186397607
## 1963 Q3 3.200105e-02 7.274405e-02 0.0933894645 0.1139998216
## 1963 Q4 6.035647e-02 7.199638e-02 0.0873717211 0.1279832317
## 1964 Q1 3.512984e-02 6.447191e-02 0.1174905905 0.1050000000
## 1964 Q2 1.833127e-02 6.788282e-02 0.0628244377 0.0287161013
## 1964 Q3 6.006993e-02 6.339419e-02 0.0289473225 -0.0099451719
## 1964 Q4 2.987929e-03 -4.690188e-02 -0.0877369607 -0.0664272938
## 1965 Q1 -3.235374e-02 -6.072628e-02 -0.0240009338 -0.0530000239
## 1965 Q2 -3.992800e-02 -1.983805e-02 -0.0585735034 -0.0641777252
## 1965 Q3 1.989288e-02 -2.253542e-02 -0.0233367289 -0.0499718416
## 1965 Q4 -4.148989e-02 -4.485170e-02 -0.0722654034 -0.0941628413
## 1966 Q1 3.109235e-03 -2.555209e-02 -0.0558465178 -0.0584701993
## 1966 Q2 -2.380817e-02 -5.523456e-02 -0.0622792005 -0.0106473321
## 1966 Q3 -2.543793e-02 -3.238576e-02 0.0183767401 0.0172999179
## 1966 Q4 -1.118874e-02 3.920875e-02 0.0364333376 0.0003757586
## 1967 Q1 5.741866e-02 5.874540e-02 0.0249247812 0.0285155308
## 1967 Q2 -3.286375e-03 -3.566966e-02 -0.0300598149 -0.0727593240
## 1967 Q3 -3.352551e-02 -2.696335e-02 -0.0675786440 -0.0235588555
## 1967 Q4 3.687221e-03 -3.372459e-02 0.0127585207 0.0081198895
## 1968 Q1 -4.154215e-02 4.969505e-03 0.0003134410 0.0180555434
## 1968 Q2 4.276720e-02 3.535501e-02 0.0463073400 0.0063277286
## 1968 Q3 -7.073400e-03 6.971741e-03 -0.0371036836 0.0831306294
## 1968 Q4 1.646154e-02 -1.953491e-02 0.1052799208 0.0838476661
## 1969 Q1 -3.465909e-02 8.970252e-02 0.0625878499 0.0680333043
## 1969 Q2 1.235001e-01 9.506996e-02 0.0942674101 0.0574743067
## 1969 Q3 -4.140226e-03 9.055237e-03 -0.0351879903 0.0125953377
## 1969 Q4 8.169693e-03 -3.650992e-02 0.0137549307 0.0073433345
## 1970 Q1 -4.347429e-02 -5.569316e-04 -0.0075548418 -0.0408887684
## 1970 Q2 3.218184e-02 1.919307e-02 -0.0174377318 -0.0003500727
## 1970 Q3 1.217650e-03 -2.786627e-02 -0.0101390396 -0.0710433823
## 1970 Q4 -2.729623e-02 -7.652898e-03 -0.0702194700 0.0246550937
## 1971 Q1 1.720489e-02 -4.406121e-02 0.0453762515 -0.0332667479
## 1971 Q2 -5.150032e-02 3.725294e-02 -0.0502615263 0.0343513085
## 1971 Q3 7.712976e-02 -4.430760e-03 0.0766899503 0.0496337513
## 1971 Q4 -6.040211e-02 1.493707e-02 -0.0218544934 -0.0061644624
## 1972 Q1 5.440756e-02 2.153982e-02 0.0337729270 0.0067216678
## 1972 Q2 -8.166172e-03 2.394453e-03 -0.0175624415 -0.0055617533
## 1972 Q3 5.787502e-03 -1.314195e-02 -0.0025942119 -0.0163251013
## 1972 Q4 -1.984992e-02 -6.777846e-03 -0.0159941736 0.1299025408
## 1973 Q1 1.653694e-03 -8.066332e-03 0.1287350906 -0.0241603089
## 1973 Q2 -7.416339e-03 1.374321e-01 -0.0086278818 0.0088944443
## 1973 Q3 1.326106e-01 -1.230172e-02 0.0039029802 -0.0799030870
## 1973 Q4 -7.849817e-02 -6.676409e-02 -0.1525376198 -0.0923675366
## 1974 Q1 -2.279525e-02 -1.079881e-01 -0.0691558044 -0.1182877308
## 1974 Q2 -9.108580e-02 -2.766827e-02 -0.1266443420 -0.1071373686
## 1974 Q3 2.946654e-02 -7.264909e-02 -0.0473770304 -0.0701816815
## 1974 Q4 -9.323637e-02 -6.269245e-02 -0.0816434464 -0.0982887548
## 1975 Q1 -9.185140e-03 -4.256457e-02 -0.0551003415 -0.0949955044
## 1975 Q2 -2.602798e-02 -4.176356e-02 -0.0798237229 -0.0441750848
## 1975 Q3 -5.019609e-02 -4.946404e-02 -0.0322997828 -0.0289522758
## 1975 Q4 -5.336273e-02 -1.125747e-02 -0.0114985576 -0.0410551763
## 1976 Q1 2.699699e-02 2.658419e-02 -0.0038337714 -0.0606277543
## 1976 Q2 6.380161e-03 -2.378184e-02 -0.0760427020 -0.0443152922
## 1976 Q3 -2.707303e-02 -7.673467e-02 -0.0462403866 -0.0466711176
## 1976 Q4 -6.387311e-02 -3.342620e-02 -0.0387079003 -0.0189294339
## 1977 Q1 1.769847e-03 1.724671e-05 0.0108893370 -0.0262573782
## 1977 Q2 -2.056736e-03 2.117233e-02 -0.0399571398 -0.0476695261
## 1977 Q3 1.604306e-02 -4.410477e-02 -0.0564729480 -0.0532080841
## 1977 Q4 -2.865398e-02 -6.798697e-02 -0.0557282162 -0.0584491715
## 1978 Q1 -4.917760e-02 -3.573261e-02 -0.0416335656 0.0398739974
## 1978 Q2 -1.392407e-03 5.511434e-03 0.0503462613 -0.0057545080
## 1978 Q3 -4.856133e-03 6.928349e-02 -0.0082473904 0.0238005716
## 1978 Q4 8.428459e-02 1.337871e-02 0.0530623624 0.0516635892
## 1979 Q1 -4.013761e-02 -1.017754e-02 0.0021916170 0.0070460173
## 1979 Q2 2.851826e-02 3.098958e-02 0.0539679700 0.0565911897
## 1979 Q3 1.852664e-02 4.117108e-02 0.0501412525 0.0608114522
## 1979 Q4 2.257385e-02 2.117357e-02 0.0320074334 0.0576693910
## 1980 Q1 1.789710e-04 1.034547e-02 0.0344676633 0.0180439990
## 1980 Q2 9.042296e-03 3.435218e-02 0.0237879668 0.1573057191
## 1980 Q3 1.472243e-02 2.547898e-02 0.1486975836 0.0551890482
## 1980 Q4 8.902635e-03 1.330970e-01 0.0502572935 0.1768692622
## 1981 Q1 1.245945e-01 3.756875e-02 0.1692959793 0.0202621420
## 1981 Q2 -4.748533e-02 8.426776e-02 -0.0584673264 -0.0083538989
## 1981 Q3 1.173887e-01 -2.635272e-02 0.0315498581 -0.1161599645
## 1981 Q4 -9.193956e-02 -4.165136e-02 -0.1884525098 -0.2855675677
## 1982 Q1 2.601697e-02 -1.246091e-01 -0.2198280524 -0.3215297676
## 1982 Q2 -1.391883e-01 -2.364199e-01 -0.3336869769 -0.4500776148
## 1982 Q3 -1.751930e-01 -2.505401e-01 -0.3803310931 -0.2834899843
## 1982 Q4 -1.614772e-01 -2.230242e-01 -0.1149153628 -0.0306888024
## 1983 Q1 -1.306019e-01 -3.097447e-02 0.0107788800 0.0317624198
## 1983 Q2 1.170459e-01 2.124554e-01 0.2225435035 0.3404279899
## 1983 Q3 8.953425e-02 9.335365e-02 0.1683747675 0.2295132107
## 1983 Q4 1.561802e-02 7.335865e-02 0.1156883033 0.1793175208
## 1984 Q1 6.627619e-02 1.175391e-01 0.1815139262 0.0862524628
## 1984 Q2 5.409295e-02 1.106749e-01 0.0180026088 0.0579139820
## 1984 Q3 6.198187e-02 -5.774724e-04 0.0166806898 0.1465038442
## 1984 Q4 -6.074919e-02 -3.080386e-02 0.0978057743 0.0458359492
## 1985 Q1 6.390279e-03 1.367307e-01 0.0716586629 0.0864984431
## 1985 Q2 1.373993e-01 8.508746e-02 0.0737459923 0.0736180284
## 1985 Q3 -3.975074e-02 -2.570237e-03 -0.0449507868 -0.0841647893
## 1985 Q4 2.067906e-02 -1.972450e-02 -0.0603255020 -0.0458355122
## 1986 Q1 -1.359085e-02 -4.426118e-02 -0.0218879044 -0.0677004982
## 1986 Q2 -3.041629e-02 -1.810133e-02 -0.0647148917 -0.1223222930
## 1986 Q3 -4.034562e-04 -4.208186e-02 -0.1012337039 -0.1094649489
## 1986 Q4 -4.223154e-02 -1.001295e-01 -0.1100797077 0.0239611337
## 1987 Q1 -7.239964e-02 -8.024022e-02 0.0592467102 0.0476956028
## 1987 Q2 -1.776867e-02 1.196958e-01 0.0999671012 0.1360763348
## 1987 Q3 1.388662e-01 1.386445e-01 0.1644571466 0.2610939736
## 1987 Q4 2.085564e-02 4.351548e-02 0.1345039648 0.1773368779
## 1988 Q1 3.330111e-02 1.344887e-01 0.1812012779 0.2042467396
## 1988 Q2 9.755089e-02 1.365467e-01 0.1550273367 0.0587995383
## 1988 Q3 4.568548e-02 1.131226e-01 -0.0080115831 0.1695910352
## 1988 Q4 8.369269e-02 -4.171843e-02 0.1324280008 0.0640606050
## 1989 Q1 -1.171832e-01 5.689564e-02 -0.0090009063 -0.0831389635
## 1989 Q2 1.610487e-01 1.039446e-01 -0.0010591566 -0.1359975341
## 1989 Q3 -3.389748e-02 -1.132386e-01 -0.2681360522 -0.3600961020
## 1989 Q4 -8.002673e-02 -2.177252e-01 -0.3096964453 -0.2743231165
## 1990 Q1 -1.650649e-01 -2.465247e-01 -0.2144462317 -0.2904648114
## 1990 Q2 -1.043661e-01 -5.934278e-02 -0.1345711797 -0.2127767993
## 1990 Q3 1.369842e-02 -7.924158e-02 -0.1615329719 -0.2505460082
## 1990 Q4 -8.155152e-02 -1.579381e-01 -0.2448861593 -0.1691170688
## 1991 Q1 -9.767305e-02 -1.879123e-01 -0.1233406553 -0.0990675166
## 1991 Q2 -1.235748e-01 -6.864743e-02 -0.0430606041 0.0151339105
## 1991 Q3 5.166749e-02 6.099501e-02 0.1250515608 0.0855811108
## 1991 Q4 2.749059e-02 8.989943e-02 0.0481967497 0.0797995834
## 1992 Q1 5.985122e-02 1.573700e-02 0.0496853821 0.1624449841
## 1992 Q2 -4.034956e-02 -1.085607e-02 0.1026619491 0.1456170049
## 1992 Q3 2.032241e-02 1.285179e-01 0.1734437582 0.0719416557
## 1992 Q4 1.124560e-01 1.564505e-01 0.0526144956 0.2561194404
## 1993 Q1 3.802435e-02 -5.458664e-02 0.1730151816 -0.0028313850
## 1993 Q2 -9.599193e-02 1.169679e-01 -0.0323565399 0.0650865969
## 1993 Q3 2.096743e-01 3.095167e-02 0.1224522873 0.2608311573
## 1993 Q4 -9.954993e-02 -6.316351e-03 0.1214967425 -0.0505943543
## 1994 Q1 8.154217e-02 2.158852e-01 0.0509343140 0.1198314148
## 1994 Q2 1.495824e-01 -2.138897e-02 0.0535538072 -0.0839580051
## 1994 Q3 -1.345394e-01 -3.475493e-02 -0.1798257543 -0.2368310266
## 1994 Q4 4.275673e-02 -9.736509e-02 -0.1511675462 -0.3421767720
## 1995 Q1 -1.338330e-01 -1.910570e-01 -0.3864800489 -0.1265536402
## 1995 Q2 -8.763383e-02 -2.838845e-01 -0.0383962915 -0.1694743202
## 1995 Q3 -2.128407e-01 1.990209e-02 -0.1088211619 -0.0801668553
## 1995 Q4 1.440840e-01 3.297446e-02 0.0698851763 0.0668588754
## 1996 Q1 -9.895576e-02 -4.263381e-02 -0.0267090193 -0.0103346086
## 1996 Q2 2.800160e-02 3.912869e-02 0.0495363750 0.1391799680
## 1996 Q3 -2.360591e-06 2.886029e-02 0.1296769762 0.0816492942
## 1996 Q4 1.490874e-02 1.000428e-01 0.0783818778 0.0981993975
## 1997 Q1 9.813113e-02 5.244899e-02 0.0659973946 0.1797681264
## 1997 Q2 -1.325704e-02 -3.214135e-03 0.1167284178 0.0699305597
## 1997 Q3 2.168898e-02 1.325373e-01 0.0747785968 0.1027011291
## 1997 Q4 1.199172e-01 6.962464e-02 0.1072791510 0.2450031647
## 1998 Q1 -3.833019e-02 -1.652163e-02 0.1151794065 0.0121758335
## 1998 Q2 7.107875e-03 1.180797e-01 0.0070961262 0.0390811439
## 1998 Q3 1.135933e-01 1.256517e-02 0.0375389225 0.0086470199
## 1998 Q4 -5.235061e-02 -4.001364e-02 -0.0739637362 -0.0580120799
## 1999 Q1 -6.147776e-03 -3.827952e-02 -0.0168193918 0.0538314443
## 1999 Q2 -3.352216e-02 -1.115785e-02 0.0587944213 0.0861927443
## 1999 Q3 3.090990e-02 7.473920e-02 0.0886251493 -0.1563209921
## 1999 Q4 7.209592e-02 9.903618e-02 -0.1644408248 -0.4114369383
## 2000 Q1 5.208887e-02 -2.128272e-01 -0.4673041399 -0.3001389748
## 2000 Q2 -2.505475e-01 -5.053587e-01 -0.3337018119 -0.3770905563
## 2000 Q3 -2.846975e-01 -1.570351e-01 -0.2096002891 -0.3137517722
## 2000 Q4 2.668633e-02 -9.635748e-04 -0.0911076232 0.0896884313
## 2001 Q1 -2.076136e-02 -1.129844e-01 0.0631855134 0.1613567518
## 2001 Q2 -9.684253e-02 7.969755e-02 0.1792124801 0.2511495527
## 2001 Q3 1.501796e-01 2.388769e-01 0.3162475947 0.3013511326
## 2001 Q4 1.445685e-01 2.136763e-01 0.1961726263 0.1590390283
## 2002 Q1 9.107351e-02 7.020772e-02 0.0397749729 0.1240540856
## 2002 Q2 3.841723e-03 -6.720682e-02 0.0184321424 -0.1187491018
## 2002 Q3 -2.713982e-02 6.603076e-02 -0.0829215823 0.1963426891
## 2002 Q4 3.986384e-02 -8.218377e-02 0.2208871457 0.0625667371
## 2003 Q1 -1.238251e-01 1.950323e-01 0.0379658188 0.0922224650
## 2003 Q2 2.657085e-01 1.090784e-01 0.1786853426 0.2259816920
## 2003 Q3 -5.784606e-02 6.628352e-02 0.0507891598 0.0987747215
## 2003 Q4 9.731096e-02 8.924929e-02 0.1440565768 0.0204441134
## 2004 Q1 5.914216e-03 5.611168e-02 -0.0732399857 0.0033349430
## 2004 Q2 5.353615e-02 -7.363495e-02 0.0020399757 0.2622937406
## 2004 Q3 -1.133499e-01 -3.128884e-02 0.2252157216 0.0070179284
## 2004 Q4 4.269288e-02 3.079398e-01 0.0922766288 0.0431172380
## 2005 Q1 2.730024e-01 5.703853e-02 0.0067277006 0.0260125422
## 2005 Q2 -1.526262e-01 -1.935609e-01 -0.1500795191 -0.1703311110
## 2005 Q3 -9.012514e-02 -6.059807e-02 -0.0709123705 -0.0003230824
## 2005 Q4 1.224639e-02 2.091585e-02 0.1081103526 0.1691463144
## 2006 Q1 -3.078828e-03 7.019610e-02 0.1248577228 0.0774769423
## 2006 Q2 7.263933e-02 1.301439e-01 0.0783820663 0.0426897949
## 2006 Q3 8.514199e-02 4.951513e-02 0.0207050740 0.1269129941
## 2006 Q4 -2.093492e-02 -6.771031e-02 0.0237900376 0.0758149287
## 2007 Q1 -4.586818e-02 4.683407e-02 0.0985194197 -0.0012882830
## 2007 Q2 8.309848e-02 1.372498e-01 0.0336426200 0.1427836343
## 2007 Q3 7.198085e-02 -2.599862e-02 0.0778844547 0.0457299980
## 2007 Q4 -7.204070e-02 2.414702e-02 -0.0066997426 -0.2177270082
## 2008 Q1 8.016897e-02 5.347319e-02 -0.1542607871 -0.2784476209
## 2008 Q2 -1.135926e-02 -2.227671e-01 -0.3380286836 -0.4292973320
## 2008 Q3 -2.132501e-01 -3.301524e-01 -0.4197184902 -0.3399588849
## 2008 Q4 -1.868232e-01 -2.596958e-01 -0.1747462275 -0.1964455092
## 2009 Q1 -8.451826e-02 6.198314e-03 -0.0114279399 -0.1156033972
## 2009 Q2 7.033035e-02 4.135148e-02 -0.0765582340 0.1670964764
## 2009 Q3 9.627761e-03 -9.401383e-02 0.1658317808 0.1913513731
## 2009 Q4 -1.100570e-01 1.390993e-01 0.1453799872 -0.0254746809
## 2010 Q1 2.460655e-01 2.735083e-01 0.1124812936 0.1312683963
## 2010 Q2 6.684919e-02 -9.978238e-02 -0.0608510559 -0.1092746576
## 2010 Q3 -1.522665e-01 -1.160186e-01 -0.1653636027 -0.0201953173
## 2010 Q4 -7.235882e-03 -4.675072e-02 0.0965176235 0.0230079522
## 2011 Q1 -3.907263e-02 1.042280e-01 0.0283312084 -0.0219131037
## 2011 Q2 1.354478e-01 6.475984e-02 0.0141557026 -0.0859638793
## 2011 Q3 -4.901492e-02 -9.187027e-02 -0.2091935536 -0.0728079339
## 2011 Q4 -5.939948e-02 -1.668267e-01 -0.0280462556 0.0523312608
## 2012 Q1 -1.079507e-01 4.906725e-02 0.1443062991 -0.0247547275
## 2012 Q2 1.232096e-01 2.153302e-01 0.0338034537 0.2620068740
## 2012 Q3 1.161515e-01 -5.758247e-02 0.1586324374 0.1819664233
## 2012 Q4 -1.378282e-01 6.951801e-02 0.0876873396 0.0529112911
## 2013 Q1 1.764193e-01 1.984542e-01 0.1694849550 0.1172640483
## 2013 Q2 6.906144e-02 3.883396e-02 -0.0007781061 NA
## 2013 Q3 -1.303157e-02 -4.398822e-02 NA NA
## 2013 Q4 -3.665988e-02 NA NA NA
## 2014 Q1 NA NA NA NAtsCV_sna## h=1 h=2 h=3 h=4
## 1956 Q1 NA NA NA NA
## 1956 Q2 NA NA NA NA
## 1956 Q3 NA NA NA NA
## 1956 Q4 NA NA NA NA
## 1957 Q1 0.072 0.042 0.012 0.025
## 1957 Q2 0.042 0.012 0.025 0.016
## 1957 Q3 0.012 0.025 0.016 0.043
## 1957 Q4 0.025 0.016 0.043 0.055
## 1958 Q1 0.016 0.043 0.055 0.019
## 1958 Q2 0.043 0.055 0.019 0.053
## 1958 Q3 0.055 0.019 0.053 0.044
## 1958 Q4 0.019 0.053 0.044 0.044
## 1959 Q1 0.053 0.044 0.044 0.048
## 1959 Q2 0.044 0.044 0.048 0.025
## 1959 Q3 0.044 0.048 0.025 0.063
## 1959 Q4 0.048 0.025 0.063 0.047
## 1960 Q1 0.025 0.063 0.047 0.067
## 1960 Q2 0.063 0.047 0.067 0.039
## 1960 Q3 0.047 0.067 0.039 -0.011
## 1960 Q4 0.067 0.039 -0.011 -0.036
## 1961 Q1 0.039 -0.011 -0.036 -0.051
## 1961 Q2 -0.011 -0.036 -0.051 0.020
## 1961 Q3 -0.036 -0.051 0.020 0.041
## 1961 Q4 -0.051 0.020 0.041 0.065
## 1962 Q1 0.020 0.041 0.065 0.037
## 1962 Q2 0.041 0.065 0.037 0.017
## 1962 Q3 0.065 0.037 0.017 0.052
## 1962 Q4 0.037 0.017 0.052 0.081
## 1963 Q1 0.017 0.052 0.081 0.123
## 1963 Q2 0.052 0.081 0.123 0.130
## 1963 Q3 0.081 0.123 0.130 0.114
## 1963 Q4 0.123 0.130 0.114 0.137
## 1964 Q1 0.130 0.114 0.137 0.105
## 1964 Q2 0.114 0.137 0.105 0.070
## 1964 Q3 0.137 0.105 0.070 0.020
## 1964 Q4 0.105 0.070 0.020 -0.008
## 1965 Q1 0.070 0.020 -0.008 -0.053
## 1965 Q2 0.020 -0.008 -0.053 -0.013
## 1965 Q3 -0.008 -0.053 -0.013 -0.003
## 1965 Q4 -0.053 -0.013 -0.003 -0.045
## 1966 Q1 -0.013 -0.003 -0.045 -0.013
## 1966 Q2 -0.003 -0.045 -0.013 0.037
## 1966 Q3 -0.045 -0.013 0.037 0.059
## 1966 Q4 -0.013 0.037 0.059 0.049
## 1967 Q1 0.037 0.059 0.049 0.056
## 1967 Q2 0.059 0.049 0.056 -0.025
## 1967 Q3 0.049 0.056 -0.025 0.022
## 1967 Q4 0.056 -0.025 0.022 0.046
## 1968 Q1 -0.025 0.022 0.046 0.056
## 1968 Q2 0.022 0.046 0.056 0.059
## 1968 Q3 0.046 0.056 0.059 0.143
## 1968 Q4 0.056 0.059 0.143 0.119
## 1969 Q1 0.059 0.143 0.119 0.101
## 1969 Q2 0.143 0.119 0.101 0.102
## 1969 Q3 0.119 0.101 0.102 0.039
## 1969 Q4 0.101 0.102 0.039 0.052
## 1970 Q1 0.102 0.039 0.052 0.009
## 1970 Q2 0.039 0.052 0.009 0.075
## 1970 Q3 0.052 0.009 0.075 -0.030
## 1970 Q4 0.009 0.075 -0.030 0.065
## 1971 Q1 0.075 -0.030 0.065 0.012
## 1971 Q2 -0.030 0.065 0.012 0.073
## 1971 Q3 0.065 0.012 0.073 0.104
## 1971 Q4 0.012 0.073 0.104 0.028
## 1972 Q1 0.073 0.104 0.028 0.078
## 1972 Q2 0.104 0.028 0.078 0.023
## 1972 Q3 0.028 0.078 0.023 0.039
## 1972 Q4 0.078 0.023 0.039 0.171
## 1973 Q1 0.023 0.039 0.171 0.036
## 1973 Q2 0.039 0.171 0.036 0.047
## 1973 Q3 0.171 0.036 0.047 -0.026
## 1973 Q4 0.036 0.047 -0.026 -0.099
## 1974 Q1 0.047 -0.026 -0.099 -0.063
## 1974 Q2 -0.026 -0.099 -0.063 -0.055
## 1974 Q3 -0.099 -0.063 -0.055 0.002
## 1974 Q4 -0.063 -0.055 0.002 -0.072
## 1975 Q1 -0.055 0.002 -0.072 -0.012
## 1975 Q2 0.002 -0.072 -0.012 0.002
## 1975 Q3 -0.072 -0.012 0.002 0.023
## 1975 Q4 -0.012 0.002 0.023 0.009
## 1976 Q1 0.002 0.023 0.009 -0.008
## 1976 Q2 0.023 0.009 -0.008 -0.015
## 1976 Q3 0.009 -0.008 -0.015 -0.020
## 1976 Q4 -0.008 -0.015 -0.020 0.027
## 1977 Q1 -0.015 -0.020 0.027 0.021
## 1977 Q2 -0.020 0.027 0.021 -0.023
## 1977 Q3 0.027 0.021 -0.023 -0.009
## 1977 Q4 0.021 -0.023 -0.009 -0.019
## 1978 Q1 -0.023 -0.009 -0.019 0.083
## 1978 Q2 -0.009 -0.019 0.083 0.041
## 1978 Q3 -0.019 0.083 0.041 0.065
## 1978 Q4 0.083 0.041 0.065 0.061
## 1979 Q1 0.041 0.065 0.061 0.027
## 1979 Q2 0.065 0.061 0.027 0.084
## 1979 Q3 0.061 0.027 0.084 0.063
## 1979 Q4 0.027 0.084 0.063 0.080
## 1980 Q1 0.084 0.063 0.080 0.050
## 1980 Q2 0.063 0.080 0.050 0.187
## 1980 Q3 0.080 0.050 0.187 0.093
## 1980 Q4 0.050 0.187 0.093 0.202
## 1981 Q1 0.187 0.093 0.202 0.055
## 1981 Q2 0.093 0.202 0.055 0.002
## 1981 Q3 0.202 0.055 0.002 -0.058
## 1981 Q4 0.055 0.002 -0.058 -0.284
## 1982 Q1 0.002 -0.058 -0.284 -0.251
## 1982 Q2 -0.058 -0.284 -0.251 -0.417
## 1982 Q3 -0.284 -0.251 -0.417 -0.196
## 1982 Q4 -0.251 -0.417 -0.196 -0.044
## 1983 Q1 -0.417 -0.196 -0.044 0.080
## 1983 Q2 -0.196 -0.044 0.080 0.215
## 1983 Q3 -0.044 0.080 0.215 0.177
## 1983 Q4 0.080 0.215 0.177 0.182
## 1984 Q1 0.215 0.177 0.182 0.085
## 1984 Q2 0.177 0.182 0.085 0.083
## 1984 Q3 0.182 0.085 0.083 0.175
## 1984 Q4 0.085 0.083 0.175 0.060
## 1985 Q1 0.083 0.175 0.060 0.124
## 1985 Q2 0.175 0.060 0.124 0.090
## 1985 Q3 0.060 0.124 0.090 -0.063
## 1985 Q4 0.124 0.090 -0.063 0.006
## 1986 Q1 0.090 -0.063 0.006 -0.043
## 1986 Q2 -0.063 0.006 -0.043 -0.088
## 1986 Q3 0.006 -0.043 -0.088 -0.067
## 1986 Q4 -0.043 -0.088 -0.067 0.054
## 1987 Q1 -0.088 -0.067 0.054 0.086
## 1987 Q2 -0.067 0.054 0.086 0.167
## 1987 Q3 0.054 0.086 0.167 0.269
## 1987 Q4 0.086 0.167 0.269 0.176
## 1988 Q1 0.167 0.269 0.176 0.203
## 1988 Q2 0.269 0.176 0.203 0.094
## 1988 Q3 0.176 0.203 0.094 0.202
## 1988 Q4 0.203 0.094 0.202 0.103
## 1989 Q1 0.094 0.202 0.103 -0.056
## 1989 Q2 0.202 0.103 -0.056 -0.074
## 1989 Q3 0.103 -0.056 -0.074 -0.314
## 1989 Q4 -0.056 -0.074 -0.314 -0.226
## 1990 Q1 -0.074 -0.314 -0.226 -0.222
## 1990 Q2 -0.314 -0.226 -0.222 -0.204
## 1990 Q3 -0.226 -0.222 -0.204 -0.214
## 1990 Q4 -0.222 -0.204 -0.214 -0.171
## 1991 Q1 -0.204 -0.214 -0.171 -0.054
## 1991 Q2 -0.214 -0.171 -0.054 0.060
## 1991 Q3 -0.171 -0.054 0.060 0.095
## 1991 Q4 -0.054 0.060 0.095 0.071
## 1992 Q1 0.060 0.095 0.071 0.161
## 1992 Q2 0.095 0.071 0.161 0.167
## 1992 Q3 0.071 0.161 0.167 0.109
## 1992 Q4 0.161 0.167 0.109 0.295
## 1993 Q1 0.167 0.109 0.295 0.018
## 1993 Q2 0.109 0.295 0.018 0.087
## 1993 Q3 0.295 0.018 0.087 0.314
## 1993 Q4 0.018 0.087 0.314 -0.030
## 1994 Q1 0.087 0.314 -0.030 0.158
## 1994 Q2 0.314 -0.030 0.158 -0.052
## 1994 Q3 -0.030 0.158 -0.052 -0.229
## 1994 Q4 0.158 -0.052 -0.229 -0.288
## 1995 Q1 -0.052 -0.229 -0.288 -0.100
## 1995 Q2 -0.229 -0.288 -0.100 -0.110
## 1995 Q3 -0.288 -0.100 -0.110 -0.027
## 1995 Q4 -0.100 -0.110 -0.027 0.154
## 1996 Q1 -0.110 -0.027 0.154 -0.015
## 1996 Q2 -0.027 0.154 -0.015 0.192
## 1996 Q3 0.154 -0.015 0.192 0.120
## 1996 Q4 -0.015 0.192 0.120 0.131
## 1997 Q1 0.192 0.120 0.131 0.208
## 1997 Q2 0.120 0.131 0.208 0.076
## 1997 Q3 0.131 0.208 0.076 0.130
## 1997 Q4 0.208 0.076 0.130 0.237
## 1998 Q1 0.076 0.130 0.237 0.018
## 1998 Q2 0.130 0.237 0.018 0.083
## 1998 Q3 0.237 0.018 0.083 0.049
## 1998 Q4 0.018 0.083 0.049 -0.040
## 1999 Q1 0.083 0.049 -0.040 0.098
## 1999 Q2 0.049 -0.040 0.098 0.126
## 1999 Q3 -0.040 0.098 0.126 -0.107
## 1999 Q4 0.098 0.126 -0.107 -0.375
## 2000 Q1 0.126 -0.107 -0.375 -0.281
## 2000 Q2 -0.107 -0.375 -0.281 -0.353
## 2000 Q3 -0.375 -0.281 -0.353 -0.219
## 2000 Q4 -0.281 -0.353 -0.219 0.184
## 2001 Q1 -0.353 -0.219 0.184 0.175
## 2001 Q2 -0.219 0.184 0.175 0.275
## 2001 Q3 0.184 0.175 0.275 0.351
## 2001 Q4 0.175 0.275 0.351 0.142
## 2002 Q1 0.275 0.351 0.142 0.102
## 2002 Q2 0.351 0.142 0.102 -0.100
## 2002 Q3 0.142 0.102 -0.100 0.197
## 2002 Q4 0.102 -0.100 0.197 0.112
## 2003 Q1 -0.100 0.197 0.112 0.132
## 2003 Q2 0.197 0.112 0.132 0.288
## 2003 Q3 0.112 0.132 0.288 0.080
## 2003 Q4 0.132 0.288 0.080 0.067
## 2004 Q1 0.288 0.080 0.067 0.017
## 2004 Q2 0.080 0.067 0.017 0.301
## 2004 Q3 0.067 0.017 0.301 0.033
## 2004 Q4 0.017 0.301 0.033 0.108
## 2005 Q1 0.301 0.033 0.108 0.047
## 2005 Q2 0.033 0.108 0.047 -0.203
## 2005 Q3 0.108 0.047 -0.203 0.087
## 2005 Q4 0.047 -0.203 0.087 0.186
## 2006 Q1 -0.203 0.087 0.186 0.113
## 2006 Q2 0.087 0.186 0.113 0.084
## 2006 Q3 0.186 0.113 0.084 0.109
## 2006 Q4 0.113 0.084 0.109 0.111
## 2007 Q1 0.084 0.109 0.111 0.043
## 2007 Q2 0.109 0.111 0.043 0.196
## 2007 Q3 0.111 0.043 0.196 0.076
## 2007 Q4 0.043 0.196 0.076 -0.189
## 2008 Q1 0.196 0.076 -0.189 -0.220
## 2008 Q2 0.076 -0.189 -0.220 -0.398
## 2008 Q3 -0.189 -0.220 -0.398 -0.287
## 2008 Q4 -0.220 -0.398 -0.287 -0.100
## 2009 Q1 -0.398 -0.287 -0.100 -0.059
## 2009 Q2 -0.287 -0.100 -0.059 0.241
## 2009 Q3 -0.100 -0.059 0.241 0.169
## 2009 Q4 -0.059 0.241 0.169 0.023
## 2010 Q1 0.241 0.169 0.023 0.151
## 2010 Q2 0.169 0.023 0.151 -0.128
## 2010 Q3 0.023 0.151 -0.128 0.005
## 2010 Q4 0.151 -0.128 0.005 0.094
## 2011 Q1 -0.128 0.005 0.094 0.012
## 2011 Q2 0.005 0.094 0.012 -0.050
## 2011 Q3 0.094 0.012 -0.050 -0.048
## 2011 Q4 0.012 -0.050 -0.048 0.113
## 2012 Q1 -0.050 -0.048 0.113 -0.018
## 2012 Q2 -0.048 0.113 -0.018 0.305
## 2012 Q3 0.113 -0.018 0.305 0.186
## 2012 Q4 -0.018 0.305 0.186 0.062
## 2013 Q1 0.305 0.186 0.062 0.180
## 2013 Q2 0.186 0.062 0.180 NA
## 2013 Q3 0.062 0.180 NA NA
## 2013 Q4 0.180 NA NA NA
## 2014 Q1 NA NA NA NAb
By comparison, the ets is better
a<-as.data.frame(tsCV_fes)
b<-as.data.frame(tsCV_sna)
mse<-function(x){
return(sum(x^2))
}
fets_sse<-apply(na.omit(a),1,mse)
fets_mse<-sum(fets_sse)/length(fets_sse)
snaive_sse<-apply(na.omit(b),1,mse)
snaive_mse<-sum(snaive_sse)/length(snaive_sse)
cbind(fets_mse,snaive_mse)## fets_mse snaive_mse
## [1,] 0.05032344 0.072166047.8.13
Compare the RMSE of each model to determine the optimal model, Determine whether to perform Box-Cox transformation by drawing a histogram
library(Metrics)## Warning: package 'Metrics' was built under R version 3.5.1##
## Attaching package: 'Metrics'## The following object is masked _by_ '.GlobalEnv':
##
## mse## The following object is masked from 'package:forecast':
##
## accuracyhist(ausbeer)
train0<-window(ausbeer,start=1956,end=2007+11/12)
test0<-window(ausbeer,start=2008)
fit_0<-ets(train0)
aus_ets_rmse<-forecast::accuracy(forecast(fit_0,h=4),test0)[2,2]
pred_snative0<-snaive(train0,h=4)
aus_sna_rmse<-rmse(pred_snative0$mean,test0)
pred_stlf0<-stlf(train0)
aus_stlf_rmse<-rmse(pred_stlf0$mean,test0)
rbind(aus_ets_rmse,aus_sna_rmse,aus_stlf_rmse)## [,1]
## aus_ets_rmse 11.32584
## aus_sna_rmse 12.03121
## aus_stlf_rmse 14.63685hist(bricksq)
train_1<-window(bricksq,start=1956,end=1991+11/12)
test_1<-window(bricksq,start=1992)
fit_1<-ets(train_1)
bri_ets_rmse<-forecast::accuracy(forecast(fit_1,h=4),test_1)[2,2]
pred_snative1<-snaive(train_1,h=4)
bri_sna_rmse<-rmse(pred_snative1$mean,test_1)
pred_stlf<-stlf(train_1)
bri_stlf_rmse<-rmse(pred_stlf$mean,test_1)
rbind(bri_ets_rmse,bri_sna_rmse,bri_stlf_rmse)## [,1]
## bri_ets_rmse 14.42046
## bri_sna_rmse 19.71040
## bri_stlf_rmse 17.41167hist(dole)
train_2<-window(dole,start=1956,end=1989+11/12)
test_2<-window(dole,start=1990)
fit_2<-ets(train_2)
dole_ets_rmse<-forecast::accuracy(predict(fit_2,h=12),test_2)[2,2]
pred_snative2<-snaive(train_2,h=12)
dole_sna_rmse<-rmse(pred_snative2$mean,test_2)
pred_stlf2<-stlf(train_2,lambda = 'auto')
dole_stlf_rmse<-rmse(pred_stlf2$mean,test_2)
rbind(dole_ets_rmse,dole_sna_rmse,dole_stlf_rmse)## [,1]
## dole_ets_rmse 65196.9
## dole_sna_rmse 76617.8
## dole_stlf_rmse 223738.9hist(a10)
train_3<-window(a10,start=1991,end=2005+11/12)## Warning in window.default(x, ...): 'start' value not changedtest_3<-window(a10,start=2006)
fit_3<-ets(train_3)
a10_ets_rmse<-forecast::accuracy(predict(fit_3,h=12),test_3)[2,2]
pred_snative3<-snaive(train_3,h=12)
a10_sna_rmse<-rmse(pred_snative3$mean,test_3)
pred_stlf3<-stlf(train_3,lambda = 'auto')
a10_stlf_rmse<-rmse(pred_stlf3$mean,test_3)
rbind(a10_ets_rmse,a10_sna_rmse,a10_stlf_rmse)## [,1]
## a10_ets_rmse 1.508425
## a10_sna_rmse 2.414377
## a10_stlf_rmse 1.963609hist(h02)
train_4<-window(h02,start=1991,end=2005+11/12)## Warning in window.default(x, ...): 'start' value not changedtest_4<-window(h02,start=2006)
fit_4<-ets(train_4)
h02_ets_rmse<-forecast::accuracy(predict(fit_4,h=12),test_4)[2,2]
pred_snative4<-snaive(train_4,h=4)
h02_sna_rmse<-rmse(pred_snative4$mean,test_4)
pred_stlf4<-stlf(train_4,lambda = 'auto')
h02_stlf_rmse<-rmse(pred_stlf4$mean,test_4)
rbind(h02_ets_rmse,h02_sna_rmse,h02_stlf_rmse)## [,1]
## h02_ets_rmse 0.07311548
## h02_sna_rmse 0.04364352
## h02_stlf_rmse 0.06169809hist(usmelec)
train_5<-window(usmelec,start=1973,end=2010+11/12)
test_5<-window(usmelec,start=2011)
fit_5<-ets(train_5)
us_ets_rmse<-forecast::accuracy(predict(fit_5,h=12),test_5)[2,2]
pred_snative<-snaive(train_5,h=12)
us_sna_rmse<-rmse(pred_snative$mean,test_5)
pred_stlf5<-stlf(train_5,lambda = 'auto')
us_stlf_rmse<-rmse(pred_stlf5$mean,test_5)
rbind(us_ets_rmse,us_sna_rmse,us_stlf_rmse)## [,1]
## us_ets_rmse 11.09911
## us_sna_rmse 10.14811
## us_stlf_rmse 20.779048.11.7
Fitting and plotting an appropriate ARIMA model for wmurders using the auto.arima() function yields an ARIMA (1,2,1) model.
arima.fit<-auto.arima(wmurders)
checkresiduals(arima.fit)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(1,2,1)
## Q* = 12.419, df = 8, p-value = 0.1335
##
## Model df: 2. Total lags used: 10Plot the forecast and the next 3 values of the time series.
## Warning: Ignoring unknown parameters: h## Scale for 'x' is already present. Adding another scale for 'x', which
## will replace the existing scale.
8.11.8
a
fit_au<-auto.arima(austa)
checkresiduals(fit_au)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(0,1,1) with drift
## Q* = 2.297, df = 5.2, p-value = 0.8266
##
## Model df: 2. Total lags used: 7.2autoplot(austa)+
autolayer(forecast(fit_au), h = 10,series = 'ARIMA')+
xlim(1980,2025)+
ylim(0,10)## Warning: Ignoring unknown parameters: h## Scale for 'x' is already present. Adding another scale for 'x', which
## will replace the existing scale.
#b
fit_arima011<-arima(austa,c(0,1,1),include.mean = FALSE)
autoplot(austa)+
autolayer(forecast(fit_arima011), h = 10,series = 'ARIMA')+
autolayer(forecast(fit_au), h = 10,series = 'ARIMA')+
xlim(1980,2025)+
ylim(0,10)## Warning: Ignoring unknown parameters: h
## Warning: Ignoring unknown parameters: h## Scale for 'x' is already present. Adding another scale for 'x', which
## will replace the existing scale.
fit_arima010<-arima(austa,c(0,1,0),include.mean = FALSE)
autoplot(austa)+
autolayer(forecast(fit_arima010), h = 10,series = 'ARIMA')+
autolayer(forecast(fit_au), h = 10,series = 'ARIMA')+
xlim(1980,2025)+
ylim(0,10)## Warning: Ignoring unknown parameters: h
## Warning: Ignoring unknown parameters: h## Scale for 'x' is already present. Adding another scale for 'x', which
## will replace the existing scale.
#c ARIMA(2,1,3) can’t work becourse there is a non-stationary AR part in CSS.
d
fit_arima001<-arima(austa,c(0,0,1))
autoplot(austa)+
autolayer(forecast(fit_arima001, h = 10),series = 'ARIMA')+
xlim(1980,2025)+
ylim(0,10)## Scale for 'x' is already present. Adding another scale for 'x', which
## will replace the existing scale.
fit_arima000<-arima(austa,c(0,0,0))
autoplot(austa)+
autolayer(forecast(fit_arima000, h = 10),series = 'ARIMA')+
xlim(1980,2025)+
ylim(0,10)## Scale for 'x' is already present. Adding another scale for 'x', which
## will replace the existing scale.
#e
fit_arima021<-arima(austa,c(0,2,1))
autoplot(austa)+
autolayer(forecast(fit_arima021), h = 10,series = 'ARIMA')+
xlim(1980,2025)+
ylim(0,10)## Warning: Ignoring unknown parameters: h## Scale for 'x' is already present. Adding another scale for 'x', which
## will replace the existing scale.
8.11.9
a
find a suitable Box-Cox transformation for the data
library(TSA)## Warning: package 'TSA' was built under R version 3.5.1##
## Attaching package: 'TSA'## The following objects are masked from 'package:stats':
##
## acf, arima## The following object is masked from 'package:utils':
##
## tarboxcox<-BoxCox.ar(mcopper,method = 'ols')
lambda<-boxcox$lambda[which.max(boxcox$loglike)]
mcopper_boxcox<-log(mcopper)b
ARIMA<-auto.arima(mcopper_boxcox)c
The plot strongly suggest an AR(1) model for this series
pacf(mcopper_boxcox)
AR<-arima(mcopper_boxcox,c(1,0,0))d
By comparision, ARIMA(0,1,1) has the smallest AIC
checkresiduals(ARIMA)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(0,1,1)
## Q* = 29.716, df = 23, p-value = 0.1577
##
## Model df: 1. Total lags used: 24e
forecast(ARIMA,h=6)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 2007 8.127651 8.050391 8.204910 8.009493 8.245808
## Feb 2007 8.127651 7.996261 8.259040 7.926708 8.328593
## Mar 2007 8.127651 7.958661 8.296640 7.869204 8.386097
## Apr 2007 8.127651 7.928022 8.327279 7.822345 8.432956
## May 2007 8.127651 7.901497 8.353805 7.781778 8.473523
## Jun 2007 8.127651 7.877771 8.377530 7.745493 8.509809f
a<-forecast(ARIMA,h=6)
a$mean<-exp(a$mean)
a$lower<-exp(a$lower)
a$upper<-exp(a$upper)
autoplot(mcopper)+
autolayer(a, series = 'ARIMA')+
autolayer(forecast(ets(mcopper), h = 6), series = 'ETS')+
xlim(2004,2008)+
ylim(0,5000)## Scale for 'x' is already present. Adding another scale for 'x', which
## will replace the existing scale.
9.7.4
a
fit1<-tslm(gasoline ~ trend + fourier(gasoline, K=1))
fit2<-tslm(gasoline ~ trend + fourier(gasoline, K=2))
fit3<-tslm(gasoline ~ trend + fourier(gasoline, K=3))
fit4<-tslm(gasoline ~ trend + fourier(gasoline, K=4))
fit5<-tslm(gasoline ~ trend + fourier(gasoline, K=5))
AIC(fit1,fit2,fit3,fit4,fit5)## df AIC
## fit1 5 1571.652
## fit2 7 1564.866
## fit3 9 1531.296
## fit4 11 1529.145
## fit5 13 1525.103b
Choose an optimal and relatively simple model
fit_final<-auto.arima(gasoline,max.q = 0,xreg=fourier(gasoline,K=3))c
Choose an optimal and relatively simple model
checkresiduals(fit_final)
##
## Ljung-Box test
##
## data: Residuals from Regression with ARIMA(4,1,0)(0,0,2)[52] errors
## Q* = 272.47, df = 92.357, p-value < 2.2e-16
##
## Model df: 12. Total lags used: 104.357142857143It is the ARIMA errors that should resemble a white noise series. #d produce forecasts for the next year
forecast(fit_final,h = 52,xreg=fourier(gasoline,K=3))## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2017.069 8.201533 7.865673 8.537393 7.687879165 8.715187
## 2017.088 8.224989 7.881552 8.568427 7.699747124 8.750232
## 2017.107 8.278681 7.924152 8.633211 7.736475537 8.820887
## 2017.126 8.349004 7.978219 8.719789 7.781937526 8.916071
## 2017.145 8.343358 7.949689 8.737028 7.741292643 8.945424
## 2017.164 8.410517 7.995919 8.825115 7.776443607 9.044590
## 2017.184 8.517510 8.088816 8.946204 7.861878398 9.173142
## 2017.203 8.482049 8.037957 8.926140 7.802869373 9.161228
## 2017.222 8.553442 8.093364 9.013521 7.849813656 9.257071
## 2017.241 8.488683 8.013078 8.964289 7.761308238 9.216059
## 2017.260 8.591637 8.101582 9.081693 7.842162308 9.341113
## 2017.279 8.596312 8.092376 9.100248 7.825608735 9.367015
## 2017.298 8.556706 8.038980 9.074432 7.764911837 9.348500
## 2017.318 8.578653 8.047447 9.109858 7.766244148 9.391061
## 2017.337 8.637889 8.093609 9.182168 7.805485364 9.470292
## 2017.356 8.671728 8.114725 9.228730 7.819866277 9.523589
## 2017.375 8.703272 8.133806 9.272738 7.832349164 9.574195
## 2017.394 8.701158 8.119473 9.282843 7.811547252 9.590769
## 2017.413 8.788652 8.195007 9.382297 7.880750425 9.696553
## 2017.433 8.796031 8.190672 9.401390 7.870213726 9.721848
## 2017.452 8.894507 8.277655 9.511359 7.951113019 9.837901
## 2017.471 8.906164 8.278025 9.534303 7.945507956 9.866820
## 2017.490 8.922566 8.283339 9.561793 7.944952998 9.900180
## 2017.509 8.908353 8.258229 9.558477 7.914073784 9.902632
## 2017.528 8.956878 8.296036 9.617719 7.946208264 9.967547
## 2017.548 8.898835 8.227447 9.570223 7.872036004 9.925634
## 2017.567 8.884054 8.202282 9.565825 7.841373671 9.926733
## 2017.586 8.860621 8.168621 9.552620 7.802298774 9.918942
## 2017.605 8.818660 8.116582 9.520738 7.744923812 9.892395
## 2017.624 8.702971 7.990957 9.414985 7.614039373 9.791903
## 2017.643 8.649025 7.927211 9.370838 7.545106141 9.752943
## 2017.663 8.608490 7.877008 9.339972 7.489785457 9.727195
## 2017.682 8.544323 7.803299 9.285347 7.411025312 9.677621
## 2017.701 8.579144 7.828699 9.329588 7.431438316 9.726849
## 2017.720 8.453750 7.694002 9.213498 7.291816063 9.615685
## 2017.739 8.519944 7.751005 9.288883 7.343952719 9.695935
## 2017.758 8.518092 7.740070 9.296114 7.328210544 9.707974
## 2017.778 8.477625 7.690625 9.264625 7.274012563 9.681237
## 2017.797 8.553162 7.757285 9.349038 7.335973842 9.770350
## 2017.816 8.548609 7.743954 9.353264 7.317995165 9.779223
## 2017.835 8.583385 7.770046 9.396724 7.339490559 9.827280
## 2017.854 8.563422 7.741491 9.385354 7.306386880 9.820458
## 2017.873 8.484703 7.654269 9.315138 7.214663429 9.754744
## 2017.893 8.490364 7.651513 9.329216 7.207451564 9.773277
## 2017.912 8.446302 7.599117 9.293487 7.150643967 9.741960
## 2017.931 8.372619 7.517182 9.228057 7.064340633 9.680898
## 2017.950 8.369353 7.505742 9.232964 7.048573935 9.690132
## 2017.969 8.340030 7.468322 9.211737 7.006867620 9.673191
## 2017.988 8.099542 7.219812 8.979272 6.754111479 9.444973
## 2018.008 8.098802 7.211122 8.986482 6.741212917 9.456391
## 2018.027 8.076206 7.180647 8.971765 6.706566242 9.445845
## 2018.046 8.046821 7.143452 8.950191 6.665236820 9.428406
## 2018.065 8.040313 7.122056 8.958569 6.635959887 9.444665
## 2018.084 8.136212 7.209022 9.063401 6.718197865 9.554225
## 2018.103 8.167842 7.231496 9.104187 6.735825146 9.599858
## 2018.123 8.242856 7.297048 9.188664 6.796367528 9.689344
## 2018.142 8.250894 7.295257 9.206530 6.789373976 9.712414
## 2018.161 8.329203 7.363881 9.294525 6.852869850 9.805536
## 2018.180 8.368533 7.394021 9.343045 6.878146007 9.858920
## 2018.199 8.416179 7.432434 9.399923 6.911671391 9.920686
## 2018.218 8.409637 7.416665 9.402609 6.891018012 9.928256
## 2018.238 8.438477 7.436357 9.440596 6.905867458 9.971086
## 2018.257 8.495452 7.484312 9.506592 6.949046476 10.041858
## 2018.276 8.482544 7.462475 9.502614 6.922482744 10.042606
## 2018.295 8.476345 7.447402 9.505288 6.902712211 10.049978
## 2018.314 8.506809 7.469064 9.544553 6.919714833 10.093902
## 2018.333 8.525041 7.478574 9.571508 6.924608361 10.125474
## 2018.353 8.554707 7.499595 9.609820 6.941051618 10.168363
## 2018.372 8.547555 7.483864 9.611245 6.920780011 10.174329
## 2018.391 8.615673 7.543471 9.687875 6.975881908 10.255464
## 2018.410 8.634468 7.553822 9.715113 6.981763249 10.287172
## 2018.429 8.698524 7.609502 9.787547 7.033008133 10.364041
## 2018.448 8.743667 7.646331 9.841003 7.065436424 10.421897
## 2018.468 8.766178 7.660590 9.871765 7.075327963 10.457027
## 2018.487 8.804831 7.691054 9.918608 7.101455498 10.508206
## 2018.506 8.810923 7.689015 9.932830 7.095113562 10.526732
## 2018.525 8.823244 7.693265 9.953223 7.095090526 10.551398
## 2018.544 8.815620 7.677626 9.953613 7.075208907 10.556031
## 2018.563 8.774539 7.628587 9.920491 7.021956326 10.527121
## 2018.583 8.740521 7.586665 9.894376 6.975851186 10.505191
## 2018.602 8.691981 7.530276 9.853686 6.915306068 10.468656
## 2018.621 8.631789 7.462287 9.801291 6.843189421 10.420388
## 2018.640 8.556502 7.379254 9.733749 6.756056782 10.356947
## 2018.659 8.531386 7.346444 9.716329 6.719173225 10.343599
## 2018.678 8.473904 7.281316 9.666492 6.649998750 10.297809
## 2018.698 8.469503 7.269319 9.669687 6.633980035 10.305026
## 2018.717 8.376280 7.168548 9.584013 6.529212433 10.223348
## 2018.736 8.432201 7.216967 9.647436 6.573660652 10.290742
## 2018.755 8.422044 7.199354 9.644735 6.552101010 10.291988
## 2018.774 8.388017 7.157916 9.618118 6.506740397 10.269294
## 2018.793 8.438004 7.200537 9.675471 6.545461386 10.330546
## 2018.813 8.459822 7.215033 9.704612 6.556080742 10.363564
## 2018.832 8.469265 7.217196 9.721334 6.554390038 10.384140
## 2018.851 8.485497 7.226190 9.744804 6.559552644 10.411441
## 2018.870 8.435433 7.168930 9.701936 6.498483180 10.372383
## 2018.889 8.406372 7.132713 9.680031 6.458478969 10.354266
## 2018.908 8.336378 7.055604 9.617153 6.377602394 10.295154
## 2018.928 8.296746 7.008895 9.584598 6.327148196 10.266345
## 2018.947 8.284419 6.989530 9.579308 6.304057671 10.264781
## 2018.966 8.225713 6.923824 9.527601 6.234645942 10.216779
## 2018.985 8.111817 6.802966 9.420667 6.110102223 10.113531
## 2019.004 8.065165 6.749388 9.380941 6.052858653 10.077470
## 2019.023 7.985708 6.663043 9.308373 5.962866011 10.008550
## 2019.043 7.971891 6.642372 9.301409 5.938567339 10.005214
## 2019.062 8.006661 6.666187 9.347135 5.956582730 10.056739
## 2019.081 8.014829 6.666796 9.362862 5.953190264 10.076467
## 2019.100 8.042997 6.687262 9.398732 5.969579502 10.116415
## 2019.119 8.085595 6.721965 9.449226 6.000102789 10.171088
## 2019.138 8.140078 6.768322 9.511833 6.042159033 10.237996
## 2019.157 8.193749 6.813942 9.573556 6.083516756 10.303981
## 2019.177 8.241779 6.854209 9.629349 6.119673330 10.363884
## 2019.196 8.285111 6.889743 9.680480 6.151079043 10.419143
## 2019.215 8.320792 6.917619 9.723964 6.174824657 10.466759
## 2019.234 8.347288 6.936352 9.758225 6.189446995 10.505130
## 2019.253 8.364919 6.946287 9.783550 6.195309002 10.534528
## 2019.272 8.376271 6.949994 9.802548 6.194968267 10.557574
## 2019.292 8.384623 6.950727 9.818519 6.191668133 10.577577
## 2019.311 8.393242 6.951766 9.834719 6.188693867 10.597791
## 2019.330 8.405410 6.956396 9.854424 6.189334228 10.621486
## 2019.349 8.423883 6.967372 9.880393 6.196342372 10.651423
## 2019.368 8.450337 6.986368 9.914307 6.211389296 10.689286
## 2019.387 8.484984 7.013592 9.956376 6.234684331 10.735284
## 2019.407 8.526524 7.047747 10.005301 6.264929974 10.788118
## 2019.426 8.572299 7.086174 10.058423 6.299467419 10.845130
## 2019.445 8.618573 7.125137 10.112009 6.334559890 10.902587
## 2019.464 8.660980 7.160268 10.161693 6.365839114 10.956122
## 2019.483 8.695083 7.187130 10.203036 6.388867340 11.001299
## 2019.502 8.716967 7.201807 10.232127 6.399730200 11.034204
## 2019.522 8.723774 7.201442 10.246106 6.395568199 11.051980
## 2019.541 8.714105 7.184635 10.243576 6.374981757 11.053229
## 2019.560 8.688251 7.151675 10.224827 6.338260591 11.038241
## 2019.579 8.648210 7.104561 10.191859 6.287403054 11.009017
## 2019.598 8.597492 7.046803 10.148181 6.225917396 10.969067
## 2019.617 8.540721 6.983023 10.098419 6.158427441 10.923014
## 2019.637 8.483096 6.918421 10.047772 6.090131870 10.876061
## 2019.656 8.429772 6.858151 10.001394 6.026184557 10.833360
## 2019.675 8.385238 6.806701 9.963776 5.971073580 10.799403
## 2019.694 8.352773 6.767350 9.938196 5.928077642 10.777468
## 2019.713 8.334050 6.741771 9.926329 5.898869669 10.769230
## 2019.732 8.328939 6.729834 9.928044 5.883318927 10.774560
## 2019.752 8.335534 6.729631 9.941437 5.879517970 10.791550
## 2019.771 8.350400 6.737729 9.963072 5.884032264 10.816768
## 2019.790 8.369019 6.749607 9.988431 5.892342207 10.845695
## 2019.809 8.386365 6.760241 10.012489 5.899422723 10.873308
## 2019.828 8.397554 6.764745 10.030363 5.900388087 10.894720
## 2019.847 8.398469 6.759002 10.037936 5.891121160 10.905818
## 2019.867 8.386297 6.740200 10.032395 5.868808155 10.903786
## 2019.886 8.359901 6.707199 10.012603 5.832311865 10.887490
## 2019.905 8.319986 6.660706 9.979265 5.782336729 10.857635
## 2019.924 8.269035 6.603203 9.934866 5.721365335 10.816704
## 2019.943 8.211025 6.538668 9.883383 5.653374867 10.768676
## 2019.962 8.150962 6.472104 9.829821 5.583369623 10.718555
## 2019.982 8.094285 6.408951 9.779619 5.516788838 10.671781
## 2020.001 8.046227 6.354441 9.738012 5.458864411 10.633589
## 2020.020 8.011200 6.312988 9.709411 5.414009079 10.608390
## 2020.039 7.992293 6.287679 9.696907 5.385311150 10.599275
## 2020.058 7.990935 6.279942 9.701927 5.374197871 10.607671
## 2020.077 8.006763 6.289416 9.724110 5.380307599 10.633218
## 2020.097 8.037722 6.314044 9.761400 5.401584055 10.673860
## 2020.116 8.080363 6.350377 9.810349 5.434577483 10.726148
## 2020.135 8.130309 6.394038 9.866580 5.474911295 10.785706
## 2020.154 8.182827 6.440293 9.925360 5.517851932 10.847802
## 2020.173 8.233425 6.484652 9.982199 5.558907103 10.907944
## 2020.192 8.278402 6.523411 10.033393 5.594374650 10.962430
## 2020.212 8.315275 6.554088 10.076462 5.621771516 11.008778
## 2020.231 8.343034 6.575673 10.110396 5.640088607 11.045980
## 2020.250 8.362196 6.588682 10.135710 5.649840608 11.074551
## 2020.269 8.374639 6.594994 10.154285 5.652906986 11.096372
## 2020.288 8.383265 6.597510 10.169021 5.652187966 11.114343
## 2020.307 8.391514 6.599669 10.183359 5.651123572 11.131905
## 2020.327 8.402814 6.604900 10.200728 5.653141727 11.152486
## 2020.346 8.420033 6.616071 10.223995 5.661110649 11.178955
## 2020.365 8.445012 6.635021 10.255002 5.676870058 11.213153
## 2020.384 8.478236 6.662237 10.294234 5.700905200 11.255566
## 2020.403 8.518698 6.696710 10.340685 5.732208759 11.305186
## 2020.422 8.563968 6.736012 10.391924 5.768350830 11.359585
## 2020.442 8.610466 6.776560 10.444371 5.805749508 11.415182
## 2020.461 8.653893 6.814058 10.493729 5.840107933 11.467679
## 2020.480 8.689787 6.844040 10.535533 5.866961315 11.512612
## 2020.499 8.714099 6.862461 10.565738 5.882262443 11.545936
## 2020.518 8.723748 6.866236 10.581260 5.882928476 11.564568
## 2020.537 8.717050 6.853683 10.580417 5.867276190 11.566824
## 2020.557 8.693987 6.824783 10.563191 5.835286739 11.552688
## 2020.576 8.656262 6.781240 10.531284 5.788662706 11.523861
## 2020.595 8.607137 6.726314 10.487960 5.730666853 11.483607
## 2020.614 8.551074 6.664469 10.437680 5.665760234 11.436388
## 2020.633 8.493214 6.600844 10.385585 5.599083442 11.387345
## 2020.652 8.438766 6.540648 10.336884 5.535845401 11.341687
## 2020.672 8.392381 6.488533 10.296229 5.480696571 11.304065
## 2020.691 8.357587 6.448027 10.267148 5.437166069 11.278009
## 2020.710 8.336367 6.421110 10.251624 5.407234654 11.265500
## 2020.729 8.328916 6.407981 10.249852 5.391098684 11.266734
## 2020.748 8.333634 6.407036 10.260232 5.387156203 11.280111
## 2020.767 8.347330 6.415086 10.279574 5.392218236 11.302442
## 2020.787 8.365641 6.427768 10.303514 5.401919853 11.329361
## 2020.806 8.383585 6.440099 10.327071 5.411280112 11.355890
## 2020.825 8.396206 6.447123 10.345288 5.415341046 11.377070
## 2020.844 8.399205 6.444542 10.353868 5.409805660 11.388605
## 2020.863 8.389505 6.429277 10.349733 5.391594945 11.387415
## 2020.882 8.365651 6.399874 10.331428 5.359254066 11.372047
## 2020.902 8.328016 6.356706 10.299327 5.313156895 11.342876
## 2020.921 8.278782 6.301953 10.255611 5.255483552 11.302080
## 2020.940 8.221688 6.239357 10.204019 5.189974248 11.253402
## 2020.959 8.161597 6.173778 10.149416 5.121490886 11.201703
## 2020.978 8.103917 6.110626 10.097208 5.055442004 11.152392
## 2020.997 8.053965 6.055217 10.052713 4.997143577 11.110786
## 2021.016 8.016341 6.012150 10.020531 4.951195917 11.081486
## 2021.036 7.994400 5.984782 10.004018 4.920954292 11.067846
## 2021.055 7.989883 5.974852 10.004914 4.908158532 11.071607
## 2021.074 8.002747 5.982317 10.023177 4.912766117 11.092728
## 2021.093 8.031222 6.005408 10.057036 4.933007074 11.129438
## 2021.112 8.072078 6.040894 10.103262 4.965650530 11.178506
## 2021.131 8.121065 6.084525 10.157604 5.006445861 11.235683
## 2021.151 8.173468 6.131586 10.215349 5.050679375 11.296256
## 2021.170 8.224709 6.177500 10.271919 5.093773231 11.355646
## 2021.189 8.270912 6.218389 10.323435 5.131848636 11.409975
## 2021.208 8.309350 6.251527 10.367173 5.162180769 11.456519
## 2021.227 8.338742 6.275632 10.401852 5.183487751 11.493996
## 2021.246 8.359337 6.290954 10.427720 5.196018155 11.522656
## 2021.266 8.372791 6.299148 10.446434 5.201428325 11.544154
## 2021.285 8.381855 6.302966 10.460744 5.202468505 11.561241
## 2021.304 8.389912 6.305790 10.474034 5.202521896 11.577302
## 2021.323 8.400433 6.311091 10.489776 5.205060047 11.595807
## 2021.342 8.416426 6.321876 10.510975 5.213088875 11.619762
## 2021.361 8.439941 6.340198 10.539685 5.228660665 11.651222
## 2021.381 8.471723 6.366799 10.576648 5.252518657 11.690928
## 2021.400 8.511032 6.400939 10.621126 5.283923148 11.738142
## 2021.419 8.555679 6.440429 10.670928 5.320684054 11.790673
## 2021.438 8.602258 6.481866 10.722651 5.359397450 11.845119
## 2021.457 8.646564 6.521041 10.772088 5.395856471 11.897272
## 2021.476 8.684120 6.553478 10.814762 5.425583619 11.942656
## 2021.496 8.710761 6.575012 10.846509 5.444415035 11.977106
## 2021.515 8.723196 6.582354 10.864039 5.449059797 11.997333
## 2021.534 8.719469 6.573544 10.865394 5.437559928 12.001378
## 2021.553 8.699252 6.548257 10.850247 5.409589039 11.988915
## 2021.572 8.663947 6.507894 10.820000 5.366548056 11.961346
## 2021.591 8.616559 6.455459 10.777658 5.311442464 11.921675
## 2021.611 8.561370 6.395236 10.727504 5.248553737 11.874186
## 2021.630 8.503442 6.332285 10.674599 5.182944374 11.823940
## 2021.649 8.448020 6.271852 10.624188 5.119857797 11.776182
## 2021.668 8.399897 6.218729 10.581065 5.064088451 11.735706
## 2021.687 8.362839 6.176683 10.548996 5.019401886 11.706277
## 2021.706 8.339128 6.147995 10.530261 4.988078844 11.690177
## 2021.726 8.329286 6.133187 10.525385 4.970642070 11.687929
## 2021.745 8.332023 6.130969 10.533076 4.965801662 11.698243
## 2021.764 8.344408 6.138411 10.550405 4.970627226 11.718189
## 2021.783 8.362251 6.151322 10.573180 4.980926324 11.743575
## 2021.802 8.380634 6.164783 10.596484 4.991782517 11.769485
## 2021.821 8.394547 6.173786 10.615308 4.998185943 11.790908
## 2021.841 8.399532 6.173871 10.625192 4.995677490 11.803386
## 2021.860 8.392257 6.161707 10.622806 4.980925822 11.803588
## 2021.879 8.370956 6.135529 10.606384 4.952164868 11.789748
## 2021.898 8.335673 6.095378 10.575968 4.909437026 11.761908
## 2021.917 8.288276 6.043124 10.533428 4.854611834 11.721939
## 2021.936 8.232254 5.982256 10.482253 4.791178230 11.673330
## 2021.956 8.172309 5.917474 10.427144 4.723837023 11.620781
## 2021.975 8.113798 5.854137 10.373458 4.657945295 11.569650
## 2021.994 8.062100 5.797624 10.326576 4.598882866 11.525317
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## 2022.032 7.997067 5.722991 10.271143 4.519168072 11.474967
## 2022.051 7.989390 5.710529 10.268251 4.504172749 11.474607
## 2022.071 7.999233 5.715597 10.282869 4.506712673 11.491753
## 2022.090 8.025120 5.736719 10.313521 4.525312635 11.524927
## 2022.109 8.064054 5.770898 10.357211 4.556974726 11.571134
## 2022.128 8.111929 5.814028 10.409830 4.597592095 11.626266
## 2022.147 8.164069 5.861432 10.466706 4.642490085 11.685648
## 2022.166 8.215830 5.908467 10.523193 4.687023331 11.744636
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## 2022.224 8.334166 6.012683 10.655648 4.783765043 11.884566
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## 2022.262 8.370801 6.039953 10.701649 4.806077242 11.935525
## 2022.281 8.380412 6.044896 10.715928 4.808548163 11.952276
## 2022.301 8.388389 6.048214 10.728565 4.809399436 11.967379
## 2022.320 8.398230 6.053404 10.743056 4.812128273 11.984331
## 2022.339 8.413062 6.063595 10.762528 4.819862444 12.006261
## 2022.358 8.435138 6.081040 10.789236 4.834855244 12.035421
## 2022.377 8.465455 6.106734 10.824176 4.858102195 12.072807
## 2022.396 8.503541 6.140206 10.866875 4.889132157 12.117949
## 2022.416 8.547453 6.179514 10.915392 4.926002396 12.168904
## 2022.435 8.593979 6.221444 10.966514 4.965500114 12.222458
## 2022.454 8.639019 6.261898 11.016141 5.003525509 12.274513
## 2022.473 8.678102 6.296403 11.059802 5.035607121 12.320598
## 2022.492 8.706966 6.320697 11.093235 5.057482386 12.356449
## 2022.511 8.722125 6.331296 11.112954 5.065667051 12.378583
## 2022.531 8.721357 6.325976 11.116738 5.057937935 12.384776
## 2022.550 8.704031 6.304107 11.103956 5.033664120 12.374399
## 2022.569 8.671243 6.266784 11.075702 4.993940850 12.348546
## 2022.588 8.625730 6.216745 11.034715 4.941505677 12.309954
## 2022.607 8.571578 6.158075 10.985080 4.880444478 12.262711
## 2022.626 8.513752 6.095740 10.931763 4.815722250 12.211781
## 2022.646 8.457509 6.034997 10.880021 4.752596549 12.162422
## 2022.665 8.407770 5.980766 10.834774 4.695987048 12.119553
## 2022.684 8.368522 5.937033 10.800010 4.649880970 12.087162
## 2022.703 8.342336 5.906372 10.778300 4.616850171 12.067822
## 2022.722 8.330060 5.889629 10.770492 4.597741949 12.062379
## 2022.741 8.330722 5.885831 10.775614 4.591583941 12.069861
## 2022.761 8.341663 5.892320 10.791005 4.595716458 12.087609
## 2022.780 8.358878 5.905093 10.812664 4.606136827 12.111620
## 2022.799 8.377538 5.919317 10.835759 4.618013366 12.137063
## 2022.818 8.392600 5.929952 10.855249 4.626305010 12.158896
## 2022.837 8.399463 5.932396 10.866531 4.626409067 12.172517
## 2022.856 8.394557 5.923078 10.866035 4.614755906 12.174357
## 2022.875 8.375811 5.899929 10.851694 4.589276004 12.162347
## 2022.895 8.342939 5.862661 10.823217 4.549680829 12.136197
## 2022.914 8.297491 5.812825 10.782157 4.497522590 12.097460
## 2022.933 8.242693 5.753647 10.731739 4.436025771 12.049361
## 2022.952 8.183067 5.689648 10.676485 4.369712136 11.996421
## 2022.971 8.123897 5.626114 10.621680 4.303866852 11.943927
## 2022.990 8.070607 5.568466 10.572748 4.243913175 11.897301
## 2023.010 8.028120 5.521630 10.534611 4.194774333 11.861466
## 2023.029 8.000289 5.489457 10.511121 4.160302344 11.840276
## 2023.048 7.989460 5.474293 10.504627 4.142844502 11.836076
## 2023.067 7.996234 5.476740 10.515728 4.143000249 11.849468
## 2023.086 8.019437 5.495623 10.543250 4.159596539 11.879277
## 2023.105 8.056317 5.528191 10.584443 4.189881722 11.922752
## 2023.125 8.102929 5.570498 10.635360 4.229910123 11.975949
## 2023.144 8.154657 5.617928 10.691386 4.275065323 12.034249
## 2023.163 8.206807 5.665787 10.747826 4.320652922 12.092960
## 2023.182 8.255190 5.709887 10.800492 4.362485314 12.147894
## 2023.201 8.296627 5.747048 10.846205 4.397382854 12.195870
## 2023.220 8.329300 5.775453 10.883147 4.423527461 12.235072
## 2023.240 8.352915 5.794806 10.911024 4.440624520 12.265205
## 2023.259 8.368652 5.806288 10.931015 4.449854514 12.287449
## 2023.278 8.378917 5.812305 10.945528 4.453623579 12.304210
## 2023.297 8.386927 5.816075 10.957779 4.455148651 12.318706
## 2023.316 8.396188 5.821102 10.971273 4.457934577 12.334441
## 2023.335 8.409932 5.830620 10.989244 4.465214665 12.354649
## 2023.355 8.430602 5.847070 11.014133 4.479430822 12.381772
## 2023.374 8.459438 5.871693 11.047182 4.501824083 12.417052
## 2023.393 8.496238 5.904287 11.088188 4.532191673 12.460284
## 2023.412 8.539313 5.943163 11.135463 4.568844667 12.509781
## 2023.431 8.585654 5.985312 11.185996 4.608773710 12.562534
## 2023.450 8.631284 6.026756 11.235812 4.648002637 12.614565
## 2023.470 8.671757 6.063051 11.280464 4.682084711 12.661430
## 2023.489 8.702730 6.089852 11.315609 4.706676988 12.698784
## 2023.508 8.720542 6.103497 11.337586 4.718117417 12.722966
## 2023.527 8.722713 6.101510 11.343916 4.713928268 12.731498
## 2023.546 8.708314 6.082958 11.333669 4.693178387 12.723449
## 2023.565 8.678130 6.048629 11.307632 4.656654616 12.699606
## 2023.585 8.634623 6.000982 11.268264 4.606817081 12.662429
## 2023.604 8.581668 5.943894 11.219441 4.547540921 12.615794
## 2023.623 8.524112 5.882212 11.166012 4.483674711 12.564550
## 2023.642 8.467208 5.821188 11.113228 4.420469799 12.513946
## 2023.661 8.415981 5.765848 11.066115 4.362951906 12.469011
## 2023.680 8.374625 5.720384 11.028866 4.315314341 12.433936
## 2023.700 8.345992 5.687651 11.004334 4.280410083 12.411575
## 2023.719 8.331252 5.668816 10.993688 4.259407896 12.403096
## 2023.738 8.329754 5.663230 10.996278 4.251657187 12.407850
## 2023.757 8.339119 5.668513 11.009725 4.254780042 12.423459
## 2023.776 8.355553 5.680871 11.030234 4.264980087 12.446125
## 2023.795 8.374327 5.695575 11.053078 4.277530337 12.471123
## 2023.815 8.390390 5.707576 11.073204 4.287379370 12.493400
## 2023.834 8.399015 5.712144 11.085887 4.289800047 12.508231
## 2023.853 8.396410 5.705488 11.087333 4.280999472 12.511821
## 2023.872 8.380211 5.685243 11.075178 4.258613694 12.501808
## 2023.891 8.349799 5.650793 11.048806 4.222025155 12.477573
## 2023.910 8.306405 5.603366 11.009444 4.172463287 12.440347
## 2023.930 8.252976 5.545910 10.960043 4.112876016 12.393077
## 2023.949 8.193839 5.482752 10.904926 4.047588841 12.340088
## 2023.968 8.134185 5.419083 10.849287 3.981794848 12.286575
## 2023.987 8.079461 5.360351 10.798572 3.920940156 12.237983
## 2024.006 8.034726 5.311612 10.757840 3.870082131 12.199369
## 2024.025 8.004057 5.276946 10.731169 3.833300675 12.174814
## 2024.045 7.990097 5.258994 10.721199 3.813235643 12.166958
## 2024.064 7.993763 5.258675 10.728851 3.810806882 12.176720
## 2024.083 8.014193 5.275125 10.753260 3.825149745 12.203235
## 2024.102 8.048892 5.305850 10.791934 3.853771226 12.244012
## 2024.121 8.094093 5.347083 10.841103 3.892903673 12.295282
## 2024.140 8.145257 5.394285 10.896230 3.938007993 12.352507
## 2024.160 8.197660 5.442731 10.952590 3.984359584 12.410961
## 2024.179 8.246984 5.488103 11.005864 4.027640262 12.466327
## 2024.198 8.289837 5.527011 11.052663 4.064459512 12.515215
## 2024.217 8.324141 5.557376 11.090907 4.092738335 12.555545
## 2024.236 8.349328 5.578628 11.120028 4.111907538 12.586748
## 2024.255 8.366325 5.591697 11.140954 4.122897038 12.609754
## 2024.275 8.377349 5.598797 11.155900 4.127920430 12.626777
## 2024.294 8.385506 5.603036 11.167975 4.130085997 12.640925
## 2024.313 8.394291 5.607909 11.180672 4.132888170 12.655693
## 2024.332 8.407026 5.616738 11.197314 4.139649016 12.674403
## 2024.351 8.426329 5.632140 11.220518 4.152985836 12.699673
## 2024.370 8.453679 5.655594 11.251764 4.174377330 12.732980
## 2024.390 8.489139 5.687164 11.291114 4.203887705 12.774390
## 2024.409 8.531280 5.725420 11.337140 4.240087167 12.822472
## 2024.428 8.577307 5.767567 11.387046 4.280181184 12.874432
## 2024.447 8.623384 5.809770 11.436997 4.320333212 12.926434
## 2024.466 8.665107 5.847625 11.482590 4.356139733 12.974075
## 2024.485 8.698072 5.876726 11.519418 4.383196014 13.012948
## 2024.505 8.718455 5.893251 11.543659 4.397678424 13.039232
## 2024.524 8.723536 5.894479 11.552593 4.396866351 13.050205
## 2024.543 8.712088 5.879183 11.544993 4.379533929 13.044642
## 2024.562 8.684589 5.847841 11.521336 4.346158151 13.023020
## 2024.581 8.643213 5.802628 11.483797 4.298913390 12.987512
## 2024.600 8.591610 5.747193 11.436027 4.241449813 12.941770
## 2024.620 8.534494 5.686250 11.382738 4.178481177 12.890507
## 2024.639 8.477090 5.625025 11.329156 4.115232368 12.838948
## 2024.658 8.424511 5.568628 11.280393 4.056815589 12.792206
## 2024.677 8.381138 5.521444 11.240832 4.007613861 12.754663
## 2024.696 8.350096 5.486596 11.213597 3.970750189 12.729443
## 2024.715 8.332871 5.465568 11.200173 3.947710433 12.718031
## 2024.734 8.329135 5.458036 11.200234 3.938168727 12.720101
## 2024.754 8.336804 5.461914 11.211694 3.940038865 12.733569
## 2024.773 8.352302 5.473626 11.230979 3.949746480 12.754858
## 2024.792 8.371028 5.488570 11.253486 3.962688798 12.779367
## 2024.811 8.387940 5.501705 11.274174 3.973824932 12.802055
## 2024.830 8.398206 5.508200 11.288212 3.978322937 12.818089
## 2024.849 8.397826 5.504053 11.291599 3.972182577 12.823470
## 2024.869 8.384152 5.486618 11.281687 3.952755691 12.815549
## 2024.888 8.356241 5.454950 11.257533 3.919098926 12.793384
## 2024.907 8.314996 5.409952 11.220039 3.872114785 12.757876
## 2024.926 8.263075 5.354284 11.171866 3.814463645 12.711687
## 2024.945 8.204593 5.292060 11.117127 3.750258297 12.658929
## 2024.964 8.144631 5.228360 11.060902 3.684579940 12.604683
## 2024.984 8.088637 5.168633 11.008641 3.622876871 12.554397
## 2025.003 8.041783 5.118052 10.965515 3.570321620 12.513245
## 2025.022 8.008363 5.080908 10.935818 3.531207068 12.485519
## 2025.041 7.991300 5.060126 10.922473 3.508456609 12.474143
## 2025.060 7.991831 5.056944 10.926719 3.503308417 12.480354
## 2025.079 8.009407 5.070810 10.948003 3.515211274 12.503602
## 2025.099 8.041803 5.099502 10.984104 3.541942215 12.541664
## 2025.118 8.085447 5.139446 11.031448 3.579927825 12.590967
## 2025.137 8.135896 5.186200 11.085592 3.624725215 12.647067
## 2025.156 8.188414 5.235027 11.141800 3.671598646 12.705229
## 2025.175 8.238568 5.281496 11.195641 3.716116146 12.761021
## 2025.194 8.282771 5.322017 11.243525 3.754688688 12.810853
## 2025.214 8.318688 5.354258 11.283119 3.784982735 12.852394
## 2025.233 8.345476 5.377373 11.313579 3.806154152 12.884798
## 2025.252 8.363806 5.392036 11.335577 3.818875334 12.908737
## 2025.271 8.375688 5.400254 11.351122 3.825154779 12.926222
## 2025.290 8.384105 5.405012 11.363197 3.827975717 12.940233
## 2025.309 8.392522 5.409775 11.375268 3.830804142 12.954239
## 2025.329 8.404333 5.417937 11.390729 3.837033710 12.971632
## 2025.348 8.422317 5.432276 11.412359 3.849443118 12.995192
## 2025.367 8.448182 5.454500 11.441865 3.869739826 13.026625
## 2025.386 8.482256 5.484937 11.479575 3.898252447 13.066260
## 2025.405 8.523373 5.522422 11.524324 3.933814476 13.112931
## 2025.424 8.568963 5.564384 11.573542 3.973856597 13.164069
## 2025.444 8.615345 5.607143 11.623547 4.014697186 13.215992
## 2025.463 8.658176 5.646356 11.669997 4.051994175 13.264359
## 2025.482 8.693010 5.677575 11.708445 4.081299805 13.304720
## 2025.501 8.715877 5.696832 11.734923 4.098645685 13.333109
## 2025.520 8.723827 5.701176 11.746479 4.101080948 13.346573
## 2025.539 8.715346 5.689093 11.741599 4.087091720 13.343601
## 2025.559 8.690601 5.660751 11.720451 4.056844974 13.324357
## 2025.578 8.651473 5.618030 11.684917 4.012222192 13.290724
## 2025.597 8.601375 5.564343 11.638407 3.956635288 13.246115
## 2025.616 8.544867 5.504251 11.585484 3.894645395 13.195089
## 2025.635 8.487128 5.442931 11.531325 3.831430835 13.142826
## 2025.654 8.433337 5.385564 11.481110 3.772170426 13.094504
## 2025.674 8.388049 5.336704 11.439394 3.721419784 13.054679
## 2025.693 8.354646 5.299733 11.409559 3.682559788 13.026732
## 2025.712 8.334924 5.276447 11.393400 3.657387757 13.012460
## 2025.731 8.328884 5.266848 11.390920 3.645904198 13.011864
## 2025.750 8.334741 5.269149 11.400332 3.646323377 13.023158
## 2025.769 8.349156 5.280014 11.418298 3.655307733 13.043004
## 2025.789 8.367671 5.294981 11.440360 3.668397721 13.066944
## 2025.808 8.385275 5.309043 11.461508 3.680583824 13.089967
## 2025.827 8.397053 5.317282 11.476825 3.686949350 13.107157
## 2025.846 8.398814 5.315507 11.482120 3.683303484 13.114324
## 2025.865 8.387635 5.300798 11.474472 3.666725211 13.108545
## 2025.884 8.362254 5.271890 11.452618 3.635949923 13.088558
## 2025.904 8.323242 5.229355 11.417129 3.591550353 13.054933
## 2025.923 8.272962 5.175557 11.370368 3.535889479 13.010035
## 2025.942 8.215300 5.114380 11.316220 3.472851789 12.957748
## 2025.961 8.155205 5.050774 11.259636 3.407387217 12.903022
## 2025.980 8.098106 4.990168 11.206044 3.344925522 12.851287
## 2025.999 8.049273 4.937832 11.160714 3.290734943 12.807811
## 2026.019 8.013195 4.898255 11.128134 3.249305556 12.777084
## 2026.038 7.993068 4.874634 11.111503 3.223834125 12.762303
## 2026.057 7.990447 4.868521 11.112373 3.215873795 12.765021
## 2026.076 8.005097 4.879684 11.130510 3.225190576 12.785004
## 2026.095 8.035076 4.906180 11.163972 3.249841898 12.820310
## 2026.114 8.077019 4.944644 11.209395 3.286464086 12.867575
## 2026.134 8.126600 4.990748 11.262451 3.330729033 12.922471
## 2026.153 8.179089 5.039766 11.318412 3.377909088 12.980270
## 2026.172 8.229960 5.087169 11.372751 3.423475862 13.036444
## 2026.191 8.275437 5.129182 11.421692 3.463655101 13.087219
## 2026.210 8.312940 5.163225 11.462655 3.495866259 13.130014
## 2026.229 8.341351 5.188179 11.494523 3.518991125 13.163711
## 2026.249 8.361079 5.204455 11.517704 3.533438742 13.188720
## 2026.268 8.373916 5.213843 11.533989 3.541000854 13.206831
## 2026.287 8.382704 5.219185 11.546222 3.544519767 13.220888
## 2026.306 8.390862 5.223902 11.557822 3.547415138 13.234309
## 2026.325 8.401839 5.231442 11.572237 3.553134710 13.250544
## 2026.344 8.418560 5.244729 11.592392 3.564603863 13.272517
## 2026.364 8.442951 5.265690 11.620213 3.583748887 13.302154
## 2026.383 8.475602 5.294914 11.656290 3.611159051 13.340045
## 2026.402 8.515612 5.331501 11.699723 3.645934101 13.385289
## 2026.421 8.560646 5.373116 11.748176 3.685739294 13.435553
## 2026.440 8.607193 5.416247 11.798139 3.727062487 13.487323
## 2026.459 8.650989 5.456632 11.845347 3.765640937 13.536338
## 2026.479 8.687564 5.489798 11.885329 3.797002715 13.578124
## 2026.498 8.712820 5.511649 11.913990 3.817052014 13.608587
## 2026.517 8.723590 5.519019 11.928162 3.822621233 13.624559
## 2026.536 8.718081 5.510112 11.926050 3.811916355 13.624246
## 2026.555 8.696151 5.484788 11.907513 3.784795422 13.607506
## 2026.574 8.659381 5.444628 11.874134 3.742841098 13.575921
## 2026.593 8.610934 5.392795 11.829073 3.689214763 13.532653
## 2026.613 8.555201 5.333679 11.776723 3.628307787 13.482094
## 2026.632 8.497294 5.272392 11.722196 3.565232055 13.429356
## 2026.651 8.442438 5.214160 11.670716 3.505212900 13.379663
## 2026.670 8.395343 5.163693 11.626994 3.452960762 13.337726
## 2026.689 8.359636 5.124617 11.594655 3.412101212 13.307171
## 2026.708 8.337418 5.099033 11.575803 3.384736154 13.290100
## 2026.728 8.329016 5.087270 11.570763 3.371192799 13.286840
## 2026.747 8.332953 5.087848 11.578058 3.369993434 13.295913
## 2026.766 8.346142 5.097682 11.594602 3.378051148 13.314233
## 2026.785 8.364284 5.112473 11.616096 3.391068044 13.337501
## 2026.804 8.382424 5.127264 11.637583 3.404086686 13.360760
## 2026.823 8.395578 5.137074 11.654082 3.412126025 13.379030
## 2026.843 8.399384 5.137539 11.661229 3.410822184 13.387946
## 2026.862 8.390660 5.125477 11.655843 3.396993749 13.384327
## 2026.881 8.367827 5.099309 11.636344 3.369060681 13.366593
## 2026.900 8.331125 5.059276 11.602973 3.327264422 13.334985
## 2026.919 8.282611 5.007435 11.557787 3.273661891 13.291561
## 2026.938 8.225928 4.947428 11.504428 3.211894575 13.239961
## 2026.958 8.165874 4.884053 11.447695 3.146762272 13.184986
## 2026.977 8.107841 4.822703 11.392980 3.083655539 13.132027
## 2026.996 8.057173 4.768720 11.345625 3.027918458 13.086427
## 2027.015 8.018539 4.726776 11.310303 2.984221705 13.052857
## 2027.034 7.995400 4.700329 11.290471 2.956024130 13.034776
## 2027.053 7.989618 4.691243 11.287993 2.945189234 13.034048
## 2027.073 8.001280 4.699604 11.302956 2.951802788 13.050758
## 2027.092 8.028732 4.723759 11.333706 2.974211886 13.083253
## 2027.111 8.068836 4.760568 11.377104 3.009277312 13.128395
## 2027.130 8.117396 4.805837 11.428954 3.052803949 13.181988
## 2027.149 8.169711 4.854865 11.484558 3.100091528 13.239331
## 2027.168 8.221176 4.903045 11.539307 3.146533121 13.295819
## 2027.188 8.267845 4.946433 11.589257 3.188183409 13.347506
## 2027.207 8.306898 4.982208 11.631588 3.222223485 13.391572
## 2027.226 8.336946 5.008981 11.664911 3.247263144 13.426629
## 2027.245 8.358130 5.026893 11.689366 3.263443408 13.452816
## 2027.264 8.372013 5.037509 11.706518 3.272328555 13.471698
## 2027.283 8.381283 5.043513 11.719052 3.276604702 13.485961
## 2027.303 8.389294 5.048263 11.730326 3.279627458 13.498961
## 2027.322 8.399531 5.055241 11.743822 3.284880900 13.514182
## 2027.341 8.415051 5.067505 11.762597 3.295421716 13.534681
## 2027.360 8.437988 5.087190 11.788786 3.313384131 13.562591
## 2027.379 8.469186 5.115139 11.823234 3.339613140 13.598759
## 2027.398 8.508014 5.150720 11.865308 3.373476513 13.642551
## 2027.418 8.552380 5.191843 11.912916 3.412882359 13.691877
## 2027.437 8.598954 5.235177 11.962730 3.454501767 13.743406
## 2027.456 8.643571 5.276558 12.010584 3.494168766 13.792973
## 2027.475 8.681754 5.311507 12.052000 3.527406122 13.836101
## 2027.494 8.709297 5.335819 12.082774 3.550008407 13.868585
## 2027.513 8.722830 5.346125 12.099534 3.558605644 13.887054
## 2027.533 8.720288 5.340359 12.100217 3.551132653 13.889443
## 2027.552 8.701223 5.318072 12.084373 3.527140890 13.875305
## 2027.571 8.666913 5.280545 12.053282 3.487909507 13.845917
## 2027.590 8.620259 5.230675 12.009843 3.436338115 13.804180
## 2027.609 8.565465 5.172669 11.958261 3.376631911 13.754299
## 2027.628 8.507558 5.111553 11.903563 3.313816780 13.701299
## 2027.648 8.451789 5.052578 11.851000 3.253144326 13.650433
## 2027.667 8.403004 5.000591 11.805418 3.199461445 13.606547
## 2027.686 8.365061 4.959447 11.770675 3.156624257 13.573498
## 2027.705 8.340357 4.931547 11.749168 3.127031158 13.553684
## 2027.724 8.329546 4.917541 11.741551 3.111335086 13.547757
## 2027.743 8.331464 4.916268 11.746660 3.108372645 13.554555
## 2027.763 8.343287 4.924904 11.761671 3.115320404 13.571255
## 2027.782 8.360898 4.939329 11.782467 3.128059876 13.593736
## 2027.801 8.379411 4.954660 11.804162 3.141706189 13.617116
## 2027.820 8.393801 4.965871 11.821732 3.151234578 13.636368
## 2027.839 8.399551 4.968444 11.830657 3.152126318 13.646975
## 2027.858 8.393231 4.958951 11.827510 3.140953087 13.645508
## 2027.878 8.372953 4.935503 11.810403 3.115826503 13.630079
## 2027.897 8.338627 4.898009 11.779244 3.076656522 13.600597
## 2027.916 8.291997 4.848215 11.735779 3.025186560 13.558807
## 2027.935 8.236447 4.789503 11.683390 2.964801369 13.508092
## 2027.954 8.176608 4.726506 11.626711 2.900132323 13.453085
## 2027.973 8.117813 4.664555 11.571071 2.836510249 13.399115
## 2027.993 8.065459 4.609048 11.521870 2.779334871 13.351584
## 2028.012 8.024382 4.564821 11.483943 2.733440122 13.315324
## 2028.031 7.998290 4.535582 11.460998 2.702534588 13.294045
## 2028.050 7.989350 4.523498 11.455203 2.688786251 13.289915
## 2028.069 7.997971 4.528976 11.466965 2.692601911 13.303339
## 2028.088 8.022795 4.550663 11.494928 2.712626467 13.332964
## 2028.108 8.060924 4.585655 11.536192 2.745959031 13.375888
## 2028.127 8.108311 4.629910 11.586713 2.788555230 13.428067
## 2028.146 8.160305 4.678773 11.641836 2.835761464 13.484848
## 2028.165 8.212237 4.727578 11.696896 2.882910577 13.541563
## 2028.184 8.260006 4.772222 11.747789 2.925901158 13.594110
## 2028.203 8.300565 4.809659 11.791470 2.961685677 13.639444
## 2028.223 8.332254 4.838230 11.826279 2.988605230 13.675903
## 2028.242 8.354944 4.857804 11.852085 3.006529330 13.703359
## 2028.261 8.369961 4.869708 11.870215 3.016785061 13.723138
## 2028.280 8.379821 4.876457 11.883186 3.021887767 13.737755
## 2028.299 8.387798 4.881326 11.894271 3.025111558 13.750485
## 2028.318 8.397395 4.887817 11.906972 3.029958924 13.764830
## 2028.338 8.411782 4.899102 11.924462 3.039601895 13.783962
## 2028.357 8.433292 4.917512 11.949071 3.056370801 13.810212
## 2028.376 8.463017 4.944141 11.981894 3.081360300 13.844674
## 2028.395 8.500596 4.978625 12.022567 3.114206743 13.886985
## 2028.414 8.544185 5.019123 12.069248 3.153068262 13.935303
## 2028.433 8.590653 5.062502 12.118804 3.194812066 13.986494
## 2028.452 8.635947 5.104710 12.167184 3.235385902 14.036508
## 2028.472 8.675602 5.141282 12.209923 3.270326023 14.080879
## 2028.491 8.705323 5.167922 12.242724 3.295335547 14.115311
## 2028.510 8.721553 5.181074 12.262032 3.306857251 14.136248
## 2028.529 8.721963 5.178409 12.265518 3.302564851 14.141362
## 2028.548 8.705805 5.159178 12.252432 3.281706801 14.129902
## 2028.567 8.674048 5.124351 12.223745 3.245255290 14.102841
## 2028.587 8.629322 5.076558 12.182086 3.195838025 14.062806
## 2028.606 8.575630 5.019801 12.131459 3.137458406 14.013801
## 2028.625 8.517891 4.959000 12.076782 3.075037152 13.960745
## 2028.644 8.461365 4.899415 12.023315 3.013831823 13.908898
## 2028.663 8.411015 4.846008 11.976022 2.958806886 13.863223
## 2028.682 8.370913 4.802852 11.938975 2.914033963 13.827792
## 2028.702 8.343745 4.772632 11.914858 2.882198666 13.805291
## 2028.721 8.330486 4.756324 11.904648 2.864277372 13.796696
## 2028.740 8.330294 4.753086 11.907503 2.859426035 13.801162
## 2028.759 8.340620 4.760367 11.920872 2.865096456 13.816143
## 2028.778 8.357541 4.774248 11.940834 2.877366589 13.837715
## 2028.797 8.376266 4.789934 11.962598 2.891444306 13.861087
## 2028.817 8.391747 4.802379 11.981115 2.902281938 13.881211
## 2028.836 8.399329 4.806927 11.991730 2.905224732 13.893433
## 2028.855 8.395351 4.799919 11.990784 2.896611942 13.894091
## 2028.874 8.377626 4.779165 11.976087 2.874255165 13.880997
## 2028.893 8.345732 4.744246 11.947219 2.837733999 13.853731
## 2028.912 8.301094 4.696584 11.905604 2.788472169 13.813716
## 2028.932 8.246827 4.639296 11.854357 2.729584969 13.764069
## 2028.951 8.187376 4.576827 11.797924 2.665517722 13.709233
## 2028.970 8.127991 4.514427 11.741556 2.601521208 13.654461
## 2028.989 8.074109 4.457531 11.690686 2.543030346 13.605187
## 2029.008 8.030707 4.411119 11.650295 2.495024303 13.566390
## 2029.027 8.001731 4.379135 11.624328 2.461448022 13.542014
## 2029.047 7.989647 4.364045 11.615249 2.444766870 13.534527
## 2029.066 7.995182 4.366576 11.623787 2.445708660 13.544655
## 2029.085 8.017285 4.385679 11.648891 2.463223063 13.571347
## 2029.104 8.053308 4.418704 11.687912 2.494660494 13.611956
## 2029.123 8.099374 4.461774 11.736974 2.536144538 13.662603
## 2029.142 8.150894 4.510301 11.791488 2.583087465 13.718701
## 2029.162 8.203162 4.559577 11.846746 2.630780558 13.775543
## 2029.181 8.251933 4.605361 11.898506 2.674982042 13.828885
## 2029.200 8.293944 4.644386 11.943503 2.712426544 13.875462
## 2029.219 8.327272 4.674730 11.979814 2.741190960 13.913353
## 2029.238 8.351511 4.695988 12.007034 2.760871003 13.942151
## 2029.257 8.367743 4.709242 12.026245 2.772547874 13.962939
## 2029.277 8.378299 4.716822 12.039777 2.778552290 13.978046
## 2029.296 8.386355 4.721903 12.050806 2.782059762 13.990650
## 2029.315 8.395413 4.727990 12.062835 2.786573171 14.004252
## 2029.334 8.408743 4.738352 12.079135 2.795363403 14.022123
## 2029.353 8.428861 4.755503 12.102220 2.810944189 14.046778
## 2029.372 8.457103 4.780780 12.133426 2.834652666 14.079553
## 2029.392 8.493373 4.814089 12.172657 2.866393120 14.120353
## 2029.411 8.536086 4.853842 12.218329 2.904579738 14.167592
## 2029.430 8.582316 4.897115 12.267517 2.946287817 14.218344
## 2029.449 8.628142 4.939987 12.316298 2.987595193 14.268689
## 2029.468 8.669132 4.978025 12.360240 3.024070244 14.314195
## 2029.487 8.700917 5.006859 12.394975 3.051343101 14.350491
## 2029.507 8.719767 5.022762 12.416772 3.065685132 14.373848
## 2029.526 8.723106 5.023156 12.423057 3.064520032 14.381692
## 2029.545 8.709885 5.006991 12.412778 3.046798066 14.372972
## 2029.564 8.680766 4.974932 12.386600 3.013182268 14.348350
## 2029.583 8.638097 4.929325 12.346869 2.966019431 14.310174
## 2029.602 8.585664 4.873956 12.297372 2.909096391 14.262232
## 2029.622 8.528264 4.813622 12.242906 2.847210002 14.209318
## 2029.641 8.471141 4.753568 12.188713 2.785603484 14.156678
## 2029.660 8.419356 4.698854 12.139858 2.729339548 14.109373
## 2029.679 8.377182 4.653753 12.100610 2.682689426 14.071674
## 2029.698 8.347580 4.621228 12.073933 2.648615529 14.046545
## 2029.717 8.331848 4.602573 12.061123 2.628414085 14.035281
## 2029.737 8.329463 4.597269 12.061658 2.621564141 14.037362
## 2029.756 8.338164 4.603052 12.073276 2.625803369 14.050525
## 2029.775 8.354242 4.616214 12.092269 2.637422389 14.071061
## 2029.794 8.373016 4.632076 12.113956 2.651741616 14.094290
## 2029.813 8.389437 4.645586 12.133288 2.663711365 14.115163
## 2029.832 8.398734 4.651975 12.145493 2.668560220 14.128908
## 2029.852 8.397029 4.647363 12.146694 2.662410419 14.131647
## 2029.871 8.381843 4.629273 12.134412 2.642783064 14.120902
## 2029.890 8.352427 4.596956 12.107898 2.608929845 14.095924
## 2029.909 8.309881 4.551511 12.068252 2.561949876 14.057813
## 2029.928 8.257040 4.495772 12.018307 2.504677346 14.009402
## 2029.947 8.198144 4.433982 11.962307 2.441354627 13.954934
## 2029.967 8.138346 4.371291 11.905401 2.377132430 13.899560
## 2029.986 8.083095 4.313149 11.853040 2.317460259 13.848729
## 2030.005 8.037496 4.264662 11.810330 2.267443876 13.807547
## 2030.024 8.005716 4.229996 11.781436 2.231250242 13.780181
## 2030.043 7.990510 4.211906 11.769114 2.211633801 13.769386
## 2030.062 7.992925 4.211440 11.774411 2.209642077 13.776208
## 2030.082 8.012222 4.227857 11.796587 2.224535287 13.799909
## 2030.101 8.046015 4.258772 11.833257 2.253927155 13.838102
## 2030.120 8.090610 4.300493 11.880728 2.294125843 13.887095
## 2030.139 8.141507 4.348516 11.934497 2.340628515 13.942385
## 2030.158 8.193973 4.398111 11.989834 2.388703839 13.999241
## 2030.177 8.243642 4.444912 12.042372 2.433986091 14.053298
## 2030.197 8.287043 4.485447 12.088640 2.473003843 14.101083
## 2030.216 8.321996 4.517535 12.126456 2.503575473 14.140416
## 2030.235 8.347819 4.540497 12.155142 2.525021810 14.170616
## 2030.254 8.365342 4.555159 12.175524 2.538170558 14.192513
## 2030.273 8.376697 4.563656 12.189737 2.545154757 14.208239
## 2030.292 8.384944 4.569048 12.200840 2.549034631 14.220853
## 2030.311 8.393569 4.574819 12.212318 2.553295416 14.233842
## 2030.331 8.405924 4.584323 12.227525 2.561289921 14.250558
## 2030.350 8.424694 4.600243 12.249144 2.575701749 14.273685
## 2030.369 8.451448 4.624151 12.278746 2.598102253 14.304795
## 2030.388 8.486359 4.656217 12.316501 2.628661803 14.344056
## 2030.407 8.528101 4.695115 12.361086 2.666055507 14.390146
## 2030.426 8.573968 4.738141 12.409794 2.707577756 14.440357
## 2030.446 8.620183 4.781519 12.458848 2.749452197 14.490915
## 2030.465 8.662367 4.820866 12.503869 2.787297748 14.537437
## 2030.484 8.696095 4.851759 12.540431 2.816690226 14.575499
## 2030.503 8.717482 4.870314 12.564651 2.833746070 14.601219
## 2030.522 8.723716 4.873717 12.573715 2.835651021 14.611781
## 2030.541 8.713454 4.860627 12.566281 2.821063161 14.605844
## 2030.561 8.687048 4.831395 12.542701 2.790335445 14.583761
## 2030.580 8.646558 4.788080 12.505035 2.745525903 14.547590
## 2030.599 8.595539 4.734239 12.456838 2.690190958 14.500887
## 2030.618 8.538646 4.674527 12.402766 2.628985440 14.448307
## 2030.637 8.481088 4.614151 12.348026 2.567118032 14.395059
## 2030.656 8.428007 4.558254 12.297760 2.509730003 14.346284
## 2030.676 8.383856 4.511289 12.256423 2.461275523 14.306436
## 2030.695 8.351863 4.476484 12.227242 2.424982285 14.278744
## 2030.714 8.333639 4.455451 12.211828 2.402461500 14.264817
## 2030.733 8.328990 4.447993 12.209986 2.393517744 14.264461
## 2030.752 8.335947 4.452145 12.219749 2.396184040 14.275710
## 2030.771 8.351030 4.464424 12.237635 2.406979036 14.295080
## 2030.791 8.369690 4.480283 12.259098 2.421355020 14.318026
## 2030.810 8.386898 4.494691 12.279105 2.434281178 14.339515
## 2030.829 8.397785 4.502780 12.292790 2.440889258 14.354681
## 2030.848 8.398272 4.500472 12.296072 2.437100755 14.359443
## 2030.867 8.385601 4.485007 12.286195 2.420156960 14.351044
## 2030.886 8.358699 4.455313 12.262084 2.388985421 14.328412
## 2030.906 8.318336 4.412161 12.224512 2.344357020 14.292316
## 2030.925 8.267057 4.358094 12.176020 2.288814307 14.245300
## 2030.944 8.208884 4.297135 12.120632 2.226380594 14.191387
## 2030.963 8.148847 4.234315 12.063379 2.162086891 14.135607
## 2030.982 8.092391 4.175077 12.009705 2.101376583 14.083406
## 2031.001 8.044729 4.124635 11.964822 2.049462861 14.039994
## 2031.021 8.010233 4.087362 11.933105 2.010719443 14.009747
## 2031.040 7.991939 4.066292 11.917586 1.988180373 13.995698
## 2031.059 7.991212 4.062791 11.919632 1.983210304 13.999213
## 2031.078 8.007625 4.076432 11.938818 1.995384542 14.019865
## 2031.097 8.039068 4.105105 11.973030 2.022591114 14.055544
## 2031.116 8.082049 4.145318 12.018779 2.061338701 14.102758
## 2031.136 8.132168 4.192672 12.071665 2.107228134 14.157108
## 2031.155 8.184692 4.242431 12.126953 2.155524614 14.213859
## 2031.174 8.235147 4.290124 12.180170 2.201755503 14.268539
## 2031.193 8.279869 4.332086 12.227652 2.242256058 14.317482
## 2031.212 8.316424 4.365883 12.266966 2.274592987 14.358256
## 2031.231 8.343859 4.390562 12.297157 2.297812369 14.389906
## 2031.251 8.362741 4.406689 12.318793 2.312481910 14.413001
## 2031.270 8.374994 4.416189 12.333798 2.320524771 14.429463
## 2031.289 8.383545 4.421990 12.345100 2.324869699 14.442221
## 2031.308 8.391846 4.427542 12.356150 2.328966311 14.454725
## 2031.327 8.403312 4.436262 12.370363 2.336232028 14.470393
## 2031.346 8.420784 4.450989 12.390580 2.349506042 14.492062
## 2031.366 8.446058 4.473519 12.418596 2.370584606 14.521531
## 2031.385 8.479567 4.504287 12.454846 2.399901129 14.559232
## 2031.404 8.520250 4.542231 12.498269 2.436395332 14.604105
## 2031.423 8.565632 4.584876 12.546388 2.477590682 14.653673
## 2031.442 8.612096 4.628605 12.595588 2.519871658 14.704321
## 2031.461 8.655331 4.669106 12.641556 2.558925816 14.751736
## 2031.481 8.690876 4.701920 12.679833 2.590293479 14.791459
## 2031.500 8.714711 4.723024 12.706398 2.609953197 14.819469
## 2031.519 8.723796 4.729381 12.718210 2.614865752 14.832725
## 2031.538 8.716503 4.719363 12.713644 2.603404222 14.829602
## 2031.557 8.692877 4.693012 12.692742 2.575611821 14.810143
## 2031.576 8.654680 4.652093 12.657267 2.533251163 14.776109
## 2031.596 8.605225 4.599917 12.610533 2.479635376 14.730815
## 2031.615 8.549007 4.540981 12.557034 2.419259723 14.678755
## 2031.634 8.491181 4.480438 12.501924 2.357278332 14.625083
## 2031.653 8.436946 4.423488 12.450404 2.298891066 14.575001
## 2031.672 8.390922 4.374750 12.407093 2.248717652 14.533126
## 2031.691 8.356589 4.337707 12.375472 2.210238495 14.502940
## 2031.711 8.335868 4.314276 12.357461 2.185373658 14.486363
## 2031.730 8.328890 4.304590 12.353190 2.174254377 14.483526
## 2031.749 8.333991 4.306985 12.360997 2.175217302 14.492765
## 2031.768 8.347933 4.318223 12.377643 2.185023544 14.510842
## 2031.787 8.366318 4.333906 12.398730 2.199276053 14.533360
## 2031.806 8.384156 4.349043 12.419268 2.212983685 14.555328
## 2031.826 8.396501 4.358690 12.434312 2.221201556 14.571800
## 2031.845 8.399091 4.358583 12.439599 2.219667496 14.578515
## 2031.864 8.388900 4.345697 12.432103 2.205354791 14.572445
## 2031.883 8.364536 4.318640 12.410432 2.176871903 14.552200
## 2031.902 8.326440 4.277852 12.375027 2.134659172 14.518220
## 2031.921 8.276852 4.225575 12.328129 2.080958136 14.472746
## 2031.941 8.219562 4.165597 12.273527 2.019557806 14.419567
## 2031.960 8.159463 4.102812 12.216114 1.955350310 14.363575
## 2031.979 8.101970 4.042635 12.161305 1.893752213 14.310188
## 2031.998 8.052385 3.990367 12.114403 1.840064506 14.264705
## 2032.017 8.015272 3.950573 12.079971 1.798851556 14.231692
## 2032.036 7.993934 3.926556 12.061311 1.773416388 14.214451
## 2032.056 7.990049 3.919994 12.060104 1.765436925 14.214661
## 2032.075 8.003510 3.930780 12.076241 1.774806863 14.232214
## 2032.094 8.032491 3.957087 12.107895 1.799698459 14.265284
## 2032.113 8.073715 3.995639 12.151791 1.836836112 14.310594
## 2032.132 8.122906 4.042159 12.203652 1.881942766 14.363868
## 2032.151 8.175343 4.091929 12.258758 1.930299416 14.420387
## 2032.170 8.226466 4.140385 12.312548 1.977344098 14.475589
## 2032.190 8.272430 4.183684 12.361177 2.019232399 14.525628
## 2032.209 8.310558 4.219149 12.401968 2.053287341 14.567829
## 2032.228 8.339623 4.245552 12.433694 2.078281608 14.600965
## 2032.247 8.359927 4.263196 12.456658 2.094517542 14.625336
## 2032.266 8.373172 4.273783 12.472561 2.103697145 14.642646
## 2032.285 8.382139 4.280094 12.484184 2.108602053 14.655676
## 2032.305 8.390225 4.285525 12.494925 2.112628428 14.667822
## 2032.324 8.400895 4.293542 12.508248 2.119240699 14.682549
## 2032.343 8.417127 4.307123 12.527131 2.131418662 14.702836
## 2032.362 8.440934 4.328280 12.553587 2.151173152 14.730695
## 2032.381 8.473006 4.357705 12.588308 2.179196396 14.766817
## 2032.400 8.512552 4.394605 12.630499 2.214695217 14.810409
## 2032.420 8.557332 4.436741 12.677924 2.255431091 14.859233
## 2032.439 8.603906 4.480672 12.727141 2.297963575 14.909849
## 2032.458 8.648048 4.522173 12.773924 2.338066581 14.958030
## 2032.477 8.685282 4.556767 12.813797 2.371263806 14.999300
## 2032.496 8.711465 4.580313 12.842618 2.393413448 15.029518
## 2032.515 8.723349 4.589560 12.857137 2.401265198 15.045432
## 2032.535 8.719028 4.582605 12.855450 2.392915447 15.045140
## 2032.554 8.698237 4.559183 12.837292 2.368099263 15.028376
## 2032.573 8.662440 4.520754 12.804126 2.328278407 14.996602
## 2032.592 8.614694 4.470379 12.759009 2.276510852 14.952877
## 2032.611 8.559317 4.412374 12.706259 2.217115436 14.901518
## 2032.630 8.501389 4.351821 12.650958 2.155171980 14.847607
## 2032.650 8.446149 4.293957 12.598342 2.095918772 14.796380
## 2032.669 8.398365 4.243549 12.553180 2.044123080 14.752606
## 2032.688 8.361754 4.204318 12.519191 2.003504454 14.720004
## 2032.707 8.338540 4.178485 12.498596 1.976284443 14.700796
## 2032.726 8.329180 4.166507 12.491853 1.962920749 14.695439
## 2032.745 8.332321 4.167032 12.497610 1.962060712 14.702580
## 2032.765 8.344979 4.177076 12.512883 1.970721343 14.719238
## 2032.784 8.362928 4.192412 12.533444 1.984674427 14.741182
## 2032.803 8.381236 4.208109 12.554363 1.998989118 14.763483
## 2032.822 8.394902 4.219166 12.570639 2.008664406 14.781140
## 2032.841 8.399498 4.221154 12.577843 2.009272262 14.789725
## 2032.860 8.391743 4.210792 12.572693 1.997530842 14.785955
## 2032.880 8.369931 4.186376 12.553486 1.971735859 14.768126
## 2032.899 8.334172 4.148014 12.520330 1.931996122 14.736348
## 2032.918 8.286398 4.097639 12.475158 1.880243980 14.692553
## 2032.937 8.230150 4.038791 12.421509 1.820019828 14.640280
## 2032.956 8.170162 3.976205 12.364119 1.756058264 14.584266
## 2032.975 8.111803 3.915250 12.308357 1.693728417 14.529878
## 2032.995 8.060442 3.861294 12.259591 1.638399073 14.482485
## 2033.014 8.020818 3.819076 12.222559 1.594808342 14.446827
## 2033.033 7.996489 3.792156 12.200823 1.566516518 14.426462
## 2033.052 7.989444 3.782520 12.196367 1.555509677 14.423378
## 2033.071 7.999895 3.790383 12.209406 1.562001933 14.437787
## 2033.090 8.026308 3.814209 12.238406 1.584458597 14.468157
## 2033.110 8.065637 3.850953 12.280321 1.619834178 14.511440
## 2033.129 8.113746 3.896478 12.331014 1.663991764 14.563500
## 2033.148 8.165951 3.946101 12.385801 1.712247452 14.619654
## 2033.167 8.217618 3.995188 12.440048 1.759968012 14.675268
## 2033.186 8.264738 4.039729 12.489747 1.803143857 14.726332
## 2033.205 8.304399 4.076813 12.531986 1.838863411 14.769935
## 2033.225 8.335104 4.104941 12.565266 1.865628536 14.804579
## 2033.244 8.356884 4.124148 12.589621 1.883472011 14.830297
## 2033.263 8.371211 4.135902 12.606521 1.893864355 14.848558
## 2033.282 8.380704 4.142824 12.618585 1.899425095 14.861983
## 2033.301 8.388688 4.148238 12.629139 1.903479325 14.873897
## 2033.320 8.398657 4.155639 12.641675 1.909520656 14.887793
## 2033.340 8.413715 4.168131 12.659300 1.920653864 14.906777
## 2033.359 8.436077 4.187928 12.684227 1.939093097 14.933061
## 2033.378 8.466689 4.215976 12.717402 1.965784014 14.967593
## 2033.397 8.505024 4.251750 12.758299 2.000201827 15.009847
## 2033.416 8.549092 4.293257 12.804927 2.040353864 15.057830
## 2033.435 8.595640 4.337246 12.854033 2.082988174 15.108291
## 2033.455 8.640545 4.379594 12.901496 2.123982280 15.157107
## 2033.474 8.679333 4.415826 12.942840 2.158862102 15.199804
## 2033.493 8.707760 4.441700 12.973821 2.183383441 15.232138
## 2033.512 8.722380 4.453767 12.990994 2.194099571 15.250661
## 2033.531 8.721022 4.449858 12.992187 2.188839614 15.253205
## 2033.550 8.703115 4.429401 12.976829 2.167033335 15.239197
## 2033.570 8.669816 4.393554 12.946078 2.129837451 15.209795
## 2033.589 8.623917 4.345108 12.902726 2.080043974 15.167790
## 2033.608 8.569545 4.288191 12.850898 2.021779218 15.117310
## 2033.627 8.511685 4.227787 12.795582 1.960029286 15.063340
## 2033.646 8.455594 4.169154 12.742033 1.900050611 15.011137
## 2033.665 8.406168 4.117188 12.695147 1.846739416 14.965596
## 2033.685 8.367351 4.075833 12.658870 1.804039884 14.930663
## 2033.704 8.341659 4.047602 12.635715 1.774466486 14.908851
## 2033.723 8.329872 4.033280 12.626465 1.758801887 14.900943
## 2033.742 8.330957 4.031830 12.630083 1.756009910 14.905903
## 2033.761 8.342196 4.040537 12.643856 1.763375874 14.921017
## 2033.780 8.359550 4.055359 12.663741 1.776857875 14.942242
## 2033.800 8.378167 4.071446 12.684888 1.791605745 14.964729
## 2033.819 8.393012 4.083762 12.702261 1.802583062 14.983440
## 2033.838 8.399508 4.087731 12.711284 1.805214404 14.993801
## 2033.857 8.394133 4.079830 12.708435 1.795976698 14.992288
## 2033.876 8.374877 4.058050 12.691703 1.772860494 14.976893
## 2033.895 8.341518 4.022169 12.660866 1.735643401 14.947392
## 2033.915 8.295671 3.973801 12.617542 1.685941640 14.905401
## 2033.934 8.240617 3.916227 12.565007 1.627033652 14.854200
## 2033.953 8.180913 3.854005 12.507821 1.563478477 14.798348
## 2033.972 8.121861 3.792436 12.451286 1.500577351 14.743145
## 2033.991 8.068877 3.736937 12.400817 1.443746549 14.694007
## 2034.010 8.026855 3.692402 12.361309 1.397880304 14.655830
## 2034.030 7.999601 3.662635 12.336567 1.366783411 14.632418
## 2034.049 7.989401 3.649924 12.328878 1.352743735 14.626058
## 2034.068 7.996792 3.654805 12.338778 1.356296561 14.637287
## 2034.087 8.020539 3.676044 12.365033 1.376207932 14.664870
## 2034.106 8.057840 3.710839 12.404841 1.409675987 14.706005
## 2034.125 8.104717 3.755211 12.454223 1.452721507 14.756713
## 2034.144 8.156540 3.804530 12.508549 1.500714953 14.812364
## 2034.164 8.208622 3.854110 12.563134 1.548970292 14.868273
## 2034.183 8.256804 3.899791 12.613817 1.593328048 14.920280
## 2034.202 8.297951 3.938439 12.657463 1.630652209 14.965249
## 2034.221 8.330296 3.968286 12.692306 1.659177591 15.001415
## 2034.240 8.353601 3.989095 12.718108 1.678664401 15.028538
## 2034.259 8.369095 4.002093 12.736097 1.690342361 15.047848
## 2034.279 8.379221 4.009726 12.748716 1.696654588 15.061787
## 2034.298 8.387216 4.015228 12.759203 1.700837790 15.073594
## 2034.317 8.396584 4.022106 12.771062 1.706396787 15.086771
## 2034.336 8.410540 4.033572 12.787507 1.716545160 15.104534
## 2034.355 8.431488 4.052032 12.810943 1.733688179 15.129287
## 2034.374 8.460621 4.078679 12.842563 1.759018423 15.162223
## 2034.394 8.497683 4.113255 12.882110 1.792279570 15.203086
## 2034.413 8.540933 4.154022 12.927844 1.831731520 15.250135
## 2034.432 8.587321 4.197928 12.976715 1.874323341 15.300319
## 2034.451 8.632845 4.240971 13.024719 1.916052891 15.349637
## 2034.470 8.673051 4.278697 13.067405 1.952467054 15.393636
## 2034.489 8.703612 4.306780 13.100444 1.979237530 15.427986
## 2034.509 8.720899 4.321590 13.120208 1.992736601 15.449061
## 2034.528 8.722484 4.320700 13.124269 1.990536594 15.454432
## 2034.547 8.707497 4.303239 13.111756 1.971766049 15.443229
## 2034.566 8.676787 4.270056 13.083517 1.937273692 15.416299
## 2034.585 8.632868 4.223666 13.042070 1.889575886 15.376160
## 2034.604 8.579661 4.167989 12.991332 1.832591134 15.326730
## 2034.624 8.522037 4.107897 12.936177 1.771192830 15.272882
## 2034.643 8.465253 4.048646 12.881860 1.710635236 15.219870
## 2034.662 8.414313 3.995240 12.833386 1.655924464 15.172701
## 2034.681 8.373371 3.951834 12.794908 1.611213937 15.135528
## 2034.700 8.345225 3.921225 12.769225 1.579301307 15.111149
## 2034.719 8.330980 3.904519 12.757441 1.561291399 15.100668
## 2034.739 8.329920 3.900998 12.758842 1.556469033 15.103371
## 2034.758 8.339611 3.908230 12.770991 1.562399152 15.116822
## 2034.777 8.356213 3.922375 12.790050 1.575242883 15.137182
## 2034.796 8.374977 3.938683 12.811271 1.590250769 15.159703
## 2034.815 8.390852 3.952103 12.829601 1.602371832 15.179332
## 2034.834 8.399135 3.957933 12.840336 1.606902308 15.191367
## 2034.854 8.396075 3.952421 12.839729 1.600092751 15.192057
## 2034.873 8.379368 3.933263 12.825472 1.579637357 15.179098
## 2034.892 8.348460 3.899906 12.797014 1.544984199 15.151936
## 2034.911 8.304648 3.853646 12.755649 1.497427681 15.111867
## 2034.930 8.250934 3.797485 12.704382 1.439971997 15.061895
## 2034.949 8.191685 3.735791 12.647578 1.376983279 15.006386
## 2034.969 8.132114 3.673776 12.590451 1.313674669 14.950553
## 2034.988 8.077664 3.616884 12.538444 1.255489405 14.899839
## 2035.007 8.033368 3.570146 12.496589 1.207459567 14.859276
## 2035.026 8.003260 3.537599 12.468922 1.173620682 14.832900
## 2035.045 7.989924 3.521824 12.458024 1.156554818 14.823293
## 2035.064 7.994215 3.523677 12.464752 1.157117905 14.831311
## 2035.084 8.015205 3.542232 12.488178 1.174383213 14.856027
## 2035.103 8.050351 3.574943 12.525758 1.205805102 14.894896
## 2035.122 8.095846 3.618005 12.573687 1.247579033 14.944113
## 2035.141 8.147135 3.666862 12.627408 1.295148971 14.999121
## 2035.160 8.199498 3.716795 12.682202 1.343794870 15.055202
## 2035.179 8.248642 3.763509 12.733775 1.389223375 15.108061
## 2035.199 8.291217 3.803656 12.778778 1.428085226 15.154350
## 2035.218 8.325197 3.835209 12.815184 1.458353172 15.192040
## 2035.237 8.350066 3.857653 12.842479 1.479513036 15.220619
## 2035.256 8.366806 3.871968 12.861643 1.492545481 15.241066
## 2035.275 8.377669 3.880409 12.874929 1.499703675 15.255634
## 2035.294 8.385788 3.886106 12.885469 1.504118997 15.267456
## 2035.314 8.394659 3.892558 12.896761 1.509289439 15.280029
## 2035.333 8.407590 3.903070 12.912110 1.518520714 15.296659
## 2035.352 8.427163 3.920225 12.934101 1.534396174 15.319929
## 2035.371 8.454810 3.945456 12.964164 1.558347700 15.351272
## 2035.390 8.490542 3.978773 13.002311 1.590386305 15.390697
## 2035.409 8.532877 4.018694 13.047060 1.629030031 15.436724
## 2035.429 8.578977 4.062382 13.095572 1.671440553 15.486513
## 2035.448 8.624976 4.105970 13.143982 1.713752188 15.536199
## 2035.467 8.666460 4.145044 13.187876 1.751551340 15.581370
## 2035.486 8.699037 4.175212 13.222861 1.780444170 15.617629
## 2035.505 8.718912 4.192680 13.245144 1.796638068 15.641186
## 2035.524 8.723414 4.194776 13.252052 1.797460050 15.649368
## 2035.544 8.711374 4.180332 13.242417 1.781742726 15.641006
## 2035.563 8.683332 4.149886 13.216778 1.750025145 15.616639
## 2035.582 8.641520 4.105672 13.177368 1.704539472 15.578501
## 2035.601 8.589635 4.051386 13.127884 1.648981993 15.530287
## 2035.620 8.532417 3.991769 13.073066 1.588094905 15.476740
## 2035.639 8.475101 3.932054 13.018147 1.527110031 15.423091
## 2035.659 8.422780 3.877337 12.968224 1.471123889 15.374437
## 2035.678 8.379804 3.831964 12.927643 1.424482935 15.335124
## 2035.697 8.349240 3.799006 12.899474 1.390257139 15.308223
## 2035.716 8.332512 3.779885 12.885140 1.369869381 15.295155
## 2035.735 8.329230 3.774210 12.884249 1.362928706 15.295531
## 2035.754 8.337247 3.779837 12.894658 1.367289859 15.307205
## 2035.774 8.352945 3.793145 12.912745 1.379332883 15.326557
## 2035.793 8.371693 3.809505 12.933881 1.394428657 15.348958
## 2035.812 8.388448 3.823873 12.953023 1.407532544 15.369363
## 2035.831 8.398396 3.831435 12.965357 1.413831996 15.382960
## 2035.850 8.397578 3.828232 12.966923 1.409366960 15.385788
## 2035.869 8.383401 3.811672 12.955130 1.391545271 15.375256
## 2035.889 8.354987 3.780876 12.929098 1.359488442 15.350486
## 2035.908 8.313304 3.736813 12.889796 1.314164661 15.312444
## 2035.927 8.261072 3.682201 12.839943 1.258292665 15.263851
## 2035.946 8.202445 3.621196 12.783695 1.196029092 15.208862
## 2035.965 8.142531 3.558904 12.726157 1.132479111 15.152583
## 2035.984 8.086778 3.500775 12.672780 1.073092266 15.100463
## 2036.003 8.040337 3.451960 12.628713 1.023019625 15.057654
## 2036.023 8.007459 3.416709 12.598210 0.986512735 15.028406
## 2036.042 7.991014 3.397891 12.584136 0.966439087 15.015588
## 2036.061 7.992174 3.396681 12.587668 0.963973441 15.020375
## 2036.080 8.010326 3.412463 12.608190 0.978501579 15.042151
## 2036.099 8.043193 3.442961 12.643425 1.007745690 15.078640
## 2036.118 8.087160 3.484561 12.689759 1.048092274 15.126228
## 2036.138 8.137764 3.532799 12.742729 1.095077731 15.180450
## 2036.157 8.190270 3.582940 12.797600 1.143967038 15.236573
## 2036.176 8.240267 3.630574 12.849961 1.190349636 15.290185
## 2036.195 8.284206 3.672150 12.896262 1.230675191 15.337737
## 2036.214 8.319803 3.705385 12.934220 1.262660534 15.376945
## 2036.233 8.346268 3.729491 12.963045 1.285516511 15.407019
## 2036.253 8.364326 3.745190 12.983462 1.299967551 15.428685
## 2036.272 8.376029 3.754535 12.997522 1.308064158 15.443993
## 2036.291 8.384384 3.760534 13.008234 1.312815883 15.455952
## 2036.310 8.392866 3.766661 13.019071 1.317695870 15.468036
## 2036.329 8.404855 3.776296 13.033414 1.326084871 15.483626
## 2036.348 8.423099 3.792187 13.054011 1.340730484 15.505468
## 2036.368 8.449260 3.815997 13.082524 1.363295201 15.535225
## 2036.387 8.483615 3.848001 13.119229 1.394055043 15.573175
## 2036.406 8.524943 3.886980 13.162906 1.431790592 15.618096
## 2036.425 8.570630 3.930319 13.210942 1.473886731 15.667374
## 2036.444 8.616962 3.974304 13.259621 1.516629685 15.717295
## 2036.463 8.659584 4.014580 13.304588 1.555663679 15.763504
## 2036.483 8.694054 4.046706 13.341402 1.586548073 15.801560
## 2036.502 8.716432 4.066740 13.366123 1.605342079 15.827521
## 2036.521 8.723811 4.071778 13.375845 1.609139850 15.838483
## 2036.540 8.714736 4.060362 13.369111 1.596484544 15.832988
## 2036.559 8.689435 4.032721 13.346149 1.567605112 15.811265
## 2036.578 8.649849 3.990796 13.308902 1.524442151 15.775255
## 2036.598 8.599438 3.938047 13.260828 1.470456261 15.728419
## 2036.617 8.542795 3.879068 13.206521 1.410240517 15.675349
## 2036.636 8.485110 3.819048 13.151171 1.348984173 15.621235
## 2036.655 8.431549 3.763154 13.099945 1.291854346 15.571244
## 2036.674 8.386636 3.715908 13.057364 1.243373558 15.529898
## 2036.693 8.353700 3.680641 13.026760 1.206872156 15.500529
## 2036.713 8.334478 3.659088 13.009868 1.184085699 15.484870
## 2036.732 8.328904 3.651184 13.006623 1.174949370 15.482859
## 2036.751 8.335132 3.655084 13.015179 1.177616722 15.492647
## 2036.770 8.349776 3.667401 13.032150 1.188701712 15.510849
## 2036.789 8.368345 3.683645 13.053046 1.203714675 15.532976
## 2036.808 8.385824 3.698799 13.072849 1.217638181 15.554010
## 2036.828 8.397310 3.707962 13.086659 1.225570886 15.569050
## 2036.847 8.398650 3.706979 13.090321 1.223358730 15.573941
## 2036.866 8.386975 3.692984 13.080967 1.208134328 15.565816
## 2036.885 8.361086 3.664774 13.057398 1.178696918 15.543475
## 2036.904 8.321621 3.622990 13.020252 1.135685372 15.507557
## 2036.923 8.271003 3.570055 12.971952 1.081522898 15.460484
## 2036.943 8.213164 3.509899 12.916429 1.020140869 15.406187
## 2036.962 8.153081 3.447501 12.858662 0.956517019 15.349646
## 2036.981 8.096190 3.388295 12.804085 0.896086490 15.296294
## 2037.000 8.047741 3.337533 12.757950 0.844099974 15.251383
## 2037.019 8.012187 3.299667 12.724707 0.805009433 15.219364
## 2037.038 7.992670 3.277839 12.707501 0.781957841 15.203381
## 2037.058 7.990680 3.273539 12.707821 0.776435697 15.204924
## 2037.077 8.005920 3.286471 12.725370 0.788145342 15.223695
## 2037.096 8.036391 3.314634 12.758148 0.815087368 15.257695
## 2037.115 8.078686 3.354623 12.802750 0.853854965 15.303518
## 2037.134 8.128453 3.402084 12.854821 0.900095622 15.356809
## 2037.153 8.180959 3.452287 12.909632 0.949078544 15.412840
## 2037.173 8.231696 3.500720 12.962672 0.996293228 15.467099
## 2037.192 8.276925 3.543647 13.010202 1.038001379 15.515848
## 2037.211 8.314113 3.578535 13.049692 1.071671214 15.556555
## 2037.230 8.342198 3.604320 13.080076 1.096239227 15.588157
## 2037.249 8.361642 3.621465 13.101819 1.112167559 15.611116
## 2037.268 8.374280 3.631806 13.116754 1.121292213 15.627268
## 2037.288 8.382985 3.638215 13.127755 1.126484937 15.639485
## 2037.307 8.391186 3.644121 13.138252 1.131176098 15.651196
## 2037.326 8.402323 3.652963 13.151682 1.138804051 15.665841
## 2037.345 8.419292 3.667639 13.170944 1.152266066 15.686317
## 2037.364 8.443976 3.690032 13.197921 1.173445527 15.714507
## 2037.383 8.476914 3.720679 13.233149 1.202879945 15.750948
## 2037.403 8.517152 3.758627 13.275676 1.239615653 15.794687
## 2037.422 8.562306 3.801493 13.323120 1.281270312 15.843342
## 2037.441 8.608831 3.845730 13.371932 1.324296773 15.893365
## 2037.460 8.652446 3.887059 13.417833 1.364415035 15.940477
## 2037.479 8.688683 3.921010 13.456355 1.397156509 15.980209
## 2037.498 8.713469 3.943512 13.483426 1.418449476 16.008488
## 2037.518 8.723680 3.951440 13.495920 1.425168576 16.022191
## 2037.537 8.717576 3.943054 13.492098 1.415575154 16.019577
## 2037.556 8.695078 3.918276 13.471881 1.389588907 16.000568
## 2037.575 8.657829 3.878746 13.436911 1.348852694 15.966805
## 2037.594 8.609040 3.827679 13.390401 1.296578949 15.921501
## 2037.613 8.553139 3.769500 13.336778 1.237194350 15.869083
## 2037.633 8.495252 3.709336 13.281167 1.175825658 15.814678
## 2037.652 8.440597 3.652406 13.228788 1.117690538 15.763503
## 2037.671 8.393855 3.603389 13.184320 1.067469931 15.720239
## 2037.690 8.358603 3.565864 13.151342 1.028741585 15.688464
## 2037.709 8.336884 3.541873 13.131895 1.003547113 15.670220
## 2037.728 8.328958 3.531676 13.126241 0.992148437 15.665768
## 2037.748 8.333287 3.533735 13.132840 0.993005708 15.673569
## 2037.767 8.346733 3.544911 13.148554 1.002980845 15.690485
## 2037.786 8.364963 3.560873 13.169052 1.017742191 15.712183
## 2037.805 8.383008 3.576652 13.189364 1.032320261 15.733695
## 2037.824 8.395898 3.587276 13.204520 1.041745248 15.750051
## 2037.843 8.399303 3.588416 13.210189 1.041686468 15.756919
## 2037.862 8.390092 3.576941 13.203242 1.029013553 15.751170
## 2037.882 8.366748 3.551335 13.182161 1.002209037 15.731286
## 2037.901 8.329578 3.511904 13.147253 0.961580695 15.697575
## 2037.920 8.280702 3.460767 13.100637 0.909247307 15.652156
## 2037.939 8.223810 3.401616 13.046005 0.848900103 15.598720
## 2037.958 8.163734 3.339281 12.988187 0.785370406 15.542098
## 2037.977 8.105874 3.279164 12.932585 0.724058277 15.487691
## 2037.997 8.055561 3.226595 12.884527 0.670294248 15.440828
## 2038.016 8.017430 3.186208 12.848652 0.628714254 15.406146
## 2038.035 7.994889 3.161413 12.828365 0.602725697 15.387052
## 2038.054 7.989740 3.154011 12.825468 0.594130450 15.385349
## 2038.073 8.002003 3.164023 12.839984 0.602950204 15.401057
## 2038.092 8.029969 3.189737 12.870201 0.627473446 15.432465
## 2038.112 8.070452 3.227970 12.912934 0.664514943 15.476389
## 2038.131 8.119228 3.274497 12.963959 0.709851559 15.528604
## 2038.150 8.171590 3.324612 13.018569 0.758776018 15.584404
## 2038.169 8.222946 3.373721 13.072172 0.806695659 15.639196
## 2038.188 8.269383 3.417912 13.120854 0.849698219 15.689068
## 2038.207 8.308130 3.454414 13.161846 0.885011731 15.731248
## 2038.227 8.337850 3.481890 13.193809 0.911300177 15.764399
## 2038.246 8.358738 3.500536 13.216940 0.928758513 15.788717
## 2038.265 8.372405 3.511961 13.232849 0.938997334 15.805813
## 2038.284 8.381570 3.518885 13.244254 0.944735198 15.818404
## 2038.303 8.389601 3.524677 13.254526 0.949341940 15.829861
## 2038.322 8.399979 3.532816 13.267142 0.956295815 15.843662
## 2038.342 8.415734 3.546333 13.285134 0.968628519 15.862839
## 2038.361 8.438959 3.567322 13.310596 0.988433622 15.889485
## 2038.380 8.470450 3.596578 13.344322 1.016505435 15.924394
## 2038.399 8.509520 3.633413 13.385626 1.052158052 15.966881
## 2038.418 8.554028 3.675688 13.432368 1.093250414 16.014805
## 2038.437 8.600607 3.720035 13.481180 1.136415917 16.064799
## 2038.457 8.645072 3.762268 13.527876 1.177467877 16.112676
## 2038.476 8.682944 3.797910 13.567978 1.211928624 16.153959
## 2038.495 8.710038 3.822774 13.597301 1.235613224 16.184462
## 2038.514 8.723023 3.833532 13.612515 1.245191002 16.200856
## 2038.533 8.719889 3.828170 13.611608 1.238650252 16.201128
## 2038.552 8.700247 3.806302 13.594193 1.215603511 16.184891
## 2038.572 8.665438 3.769267 13.561609 1.177390960 16.153485
## 2038.591 8.618414 3.720019 13.516809 1.126965426 16.109863
## 2038.610 8.563419 3.662801 13.464038 1.068570626 16.058268
## 2038.629 8.505499 3.602658 13.408339 1.007251205 16.003746
## 2038.648 8.449900 3.544838 13.354961 0.948255147 15.951544
## 2038.667 8.401444 3.494162 13.308726 0.896403566 15.906484
## 2038.687 8.363942 3.454441 13.273443 0.855507657 15.872376
## 2038.706 8.339734 3.428014 13.251453 0.827907237 15.851560
## 2038.725 8.329408 3.415471 13.243344 0.814190248 15.844625
## 2038.744 8.331737 3.415584 13.247890 0.813130001 15.850344
## 2038.763 8.343844 3.425476 13.262212 0.821849405 15.865839
## 2038.782 8.361574 3.440992 13.282156 0.836192714 15.886955
## 2038.802 8.380025 3.457229 13.302821 0.851258929 15.908791
## 2038.821 8.394180 3.469172 13.319188 0.862030380 15.926329
## 2038.840 8.399549 3.472330 13.326768 0.864017835 15.935080
## 2038.859 8.392753 3.463323 13.322182 0.853841238 15.931664
## 2038.878 8.371964 3.440325 13.303602 0.829673257 15.914254
## 2038.897 8.337158 3.403311 13.271005 0.791490111 15.882825
## 2038.917 8.290142 3.354088 13.226196 0.741098463 15.839185
## 2038.936 8.234353 3.296093 13.172613 0.681935471 15.786771
## 2038.955 8.174458 3.233992 13.114924 0.618667641 15.730248
## 2038.974 8.115801 3.173131 13.058471 0.556639387 15.674962
## 2038.993 8.063772 3.118899 13.008646 0.501240945 15.626304
## 2039.012 8.023175 3.076099 12.970250 0.457275029 15.589074
## 2039.032 7.997668 3.048390 12.946945 0.428401326 15.566934
## 2039.051 7.989359 3.037881 12.940837 0.416727461 15.561990
## 2039.070 7.998591 3.044914 12.952268 0.422596001 15.574586
## 2039.089 8.023949 3.068074 12.979825 0.444591695 15.603307
## 2039.108 8.062483 3.104410 13.020556 0.479764936 15.645201
## 2039.127 8.110117 3.149848 13.070387 0.524039814 15.696195
## 2039.147 8.162187 3.199722 13.124652 0.572751868 15.751622
## 2039.166 8.214036 3.249377 13.178696 0.621244514 15.806828
## 2039.185 8.261593 3.294740 13.228446 0.665446335 15.857739
## 2039.204 8.301854 3.332809 13.270900 0.702354703 15.901354
## 2039.223 8.333216 3.361978 13.304453 0.730364146 15.936067
## 2039.242 8.355601 3.382173 13.329029 0.749398710 15.961803
## 2039.262 8.370384 3.394767 13.346002 0.760833655 15.979935
## 2039.281 8.380118 3.402311 13.357925 0.767219622 15.993016
## 2039.300 8.388093 3.408098 13.368087 0.771848413 16.004337
## 2039.319 8.397809 3.415628 13.379990 0.778220285 16.017398
## 2039.338 8.412417 3.428050 13.396784 0.789485510 16.035349
## 2039.357 8.434209 3.447657 13.420762 0.807936198 16.060483
## 2039.377 8.464231 3.475495 13.452967 0.834617622 16.093844
## 2039.396 8.502064 3.511145 13.492983 0.869112441 16.135016
## 2039.415 8.545817 3.552716 13.538919 0.909528271 16.182107
## 2039.434 8.592317 3.597035 13.587599 0.952692121 16.231942
## 2039.453 8.637487 3.640024 13.634949 0.994527673 16.280446
## 2039.472 8.676859 3.677217 13.676500 1.030566995 16.323151
## 2039.492 8.706153 3.704334 13.707973 1.056530037 16.355777
## 2039.511 8.721849 3.717852 13.725846 1.068895789 16.374802
## 2039.530 8.721671 3.715498 13.727844 1.065389343 16.377953
## 2039.549 8.704928 3.696579 13.713277 1.045319416 16.364537
## 2039.568 8.672654 3.662131 13.683177 1.009719814 16.335588
## 2039.587 8.627532 3.614835 13.640229 0.961273393 16.293790
## 2039.607 8.573606 3.558737 13.588476 0.904025130 16.243187
## 2039.626 8.515821 3.498779 13.532862 0.842918298 16.188723
## 2039.645 8.459433 3.440221 13.478645 0.783210901 16.135655
## 2039.664 8.409386 3.388004 13.430767 0.729845434 16.088926
## 2039.683 8.369709 3.346159 13.393259 0.686851671 16.052566
## 2039.702 8.343031 3.317313 13.368750 0.656858590 16.029204
## 2039.721 8.330265 3.302380 13.358150 0.640778132 16.019752
## 2039.741 8.330502 3.300450 13.360553 0.637702051 16.023301
## 2039.760 8.341137 3.308921 13.373353 0.645026222 16.037248
## 2039.779 8.358209 3.323828 13.392589 0.658787994 16.057629
## 2039.798 8.376904 3.340360 13.413448 0.674175103 16.079633
## 2039.817 8.392179 3.353473 13.430885 0.686142799 16.098215
## 2039.836 8.399403 3.358536 13.440271 0.690062011 16.108745
## 2039.856 8.394963 3.351935 13.437991 0.682317281 16.107608
## 2039.875 8.376728 3.331540 13.421915 0.660779495 16.092676
## 2039.894 8.344344 3.296998 13.391690 0.625094179 16.063593
## 2039.913 8.299299 3.249795 13.348802 0.576749437 16.021848
## 2039.932 8.244763 3.193103 13.296424 0.518915430 15.970611
## 2039.951 8.185221 3.131405 13.239037 0.456076078 15.914366
## 2039.971 8.125940 3.069969 13.181912 0.393499789 15.858381
## 2039.990 8.072351 3.014226 13.130476 0.336616097 15.808086
## 2040.009 8.029404 2.969126 13.089682 0.290376602 15.768432
## 2040.028 8.000999 2.938569 13.063430 0.258680102 15.743318
## 2040.047 7.989542 2.924961 13.054124 0.243933122 15.735151
## 2040.066 7.995697 2.928965 13.062429 0.246799341 15.744595
## 2040.086 8.018352 2.949471 13.087233 0.266166985 15.770537
## 2040.105 8.054806 2.983776 13.125836 0.299335225 15.810277
## 2040.124 8.101148 3.027971 13.174326 0.342392710 15.859904
## 2040.143 8.152776 3.077451 13.228100 0.390737019 15.914814
## 2040.162 8.204986 3.127516 13.282456 0.439666244 15.970307
## 2040.181 8.253566 3.173951 13.333181 0.484965400 16.022166
## 2040.201 8.295291 3.213532 13.377050 0.523411779 16.067171
## 2040.220 8.328292 3.244390 13.412194 0.553134707 16.103449
## 2040.239 8.352218 3.266174 13.438262 0.573784792 16.130651
## 2040.258 8.368201 3.280016 13.456386 0.586493061 16.149909
## 2040.277 8.378610 3.288284 13.468935 0.593628236 16.163591
## 2040.296 8.386640 3.294175 13.479105 0.598386906 16.174894
## 2040.316 8.395797 3.301194 13.490401 0.604273323 16.187321
## 2040.335 8.409333 3.312592 13.506075 0.614539953 16.204127
## 2040.354 8.429726 3.330848 13.528604 0.631664908 16.227787
## 2040.373 8.458265 3.357251 13.559279 0.656937466 16.259593
## 2040.392 8.494801 3.391652 13.597950 0.690208442 16.299394
## 2040.411 8.537697 3.432414 13.642980 0.729840394 16.345554
## 2040.431 8.583985 3.476569 13.691402 0.772866097 16.395104
## 2040.450 8.629716 3.520168 13.739265 0.815336095 16.444097
## 2040.469 8.670451 3.558771 13.782131 0.852810869 16.488091
## 2040.488 8.701832 3.588021 13.815643 0.880933664 16.522730
## 2040.507 8.720164 3.604224 13.836104 0.896008841 16.544319
## 2040.526 8.722920 3.604851 13.840989 0.895509171 16.550331
## 2040.546 8.709109 3.588913 13.829306 0.878444120 16.539775
## 2040.565 8.679457 3.557133 13.801781 0.845538677 16.513375
## 2040.584 8.636366 3.511916 13.760816 0.799196488 16.473536
## 2040.603 8.583669 3.457094 13.710244 0.743248849 16.424089
## 2040.622 8.526188 3.397488 13.654887 0.682518864 16.369857
## 2040.641 8.469171 3.338348 13.599994 0.622254249 16.316087
## 2040.661 8.417662 3.284717 13.550608 0.567499638 16.267825
## 2040.680 8.375895 3.240828 13.510962 0.522487656 16.229303
## 2040.699 8.346777 3.209589 13.483965 0.490126251 16.203429
## 2040.718 8.331541 3.192233 13.470849 0.471648034 16.191435
## 2040.737 8.329601 3.188174 13.471028 0.466467204 16.192736
## 2040.756 8.338637 3.195092 13.482182 0.472263470 16.205011
## 2040.776 8.354896 3.209233 13.500558 0.485283668 16.224507
## 2040.795 8.373673 3.225894 13.521452 0.500824149 16.246522
## 2040.814 8.389918 3.240024 13.539813 0.513833966 16.266002
## 2040.833 8.398882 3.246873 13.550891 0.519563469 16.278200
## 2040.852 8.396728 3.242605 13.550852 0.514177032 16.279280
## 2040.871 8.381036 3.224800 13.537272 0.495253042 16.266819
## 2040.891 8.351122 3.192773 13.509470 0.462108521 16.240135
## 2040.910 8.308150 3.147690 13.468609 0.415907842 16.200392
## 2040.929 8.255012 3.092442 13.417582 0.359542301 16.150481
## 2040.948 8.195992 3.031312 13.360672 0.297296180 16.094688
## 2040.967 8.136263 2.969474 13.303051 0.234341797 16.038183
## 2040.986 8.081272 2.912376 13.250168 0.176127399 15.986416
## 2041.006 8.036102 2.865099 13.207105 0.127734953 15.944468
## 2041.025 8.004876 2.831767 13.177985 0.093288215 15.916464
## 2041.044 7.990292 2.815078 13.165506 0.075484543 15.905099
## 2041.063 7.993333 2.816015 13.170652 0.075307688 15.911359
## 2041.082 8.013198 2.833776 13.192620 0.091955362 15.934441
## 2041.101 8.047446 2.865921 13.228971 0.122987722 15.971905
## 2041.121 8.092348 2.908721 13.275975 0.164674723 16.020021
## 2041.140 8.143381 2.957653 13.329109 0.212495044 16.074267
## 2041.159 8.195818 3.007991 13.383646 0.261720356 16.129916
## 2041.178 8.245317 3.055390 13.435244 0.308008389 16.182626
## 2041.197 8.288446 3.096420 13.480471 0.347927941 16.228964
## 2041.216 8.323074 3.128951 13.517198 0.379348589 16.266800
## 2041.236 8.348579 3.152359 13.544799 0.401646173 16.295511
## 2041.255 8.365838 3.167522 13.564153 0.415699661 16.315975
## 2041.274 8.377025 3.176614 13.577435 0.423682636 16.330367
## 2041.293 8.385224 3.182719 13.587729 0.428679699 16.341769
## 2041.312 8.393927 3.189329 13.598526 0.434181062 16.353674
## 2041.331 8.406471 3.199780 13.613162 0.443524529 16.369417
## 2041.351 8.425506 3.216724 13.634289 0.459360854 16.391652
## 2041.370 8.452558 3.241685 13.663432 0.483215266 16.421901
## 2041.389 8.487745 3.274781 13.700708 0.515205127 16.460284
## 2041.408 8.529688 3.314635 13.744740 0.553953038 16.505422
## 2041.427 8.575637 3.358496 13.792778 0.596708741 16.554565
## 2041.446 8.621786 3.402558 13.841015 0.639665504 16.603907
## 2041.466 8.663743 3.442428 13.885058 0.678430682 16.649055
## 2041.485 8.697091 3.473691 13.920492 0.708589461 16.685594
## 2041.504 8.717979 3.492493 13.943464 0.726287924 16.709669
## 2041.523 8.723637 3.496067 13.951206 0.728758398 16.718515
## 2041.542 8.712781 3.483128 13.942434 0.714716869 16.710845
## 2041.561 8.685828 3.454092 13.917563 0.684578329 16.687077
## 2041.580 8.644892 3.411075 13.878709 0.640459135 16.649325
## 2041.600 8.593578 3.357680 13.829476 0.585962863 16.601193
## 2041.619 8.536570 3.298592 13.774549 0.525774176 16.547367
## 2041.638 8.479086 3.239029 13.719144 0.465110278 16.493062
## 2041.657 8.426253 3.184117 13.668389 0.409098323 16.443408
## 2041.676 8.382489 3.138276 13.626702 0.362157267 16.402821
## 2041.695 8.350971 3.104681 13.597261 0.327462725 16.374479
## 2041.715 8.333246 3.084880 13.581612 0.306563367 16.359929
## 2041.734 8.329055 3.078614 13.579496 0.299198537 16.358911
## 2041.753 8.336370 3.083855 13.588885 0.303341491 16.369399
## 2041.772 8.351664 3.097075 13.606252 0.315464093 16.387863
## 2041.791 8.370360 3.113699 13.627021 0.330990740 16.409730
## 2041.810 8.387423 3.128690 13.646156 0.344885181 16.429961
## 2041.830 8.398002 3.137198 13.658806 0.352296931 16.443707
## 2041.849 8.398058 3.135184 13.660932 0.349186298 16.446929
## 2041.868 8.384886 3.119942 13.649829 0.332849632 16.436922
## 2041.887 8.357479 3.090467 13.624491 0.302278950 16.412678
## 2041.906 8.316673 3.047593 13.585753 0.258311013 16.375035
## 2041.925 8.265071 2.993924 13.536217 0.203547491 16.326594
## 2041.945 8.206740 2.933527 13.479952 0.142056763 16.271422
## 2041.964 8.146737 2.871459 13.422015 0.078895334 16.214578
## 2041.983 8.090508 2.813166 13.367851 0.019509475 16.161507
## 2042.002 8.043247 2.763841 13.322654 -0.030907547 16.117402
## 2042.021 8.009288 2.727819 13.290757 -0.068022097 16.086598
## 2042.040 7.991608 2.708077 13.275139 -0.088855234 16.072072
## 2042.060 7.991510 2.705918 13.277103 -0.092105456 16.075126
## 2042.079 8.008506 2.720854 13.296159 -0.078260774 16.095273
## 2042.098 8.040428 2.750716 13.330141 -0.049488846 16.130345
## 2042.117 8.083743 2.791972 13.375515 -0.009322330 16.176809
## 2042.136 8.134031 2.840201 13.427860 0.037817390 16.230244
## 2042.155 8.186554 2.890668 13.482441 0.087194674 16.285914
## 2042.175 8.236862 2.938919 13.534805 0.134356882 16.339366
## 2042.194 8.281325 2.981327 13.581324 0.175676875 16.386974
## 2042.213 8.317562 3.015509 13.619616 0.208771098 16.426353
## 2042.232 8.344673 3.040565 13.648781 0.232740516 16.456606
## 2042.251 8.363278 3.057117 13.669439 0.248205304 16.478351
## 2042.270 8.375343 3.067130 13.683557 0.257131538 16.493555
## 2042.290 8.383825 3.073560 13.694090 0.262475534 16.505175
## 2042.309 8.392182 3.079866 13.704497 0.267695471 16.516668
## 2042.328 8.403819 3.089453 13.718185 0.276197251 16.531440
## 2042.347 8.421546 3.105130 13.737961 0.290790131 16.552301
## 2042.366 8.447115 3.128651 13.765579 0.313226184 16.581003
## 2042.385 8.480907 3.160395 13.801418 0.343886515 16.617927
## 2042.405 8.521808 3.199250 13.844367 0.381657690 16.661959
## 2042.424 8.567297 3.242692 13.891901 0.424016616 16.710577
## 2042.443 8.613723 3.287073 13.940373 0.467314447 16.760131
## 2042.462 8.656759 3.328064 13.985453 0.507223583 16.806294
## 2042.481 8.691951 3.361212 14.022689 0.539289943 16.844612
## 2042.500 8.715304 3.382522 14.048085 0.559518271 16.871089
## 2042.520 8.723822 3.388998 14.058646 0.564913263 16.882731
## 2042.539 8.715935 3.379070 14.052800 0.553904374 16.877966
## 2042.558 8.691748 3.352843 14.030654 0.526596682 16.856900
## 2042.577 8.653084 3.312138 13.994029 0.484812363 16.821355
## 2042.596 8.603304 3.260319 13.946289 0.431914192 16.774694
## 2042.615 8.546938 3.201915 13.891961 0.372430721 16.721445
## 2042.635 8.489152 3.142092 13.836213 0.311528584 16.666776
## 2042.654 8.435136 3.086039 13.784233 0.254397536 16.615875
## 2042.673 8.389478 3.038345 13.740611 0.205625621 16.573330
## 2042.692 8.355609 3.002440 13.708777 0.168643761 16.542574
## 2042.711 8.335387 2.980184 13.690590 0.145310801 16.525464
## 2042.730 8.328879 2.971642 13.686116 0.135692623 16.522066
## 2042.750 8.334360 2.975091 13.693630 0.138064555 16.530656
## 2042.769 8.348542 2.987240 13.709844 0.149138108 16.547945
## 2042.788 8.366995 3.003662 13.730328 0.164484632 16.569505
## 2042.807 8.384719 3.019355 13.750083 0.179103395 16.590335
## 2042.826 8.396783 3.029390 13.764177 0.188063212 16.605504
## 2042.845 8.398961 3.029538 13.768383 0.187137250 16.610784
## 2042.865 8.388277 3.016826 13.759728 0.173351402 16.603202
## 2042.884 8.363404 2.989925 13.736882 0.145377486 16.581430
## 2042.903 8.324848 2.949343 13.700353 0.103722075 16.545974
## 2042.922 8.274912 2.897381 13.652443 0.050687609 16.499136
## 2042.941 8.217433 2.837876 13.596989 -0.009888861 16.444755
## 2042.960 8.157332 2.775751 13.538913 -0.073085919 16.387750
## 2042.980 8.100033 2.716428 13.483638 -0.133479977 16.333546
## 2042.999 8.050821 2.665193 13.436449 -0.185786009 16.287428
## 2043.018 8.014223 2.626573 13.401873 -0.225476447 16.25392310.8.4
8-step-ahead optimally reconciled coherent forecasts using ARIMA base forecasts for the visnights
library(hts)## Warning: package 'hts' was built under R version 3.5.1library(tibble)## Warning: package 'tibble' was built under R version 3.5.1library(stringr)## Warning: package 'stringr' was built under R version 3.5.1library(tidyr)## Warning: package 'tidyr' was built under R version 3.5.1library(dplyr)## Warning: package 'dplyr' was built under R version 3.5.1##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionvisnights_gts <- gts(visnights,characters = c(3,5),
gnames = c("State", "region" ))
fore_a<-forecast(visnights_gts, method="bu", fmethod="arima")
visnights_fc<-forecast(visnights_gts)
fcsts <- aggts(visnights_fc, levels=0:2)
groups <- aggts(visnights_gts, levels=0:2)
visnights_fc <- ts(rbind(groups, fcsts),
start=start(groups), frequency=4)
p1 <- autoplot(visnights_fc[,"Total"]) +
ggtitle(" Australian domestic tourism population") +
xlab("Year") + ylab("Total number of tourists ('000)") +
geom_vline(xintercept=2017)
cols <- sample(scales::hue_pal(h=c(15,375),
c=100,l=65,h.start=0,direction = 1)(NCOL(groups)))
p2 <- as_tibble(visnights_fc[,-1]) %>%
gather(Series) %>%
mutate(Date = rep(time(visnights_fc), NCOL(visnights_fc)-1),
Group = str_extract(Series, "([A-Za-z ]*)")) %>%
ggplot(aes(x=Date, y=value, group=Series, colour=Series)) +
geom_line() +
xlab("Year") + ylab("Number of tourists ('000)") +
scale_colour_manual(values = cols) +
facet_grid(. ~ Group, scales="free_y") +
scale_x_continuous(breaks=seq(1999,2018,by=2)) +
theme(axis.text.x = element_text(angle=90, hjust=1)) +
geom_vline(xintercept=2017)## Warning: package 'bindrcpp' was built under R version 3.5.1gridExtra::grid.arrange(p1, p2, ncol=1)
11.5.3
Experiment with using nnetar() on other data we have considered in previous chapters
fit_a <- nnetar(ausbeer, lambda=0)
autoplot(forecast(ausbeer,h=8))
fit_b <- nnetar(gasoline, lambda=0)
autoplot(forecast(gasoline,h=52))
fit_c <- nnetar(mcopper, lambda=0)
autoplot(forecast(mcopper,h=12))