Represent Result of Dynamic Linear Model for Time Series Data in ggplot.
# Create Data.frame for ggplot
xts(ridership.ts, order = as.Date(ridership.ts) )
[,1]
1991-01-01 1708.917
1991-02-01 1620.586
1991-03-01 1972.715
1991-04-01 1811.665
1991-05-01 1974.964
1991-06-01 1862.356
1991-07-01 1939.860
1991-08-01 2013.264
1991-09-01 1595.657
1991-10-01 1724.924
1991-11-01 1675.667
1991-12-01 1813.863
1992-01-01 1614.827
1992-02-01 1557.088
1992-03-01 1891.223
1992-04-01 1955.981
1992-05-01 1884.714
1992-06-01 1623.042
1992-07-01 1903.309
1992-08-01 1996.712
1992-09-01 1703.897
1992-10-01 1810.000
1992-11-01 1861.601
1992-12-01 1875.122
1993-01-01 1705.259
1993-02-01 1618.535
1993-03-01 1836.709
1993-04-01 1957.043
1993-05-01 1917.185
1993-06-01 1882.398
1993-07-01 1933.009
1993-08-01 1996.167
1993-09-01 1672.841
1993-10-01 1752.827
1993-11-01 1720.377
1993-12-01 1734.292
1994-01-01 1563.365
1994-02-01 1573.959
1994-03-01 1902.639
1994-04-01 1833.888
1994-05-01 1831.049
1994-06-01 1775.755
1994-07-01 1867.508
1994-08-01 1906.608
1994-09-01 1685.632
1994-10-01 1778.546
1994-11-01 1775.995
1994-12-01 1783.350
1995-01-01 1548.415
1995-02-01 1496.925
1995-03-01 1798.316
1995-04-01 1732.895
1995-05-01 1772.345
1995-06-01 1761.207
1995-07-01 1791.655
1995-08-01 1874.820
1995-09-01 1571.309
1995-10-01 1646.948
1995-11-01 1672.631
1995-12-01 1656.845
1996-01-01 1381.758
1996-02-01 1360.852
1996-03-01 1558.575
1996-04-01 1608.420
1996-05-01 1696.696
1996-06-01 1693.183
1996-07-01 1835.516
1996-08-01 1942.573
1996-09-01 1551.401
1996-10-01 1686.508
1996-11-01 1576.204
1996-12-01 1700.433
1997-01-01 1396.588
1997-02-01 1371.690
1997-03-01 1707.522
1997-04-01 1654.604
1997-05-01 1762.903
1997-06-01 1775.800
1997-07-01 1934.219
1997-08-01 2008.055
1997-09-01 1615.924
1997-10-01 1773.910
1997-11-01 1732.368
1997-12-01 1796.626
1998-01-01 1570.330
1998-02-01 1412.691
1998-03-01 1754.641
1998-04-01 1824.932
1998-05-01 1843.289
1998-06-01 1825.964
1998-07-01 1968.172
1998-08-01 1921.645
1998-09-01 1669.597
1998-10-01 1791.474
1998-11-01 1816.714
1998-12-01 1846.754
1999-01-01 1599.427
1999-02-01 1548.804
1999-03-01 1832.333
1999-04-01 1839.720
1999-05-01 1846.498
1999-06-01 1864.852
1999-07-01 1965.743
1999-08-01 1949.002
1999-09-01 1607.373
1999-10-01 1803.664
1999-11-01 1850.309
1999-12-01 1836.435
2000-01-01 1541.660
2000-02-01 1616.928
2000-03-01 1919.538
2000-04-01 1971.493
2000-05-01 1992.301
2000-06-01 2009.763
2000-07-01 2053.996
2000-08-01 2097.471
2000-09-01 1823.706
2000-10-01 1976.997
2000-11-01 1981.408
2000-12-01 2000.153
2001-01-01 1683.148
2001-02-01 1663.404
2001-03-01 2007.928
2001-04-01 2023.792
2001-05-01 2047.008
2001-06-01 2072.913
2001-07-01 2126.717
2001-08-01 2202.638
2001-09-01 1707.693
2001-10-01 1950.716
2001-11-01 1973.614
2001-12-01 1984.729
2002-01-01 1759.629
2002-02-01 1770.595
2002-03-01 2019.912
2002-04-01 2048.398
2002-05-01 2068.763
2002-06-01 1994.267
2002-07-01 2075.258
2002-08-01 2026.560
2002-09-01 1734.155
2002-10-01 1916.771
2002-11-01 1858.345
2002-12-01 1996.352
2003-01-01 1778.033
2003-02-01 1749.489
2003-03-01 2066.466
2003-04-01 2098.899
2003-05-01 2104.911
2003-06-01 2129.671
2003-07-01 2223.349
2003-08-01 2174.360
2003-09-01 1931.406
2003-10-01 2121.470
2003-11-01 2076.054
2003-12-01 2140.677
2004-01-01 1831.508
2004-02-01 1838.006
2004-03-01 2132.446
ridership <- as.data.frame(ridership.ts)
names(ridership) <- "Observation"
ridership$Date <- as.Date(ridership.ts) # add Date column in last
cln <- ncol(ridership)
ridership <- ridership[, c(cln, 1:(cln-1))] # change the order of the column. That is first column "Date"
row.names(ridership) <- NULL
train <- ridership[1:123,]
valid <- ridership[-(1:123),]
nValid <- nrow(valid)
library(dlm)
model1 <- function(p){
mod <- dlmModPoly(2) + # local trend linear
dlmModSeas(12)
V(mod) <- exp(p[1])
diag(W(mod))[1:3] <- exp(p[2:4])
return(mod)
}
mle1 <- dlmMLE(train.ts, parm=c(0.1,0.1,1,1), build=model1 ) # Estimation parameter through MLE. parameter=variance of error
ifelse(mle1$convergence==0, print("converge"), print("did not converge") ) # Check converge
[1] "converge"
[1] "converge"
modelfit1 <- model1(mle1$par) # Fitting the DLM model
V(modelfit1)
[,1]
[1,] 1413.91
W(modelfit1)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 1143.988 0.0000000 0.00000 0 0 0 0 0 0 0
[2,] 0.000 0.2040261 0.00000 0 0 0 0 0 0 0
[3,] 0.000 0.0000000 14.47867 0 0 0 0 0 0 0
[4,] 0.000 0.0000000 0.00000 0 0 0 0 0 0 0
[5,] 0.000 0.0000000 0.00000 0 0 0 0 0 0 0
[6,] 0.000 0.0000000 0.00000 0 0 0 0 0 0 0
[7,] 0.000 0.0000000 0.00000 0 0 0 0 0 0 0
[8,] 0.000 0.0000000 0.00000 0 0 0 0 0 0 0
[9,] 0.000 0.0000000 0.00000 0 0 0 0 0 0 0
[10,] 0.000 0.0000000 0.00000 0 0 0 0 0 0 0
[11,] 0.000 0.0000000 0.00000 0 0 0 0 0 0 0
[12,] 0.000 0.0000000 0.00000 0 0 0 0 0 0 0
[13,] 0.000 0.0000000 0.00000 0 0 0 0 0 0 0
[,11] [,12] [,13]
[1,] 0 0 0
[2,] 0 0 0
[3,] 0 0 0
[4,] 0 0 0
[5,] 0 0 0
[6,] 0 0 0
[7,] 0 0 0
[8,] 0 0 0
[9,] 0 0 0
[10,] 0 0 0
[11,] 0 0 0
[12,] 0 0 0
[13,] 0 0 0
List of 9
$ y : Time-Series [1:123] from 1991 to 2001: 1709 1621 1973 1812 1975 ...
$ mod:List of 10
..- attr(*, "class")= chr "dlm"
$ m : Time-Series [1:124, 1:13] from 1991 to 2001: 0 263 1477 2060 2237 ...
$ U.C:List of 124
$ D.C: num [1:124, 1:13] 3162.3 26.6 23.4 21.4 19.3 ...
$ a : Time-Series [1:123, 1:13] from 1991 to 2001: 0 394 2049 2636 2683 ...
$ U.R:List of 123
$ D.R: num [1:123, 1:13] 10916 8163 5192 4498 4165 ...
$ f : Time-Series [1:123] from 1991 to 2001: 0 263 1954 2542 2578 ...
- attr(*, "class")= chr "dlmFiltered"
filter <- data.frame(Date=train$Date, Observation=filtering$f) # Observation = Fitted filtering
# train + filter data.frame
train_filter <- rbind(
train,
filter
)
train_filter$Group <- c(rep("Data",length(train.ts)), rep("filtered",length(train.ts)))
# pal <- choose_palette()
ggplot(train_filter, aes(x=Date, y=Observation, col=Group, size=Group))+geom_line()+
scale_color_manual(values=c("#1E2B5A","#EA9AEB"))+
scale_size_manual(values=c(1.5,1)) +
theme_classic()+
theme(legend.title = element_blank())
cov.filter <- dlmSvd2var(filtering$U.C, filtering$D.C) # covariance of state
sd.filter <- t(sapply(cov.filter, FUN=function(x) sqrt(diag(x)))) # standard deviation of state
cov.filter # covariance of state for each observation
[[1]]
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
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[13,] 0e+00 0e+00 0e+00 0e+00 0e+00 0e+00 0e+00 0e+00 0e+00 0e+00
[,11] [,12] [,13]
[1,] 0e+00 0e+00 0e+00
[2,] 0e+00 0e+00 0e+00
[3,] 0e+00 0e+00 0e+00
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[5,] 0e+00 0e+00 0e+00
[6,] 0e+00 0e+00 0e+00
[7,] 0e+00 0e+00 0e+00
[8,] 0e+00 0e+00 0e+00
[9,] 0e+00 0e+00 0e+00
[10,] 0e+00 0e+00 0e+00
[11,] 1e+07 0e+00 0e+00
[12,] 0e+00 1e+07 0e+00
[13,] 0e+00 0e+00 1e+07
[[2]]
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[5,] -71.6539961 -67.6518702 -54.1433174
[6,] -66.3611666 -71.7100851 -67.5468351
[7,] -51.5029521 -66.1797663 -71.3545193
[8,] -27.2294201 -51.0907288 -65.5839698
[9,] 4.7640551 -26.5859177 -50.2551947
[10,] 17.3077439 5.6443182 -25.5042069
[11,] 314.9415473 18.4358079 6.9858001
[12,] 18.4358079 316.3332933 20.0569862
[13,] 6.9858001 20.0569862 318.1965042
[[122]]
[,1] [,2] [,3] [,4] [,5]
[1,] 966.276129 13.3845797 -187.867849 -89.428013 -41.4149911
[2,] 13.384580 16.8881068 -1.990434 -1.095821 -0.6450321
[3,] -187.867849 -1.9904339 316.114708 18.798274 6.9982260
[4,] -89.428013 -1.0958213 18.798274 315.502878 18.3022036
[5,] -41.414991 -0.6450321 6.998226 18.302204 315.1689217
[6,] -2.496034 -0.2351919 -24.811126 5.909292 17.7133169
[7,] 28.011698 0.1064895 -49.021433 -26.000186 5.2925841
[8,] 49.471226 0.3681583 -63.844616 -50.206931 -26.5953974
[9,] 61.549381 0.5463902 -69.118907 -65.007772 -50.7713908
[10,] 64.103777 0.6410038 -64.824864 -70.264475 -65.5435564
[11,] 57.080747 0.6533943 -50.964281 -65.969730 -70.7794772
[12,] 40.532634 0.5870879 -27.652263 -52.143686 -66.4805808
[13,] 15.411727 0.4592019 3.578961 -28.989711 -52.7098657
[,6] [,7] [,8] [,9] [,10]
[1,] -2.4960345 28.0116978 49.4712255 61.5493813 64.1037765
[2,] -0.2351919 0.1064895 0.3681583 0.5463902 0.6410038
[3,] -24.8111264 -49.0214327 -63.8446159 -69.1189066 -64.8248635
[4,] 5.9092920 -26.0001864 -50.2069313 -65.0077721 -70.2644747
[5,] 17.7133169 5.2925841 -26.5953974 -50.7713908 -65.5435564
[6,] 314.4971092 17.0931006 4.7588347 -27.0349953 -51.1176587
[7,] 17.0931006 313.9538127 16.6513989 4.4248108 -27.2615583
[8,] 4.7588347 16.6513989 313.6187575 16.4273318 4.3120220
[9,] -27.0349953 4.4248108 16.4273318 313.5071939 16.4290597
[10,] -51.1176587 -27.2615583 4.3120220 16.4290597 313.6247332
[11,] -65.8015985 -51.2401049 -27.2647398 4.4270544 16.6641266
[12,] -70.9606049 -65.8276814 -51.1379789 -27.0372126 4.7823651
[13,] -66.6295413 -70.9235200 -65.6416535 -50.8073399 -26.5589215
[,11] [,12] [,13]
[1,] 57.0807474 40.5326338 15.4117273
[2,] 0.6533943 0.5870879 0.4592019
[3,] -50.9642806 -27.6522627 3.5789612
[4,] -65.9697301 -52.1436859 -28.9897109
[5,] -70.7794772 -66.4805808 -52.7098657
[6,] -65.8015985 -70.9606049 -66.6295413
[7,] -51.2401049 -65.8276814 -70.9235200
[8,] -27.2647398 -51.1379789 -65.6416535
[9,] 4.4270544 -27.0372126 -50.8073399
[10,] 16.6641266 4.7823651 -26.5589215
[11,] 313.9832788 17.1524329 5.4153292
[12,] 17.1524329 314.6144853 17.9535909
[13,] 5.4153292 17.9535909 315.6223175
[[123]]
[,1] [,2] [,3] [,4] [,5]
[1,] 966.036806 13.28837945 -188.307514 -88.0369699 -40.769115
[2,] 13.288379 16.84929216 -2.168447 -0.5345982 -0.384431
[3,] -188.307514 -2.16844746 315.295290 21.3727209 8.193804
[4,] -88.036970 -0.53459815 21.372721 307.3869398 14.533472
[5,] -40.769115 -0.38443100 8.193804 14.5334723 313.418894
[6,] -2.095733 -0.07362477 -24.069608 3.5726262 16.628293
[7,] 28.262949 0.20796116 -48.555394 -27.4678393 4.611096
[8,] 49.583025 0.41339261 -63.636441 -50.8613238 -26.899244
[9,] 61.521268 0.53518530 -69.169608 -64.8459463 -50.696229
[10,] 63.934091 0.57267442 -65.137672 -69.2764810 -65.084770
[11,] 56.767046 0.52693858 -51.543879 -64.1410399 -69.930326
[12,] 40.070840 0.40084533 -28.506358 -49.4502438 -65.229900
[13,] 14.795793 0.21070978 2.438957 -25.3958671 -51.041104
[,6] [,7] [,8] [,9] [,10]
[1,] -2.09573336 28.2629487 49.5830248 61.5212679 63.9340912
[2,] -0.07362477 0.2079612 0.4133926 0.5351853 0.5726744
[3,] -24.06960762 -48.5553939 -63.6364409 -69.1696081 -65.1376722
[4,] 3.57262616 -27.4678393 -50.8613238 -64.8459463 -69.2764810
[5,] 16.62829315 4.6110962 -26.8992445 -50.6962288 -65.0847704
[6,] 313.82439313 16.6705721 4.5704372 -26.9884140 -50.8332379
[7,] 16.67057207 313.6884119 16.5330420 4.4540320 -27.0829605
[8,] 4.57043716 16.5330420 313.5659472 16.4403095 4.3915927
[9,] -26.98841396 4.4540320 16.4403095 313.5038756 16.4092534
[10,] -50.83323787 -27.0829605 4.3915927 16.4092534 313.5043490
[11,] -65.27515953 -50.9095043 -27.1174022 4.3904766 16.4414057
[12,] -70.18522628 -65.3407292 -50.9209325 -27.0910354 4.4543894
[13,] -65.59495828 -70.2737633 -65.3520132 -50.8791043 -26.9964748
[,11] [,12] [,13]
[1,] 56.7670459 40.0708399 14.7957932
[2,] 0.5269386 0.4008453 0.2107098
[3,] -51.5438793 -28.5063578 2.4389574
[4,] -64.1410399 -49.4502438 -25.3958671
[5,] -69.9303261 -65.2299000 -51.0411041
[6,] -65.2751595 -70.1852263 -65.5949583
[7,] -50.9095043 -65.3407292 -70.2737633
[8,] -27.1174022 -50.9209325 -65.3520132
[9,] 4.3904766 -27.0910354 -50.8791043
[10,] 16.4414057 4.4543894 -26.9964748
[11,] 313.5711554 16.5455013 4.6055758
[12,] 16.5455013 313.7206335 16.7610080
[13,] 4.6055758 16.7610080 314.0311379
[[124]]
[,1] [,2] [,3] [,4] [,5]
[1,] 965.721177 13.38903139 -187.533594 -87.4681262 -42.340453
[2,] 13.389031 16.81713908 -2.415502 -0.7158934 0.117325
[3,] -187.533594 -2.41550197 313.396312 19.9784976 12.050323
[4,] -87.468126 -0.71589343 19.978498 306.3617617 17.363250
[5,] -42.340453 0.11732499 12.050323 17.3632496 305.587442
[6,] -2.815457 0.15618949 -22.303249 4.8688683 13.041203
[7,] 27.820322 0.34927152 -47.469239 -26.6705131 2.405309
[8,] 49.306113 0.50176656 -62.957114 -50.3623619 -28.278828
[9,] 61.398938 0.57418405 -68.869747 -64.6253437 -51.305155
[10,] 63.966908 0.56212537 -65.218617 -69.3352954 -64.920309
[11,] 56.957003 0.46621224 -52.010479 -64.4829063 -68.982614
[12,] 40.420895 0.28900450 -29.365828 -50.0805322 -63.484273
[13,] 15.311310 0.04604898 1.173497 -26.3242820 -48.470903
[,6] [,7] [,8] [,9] [,10]
[1,] -2.8154566 27.8203221 49.3061125 61.398938 63.9669075
[2,] 0.1561895 0.3492715 0.5017666 0.574184 0.5621254
[3,] -22.3032486 -47.4692390 -62.9571138 -68.869747 -65.2186167
[4,] 4.8688683 -26.6705131 -50.3623619 -64.625344 -69.3352954
[5,] 13.0412026 2.4053095 -28.2788276 -51.305155 -64.9203088
[6,] 312.1813988 15.6602721 3.9385677 -27.267300 -50.7578953
[7,] 15.6602721 313.0671660 16.1444918 4.282528 -27.0366521
[8,] 3.9385677 16.1444918 313.3229198 16.333018 4.4205187
[9,] -27.2673004 4.2825282 16.3330184 313.456480 16.4219686
[10,] -50.7578953 -27.0366521 4.4205187 16.421969 313.5008026
[11,] -64.8410756 -50.6426095 -26.9505223 4.464072 16.4213971
[12,] -69.3856886 -64.8491210 -50.6135158 -26.955416 4.4176169
[13,] -64.4177616 -69.5499372 -64.8993648 -50.679386 -27.0505651
[,11] [,12] [,13]
[1,] 56.9570027 40.4208950 15.31131044
[2,] 0.4662122 0.2890045 0.04604898
[3,] -52.0104791 -29.3658278 1.17349725
[4,] -64.4829063 -50.0805322 -26.32428195
[5,] -68.9826144 -63.4842727 -48.47090279
[6,] -64.8410756 -69.3856886 -64.41776159
[7,] -50.6426095 -64.8491210 -69.54993715
[8,] -26.9505223 -50.6135158 -64.89936479
[9,] 4.4640725 -26.9554160 -50.67938613
[10,] 16.4213971 4.4176169 -27.05056515
[11,] 313.4563591 16.3341524 4.29445832
[12,] 16.3341524 313.3314534 16.18806864
[13,] 4.2944583 16.1880686 313.18764641
sd.filter # standard deviation of state for each observation
[,1] [,2] [,3] [,4] [,5]
[1,] 3162.27766 3162.277660 3162.27766 3162.27766 3162.27766
[2,] 4113.87042 3038.220640 4113.98938 3038.22061 3038.22061
[3,] 2625.42959 1801.015721 2625.26948 2047.23919 3031.36997
[4,] 2284.37701 1318.892225 2284.30074 1615.49216 1865.51736
[5,] 2119.56369 1008.582596 2119.53256 1455.99070 1334.54148
[6,] 1997.11776 793.622816 1997.12254 1397.52159 1084.80240
[7,] 1888.83047 640.132282 1888.86873 1364.35046 987.09583
[8,] 1789.11461 527.659646 1789.18534 1332.98784 953.56773
[9,] 1696.89263 443.175813 1696.99507 1298.41328 940.63900
[10,] 1611.71799 378.236734 1611.85143 1260.75282 931.21763
[11,] 1533.09138 327.270890 1533.25517 1221.15465 919.75126
[12,] 1460.42870 286.524037 1460.62228 1180.68938 905.00273
[13,] 1393.12141 253.409760 1393.34431 1140.13910 887.17753
[14,] 44.33686 10.774087 42.98777 54.06646 52.38337
[15,] 43.45089 10.111525 42.41863 42.33829 50.44231
[16,] 43.41142 9.786026 41.67484 41.06055 41.59019
[17,] 43.14054 9.537315 41.02575 39.93237 39.94060
[18,] 42.85671 9.311812 40.58949 39.11124 38.62976
[19,] 42.68598 9.092207 40.36717 38.60483 37.72588
[20,] 42.62970 8.871743 40.29811 38.35709 37.18323
[21,] 42.62822 8.647442 40.29595 38.28144 36.92127
[22,] 42.59865 8.417799 40.26556 38.27899 36.84178
[23,] 42.45018 8.181877 40.10896 38.24717 36.83921
[24,] 42.08797 7.938930 39.72532 38.08223 36.80613
[25,] 41.41105 7.688235 39.00692 37.67769 36.63456
[26,] 37.36817 7.286128 32.56169 36.22601 35.78038
[27,] 37.24440 7.121925 32.14283 32.02603 35.29172
[28,] 37.20007 7.006825 31.70574 31.39188 31.58967
[29,] 37.06763 6.906669 31.37595 30.85190 30.86232
[30,] 36.94503 6.811295 31.16973 30.47969 30.27965
[31,] 36.87426 6.716966 31.06819 30.25697 29.88923
[32,] 36.85103 6.622168 31.03700 30.14970 29.65894
[33,] 36.85031 6.526248 31.03596 30.11713 29.54887
[34,] 36.83942 6.428902 31.02304 30.11606 29.51564
[35,] 36.78325 6.329968 30.95576 30.10266 29.51457
[36,] 36.64576 6.229339 30.79110 30.03303 29.50079
[37,] 36.38702 6.126778 30.47915 29.86226 29.42921
[38,] 34.85130 5.986746 27.49154 29.36238 29.14934
[39,] 34.80768 5.908622 27.21556 27.07325 28.89771
[40,] 34.77674 5.847992 26.93130 26.70004 26.79250
[41,] 34.69974 5.793119 26.72670 26.37064 26.38209
[42,] 34.63098 5.740105 26.60193 26.14822 26.03590
[43,] 34.59167 5.687463 26.54122 26.01678 25.80667
[44,] 34.57864 5.634593 26.52262 25.95389 25.67257
[45,] 34.57816 5.581239 26.52198 25.93484 25.60878
[46,] 34.57267 5.527291 26.51451 25.93420 25.58958
[47,] 34.54343 5.472699 26.47537 25.92645 25.58897
[48,] 34.47134 5.417432 26.37952 25.88611 25.58100
[49,] 34.33292 5.361302 26.19278 25.78698 25.53958
[50,] 33.52665 5.290043 24.42001 25.51983 25.39702
[51,] 33.50460 5.244709 24.23375 24.07167 25.23356
[52,] 33.48276 5.207928 24.03221 23.83189 23.87516
[53,] 33.43224 5.174061 23.89037 23.60534 23.61707
[54,] 33.38778 5.141131 23.80505 23.45395 23.38223
[55,] 33.36239 5.108374 23.76380 23.36522 23.22767
[56,] 33.35386 5.075483 23.75117 23.32294 23.13780
[57,] 33.35348 5.042332 23.75072 23.31016 23.09521
[58,] 33.35029 5.008869 23.74577 23.30973 23.08243
[59,] 33.33262 4.975071 23.71968 23.30456 23.08202
[60,] 33.28859 4.940927 23.65568 23.27769 23.07671
[61,] 33.20157 4.906268 23.52602 23.21138 23.04907
[62,] 32.71215 4.864010 22.35601 23.03813 22.96121
[63,] 32.69860 4.835188 22.22737 22.04940 22.84180
[64,] 32.68256 4.811289 22.07612 21.88768 21.90384
[65,] 32.64664 4.789096 21.97083 21.72054 21.73197
[66,] 32.61523 4.767444 21.90808 21.60937 21.56019
[67,] 32.59724 4.745881 21.87788 21.54463 21.44736
[68,] 32.59108 4.724225 21.86863 21.51390 21.38211
[69,] 32.59075 4.702404 21.86829 21.50462 21.35130
[70,] 32.58875 4.680390 21.86475 21.50431 21.34206
[71,] 32.57710 4.658171 21.84595 21.50059 21.34177
[72,] 32.54765 4.635741 21.79967 21.48119 21.33794
[73,] 32.48739 4.612948 21.70150 21.43307 21.31794
[74,] 32.16450 4.585841 20.88262 21.30782 21.25771
[75,] 32.15511 4.566584 20.79241 20.60146 21.16412
[76,] 32.14292 4.550452 20.67445 20.48924 20.48919
[77,] 32.11592 4.535414 20.59259 20.36024 20.37090
[78,] 32.09232 4.520711 20.54416 20.27442 20.23909
[79,] 32.07874 4.506053 20.52093 20.22473 20.15236
[80,] 32.07399 4.491322 20.51381 20.20123 20.10245
[81,] 32.07370 4.476473 20.51355 20.19414 20.07896
[82,] 32.07238 4.461489 20.51087 20.19390 20.07194
[83,] 32.06424 4.446363 20.49663 20.19108 20.07171
[84,] 32.04331 4.431091 20.46135 20.17633 20.06879
[85,] 31.99880 4.415537 20.38274 20.13947 20.05356
[86,] 31.77397 4.397338 19.78823 20.04251 20.00933
[87,] 31.76693 4.384086 19.72439 19.52275 19.93253
[88,] 31.75740 4.372947 19.62989 19.44352 19.43345
[89,] 31.73625 4.362550 19.56409 19.34085 19.35040
[90,] 31.71773 4.352373 19.52539 19.27222 19.24594
[91,] 31.70700 4.342214 19.50688 19.23269 19.17679
[92,] 31.70317 4.331993 19.50121 19.21405 19.13718
[93,] 31.70289 4.321680 19.50099 19.20845 19.11860
[94,] 31.70199 4.311264 19.49891 19.20826 19.11307
[95,] 31.69607 4.300741 19.48772 19.20603 19.11289
[96,] 31.68053 4.290106 19.45976 19.19440 19.11059
[97,] 31.64609 4.279242 19.39438 19.16505 19.09855
[98,] 31.48352 4.266671 18.95193 19.08646 19.06445
[99,] 31.47795 4.257378 18.90653 18.69602 18.99944
[100,] 31.47034 4.249577 18.82934 18.63949 18.62321
[101,] 31.45326 4.242301 18.77511 18.55599 18.56418
[102,] 31.43823 4.235173 18.74336 18.49964 18.47948
[103,] 31.42946 4.228046 18.72822 18.46732 18.42284
[104,] 31.42626 4.220865 18.72358 18.45214 18.39053
[105,] 31.42600 4.213609 18.72340 18.44759 18.37543
[106,] 31.42537 4.206270 18.72173 18.44743 18.37094
[107,] 31.42093 4.198846 18.71269 18.44563 18.37080
[108,] 31.40898 4.191329 18.68987 18.43620 18.36893
[109,] 31.38142 4.183620 18.63401 18.41211 18.35915
[110,] 31.26053 4.174771 18.29879 18.34636 18.33187
[111,] 31.25596 4.168173 18.26644 18.04853 18.27565
[112,] 31.24978 4.162660 18.20248 18.00793 17.98791
[113,] 31.23565 4.157530 18.15691 17.93895 17.94563
[114,] 31.22314 4.152501 18.13031 17.89173 17.87582
[115,] 31.21579 4.147464 18.11767 17.86475 17.82844
[116,] 31.21304 4.142379 18.11380 17.85212 17.80150
[117,] 31.21278 4.137230 18.11364 17.84834 17.78896
[118,] 31.21234 4.132012 18.11227 17.84821 17.78525
[119,] 31.20892 4.126723 18.10482 17.84672 17.78513
[120,] 31.19948 4.121356 18.08573 17.83891 17.78358
[121,] 31.17684 4.115826 18.03711 17.81864 17.77546
[122,] 31.08498 4.109514 17.77961 17.76240 17.75300
[123,] 31.08113 4.104789 17.75656 17.53245 17.70364
[124,] 31.07605 4.100871 17.70300 17.50319 17.48106
[,6] [,7] [,8] [,9] [,10]
[1,] 3162.27766 3162.27766 3162.27766 3162.27766 3162.27766
[2,] 3038.22061 3038.22061 3038.22061 3038.22061 3038.22061
[3,] 3031.36997 3031.36997 3031.36997 3031.36997 3031.36997
[4,] 3029.64670 3029.64670 3029.64670 3029.64670 3029.64670
[5,] 1864.52771 3028.81903 3028.81903 3028.81903 3028.81903
[6,] 1288.13660 1844.37302 3028.37504 3028.37504 3028.37504
[7,] 951.89989 1287.11425 1796.45194 3028.12283 3028.12283
[8,] 773.35334 918.17754 1282.30707 1732.93376 3027.97230
[9,] 690.06386 678.11340 914.18374 1266.50360 1663.08707
[10,] 656.30155 529.13684 644.08942 913.99205 1241.67673
[11,] 644.00362 444.30815 440.27698 635.63756 909.98560
[12,] 638.87150 401.51568 279.43896 398.40661 634.95840
[13,] 634.56283 382.74694 135.78830 135.78391 382.73742
[14,] 51.04159 50.06430 49.47000 49.27063 49.47030
[15,] 49.97010 49.61684 49.38181 49.26447 49.26455
[16,] 49.13562 49.18360 49.22046 49.24363 49.25156
[17,] 40.93872 48.54608 48.89016 49.11987 49.23485
[18,] 39.12270 40.50114 48.29816 48.80941 49.11457
[19,] 37.72898 38.61743 40.27824 48.22688 48.80589
[20,] 36.78429 37.18736 38.37008 40.20905 48.22416
[21,] 36.22192 36.22293 36.92573 38.29455 40.20694
[22,] 35.95161 35.65019 35.95293 36.84635 38.29217
[23,] 35.86979 35.37525 35.37556 35.87121 36.84383
[24,] 35.86716 35.29212 35.09850 35.29252 35.86862
[25,] 35.83314 35.28946 35.01476 35.01486 35.28991
[26,] 35.42215 35.16545 35.01118 34.95976 35.01132
[27,] 35.16084 35.05844 34.99110 34.95774 34.95777
[28,] 34.83187 34.88774 34.92448 34.94720 34.95490
[29,] 31.25994 34.59710 34.77364 34.88780 34.94508
[30,] 30.49429 31.05341 34.49249 34.74102 34.88628
[31,] 29.89333 30.27298 30.95164 34.46180 34.73983
[32,] 29.49444 29.66438 30.16615 30.92039 34.46074
[33,] 29.26029 29.26162 29.55471 30.13369 30.91937
[34,] 29.14871 29.02559 29.15043 29.52158 30.13265
[35,] 29.11512 28.91325 28.91363 29.11694 29.52054
[36,] 29.11406 28.87949 28.80098 28.87998 29.11591
[37,] 29.09990 28.87846 28.76725 28.76735 28.87897
[38,] 28.97089 28.84339 28.76703 28.74163 28.76715
[39,] 28.84046 28.78995 28.75693 28.74067 28.74069
[40,] 28.64574 28.69131 28.71766 28.73394 28.73947
[41,] 26.58950 28.51131 28.62654 28.69740 28.73300
[42,] 26.16455 26.46502 28.45017 28.60781 28.69666
[43,] 25.81149 26.03473 26.40430 28.43212 28.60718
[44,] 25.57927 25.67896 25.97232 26.38570 28.43154
[45,] 25.44392 25.44548 25.61564 25.95339 26.38508
[46,] 25.37972 25.30959 25.38173 25.59655 25.95277
[47,] 25.36046 25.24521 25.24566 25.36259 25.59595
[48,] 25.35987 25.22596 25.18125 25.22652 25.36201
[49,] 25.35164 25.22539 25.16210 25.16221 25.22598
[50,] 25.28747 25.20902 25.16208 25.14649 25.16221
[51,] 25.20603 25.17558 25.15569 25.14593 25.14595
[52,] 25.07049 25.10945 25.12908 25.14117 25.14527
[53,] 23.73607 24.98134 25.06667 25.11591 25.14064
[54,] 23.47114 23.65148 24.94035 25.05423 25.11547
[55,] 23.23308 23.38419 23.61039 24.92821 25.05384
[56,] 23.07724 23.14498 23.34244 23.59781 24.92783
[57,] 22.98692 22.98867 23.10290 23.32978 23.59738
[58,] 22.94423 22.89819 22.94649 23.09024 23.32937
[59,] 22.93147 22.85549 22.85600 22.93385 23.08984
[60,] 22.93108 22.84277 22.81333 22.84340 22.93347
[61,] 22.92555 22.84242 22.80074 22.80086 22.84306
[62,] 22.88666 22.83290 22.80072 22.79004 22.80085
[63,] 22.82983 22.80956 22.79621 22.78967 22.78969
[64,] 22.72660 22.76141 22.77674 22.78609 22.78926
[65,] 21.80219 22.66255 22.73072 22.76739 22.78575
[66,] 21.62672 21.74054 22.63286 22.72178 22.76710
[67,] 21.45326 21.56393 21.71063 22.62406 22.72151
[68,] 21.33985 21.38994 21.53376 21.70147 22.62379
[69,] 21.27445 21.27637 21.35968 21.52461 21.70115
[70,] 21.24365 21.21097 21.24612 21.35058 21.52431
[71,] 21.23447 21.18022 21.18077 21.23708 21.35030
[72,] 21.23420 21.17109 21.15007 21.17177 21.23682
[73,] 21.23018 21.17085 21.14107 21.14120 21.17154
[74,] 21.20389 21.16461 21.14104 21.13322 21.14118
[75,] 21.16145 21.14722 21.13764 21.13296 21.13297
[76,] 21.07824 21.11023 21.12268 21.13015 21.13268
[77,] 20.41170 21.02984 21.08704 21.11567 21.12992
[78,] 20.29138 20.36468 21.00726 21.08028 21.11546
[79,] 20.15865 20.24378 20.34185 21.00056 21.08008
[80,] 20.07165 20.11082 20.22087 20.33486 21.00036
[81,] 20.02173 20.02381 20.08792 20.21392 20.33461
[82,] 19.99830 19.97394 20.00097 20.08104 20.21369
[83,] 19.99134 19.95058 19.95118 19.99415 20.08083
[84,] 19.99114 19.94368 19.92788 19.94441 19.99396
[85,] 19.98806 19.94350 19.92109 19.92123 19.94425
[86,] 19.96902 19.93908 19.92105 19.91507 19.92120
[87,] 19.93583 19.92555 19.91839 19.91488 19.91489
[88,] 19.86616 19.89603 19.90649 19.91261 19.91468
[89,] 19.37260 19.82830 19.87787 19.90103 19.91244
[90,] 19.28835 19.33558 19.81053 19.87257 19.90087
[91,] 19.18337 19.25104 19.31757 19.80525 19.87242
[92,] 19.11412 19.14598 19.23303 19.31205 19.80510
[93,] 19.07457 19.07679 19.12804 19.22757 19.31186
[94,] 19.05607 19.03731 19.05892 19.12264 19.22739
[95,] 19.05060 19.01889 19.01953 19.05359 19.12248
[96,] 19.05044 19.01347 19.00117 19.01425 19.05344
[97,] 19.04799 19.01334 18.99585 18.99599 19.01413
[98,] 19.03353 19.01004 18.99581 18.99108 18.99596
[99,] 19.00667 18.99918 18.99366 18.99093 18.99095
[100,] 18.94679 18.97490 18.98394 18.98906 18.99079
[101,] 18.57437 18.91643 18.96031 18.97957 18.98893
[102,] 18.51460 18.54453 18.90210 18.95604 18.97945
[103,] 18.42962 18.48463 18.52998 18.89783 18.95593
[104,] 18.37297 18.39965 18.47012 18.52552 18.89771
[105,] 18.34074 18.34308 18.38522 18.46571 18.52536
[106,] 18.32573 18.31095 18.32874 18.38088 18.46556
[107,] 18.32130 18.29601 18.29668 18.32446 18.38075
[108,] 18.32117 18.29163 18.28180 18.29245 18.32435
[109,] 18.31918 18.29152 18.27751 18.27766 18.29236
[110,] 18.30779 18.28897 18.27747 18.27365 18.27763
[111,] 18.28546 18.28003 18.27570 18.27353 18.27355
[112,] 18.23306 18.25960 18.26760 18.27196 18.27342
[113,] 17.94804 18.20825 18.24763 18.26403 18.27186
[114,] 17.90529 17.92357 18.19647 18.24413 18.26394
[115,] 17.83533 17.88077 17.91159 18.19296 18.24404
[116,] 17.78800 17.81086 17.86884 17.90791 18.19286
[117,] 17.76115 17.76361 17.79902 17.86521 17.90778
[118,] 17.74870 17.73686 17.75186 17.79546 17.86510
[119,] 17.74504 17.72448 17.72519 17.74836 17.79535
[120,] 17.74493 17.72087 17.71286 17.72173 17.74826
[121,] 17.74328 17.72078 17.70932 17.70948 17.72165
[122,] 17.73407 17.71874 17.70928 17.70613 17.70945
[123,] 17.71509 17.71125 17.70779 17.70604 17.70605
[124,] 17.66866 17.69370 17.70093 17.70470 17.70595
[,11] [,12] [,13]
[1,] 3162.27766 3162.27766 3162.27766
[2,] 3038.22061 3038.22061 3038.22061
[3,] 3031.36997 3031.36997 3031.36997
[4,] 3029.64670 3029.64670 3029.64670
[5,] 3028.81903 3028.81903 3028.81903
[6,] 3028.37504 3028.37504 3028.37504
[7,] 3028.12283 3028.12283 3028.12283
[8,] 3027.97230 3027.97230 3027.97230
[9,] 3027.87830 3027.87830 3027.87830
[10,] 1592.17524 3027.81718 3027.81718
[11,] 1210.84600 1522.89185 3027.77598
[12,] 900.84767 1176.42398 1456.49234
[13,] 634.54354 887.14361 1140.08568
[14,] 50.06473 51.04184 52.38300
[15,] 49.38201 49.61705 49.97013
[16,] 49.24373 49.22061 49.18371
[17,] 49.23488 49.11997 48.89026
[18,] 49.21604 49.11461 48.80947
[19,] 49.09329 49.09330 48.80590
[20,] 48.78283 48.96778 48.78281
[21,] 48.19993 48.65501 48.65498
[22,] 40.17654 48.06983 48.33935
[23,] 38.26041 40.01961 47.75003
[24,] 36.81080 38.09555 39.63507
[25,] 35.83464 36.63927 37.69116
[26,] 35.16583 35.42323 35.78367
[27,] 34.99122 35.05875 35.16179
[28,] 34.94724 34.92459 34.88804
[29,] 34.94510 34.88784 34.77376
[30,] 34.93469 34.88630 34.74109
[31,] 34.87529 34.87529 34.73987
[32,] 34.72844 34.81528 34.72847
[33,] 34.44913 34.66796 34.66798
[34,] 30.90642 34.38814 34.52002
[35,] 30.11927 30.83887 34.23918
[36,] 29.50677 30.04968 30.67357
[37,] 29.10176 29.43520 29.87910
[38,] 28.84384 28.97242 29.15448
[39,] 28.75704 28.79035 28.84203
[40,] 28.73397 28.71776 28.69184
[41,] 28.73300 28.69742 28.62678
[42,] 28.72628 28.69666 28.60797
[43,] 28.68972 28.68971 28.60732
[44,] 28.60011 28.65294 28.60025
[45,] 28.42441 28.56320 28.56335
[46,] 26.37756 28.38738 28.47344
[47,] 25.94503 26.33817 28.29728
[48,] 25.58799 25.90473 26.24193
[49,] 25.35378 25.54658 25.80593
[50,] 25.20954 25.28935 25.40362
[51,] 25.15580 25.17606 25.20816
[52,] 25.14119 25.12918 25.11025
[53,] 25.14064 25.11592 25.06710
[54,] 25.13587 25.11546 25.05458
[55,] 25.11059 25.11059 25.05417
[56,] 25.04892 25.08524 25.04925
[57,] 24.92291 25.02355 25.02389
[58,] 23.59237 24.89754 24.96218
[59,] 23.32421 23.56605 24.83608
[60,] 23.08453 23.29736 23.50190
[61,] 22.92794 23.05689 23.23163
[62,] 22.83349 22.88883 22.96911
[63,] 22.79632 22.81011 22.83255
[64,] 22.78610 22.77685 22.76256
[65,] 22.78574 22.76739 22.73143
[66,] 22.78215 22.76709 22.72239
[67,] 22.76345 22.76345 22.72210
[68,] 22.71785 22.74473 22.71845
[69,] 22.62016 22.69915 22.69975
[70,] 21.69754 22.60150 22.65422
[71,] 21.52059 21.67852 22.55659
[72,] 21.34646 21.50122 21.63240
[73,] 21.23279 21.32647 21.45398
[74,] 21.16525 21.20631 21.26679
[75,] 21.13776 21.14783 21.16478
[76,] 21.13016 21.12280 21.11177
[77,] 21.12991 21.11568 21.08809
[78,] 21.12709 21.11546 21.08121
[79,] 21.11261 21.11262 21.08100
[80,] 21.07723 21.09814 21.07816
[81,] 20.99754 21.06281 21.06373
[82,] 20.33187 20.98318 21.02846
[83,] 20.21086 20.31741 20.94891
[84,] 20.07790 20.19616 20.28262
[85,] 19.99087 20.06268 20.16054
[86,] 19.93978 19.97168 20.01949
[87,] 19.91852 19.92623 19.93979
[88,] 19.91261 19.90662 19.89801
[89,] 19.91243 19.90104 19.87930
[90,] 19.91015 19.90088 19.87387
[91,] 19.89858 19.89859 19.87371
[92,] 19.87014 19.88703 19.87144
[93,] 19.80285 19.85864 19.85993
[94,] 19.30970 19.79143 19.83161
[95,] 19.22516 19.29831 19.76450
[96,] 19.12016 19.21358 19.27122
[97,] 19.05099 19.10815 19.18586
[98,] 19.01078 19.03641 19.07559
[99,] 18.99379 18.99993 19.01126
[100,] 18.98906 18.98410 18.97735
[101,] 18.98892 18.97960 18.96216
[102,] 18.98705 18.97947 18.95776
[103,] 18.97756 18.97759 18.95763
[104,] 18.95405 18.96813 18.95577
[105,] 18.89587 18.94467 18.94636
[106,] 18.52361 18.88657 18.92298
[107,] 18.46376 18.51440 18.86498
[108,] 18.37887 18.45440 18.49284
[109,] 18.32234 18.36915 18.43235
[110,] 18.28976 18.31089 18.34388
[111,] 18.27584 18.28086 18.29069
[112,] 18.27196 18.26780 18.26253
[113,] 18.27185 18.26409 18.24992
[114,] 18.27027 18.26398 18.24628
[115,] 18.26235 18.26241 18.24617
[116,] 18.24247 18.25451 18.24462
[117,] 18.19132 18.23468 18.23676
[118,] 17.90633 18.18361 18.21701
[119,] 17.86360 17.89874 18.16605
[120,] 17.79379 17.85588 17.88129
[121,] 17.74659 17.78576 17.83806
[122,] 17.71957 17.73738 17.76576
[123,] 17.70794 17.71216 17.72092
[124,] 17.70470 17.70117 17.69711
filter$level <-filtering$m[-1,1]
filter$level_sd <- sd.filter[-1,1]
filter$slope <- filtering$m[-1,2]
filter$slope_sd <- sd.filter[-1,2]
filter$season1 <- filtering$m[-1,3]
filter$season1_sd <- sd.filter[-1,3] # Seasonal represent 3~13 columns. However, just for Season1.
filter
Date Observation level level_sd slope
1 1991-01-01 0.000 262.9201 4113.87042 131.45255272
2 1991-02-01 262.920 1476.8170 2625.42959 571.74869261
3 1991-03-01 1953.806 2060.3500 2284.37701 576.13321930
4 1991-04-01 2542.089 2237.2100 2119.56369 446.14979550
5 1991-05-01 2578.135 2385.8677 1997.11776 362.60860175
6 1991-06-01 2636.291 2401.0946 1888.83047 278.08632437
7 1991-07-01 2559.954 2424.6579 1789.11461 223.86256634
8 1991-08-01 2524.776 2455.6739 1696.89263 187.47536594
9 1991-09-01 2516.373 2322.8861 1611.71799 133.37877892
10 1991-10-01 2324.983 2262.8372 1533.09138 103.85035440
11 1991-11-01 2232.946 2199.5074 1460.42870 80.59240254
12 1991-12-01 2144.421 2187.4420 1393.12141 68.76012370
13 1992-01-01 2533.874 1758.4610 44.33686 -7.79796175
14 1992-02-01 1526.730 1764.1211 43.45089 -6.52855213
15 1992-03-01 1898.173 1754.4566 43.41142 -6.74330687
16 1992-04-01 1733.055 1850.1970 43.14054 -0.71397510
17 1992-05-01 2025.580 1784.2186 42.85671 -4.31300134
18 1992-06-01 1832.052 1682.8321 42.68598 -9.52788355
19 1992-07-01 1788.410 1726.6984 42.62970 -6.68851443
20 1992-08-01 1927.770 1752.0489 42.62822 -4.99112401
21 1992-09-01 1550.400 1818.3872 42.59865 -1.21683610
22 1992-10-01 1768.653 1836.3837 42.45018 -0.20048676
23 1992-11-01 1743.813 1890.9173 42.08797 2.69491388
24 1992-12-01 1950.380 1858.6399 41.41105 0.84468898
25 1993-01-01 1717.550 1853.2098 37.36817 0.46507386
26 1993-02-01 1636.483 1844.8289 37.24440 0.07146788
27 1993-03-01 1968.354 1779.4310 37.20007 -2.40382931
28 1993-04-01 1845.458 1832.8733 37.06763 -0.43847445
29 1993-05-01 1954.275 1813.8261 36.94503 -1.07437031
30 1993-06-01 1745.782 1881.3442 36.87426 1.24290741
31 1993-07-01 2004.339 1846.7681 36.85103 0.03774376
32 1993-08-01 2046.631 1821.4646 36.85031 -0.81391069
33 1993-09-01 1662.751 1825.7177 36.83942 -0.64362476
34 1993-10-01 1785.547 1808.6432 36.78325 -1.19601945
35 1993-11-01 1768.711 1783.1737 36.64576 -2.01246514
36 1993-12-01 1815.898 1740.1727 36.38702 -3.39252386
37 1994-01-01 1586.540 1724.6768 34.85130 -3.84210018
38 1994-02-01 1494.872 1761.5921 34.80768 -2.61091076
39 1994-03-01 1839.784 1791.5335 34.77674 -1.72462862
40 1994-04-01 1887.641 1761.8825 34.69974 -2.45216408
41 1994-05-01 1871.495 1738.3925 34.63098 -2.99073653
42 1994-06-01 1713.669 1767.7061 34.59167 -2.16900680
43 1994-07-01 1884.130 1756.8877 34.57864 -2.38860650
44 1994-08-01 1951.432 1731.1753 34.57816 -2.98070317
45 1994-09-01 1581.900 1782.1713 34.57267 -1.60972790
46 1994-10-01 1745.575 1797.7186 34.54343 -1.17363164
47 1994-11-01 1755.732 1807.0900 34.47134 -0.90538924
48 1994-12-01 1821.999 1786.0676 34.33292 -1.41802003
49 1995-01-01 1624.354 1744.2986 33.52665 -2.52236342
50 1995-02-01 1524.911 1727.0069 33.50460 -2.87163367
51 1995-03-01 1815.238 1715.1770 33.48276 -3.06865354
52 1995-04-01 1795.564 1678.8704 33.43224 -3.77808015
53 1995-05-01 1769.381 1676.6656 33.38778 -3.74488235
54 1995-06-01 1657.413 1728.0172 33.36239 -2.58668595
55 1995-07-01 1840.032 1699.7502 33.35386 -3.12611529
56 1995-08-01 1879.097 1694.3534 33.35348 -3.17383166
57 1995-09-01 1565.207 1694.4186 33.35029 -3.10570744
58 1995-10-01 1668.964 1679.6253 33.33262 -3.35176305
59 1995-11-01 1649.227 1688.6987 33.28859 -3.08992098
60 1995-12-01 1704.771 1660.1596 33.20157 -3.62745462
61 1996-01-01 1486.364 1600.2927 32.71215 -4.87900910
62 1996-02-01 1371.718 1589.5958 32.69860 -4.99548293
63 1996-03-01 1670.654 1524.4643 32.68256 -6.13727336
64 1996-04-01 1584.176 1531.3518 32.64664 -5.89520732
65 1996-05-01 1612.430 1570.7460 32.61523 -5.06002885
66 1996-06-01 1569.849 1631.9787 32.59724 -3.84028550
67 1996-07-01 1742.445 1678.1645 32.59108 -2.91990304
68 1996-08-01 1867.054 1715.8364 32.59075 -2.17252892
69 1996-09-01 1601.545 1686.7106 32.58875 -2.66930436
70 1996-10-01 1668.318 1693.8193 32.57710 -2.48888869
71 1996-11-01 1679.321 1635.8979 32.54765 -3.51282856
72 1996-12-01 1651.080 1658.9220 32.48739 -3.02136177
73 1997-01-01 1469.337 1616.4701 32.16450 -3.78013311
74 1997-02-01 1383.042 1606.5541 32.15511 -3.88939285
75 1997-03-01 1660.411 1628.1683 32.14292 -3.45290398
76 1997-04-01 1678.985 1611.5050 32.11592 -3.67547497
77 1997-05-01 1693.031 1645.6997 32.09232 -3.04087550
78 1997-06-01 1657.212 1706.9367 32.07874 -1.96513254
79 1997-07-01 1833.422 1759.6055 32.07399 -1.05048085
80 1997-08-01 1970.264 1779.0382 32.07370 -0.70724177
81 1997-09-01 1667.501 1750.3748 32.07238 -1.17623032
82 1997-10-01 1743.867 1765.4836 32.06424 -0.90272078
83 1997-11-01 1739.239 1760.8562 32.04331 -0.96535183
84 1997-12-01 1786.765 1765.2376 31.99880 -0.87519812
85 1998-01-01 1564.790 1767.3825 31.77397 -0.82282458
86 1998-02-01 1527.906 1703.8718 31.76693 -1.84944592
87 1998-03-01 1756.789 1700.8521 31.75740 -1.86804190
88 1998-04-01 1737.385 1746.7049 31.73625 -1.11855857
89 1998-05-01 1827.893 1753.9805 31.71773 -0.98722553
90 1998-06-01 1775.680 1780.4087 31.70700 -0.55856121
91 1998-07-01 1920.357 1805.9195 31.70317 -0.15071179
92 1998-08-01 2026.409 1748.6509 31.70289 -1.04521064
93 1998-09-01 1630.716 1768.8039 31.70199 -0.71285595
94 1998-10-01 1767.309 1781.2666 31.69607 -0.50604801
95 1998-11-01 1755.758 1813.9979 31.68053 0.01628816
96 1998-12-01 1844.312 1815.3460 31.64609 0.03727573
97 1999-01-01 1620.497 1803.8484 31.48352 -0.14827519
98 1999-02-01 1549.632 1803.2476 31.47795 -0.15525221
99 1999-03-01 1851.400 1792.6595 31.47034 -0.31243207
100 1999-04-01 1837.085 1793.7894 31.45326 -0.29089409
101 1999-05-01 1873.979 1778.4517 31.43823 -0.51497418
102 1999-06-01 1803.831 1811.3484 31.42946 -0.01751819
103 1999-07-01 1959.681 1814.6498 31.42626 0.03193061
104 1999-08-01 2021.165 1775.1704 31.42600 -0.55734745
105 1999-09-01 1658.950 1746.3727 31.42537 -0.97899458
106 1999-10-01 1743.359 1778.4140 31.42093 -0.48543394
107 1999-11-01 1759.353 1827.7355 31.40898 0.25991971
108 1999-12-01 1860.963 1814.5611 31.38142 0.05835557
109 2000-01-01 1619.308 1771.9747 31.26053 -0.59118735
110 2000-02-01 1518.112 1825.5935 31.25596 0.20888106
111 2000-03-01 1872.280 1851.7465 31.24978 0.58501496
112 2000-04-01 1899.065 1892.1105 31.23565 1.15780202
113 2000-05-01 1970.615 1905.1803 31.22314 1.32899841
114 2000-06-01 1941.086 1944.2326 31.21579 1.87115667
115 2000-07-01 2099.478 1921.1216 31.21304 1.51184903
116 2000-08-01 2122.863 1908.6864 31.21278 1.31105755
117 2000-09-01 1784.138 1931.7311 31.21234 1.62427601
118 2000-10-01 1932.182 1957.9722 31.20892 1.97941456
119 2000-11-01 1950.244 1977.0708 31.19948 2.22666702
120 2000-12-01 2009.787 1974.0040 31.17684 2.15003000
121 2001-01-01 1766.339 1930.3543 31.08498 1.47962479
122 2001-02-01 1682.651 1921.2473 31.08113 1.32825739
123 2001-03-01 1967.525 1944.8125 31.07605 1.64182974
slope_sd season1 season1_sd
1 3038.220640 1445.9782703 4113.98938
2 1801.015721 143.7352725 2625.26948
3 1318.892225 -87.6359341 2284.30074
4 1008.582596 -425.4996685 2119.53256
5 793.622816 -410.8614818 1997.12254
6 640.132282 -538.6793347 1888.86873
7 527.659646 -484.7468721 1789.18534
8 443.175813 -442.3652688 1696.99507
9 378.236734 -727.1449006 1611.85143
10 327.270890 -537.8560730 1533.25517
11 286.524037 -523.7855314 1460.62228
12 253.409760 -373.5454858 1393.34431
13 10.774087 -143.4933161 42.98777
14 10.111525 -212.3778196 42.41863
15 9.786026 138.1708580 41.67484
16 9.537315 59.7619129 41.02575
17 9.311812 129.6817089 40.58949
18 9.092207 -16.4587139 40.36717
19 8.871743 152.7877762 40.29811
20 8.647442 230.3685279 40.29595
21 8.417799 -146.3164463 40.26556
22 8.181877 -34.9567937 40.10896
23 7.938930 -53.7386506 39.72532
24 7.688235 32.0863375 39.00692
25 7.286128 -145.2878076 32.56169
26 7.121925 -221.5446790 32.14283
27 7.006825 93.2331456 31.70574
28 6.906669 93.5328089 31.37595
29 6.811295 113.5513215 31.16973
30 6.716966 -36.4930956 31.06819
31 6.622168 105.8452432 31.03700
32 6.526248 188.5719385 31.03596
33 6.428902 -155.6499781 31.02304
34 6.329968 -46.8235668 30.95576
35 6.229339 -49.5124028 30.79110
36 6.126778 16.5478228 30.47915
37 5.986746 -154.8227923 27.49154
38 5.908622 -211.5445695 27.21556
39 5.847992 91.8331585 26.93130
40 5.793119 88.5250677 26.72670
41 5.740105 105.0911583 26.60193
42 5.687463 -11.0395478 26.54122
43 5.634593 115.7308994 26.52262
44 5.581239 189.2139325 26.52198
45 5.527291 -128.4318968 26.51451
46 5.472699 -29.3095243 26.47537
47 5.417432 -37.3250324 26.37952
48 5.361302 9.1652180 26.19278
49 5.290043 -172.3045139 24.42001
50 5.244709 -221.0397644 24.23375
51 5.207928 88.6502148 24.03221
52 5.174061 74.4605997 23.89037
53 5.141131 94.7126076 23.80505
54 5.108374 -0.6640731 23.76380
55 5.075483 107.6837589 23.75117
56 5.042332 181.8617503 23.75072
57 5.008869 -125.0997161 23.74577
58 4.975071 -25.4964015 23.71968
59 4.940927 -23.7013532 23.65568
60 4.906268 12.3172866 23.52602
61 4.864010 -184.2147227 22.35601
62 4.835188 -225.0934080 22.22737
63 4.811289 71.9561763 22.07612
64 4.789096 68.8749257 21.97083
65 4.767444 97.4696759 21.90808
66 4.745881 19.5195821 21.87788
67 4.724225 125.8952139 21.86863
68 4.702404 201.2123849 21.86829
69 4.680390 -118.3616131 21.86475
70 4.658171 -13.4594157 21.84595
71 4.635741 -24.8419169 21.79967
72 4.612948 24.8299907 21.70150
73 4.585841 -195.2017373 20.88262
74 4.566584 -230.9539355 20.79241
75 4.550452 63.0675907 20.67445
76 4.535414 51.5320620 20.59259
77 4.520711 93.0336323 20.54416
78 4.506053 27.8417756 20.52093
79 4.491322 139.7463137 20.51381
80 4.476473 215.9442994 20.51355
81 4.461489 -116.6093318 20.51087
82 4.446363 -1.9662123 20.49663
83 4.431091 -26.1113393 20.46135
84 4.415537 27.9771395 20.38274
85 4.397338 -198.9736169 19.78823
86 4.384086 -250.8120176 19.72439
87 4.372947 54.5436204 19.62989
88 4.362550 47.4584612 19.56409
89 4.352373 83.8972192 19.52539
90 4.342214 27.8821334 19.50688
91 4.331993 145.4468477 19.50121
92 4.321680 209.8155931 19.50099
93 4.311264 -112.8724300 19.49891
94 4.300741 1.7141435 19.48772
95 4.290106 -18.7083317 19.45976
96 4.279242 30.5496682 19.39438
97 4.266671 -197.0032384 18.95193
98 4.257378 -254.1500154 18.90653
99 4.249577 46.4491255 18.82934
100 4.242301 44.9940423 18.77511
101 4.235173 77.8151376 18.74336
102 4.228046 31.8119758 18.72822
103 4.220865 148.9383869 18.72358
104 4.213609 199.4840194 18.72340
105 4.206270 -120.6649685 18.72173
106 4.198846 3.8126936 18.71269
107 4.191329 -9.7602375 18.68987
108 4.183620 30.5935124 18.63401
109 4.174771 -202.6785842 18.29879
110 4.168173 -244.0187607 18.26644
111 4.162660 50.8567600 18.20248
112 4.157530 53.4219439 18.15691
113 4.152501 79.3473500 18.13031
114 4.147464 40.9136560 18.11767
115 4.142379 149.1771108 18.11380
116 4.137230 197.8861984 18.11364
117 4.132012 -122.2080930 18.11227
118 4.126723 2.9608197 18.10482
119 4.121356 -6.8336449 18.08573
120 4.115826 29.6025600 18.03711
121 4.109514 -217.3605600 17.77961
122 4.104789 -250.9119895 17.75656
123 4.100871 48.5460765 17.70300
ggplot(filter, aes(x = Date, y = level)) + geom_line(color = "orange") +
geom_line(aes(y = slope),color = "green") + geom_line(aes(y = season1),color = "blue") +
geom_ribbon(aes(ymin = level - 1.96 * level_sd, ymax = level + 1.96 * level_sd), alpha = 0.3, fill = "orange") +
geom_ribbon(aes(ymin = slope - 1.96 * slope_sd, ymax = slope + 1.96 * slope_sd), alpha = 0.1, fill = "green") +
geom_ribbon(aes(ymin = season1 - 1.96 * season1_sd, ymax = season1 + 1.96 * season1_sd), alpha = 0.1, fill = "blue") +
theme(axis.title = element_blank())
List of 3
$ s : Time-Series [1:124, 1:13] from 1991 to 2001: 1900 1899 1885 1878 1835 ...
$ U.S:List of 124
$ D.S: num [1:124, 1:13] 46.6 32.5 29.1 28.3 28 ...
theta <- modelfit1$GG%*%t(smoothing$s[1:length(train.ts),]) #s0-s[t] : total t+1
fitted_smooth <- data.frame(Date=train$Date, Observation=as.numeric(modelfit1$FF%*%theta))
# train + smooth data.frame
train_smooth <- rbind(
train,
fitted_smooth
)
train_smooth$Group <- c(rep("Data",length(train.ts)), rep("smoothed",length(train.ts)))
ggplot(train_smooth, aes(x=Date, y=Observation, col=Group, size=Group))+geom_line()+
scale_color_manual(values = c("black", "cyan")) +
scale_size_manual(values=c(1.5,1)) +
theme_classic()+
theme(legend.title = element_blank())
cov.smooth <- dlmSvd2var(smoothing$U.S, smoothing$D.S) # covariance of state for each observation
sd.smooth <- t(sapply(cov.smooth, FUN=function(x) sqrt(diag(x)))) # standard deviation of state for each observation
cov.smooth
[[1]]
[,1] [,2] [,3] [,4] [,5]
[1,] 2152.842465 -30.19959710 5.04258606 15.35376591 40.7014344
[2,] -30.199597 16.81703545 -0.06923674 -0.04597259 -0.2888346
[3,] 5.042586 -0.06923674 342.89709656 2.07238075 4.4629556
[4,] 15.353766 -0.04597259 2.07238075 342.13821232 1.7118495
[5,] 40.701434 -0.28883459 4.46295560 1.71184951 342.2819614
[6,] 57.411623 -0.46598389 -27.06032078 4.29429302 1.8579546
[7,] 64.516207 -0.56187570 -50.86304420 -27.05065829 4.4175418
[8,] 61.960962 -0.57395027 -65.25574678 -50.67942992 -26.9553612
[9,] 49.798274 -0.50158575 -70.09775845 -64.89932615 -50.6133654
[10,] 28.164405 -0.34918115 -65.34152789 -69.54977738 -64.8488502
[11,] -2.658219 -0.15623329 -50.71699091 -64.41740937 -69.3852378
[12,] -42.214014 -0.11758203 -25.00471217 -48.47012899 -63.4834201
[13,] -88.165309 0.71552606 3.30934028 -26.32391195 -50.0800540
[,6] [,7] [,8] [,9] [,10]
[1,] 57.4116234 64.5162068 61.9609621 49.7982738 28.1644052
[2,] -0.4659839 -0.5618757 -0.5739503 -0.5015858 -0.3491812
[3,] -27.0603208 -50.8630442 -65.2557468 -70.0977585 -65.3415279
[4,] 4.2942930 -27.0506583 -50.6794299 -64.8993262 -69.5497774
[5,] 1.8579546 4.4175418 -26.9553612 -50.6133654 -64.8488502
[6,] 342.4069476 1.9453267 4.4641600 -26.9502817 -50.6422592
[7,] 1.9453267 342.4515419 1.9460605 4.4207681 -27.0362598
[8,] 4.4641600 1.9460605 342.4073695 1.8572366 4.2828693
[9,] -26.9502817 4.4207681 1.8572366 342.2738883 1.6687296
[10,] -50.6422592 -27.0362598 4.2828693 1.6687296 342.0180681
[11,] -64.8405735 -50.7573924 -27.2668495 3.9388577 1.1843286
[12,] -68.9817662 -64.9195504 -51.3045727 -28.2785109 2.4052026
[13,] -64.4823709 -69.3347580 -64.6248600 -50.3619876 -26.6703064
[,11] [,12] [,13]
[1,] -2.6582188 -42.214014 -88.1653089
[2,] -0.1562333 -0.117582 0.7155261
[3,] -50.7169909 -25.004712 3.3093403
[4,] -64.4174094 -48.470129 -26.3239120
[5,] -69.3852378 -63.483420 -50.0800540
[6,] -64.8405735 -68.981766 -64.4823709
[7,] -50.7573924 -64.919550 -69.3347580
[8,] -27.2668495 -51.304573 -64.6248600
[9,] 3.9388577 -28.278511 -50.3619876
[10,] 1.1843286 2.405203 -26.6703064
[11,] 341.1319277 -1.435561 4.8687830
[12,] -1.4355608 334.536049 2.8867413
[13,] 4.8687830 2.886741 335.3121541
[[2]]
[,1] [,2] [,3] [,4] [,5]
[1,] 965.627369 -13.38603669 -187.518405 4.97392618 15.3095498
[2,] -13.386037 16.61301127 2.414913 -0.06923685 -0.0459729
[3,] -187.518405 2.41491283 313.397484 1.59832320 1.1739446
[4,] 4.973926 -0.06923685 1.598323 342.89709656 2.0723807
[5,] 15.309550 -0.04597290 1.173945 2.07238075 342.1382123
[6,] 40.417256 -0.28883542 -29.365902 4.46295560 1.7118495
[7,] 56.952207 -0.46598507 -52.011016 -27.06032078 4.2942930
[8,] 63.961712 -0.56187703 -65.219475 -50.86304420 -27.0506583
[9,] 61.394100 -0.57395154 -68.870782 -65.25574678 -50.6794299
[10,] 49.302385 -0.50158678 -62.958177 -70.09775845 -64.8993262
[11,] 27.818446 -0.34918173 -47.470179 -65.34152789 -69.5497774
[12,] -2.814756 -0.15623323 -22.303890 -50.71699091 -64.4174094
[13,] -42.336426 -0.11758117 12.050225 -25.00471217 -48.4701290
[,6] [,7] [,8] [,9] [,10]
[1,] 40.4172560 56.9522073 63.961712 61.3941001 49.3023850
[2,] -0.2888354 -0.4659851 -0.561877 -0.5739515 -0.5015868
[3,] -29.3659017 -52.0110161 -65.219475 -68.8707822 -62.9581771
[4,] 4.4629556 -27.0603208 -50.863044 -65.2557468 -70.0977585
[5,] 1.7118495 4.2942930 -27.050658 -50.6794299 -64.8993262
[6,] 342.2819614 1.8579546 4.417542 -26.9553612 -50.6133654
[7,] 1.8579546 342.4069476 1.945327 4.4641600 -26.9502817
[8,] 4.4175418 1.9453267 342.451542 1.9460605 4.4207681
[9,] -26.9553612 4.4641600 1.946060 342.4073695 1.8572366
[10,] -50.6133654 -26.9502817 4.420768 1.8572366 342.2738883
[11,] -64.8488502 -50.6422592 -27.036260 4.2828693 1.6687296
[12,] -69.3852378 -64.8405735 -50.757392 -27.2668495 3.9388577
[13,] -63.4834201 -68.9817662 -64.919550 -51.3045727 -28.2785109
[,11] [,12] [,13]
[1,] 27.8184460 -2.8147562 -42.3364255
[2,] -0.3491817 -0.1562332 -0.1175812
[3,] -47.4701787 -22.3038896 12.0502255
[4,] -65.3415279 -50.7169909 -25.0047122
[5,] -69.5497774 -64.4174094 -48.4701290
[6,] -64.8488502 -69.3852378 -63.4834201
[7,] -50.6422592 -64.8405735 -68.9817662
[8,] -27.0362598 -50.7573924 -64.9195504
[9,] 4.2828693 -27.2668495 -51.3045727
[10,] 1.6687296 3.9388577 -28.2785109
[11,] 342.0180681 1.1843286 2.4052026
[12,] 1.1843286 341.1319277 -1.4355608
[13,] 2.4052026 -1.4355608 334.5360486
[[3]]
[,1] [,2] [,3] [,4] [,5]
[1,] 719.274679 -5.70433158 -141.3528315 -83.277095 10.22283971
[2,] -5.704332 16.41214392 0.7252711 2.396756 -0.07018542
[3,] -141.352832 0.72527112 306.3566437 19.979455 3.30938607
[4,] -83.277095 2.39675638 19.9794546 313.397484 1.59832320
[5,] 10.222840 -0.07018542 3.3093861 1.598323 342.89709656
[6,] 28.602045 -0.04835208 -26.3239180 1.173945 2.07238075
[7,] 49.074671 -0.29043179 -50.0801071 -29.365902 4.46295560
[8,] 60.490683 -0.46670043 -64.4824577 -52.011016 -27.06032078
[9,] 62.389565 -0.56169818 -69.3348648 -65.219475 -50.86304420
[10,] 54.777891 -0.57287520 -64.6249728 -68.870782 -65.25574678
[11,] 37.757663 -0.49961830 -50.3620925 -62.958177 -70.09775845
[12,] 11.572369 -0.34634716 -26.6703871 -47.470179 -65.34152789
[13,] -23.294662 -0.15260853 4.8687992 -22.303890 -50.71699091
[,6] [,7] [,8] [,9] [,10]
[1,] 28.60204466 49.0746712 60.4906827 62.3895650 54.7778913
[2,] -0.04835208 -0.2904318 -0.4667004 -0.5616982 -0.5728752
[3,] -26.32391800 -50.0801071 -64.4824577 -69.3348648 -64.6249728
[4,] 1.17394459 -29.3659017 -52.0110161 -65.2194755 -68.8707822
[5,] 2.07238075 4.4629556 -27.0603208 -50.8630442 -65.2557468
[6,] 342.13821232 1.7118495 4.2942930 -27.0506583 -50.6794299
[7,] 1.71184951 342.2819614 1.8579546 4.4175418 -26.9553612
[8,] 4.29429302 1.8579546 342.4069476 1.9453267 4.4641600
[9,] -27.05065829 4.4175418 1.9453267 342.4515419 1.9460605
[10,] -50.67942992 -26.9553612 4.4641600 1.9460605 342.4073695
[11,] -64.89932615 -50.6133654 -26.9502817 4.4207681 1.8572366
[12,] -69.54977738 -64.8488502 -50.6422592 -27.0362598 4.2828693
[13,] -64.41740937 -69.3852378 -64.8405735 -50.7573924 -27.2668495
[,11] [,12] [,13]
[1,] 37.7576632 11.5723694 -23.2946620
[2,] -0.4996183 -0.3463472 -0.1526085
[3,] -50.3620925 -26.6703871 4.8687992
[4,] -62.9581771 -47.4701787 -22.3038896
[5,] -70.0977585 -65.3415279 -50.7169909
[6,] -64.8993262 -69.5497774 -64.4174094
[7,] -50.6133654 -64.8488502 -69.3852378
[8,] -26.9502817 -50.6422592 -64.8405735
[9,] 4.4207681 -27.0362598 -50.7573924
[10,] 1.8572366 4.2828693 -27.2668495
[11,] 342.2738883 1.6687296 3.9388577
[12,] 1.6687296 342.0180681 1.1843286
[13,] 3.9388577 1.1843286 341.1319277
[[4]]
[,1] [,2] [,3] [,4] [,5]
[1,] 666.779528 -2.22639265 -131.71564276 -61.7321171 -30.267741
[2,] -2.226393 16.21581660 -0.09731017 0.7209414 2.369573
[3,] -131.715643 -0.09731017 305.58051692 17.3645346 12.050264
[4,] -61.732117 0.72094141 17.36453458 306.3566437 19.979455
[5,] -30.267741 2.36957336 12.05026375 19.9794546 313.397484
[6,] 26.419660 -0.07403516 -25.00471594 3.3093861 1.598323
[7,] 43.735381 -0.05343887 -48.47013797 -26.3239180 1.173945
[8,] 56.916997 -0.29347861 -63.48343096 -50.0801071 -29.365902
[9,] 60.798718 -0.46755397 -68.98177873 -64.4824577 -52.011016
[10,] 55.198190 -0.56033695 -64.91956453 -69.3348648 -65.219475
[11,] 40.195535 -0.56930033 -51.30458819 -64.6249728 -68.870782
[12,] 16.016730 -0.49386151 -28.27852555 -50.3620925 -62.958177
[13,] -16.886602 -0.33849880 2.40524651 -26.6703871 -47.470179
[,6] [,7] [,8] [,9] [,10]
[1,] 26.41965962 43.73538103 56.9169971 60.798718 55.198190
[2,] -0.07403516 -0.05343887 -0.2934786 -0.467554 -0.560337
[3,] -25.00471594 -48.47013797 -63.4834310 -68.981779 -64.919565
[4,] 3.30938607 -26.32391800 -50.0801071 -64.482458 -69.334865
[5,] 1.59832320 1.17394459 -29.3659017 -52.011016 -65.219475
[6,] 342.89709656 2.07238075 4.4629556 -27.060321 -50.863044
[7,] 2.07238075 342.13821232 1.7118495 4.294293 -27.050658
[8,] 4.46295560 1.71184951 342.2819614 1.857955 4.417542
[9,] -27.06032078 4.29429302 1.8579546 342.406948 1.945327
[10,] -50.86304420 -27.05065829 4.4175418 1.945327 342.451542
[11,] -65.25574678 -50.67942992 -26.9553612 4.464160 1.946060
[12,] -70.09775845 -64.89932615 -50.6133654 -26.950282 4.420768
[13,] -65.34152789 -69.54977738 -64.8488502 -50.642259 -27.036260
[,11] [,12] [,13]
[1,] 40.1955354 16.0167302 -16.8866019
[2,] -0.5693003 -0.4938615 -0.3384988
[3,] -51.3045882 -28.2785255 2.4052465
[4,] -64.6249728 -50.3620925 -26.6703871
[5,] -68.8707822 -62.9581771 -47.4701787
[6,] -65.2557468 -70.0977585 -65.3415279
[7,] -50.6794299 -64.8993262 -69.5497774
[8,] -26.9553612 -50.6133654 -64.8488502
[9,] 4.4641600 -26.9502817 -50.6422592
[10,] 1.9460605 4.4207681 -27.0362598
[11,] 342.4073695 1.8572366 4.2828693
[12,] 1.8572366 342.2738883 1.6687296
[13,] 4.2828693 1.6687296 342.0180681
[[5]]
[,1] [,2] [,3] [,4] [,5]
[1,] 655.3823203 -0.74723685 -130.9784249 -55.77899950 -18.0133175
[2,] -0.7472369 16.02461603 -0.1199231 -0.09497039 0.7089432
[3,] -130.9784249 -0.11992307 312.1764391 13.04219551 4.8688268
[4,] -55.7789995 -0.09497039 13.0421955 305.58051692 17.3645346
[5,] -18.0133175 0.70894318 4.8688268 17.36453458 306.3566437
[6,] 7.9747622 2.33599253 -22.3039029 12.05026375 19.9794546
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[117,] 25.54528 3.934808 17.70595 17.70470 17.70117 17.69711 17.71323
[118,] 25.54463 3.957037 17.70470 17.70595 17.70470 17.70117 17.69711
[119,] 25.54467 3.979797 17.70093 17.70470 17.70595 17.70470 17.70117
[120,] 25.55310 4.003088 17.69370 17.70093 17.70470 17.70595 17.70470
[121,] 25.60046 4.026899 17.66866 17.69370 17.70093 17.70470 17.70595
[122,] 25.82217 4.051203 17.48106 17.66866 17.69370 17.70093 17.70470
[123,] 26.81966 4.075919 17.50319 17.48106 17.66866 17.69370 17.70093
[124,] 31.07605 4.100871 17.70300 17.50319 17.48106 17.66866 17.69370
[,8] [,9] [,10] [,11] [,12] [,13]
[1,] 18.50425 18.50065 18.49373 18.46976 18.29033 18.31153
[2,] 18.50545 18.50425 18.50065 18.49373 18.46976 18.29033
[3,] 18.50424 18.50545 18.50425 18.50065 18.49373 18.46976
[4,] 18.50086 18.50424 18.50545 18.50425 18.50065 18.49373
[5,] 18.49698 18.50086 18.50424 18.50545 18.50425 18.50065
[6,] 18.51748 18.49698 18.50086 18.50424 18.50545 18.50425
[7,] 17.70304 18.51748 18.49698 18.50086 18.50424 18.50545
[8,] 17.50305 17.70304 18.51748 18.49698 18.50086 18.50424
[9,] 17.48086 17.50305 17.70304 18.51748 18.49698 18.50086
[10,] 17.66852 17.48086 17.50305 17.70304 18.51748 18.49698
[11,] 17.69357 17.66852 17.48086 17.50305 17.70304 18.51748
[12,] 17.70080 17.69357 17.66852 17.48086 17.50305 17.70304
[13,] 17.70457 17.70080 17.69357 17.66852 17.48086 17.50305
[14,] 17.70582 17.70457 17.70080 17.69357 17.66852 17.48086
[15,] 17.70456 17.70582 17.70457 17.70080 17.69357 17.66852
[16,] 17.70103 17.70456 17.70582 17.70457 17.70080 17.69357
[17,] 17.69697 17.70103 17.70456 17.70582 17.70457 17.70080
[18,] 17.71309 17.69697 17.70103 17.70456 17.70582 17.70457
[19,] 17.04370 17.71309 17.69697 17.70103 17.70456 17.70582
[20,] 16.90739 17.04370 17.71309 17.69697 17.70103 17.70456
[21,] 16.88332 16.90739 17.04370 17.71309 17.69697 17.70103
[22,] 17.08295 16.88332 16.90739 17.04370 17.71309 17.69697
[23,] 17.10750 17.08295 16.88332 16.90739 17.04370 17.71309
[24,] 17.11475 17.10750 17.08295 16.88332 16.90739 17.04370
[25,] 17.11862 17.11475 17.10750 17.08295 16.88332 16.90739
[26,] 17.11991 17.11862 17.11475 17.10750 17.08295 16.88332
[27,] 17.11861 17.11991 17.11862 17.11475 17.10750 17.08295
[28,] 17.11495 17.11861 17.11991 17.11862 17.11475 17.10750
[29,] 17.11054 17.11495 17.11861 17.11991 17.11862 17.11475
[30,] 17.12041 17.11054 17.11495 17.11861 17.11991 17.11862
[31,] 16.56894 17.12041 17.11054 17.11495 17.11861 17.11991
[32,] 16.48123 16.56894 17.12041 17.11054 17.11495 17.11861
[33,] 16.45845 16.48123 16.56894 17.12041 17.11054 17.11495
[34,] 16.67613 16.45845 16.48123 16.56894 17.12041 17.11054
[35,] 16.69895 16.67613 16.45845 16.48123 16.56894 17.12041
[36,] 16.70607 16.69895 16.67613 16.45845 16.48123 16.56894
[37,] 16.70999 16.70607 16.69895 16.67613 16.45845 16.48123
[38,] 16.71131 16.70999 16.70607 16.69895 16.67613 16.45845
[39,] 16.70998 16.71131 16.70999 16.70607 16.69895 16.67613
[40,] 16.70622 16.70998 16.71131 16.70999 16.70607 16.69895
[41,] 16.70134 16.70622 16.70998 16.71131 16.70999 16.70607
[42,] 16.70423 16.70134 16.70622 16.70998 16.71131 16.70999
[43,] 16.24744 16.70423 16.70134 16.70622 16.70998 16.71131
[44,] 16.19629 16.24744 16.70423 16.70134 16.70622 16.70998
[45,] 16.17831 16.19629 16.24744 16.70423 16.70134 16.70622
[46,] 16.42143 16.17831 16.19629 16.24744 16.70423 16.70134
[47,] 16.44126 16.42143 16.17831 16.19629 16.24744 16.70423
[48,] 16.44809 16.44126 16.42143 16.17831 16.19629 16.24744
[49,] 16.45204 16.44809 16.44126 16.42143 16.17831 16.19629
[50,] 16.45338 16.45204 16.44809 16.44126 16.42143 16.17831
[51,] 16.45203 16.45338 16.45204 16.44809 16.44126 16.42143
[52,] 16.44818 16.45203 16.45338 16.45204 16.44809 16.44126
[53,] 16.44278 16.44818 16.45203 16.45338 16.45204 16.44809
[54,] 16.43878 16.44278 16.44818 16.45203 16.45338 16.45204
[55,] 16.05708 16.43878 16.44278 16.44818 16.45203 16.45338
[56,] 16.03297 16.05708 16.43878 16.44278 16.44818 16.45203
[57,] 16.02381 16.03297 16.05708 16.43878 16.44278 16.44818
[58,] 16.30136 16.02381 16.03297 16.05708 16.43878 16.44278
[59,] 16.31699 16.30136 16.02381 16.03297 16.05708 16.43878
[60,] 16.32342 16.31699 16.30136 16.02381 16.03297 16.05708
[61,] 16.32736 16.32342 16.31699 16.30136 16.02381 16.03297
[62,] 16.32871 16.32736 16.32342 16.31699 16.30136 16.02381
[63,] 16.32735 16.32871 16.32736 16.32342 16.31699 16.30136
[64,] 16.32344 16.32735 16.32871 16.32736 16.32342 16.31699
[65,] 16.31751 16.32344 16.32735 16.32871 16.32736 16.32342
[66,] 16.30723 16.31751 16.32344 16.32735 16.32871 16.32736
[67,] 15.98435 16.30723 16.31751 16.32344 16.32735 16.32871
[68,] 15.97980 15.98435 16.30723 16.31751 16.32344 16.32735
[69,] 15.98426 15.97980 15.98435 16.30723 16.31751 16.32344
[70,] 16.30724 15.98426 15.97980 15.98435 16.30723 16.31751
[71,] 16.31752 16.30724 15.98426 15.97980 15.98435 16.30723
[72,] 16.32346 16.31752 16.30724 15.98426 15.97980 15.98435
[73,] 16.32737 16.32346 16.31752 16.30724 15.98426 15.97980
[74,] 16.32872 16.32737 16.32346 16.31752 16.30724 15.98426
[75,] 16.32736 16.32872 16.32737 16.32346 16.31752 16.30724
[76,] 16.32342 16.32736 16.32872 16.32737 16.32346 16.31752
[77,] 16.31699 16.32342 16.32736 16.32872 16.32737 16.32346
[78,] 16.30138 16.31699 16.32342 16.32736 16.32872 16.32737
[79,] 16.02391 16.30138 16.31699 16.32342 16.32736 16.32872
[80,] 16.03299 16.02391 16.30138 16.31699 16.32342 16.32736
[81,] 16.05700 16.03299 16.02391 16.30138 16.31699 16.32342
[82,] 16.43881 16.05700 16.03299 16.02391 16.30138 16.31699
[83,] 16.44282 16.43881 16.05700 16.03299 16.02391 16.30138
[84,] 16.44823 16.44282 16.43881 16.05700 16.03299 16.02391
[85,] 16.45207 16.44823 16.44282 16.43881 16.05700 16.03299
[86,] 16.45341 16.45207 16.44823 16.44282 16.43881 16.05700
[87,] 16.45207 16.45341 16.45207 16.44823 16.44282 16.43881
[88,] 16.44812 16.45207 16.45341 16.45207 16.44823 16.44282
[89,] 16.44129 16.44812 16.45207 16.45341 16.45207 16.44823
[90,] 16.42147 16.44129 16.44812 16.45207 16.45341 16.45207
[91,] 16.17843 16.42147 16.44129 16.44812 16.45207 16.45341
[92,] 16.19634 16.17843 16.42147 16.44129 16.44812 16.45207
[93,] 16.24738 16.19634 16.17843 16.42147 16.44129 16.44812
[94,] 16.70430 16.24738 16.19634 16.17843 16.42147 16.44129
[95,] 16.70141 16.70430 16.24738 16.19634 16.17843 16.42147
[96,] 16.70629 16.70141 16.70430 16.24738 16.19634 16.17843
[97,] 16.71005 16.70629 16.70141 16.70430 16.24738 16.19634
[98,] 16.71137 16.71005 16.70629 16.70141 16.70430 16.24738
[99,] 16.71005 16.71137 16.71005 16.70629 16.70141 16.70430
[100,] 16.70612 16.71005 16.71137 16.71005 16.70629 16.70141
[101,] 16.69901 16.70612 16.71005 16.71137 16.71005 16.70629
[102,] 16.67621 16.69901 16.70612 16.71005 16.71137 16.71005
[103,] 16.45859 16.67621 16.69901 16.70612 16.71005 16.71137
[104,] 16.48131 16.45859 16.67621 16.69901 16.70612 16.71005
[105,] 16.56889 16.48131 16.45859 16.67621 16.69901 16.70612
[106,] 17.12051 16.56889 16.48131 16.45859 16.67621 16.69901
[107,] 17.11064 17.12051 16.56889 16.48131 16.45859 16.67621
[108,] 17.11505 17.11064 17.12051 16.56889 16.48131 16.45859
[109,] 17.11871 17.11505 17.11064 17.12051 16.56889 16.48131
[110,] 17.12000 17.11871 17.11505 17.11064 17.12051 16.56889
[111,] 17.11871 17.12000 17.11871 17.11505 17.11064 17.12051
[112,] 17.11484 17.11871 17.12000 17.11871 17.11505 17.11064
[113,] 17.10759 17.11484 17.11871 17.12000 17.11871 17.11505
[114,] 17.08305 17.10759 17.11484 17.11871 17.12000 17.11871
[115,] 16.88348 17.08305 17.10759 17.11484 17.11871 17.12000
[116,] 16.90750 16.88348 17.08305 17.10759 17.11484 17.11871
[117,] 17.04366 16.90750 16.88348 17.08305 17.10759 17.11484
[118,] 17.71323 17.04366 16.90750 16.88348 17.08305 17.10759
[119,] 17.69711 17.71323 17.04366 16.90750 16.88348 17.08305
[120,] 17.70117 17.69711 17.71323 17.04366 16.90750 16.88348
[121,] 17.70470 17.70117 17.69711 17.71323 17.04366 16.90750
[122,] 17.70595 17.70470 17.70117 17.69711 17.71323 17.04366
[123,] 17.70470 17.70595 17.70470 17.70117 17.69711 17.71323
[124,] 17.70093 17.70470 17.70595 17.70470 17.70117 17.69711
fitted_smooth$level <-smoothing$s[-1,1]
fitted_smooth$level_sd <- sd.smooth[-1,1]
fitted_smooth$slope <- smoothing$s[-1,2]
fitted_smooth$slope_sd <- sd.smooth[-1,2]
fitted_smooth$season1 <- smoothing$s[-1,3]
fitted_smooth$season1_sd <- sd.smooth[-1,3] # Seasonal represent 3~13 columns. However, just for Season1.
# Smoothing estimation error
ggplot(fitted_smooth, aes(x = Date, y = level)) + geom_line(color = "orange") +
geom_line(aes(y = slope),color = "green") + geom_line(aes(y = season1),color = "blue") +
geom_ribbon(aes(ymin = level - 1.96 * level_sd, ymax = level + 1.96 * level_sd), alpha = 0.3, fill = "orange") +
geom_ribbon(aes(ymin = slope - 1.96 * slope_sd, ymax = slope + 1.96 * slope_sd), alpha = 0.1, fill = "green") +
geom_ribbon(aes(ymin = season1 - 1.96 * season1_sd, ymax = season1 + 1.96 * season1_sd), alpha = 0.1, fill = "blue") +
theme(axis.title = element_blank())
forecast <- dlmForecast(filtering, nAhead = nValid) # Forecast(filtering model)
# data.frame
x <- index(ridership.ts)
forecast_df <- rbind(
data.frame(Date=x, Observation=as.numeric(ridership$Observation), sd=0, series="Data"),
data.frame(Date=x[1:123], Observation=filtering$f, sd=0, series="filtered"), # Filtered
data.frame(Date=x[-(1:123)], Observation=c(forecast$f), sd=sqrt(unlist(forecast$Q)), series="forecast") # Forecast
)
ggplot(forecast_df, aes(x = Date, y = Observation,col=series, size=series)) + geom_line() +
scale_color_manual(values = c("black", "cyan", "red")) +
scale_size_manual(values = c(1.5,0.5,1)) +
geom_ribbon(aes(ymin = Observation - 1.96 * sd, ymax = Observation + 1.96 * sd), alpha = 0.3, fill="orange", show.legend = FALSE, colour=FALSE) +
scale_x_continuous(breaks = seq(min(x)+1, max(x), by = 2))
Text and figures are licensed under Creative Commons Attribution CC BY 4.0. The figures that have been reused from other sources don't fall under this license and can be recognized by a note in their caption: "Figure from ...".