Bayesian Time Series
Nana Boateng
#install.packages("bsts")
#install.packages("CausalImpact")
library(bsts)
require(RColorBrewer)
library(foreach)
library(doParallel)
library(viridis)
library(RColorBrewer)
library(tidyverse)
library(ggthemes)
# Calculate the number of cores
no_cores <- detectCores() - 1
cl<-makeCluster(no_cores)
registerDoParallel(cl)
#bsts::Structural time series models
A structural time series model is defined by two equations. The observation equation relates the observed data \(y_{t}\) to a vector of latent variables \(\alpha_{t}\) known as the “state.”
\[\begin{equation} y_{t}=Z_{t}^{T}\alpha_{t}+\epsilon_{t} \end{equation}\]The transition equation describes how the latent state evolves through time.
\[\begin{equation} \alpha_{t+1}=T_{t}\alpha_{t}+R_{t}\eta_{t} \end{equation}\]The error terms \(\eta_{t}\) and \(\epsilon_{t}\) are Gaussian and independent of everything else. The arrays \(Z_{t},T_{t}\) and \(R_{t}\) are structural parameters. They may contain parameters in the statistical sense, but often they simply contain strategically placed 0’s and 1’s indicating which bits of \(\alpha_{t}\) are relevant for a particular computation.
The simplest useful model is the “local level model,” in which the vector \(\alpha_{t}\) is just a scalar \(\mu_{t}\) .The local level model is a random walk observed in noise.
\[\begin{equation} y_{t}=\mu_{t}+\epsilon_{t} \end{equation}\] \[\begin{equation} \mu_{t+1}=\mu_{t}+\eta_{t} \end{equation}\]Nowcasting
# load the bsts package
data(iclaims) # bring the initial.claims data into scope
initial.claims%>%head()%>%tk_tbl()
autoplot(initial.claims$iclaimsNSA)+theme_economist()+ylab("iclaimsNSA")+xlab("Time")+(scale_x_date(breaks=date_breaks("4 month"),labels=date_format("%b %d %y")))+ theme(axis.text.x=element_text(angle=50, size=10, vjust=0.5))
adf.test(initial.claims$iclaimsNSA)
Augmented Dickey-Fuller Test
data: initial.claims$iclaimsNSA
Dickey-Fuller = -3.4741, Lag order = 7, p-value = 0.04497
alternative hypothesis: stationary#weekly data,number of seasons=52
ss<-AddLocalLinearTrend(list(), initial.claims$iclaimsNSA)
ss<- AddSeasonal(ss, initial.claims$iclaimsNSA, nseasons = 52)
model1 <- bsts(initial.claims$iclaimsNSA,state.specification = ss,niter = 1000)=-=-=-=-= Iteration 0 Sun Sep 10 22:50:25 2017 =-=-=-=-=
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=-=-=-=-= Iteration 800 Sun Sep 10 22:50:46 2017 =-=-=-=-=
=-=-=-=-= Iteration 900 Sun Sep 10 22:50:48 2017 =-=-=-=-=#bsts::PlotSeasonalEffect(model1)
plot(model1)
plot(model1, "components") # plot(model1, "comp") works too!
plot(model1, "help")
plot(model1,"forecast.distribution")
plot(model1,"prediction.errors")
plot(model1,"state")
pred1 <- predict(model1, horizon = 12,quantiles = c(.025, .975))
plot(pred1, plot.original = 156)
pred1%>%head$mean
[1] -0.04825910 -0.51923555 -0.37806224 -0.42461246 -0.05047353 -0.37551402 0.36423551
[8] -0.22978498 1.21517240 0.42691357 0.29233626 0.87398357
$median
[1] -0.05148937 -0.56947639 -0.38498958 -0.46509774 -0.01645982 -0.33368530 0.34163448
[8] -0.22740426 1.13110127 0.51395751 0.38347120 0.90173648
$interval
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
2.5% -0.5909969 -1.04319893 -1.0530661 -1.1078638 -0.7839605 -1.3776313 -0.4236237
97.5% 0.4859783 0.04063862 0.2103079 0.3957649 0.6363574 0.3922725 1.3300442
[,8] [,9] [,10] [,11] [,12]
2.5% -1.3209674 0.1733181 -0.7321765 -0.7967837 -0.5690577
97.5% 0.7581783 2.2578965 1.3548215 1.5529610 2.0524604
$distribution
[,1] [,2] [,3] [,4] [,5] [,6]
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[3,] 0.3229786185 -0.44888351 -1.006951035 -0.498419303 0.114930298 -0.221880108
[4,] 0.0397161327 -0.59356124 -0.508507434 -0.876451603 0.121088179 -0.252364517
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[,7] [,8] [,9] [,10] [,11] [,12]
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$original.series
2004-01-04 2004-01-11 2004-01-18 2004-01-25 2004-02-01 2004-02-08 2004-02-15 2004-02-22
2.536 0.882 -0.077 0.135 0.373 -0.437 -0.556 -0.432
2004-02-29 2004-03-07 2004-03-14 2004-03-21 2004-03-28 2004-04-04 2004-04-11 2004-04-18
-0.460 -0.698 -0.765 -0.833 -0.767 -0.356 -0.496 -0.684
2004-04-25 2004-05-02 2004-05-09 2004-05-16 2004-05-23 2004-05-30 2004-06-06 2004-06-13
-0.953 -0.869 -0.831 -0.858 -0.769 -0.732 -0.682 -0.606
2004-06-20 2004-06-27 2004-07-04 2004-07-11 2004-07-18 2004-07-25 2004-08-01 2004-08-08
-0.639 -0.363 0.473 0.030 -0.688 -0.963 -0.879 -1.132
2004-08-15 2004-08-22 2004-08-29 2004-09-05 2004-09-12 2004-09-19 2004-09-26 2004-10-03
-1.031 -1.014 -1.026 -1.242 -1.018 -0.957 -0.985 -0.463
2004-10-10 2004-10-17 2004-10-24 2004-10-31 2004-11-07 2004-11-14 2004-11-21 2004-11-28
-0.983 -0.649 -0.756 -0.350 -0.700 -0.310 -0.622 0.730
2004-12-05 2004-12-12 2004-12-19 2004-12-26 2005-01-02 2005-01-09 2005-01-16 2005-01-23
-0.181 -0.144 0.492 1.325 2.677 0.679 -0.269 -0.233
2005-01-30 2005-02-06 2005-02-13 2005-02-20 2005-02-27 2005-03-06 2005-03-13 2005-03-20
-0.386 -0.723 -0.771 -0.886 -0.521 -0.742 -0.887 -0.881
2005-03-27 2005-04-03 2005-04-10 2005-04-17 2005-04-24 2005-05-01 2005-05-08 2005-05-15
-0.849 -0.454 -0.932 -0.806 -0.886 -0.828 -1.021 -1.010
2005-05-22 2005-05-29 2005-06-05 2005-06-12 2005-06-19 2005-06-26 2005-07-03 2005-07-10
-0.767 -0.894 -0.664 -0.895 -0.923 -0.564 0.321 -0.145
2005-07-17 2005-07-24 2005-07-31 2005-08-07 2005-08-14 2005-08-21 2005-08-28 2005-09-04
-0.849 -1.142 -1.072 -1.184 -1.229 -1.232 -1.056 -0.607
2005-09-11 2005-09-18 2005-09-25 2005-10-02 2005-10-09 2005-10-16 2005-10-23 2005-10-30
-0.396 -0.872 -0.682 -0.097 -0.777 -0.763 -0.854 -0.447
2005-11-06 2005-11-13 2005-11-20 2005-11-27 2005-12-04 2005-12-11 2005-12-18 2005-12-25
-0.950 -0.196 -0.887 0.474 0.008 -0.282 0.375 0.750
2006-01-01 2006-01-08 2006-01-15 2006-01-22 2006-01-29 2006-02-05 2006-02-12 2006-02-19
1.451 0.432 -0.646 -0.639 -0.614 -0.716 -1.074 -1.048
2006-02-26 2006-03-05 2006-03-12 2006-03-19 2006-03-26 2006-04-02 2006-04-09 2006-04-16
-0.788 -0.851 -1.077 -1.111 -1.212 -0.675 -1.084 -0.881
2006-04-23 2006-04-30 2006-05-07 2006-05-14 2006-05-21 2006-05-28 2006-06-04 2006-06-11
-0.984 -0.652 -0.902 -1.007 -0.869 -1.156 -0.930 -1.004
2006-06-18 2006-06-25 2006-07-02 2006-07-09 2006-07-16 2006-07-23 2006-07-30 2006-08-06
-0.915 -0.764 0.242 -0.123 -0.903 -1.159 -1.022 -1.192
2006-08-13 2006-08-20 2006-08-27 2006-09-03 2006-09-10 2006-09-17 2006-09-24 2006-10-01
-1.226 -1.236 -1.163 -1.333 -1.096 -1.146 -1.253 -0.737
2006-10-08 2006-10-15 2006-10-22 2006-10-29 2006-11-05 2006-11-12 2006-11-19 2006-11-26
-1.054 -0.881 -0.795 -0.569 -0.927 -0.206 -0.597 0.512
2006-12-03 2006-12-10 2006-12-17 2006-12-24 2006-12-31 2007-01-07 2007-01-14 2007-01-21
-0.061 -0.259 0.304 0.963 1.017 1.023 -0.207 -0.275
2007-01-28 2007-02-04 2007-02-11 2007-02-18 2007-02-25 2007-03-04 2007-03-11 2007-03-18
-0.460 -0.248 -0.752 -0.814 -0.626 -0.814 -1.006 -1.040
2007-03-25 2007-04-01 2007-04-08 2007-04-15 2007-04-22 2007-04-29 2007-05-06 2007-05-13
-1.086 -0.555 -0.646 -0.770 -1.091 -1.028 -1.172 -1.066
2007-05-20 2007-05-27 2007-06-03 2007-06-10 2007-06-17 2007-06-24 2007-07-01 2007-07-08
-1.040 -1.127 -0.784 -0.885 -0.870 -0.802 0.235 -0.064
2007-07-15 2007-07-22 2007-07-29 2007-08-05 2007-08-12 2007-08-19 2007-08-26 2007-09-02
-0.819 -1.181 -1.065 -1.102 -1.180 -1.104 -1.181 -1.286
2007-09-09 2007-09-16 2007-09-23 2007-09-30 2007-10-07 2007-10-14 2007-10-21 2007-10-28
-1.141 -1.268 -1.199 -0.820 -0.747 -0.737 -0.775 -0.576
2007-11-04 2007-11-11 2007-11-18 2007-11-25 2007-12-02 2007-12-09 2007-12-16 2007-12-23
-0.347 -0.600 -0.592 0.636 0.284 0.018 0.577 1.033
2007-12-30 2008-01-06 2008-01-13 2008-01-20 2008-01-27 2008-02-03 2008-02-10 2008-02-17
1.164 1.387 0.216 -0.190 -0.095 -0.119 -0.576 -0.539
2008-02-24 2008-03-02 2008-03-09 2008-03-16 2008-03-23 2008-03-30 2008-04-06 2008-04-13
-0.404 -0.439 -0.487 -0.661 -0.432 -0.299 -0.177 -0.554
2008-04-20 2008-04-27 2008-05-04 2008-05-11 2008-05-18 2008-05-25 2008-06-01 2008-06-08
-0.470 -0.491 -0.579 -0.630 -0.569 -0.796 -0.159 -0.369
2008-06-15 2008-06-22 2008-06-29 2008-07-06 2008-07-13 2008-07-20 2008-07-27 2008-08-03
-0.291 -0.199 0.094 0.752 0.111 -0.149 -0.081 -0.161
2008-08-10 2008-08-17 2008-08-24 2008-08-31 2008-09-07 2008-09-14 2008-09-21 2008-09-28
-0.432 -0.413 -0.270 -0.485 -0.082 0.058 0.010 0.316
2008-10-05 2008-10-12 2008-10-19 2008-10-26 2008-11-02 2008-11-09 2008-11-16 2008-11-23
0.558 0.222 0.516 0.666 1.316 1.079 1.928 1.293
2008-11-30 2008-12-07 2008-12-14 2008-12-21 2008-12-28 2009-01-04 2009-01-11 2009-01-18
3.267 2.112 2.906 2.882 3.014 5.002 3.298 2.026
2009-01-25 2009-02-01 2009-02-08 2009-02-15 2009-02-22 2009-03-01 2009-03-08 2009-03-15
2.574 2.822 2.024 1.898 2.253 2.313 1.858 1.760
2009-03-22 2009-03-29 2009-04-05 2009-04-12 2009-04-19 2009-04-26 2009-05-03 2009-05-10
1.841 2.054 1.941 1.817 1.701 1.288 1.586 1.325
2009-05-17 2009-05-24 2009-05-31 2009-06-07 2009-06-14 2009-06-21 2009-06-28 2009-07-05
1.302 0.967 1.681 1.516 1.604 1.524 1.724 2.529
2009-07-12 2009-07-19 2009-07-26 2009-08-02 2009-08-09 2009-08-16 2009-08-23 2009-08-30
1.766 1.108 0.707 0.845 0.626 0.619 0.615 0.699
2009-09-06 2009-09-13 2009-09-20 2009-09-27 2009-10-04 2009-10-11 2009-10-18 2009-10-25
0.208 0.445 0.518 0.577 1.086 0.654 0.958 0.855
2009-11-01 2009-11-08 2009-11-15 2009-11-22 2009-11-29 2009-12-06 2009-12-13 2009-12-20
1.293 0.781 1.379 0.628 2.494 1.509 1.595 1.510
2009-12-27 2010-01-03 2010-01-10 2010-01-17 2010-01-24 2010-01-31 2010-02-07 2010-02-14
2.301 3.845 2.371 1.031 1.305 1.074 0.805 0.594
2010-02-21 2010-02-28 2010-03-07 2010-03-14 2010-03-21 2010-03-28 2010-04-04 2010-04-11
0.740 0.634 0.425 0.195 0.192 0.266 1.089 0.405
2010-04-18 2010-04-25 2010-05-02 2010-05-09 2010-05-16 2010-05-23 2010-05-30 2010-06-06
0.338 0.074 0.206 0.208 0.175 0.246 0.069 0.506
2010-06-13 2010-06-20 2010-06-27 2010-07-04 2010-07-11 2010-07-18 2010-07-25 2010-08-01
0.319 0.475 0.702 1.105 0.982 0.200 0.098 0.305
2010-08-08 2010-08-15 2010-08-22 2010-08-29 2010-09-05 2010-09-12 2010-09-19 2010-09-26
0.128 -0.054 -0.070 -0.081 -0.435 -0.077 -0.163 -0.153
2010-10-03 2010-10-10 2010-10-17 2010-10-24 2010-10-31 2010-11-07 2010-11-14 2010-11-21
0.633 0.026 0.154 0.266 0.545 0.164 0.652 0.194
2010-11-28 2010-12-05 2010-12-12 2010-12-19 2010-12-26 2011-01-02 2011-01-09 2011-01-16
1.721 0.891 0.924 1.191 1.661 3.382 1.403 0.839
2011-01-23 2011-01-30 2011-02-06 2011-02-13 2011-02-20 2011-02-27 2011-03-06 2011-03-13
0.652 0.439 0.295 -0.089 -0.329 0.144 -0.171 -0.323
2011-03-20 2011-03-27 2011-04-03 2011-04-10 2011-04-17 2011-04-24 2011-05-01 2011-05-08
-0.297 -0.329 0.504 -0.081 -0.028 0.221 0.059 -0.260
2011-05-15 2011-05-22 2011-05-29 2011-06-05 2011-06-12 2011-06-19 2011-06-26 2011-07-03
-0.127 -0.084 -0.214 0.085 0.029 0.138 0.306 0.733
2011-07-10 2011-07-17 2011-07-24 2011-07-31 2011-08-07 2011-08-14 2011-08-21 2011-08-28
0.699 -0.193 -0.441 -0.324 -0.398 -0.408 -0.480 -0.375
2011-09-04 2011-09-11 2011-09-18 2011-09-25 2011-10-02 2011-10-09 2011-10-16 2011-10-23
-0.550 -0.329 -0.557 -0.518 0.132 -0.296 -0.123 -0.189
2011-10-30 2011-11-06 2011-11-13 2011-11-20 2011-11-27 2011-12-04 2011-12-11 2011-12-18
0.102 -0.248 0.434 -0.163 1.218 0.396 0.266 0.943
2011-12-25 2012-01-01 2012-01-08 2012-01-15 2012-01-22 2012-01-29 2012-02-05 2012-02-12
1.318 2.256 1.188 0.229 0.276 0.091 -0.230 -0.392
2012-02-19 2012-02-26 2012-03-04 2012-03-11 2012-03-18 2012-03-25 2012-04-01 2012-04-08
-0.502 -0.200 -0.450 -0.632 -0.598 -0.667 -0.008 -0.182
2012-04-15 2012-04-22 2012-04-29 2012-05-06 2012-05-13 2012-05-20 2012-05-27 2012-06-03
-0.180 -0.509 -0.442 -0.583 -0.536 -0.396 -0.589 -0.127
2012-06-10 2012-06-17 2012-06-24 2012-07-01 2012-07-08 2012-07-15 2012-07-22 2012-07-29
-0.234 -0.182 -0.187 0.452 0.568 -0.444 -0.690 -0.626
2012-08-05 2012-08-12 2012-08-19 2012-08-26 2012-09-02 2012-09-09 2012-09-16 2012-09-23
-0.648 -0.700 -0.694 -0.720 -0.807 -0.536 -0.772 -0.816 Regression with spike and slab priors
# Fit a bsts model with expected model size 1, the default.
model2 <- bsts(iclaimsNSA ~ .,
state.specification = ss,
niter = 1000,
data = initial.claims)=-=-=-=-= Iteration 0 Sun Sep 10 22:51:04 2017 =-=-=-=-=
=-=-=-=-= Iteration 100 Sun Sep 10 22:51:07 2017 =-=-=-=-=
=-=-=-=-= Iteration 200 Sun Sep 10 22:51:10 2017 =-=-=-=-=
=-=-=-=-= Iteration 300 Sun Sep 10 22:51:12 2017 =-=-=-=-=
=-=-=-=-= Iteration 400 Sun Sep 10 22:51:15 2017 =-=-=-=-=
=-=-=-=-= Iteration 500 Sun Sep 10 22:51:17 2017 =-=-=-=-=
=-=-=-=-= Iteration 600 Sun Sep 10 22:51:20 2017 =-=-=-=-=
=-=-=-=-= Iteration 700 Sun Sep 10 22:51:22 2017 =-=-=-=-=
=-=-=-=-= Iteration 800 Sun Sep 10 22:51:25 2017 =-=-=-=-=
=-=-=-=-= Iteration 900 Sun Sep 10 22:51:27 2017 =-=-=-=-=# Fit a bsts model with expected model size 5, to include more coefficients.
model3 <- bsts(iclaimsNSA ~ .,
state.specification = ss,
niter = 1000,
data = initial.claims,
expected.model.size = 5) # Passed to SpikeSlabPrior.=-=-=-=-= Iteration 0 Sun Sep 10 22:51:30 2017 =-=-=-=-=
=-=-=-=-= Iteration 100 Sun Sep 10 22:51:32 2017 =-=-=-=-=
=-=-=-=-= Iteration 200 Sun Sep 10 22:51:35 2017 =-=-=-=-=
=-=-=-=-= Iteration 300 Sun Sep 10 22:51:38 2017 =-=-=-=-=
=-=-=-=-= Iteration 400 Sun Sep 10 22:51:40 2017 =-=-=-=-=
=-=-=-=-= Iteration 500 Sun Sep 10 22:51:43 2017 =-=-=-=-=
=-=-=-=-= Iteration 600 Sun Sep 10 22:51:46 2017 =-=-=-=-=
=-=-=-=-= Iteration 700 Sun Sep 10 22:51:48 2017 =-=-=-=-=
=-=-=-=-= Iteration 800 Sun Sep 10 22:51:51 2017 =-=-=-=-=
=-=-=-=-= Iteration 900 Sun Sep 10 22:51:54 2017 =-=-=-=-=plot(model2, "comp")
#plot(model1 , "coef")
plot(model2, "coef")
plot(model3, "coef")
Model diagnostics
bsts.prediction.errors(model1)$in.sample
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,] 0 -1.654 -2.550912 0.6662130 1.285351 0.3721326 0.6179464 0.9705945 0.9867260
[2,] 0 -1.654 -2.551007 0.6628572 1.286423 0.3749818 0.6193339 0.9726242 0.9898098
[,10] [,11] [,12] [,13] [,14] [,15] [,16] [,17]
[1,] 0.7561412 0.7599190 0.7492750 0.8634926 1.278714 1.0171989 0.7557369 0.4695454
[2,] 0.7596028 0.7630708 0.7525908 0.8669656 1.282731 1.0228302 0.7612822 0.4746532
[,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
[1,] 0.5983028 0.6525759 0.6242636 0.7121293 0.7272729 0.7468378 0.7841976 0.7014166
[2,] 0.6025846 0.6571479 0.6291320 0.7171154 0.7327956 0.7527966 0.7906454 0.7084680
[,26] [,27] [,28] [,29] [,30] [,31] [,32]
[1,] 0.9363740 1.690551 1.0388730 0.1928867 -0.07998115 0.056687520 -0.1591270
[2,] 0.9436129 1.699132 1.0514891 0.2050287 -0.07078082 0.063602914 -0.1530963
[,33] [,34] [,35] [,36] [,37] [,38] [,39]
[1,] 0.0073136148 0.07046592 0.09367185 -0.09146155 0.1871560 0.2693364 0.2470854
[2,] 0.0113632760 0.07391254 0.09660849 -0.08896789 0.1883426 0.2710410 0.2491862
[,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47]
[1,] 0.7752208 0.1935002 0.5277055 0.3864605 0.7684208 0.3475114 0.7088938 0.3285702
[2,] 0.7774320 0.1984830 0.5313575 0.3916266 0.7736725 0.3550333 0.7156434 0.3372222
[,48] [,49] [,50] [,51] [,52] [,53] [,54] [,55]
[1,] 1.649658 0.5524044 0.5079343 1.0805363 1.780028 0.7165634 -0.3315167 -0.2942596
[2,] 1.657438 0.5668944 0.5204317 1.0929163 1.795392 0.7150640 -0.3185977 -0.2948587
[,56] [,57] [,58] [,59] [,60] [,61] [,62]
[1,] -0.4105466 -0.6990620 -0.0128657895 0.1591252 -0.1585680 0.2047932 0.1635664
[2,] -0.4195727 -0.7188222 -0.0488583580 0.1490301 -0.1483492 0.2082270 0.1782972
[,63] [,64] [,65] [,66] [,67] [,68]
[1,] -0.034910069 -0.020319282 -0.07213258 -0.09056140 -0.4162621 0.0002281004
[2,] -0.018037570 -0.013276096 -0.06994438 -0.09298071 -0.4212968 -0.0205918627
[,69] [,70] [,71] [,72] [,73] [,74] [,75]
[1,] 0.2597596 0.1579027 -0.1767473 -0.1353897 0.08617630 -0.07114269 0.1113141
[2,] 0.2518099 0.1686814 -0.1633124 -0.1382416 0.07707264 -0.07206140 0.1072059
[,76] [,77] [,78] [,79] [,80] [,81]
[1,] -0.2049619 -0.1665900 0.0103357957 0.10389045 0.05350177 0.027365126
[2,] -0.2022817 -0.1756208 -0.0022796827 0.09908018 0.05776896 0.033642169
[,82] [,83] [,84] [,85] [,86] [,87] [,88]
[1,] -0.0116242492 -0.02705284 0.1261570 -0.03755085 -0.06321654 0.1632815 0.8179786
[2,] -0.0061489352 -0.02428907 0.1265884 -0.03128147 -0.06233651 0.1598382 0.8229848
[,89] [,90] [,91] [,92] [,93] [,94] [,95]
[1,] 0.5847162 -0.2568720 -0.08661340 0.06661877 -0.07990682 -0.3747443 -0.2270228
[2,] 0.6236945 -0.2153381 -0.08670817 0.05451687 -0.08807566 -0.3865013 -0.2537442
[,96] [,97] [,98] [,99] [,100] [,101]
[1,] -0.0618648707 -0.12293327 0.3204830 -0.08139066 -0.08062836 0.4233057
[2,] -0.0876160483 -0.13906612 0.3063801 -0.07421310 -0.08309564 0.4155037
[,102] [,103] [,104] [,105] [,106] [,107] [,108]
[1,] 0.0291556013 -0.02524838 -0.4697002 -0.9746072 0.2817723 0.2242253 -1.906080e-03
[2,] 0.0429366870 -0.01992617 -0.4713493 -0.9977306 0.2324141 0.2230503 1.991867e-02
[,109] [,110] [,111] [,112] [,113] [,114]
[1,] 0.0488356502 0.3143297 -0.053073072 0.015864005 -0.03380278 0.1372875
[2,] 0.0705057584 0.3269518 -0.033398574 0.026326097 -0.02962409 0.1373982
[,115] [,116] [,117] [,118] [,119] [,120] [,121]
[1,] 0.0159931112 -0.07563510 -0.2326592 -0.07305921 0.010440173 0.1446167 0.1357452
[2,] 0.0239779735 -0.07022985 -0.2327602 -0.08151755 0.007860764 0.1423319 0.1371903
[,122] [,123] [,124] [,125] [,126] [,127] [,128]
[1,] 0.3428243 0.13824959 -0.10000799 -0.2144379 -0.3350799 -0.2329910 -0.0049147966
[2,] 0.3492948 0.16044391 -0.08252829 -0.2146214 -0.3480758 -0.2576880 -0.0251380472
[,129] [,130] [,131] [,132] [,133] [,134] [,135]
[1,] 0.1556999 -0.06047924 0.05977379 0.1686972 0.04652690 0.03210166 0.08244102
[2,] 0.1479923 -0.05711685 0.05776211 0.1703010 0.05557584 0.03929764 0.08716129
[,136] [,137] [,138] [,139] [,140] [,141] [,142]
[1,] 0.013095524 -0.0057210490 0.0179291682 0.02238556 -0.4456188 -0.2972115 0.2323963
[2,] 0.018164543 -0.0001011270 0.0225834218 0.02149857 -0.4563497 -0.3265906 0.2108285
[,143] [,144] [,145] [,146] [,147] [,148] [,149]
[1,] -0.02709142 -0.1312910 0.2255716 0.3066954 0.3551000 0.0340930069 0.09409030
[2,] -0.02529581 -0.1333509 0.2184425 0.3163288 0.3739942 0.0570211472 0.10354604
[,150] [,151] [,152] [,153] [,154] [,155]
[1,] 0.07629213 0.3130451 -0.0014052883 -0.12515041 -0.061104946 -0.2000999
[2,] 0.07833252 0.3168706 0.0117101461 -0.12835545 -0.073300241 -0.2100250
[,156] [,157] [,158] [,159] [,160] [,161] [,162]
[1,] -1.563152e-02 -0.7273461 0.5442683 0.4698630 0.18106825 -0.14870730 0.1775281
[2,] -2.200214e-02 -0.7214014 0.5089197 0.4764500 0.21171606 -0.12210166 0.1821382
[,163] [,164] [,165] [,166] [,167] [,168] [,169]
[1,] -0.0082033635 -0.1458645 -0.2022944 -0.2550786 -0.1634284 -0.12221011 -0.05129996
[2,] 0.0037494917 -0.1396420 -0.2069478 -0.2696682 -0.1816502 -0.13457855 -0.05637002
[,170] [,171] [,172] [,173] [,174] [,175] [,176]
[1,] -0.013214443 0.3489660 0.0136095455 -0.2357794 -0.4122460 -0.1931126 0.09359466
[2,] -0.013826638 0.3502220 0.0283806882 -0.2313762 -0.4263267 -0.2190698 0.07609879
[,177] [,178] [,179] [,180] [,181] [,182] [,183]
[1,] -0.05321470 0.07927028 0.1833980 0.1320655 1.763426e-02 -0.10357986 -0.027073001
[2,] -0.05199050 0.08324626 0.1906726 0.1420941 2.608168e-02 -0.09832617 -0.030832545
[,184] [,185] [,186] [,187] [,188] [,189] [,190]
[1,] 0.10039659 0.1229474 -0.0130959597 -0.04358380 0.09976664 0.04238708 0.07984149
[2,] 0.09526345 0.1255759 -0.0054602102 -0.04076043 0.09886614 0.04558023 0.08143419
[,191] [,192] [,193] [,194] [,195] [,196] [,197]
[1,] -0.1474734 -0.1729282 -0.2138153 -0.16814327 0.07543130 -0.011808001 0.4595829
[2,] -0.1445387 -0.1752286 -0.2237153 -0.18713184 0.06025909 -0.013151339 0.4563265
[,198] [,199] [,200] [,201] [,202] [,203] [,204]
[1,] 0.2993841 0.07463073 -0.0261662901 0.5843772 -0.4553964 -0.067626437 0.13214224
[2,] 0.3156416 0.09458995 -0.0115685561 0.5886590 -0.4318509 -0.083295305 0.11780666
[,205] [,206] [,207] [,208] [,209] [,210] [,211]
[1,] 0.3062579 0.11520431 -0.04206855 -0.3081470 -0.2203372 0.2763856 0.3996568
[2,] 0.3032126 0.12645490 -0.02831976 -0.2979416 -0.2204825 0.2547500 0.3941306
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[,419] [,420] [,421] [,422] [,423] [,424] [,425]
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[,446] [,447] [,448] [,449] [,450] [,451] [,452]
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[ reached getOption("max.print") -- omitted 70 rows ]
attr(,"timestamps")
[1] "2004-01-04" "2004-01-11" "2004-01-18" "2004-01-25" "2004-02-01" "2004-02-08"
[7] "2004-02-15" "2004-02-22" "2004-02-29" "2004-03-07" "2004-03-14" "2004-03-21"
[13] "2004-03-28" "2004-04-04" "2004-04-11" "2004-04-18" "2004-04-25" "2004-05-02"
[19] "2004-05-09" "2004-05-16" "2004-05-23" "2004-05-30" "2004-06-06" "2004-06-13"
[25] "2004-06-20" "2004-06-27" "2004-07-04" "2004-07-11" "2004-07-18" "2004-07-25"
[31] "2004-08-01" "2004-08-08" "2004-08-15" "2004-08-22" "2004-08-29" "2004-09-05"
[37] "2004-09-12" "2004-09-19" "2004-09-26" "2004-10-03" "2004-10-10" "2004-10-17"
[43] "2004-10-24" "2004-10-31" "2004-11-07" "2004-11-14" "2004-11-21" "2004-11-28"
[49] "2004-12-05" "2004-12-12" "2004-12-19" "2004-12-26" "2005-01-02" "2005-01-09"
[55] "2005-01-16" "2005-01-23" "2005-01-30" "2005-02-06" "2005-02-13" "2005-02-20"
[61] "2005-02-27" "2005-03-06" "2005-03-13" "2005-03-20" "2005-03-27" "2005-04-03"
[67] "2005-04-10" "2005-04-17" "2005-04-24" "2005-05-01" "2005-05-08" "2005-05-15"
[73] "2005-05-22" "2005-05-29" "2005-06-05" "2005-06-12" "2005-06-19" "2005-06-26"
[79] "2005-07-03" "2005-07-10" "2005-07-17" "2005-07-24" "2005-07-31" "2005-08-07"
[85] "2005-08-14" "2005-08-21" "2005-08-28" "2005-09-04" "2005-09-11" "2005-09-18"
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[157] "2006-12-31" "2007-01-07" "2007-01-14" "2007-01-21" "2007-01-28" "2007-02-04"
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[169] "2007-03-25" "2007-04-01" "2007-04-08" "2007-04-15" "2007-04-22" "2007-04-29"
[175] "2007-05-06" "2007-05-13" "2007-05-20" "2007-05-27" "2007-06-03" "2007-06-10"
[181] "2007-06-17" "2007-06-24" "2007-07-01" "2007-07-08" "2007-07-15" "2007-07-22"
[187] "2007-07-29" "2007-08-05" "2007-08-12" "2007-08-19" "2007-08-26" "2007-09-02"
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[205] "2007-12-02" "2007-12-09" "2007-12-16" "2007-12-23" "2007-12-30" "2008-01-06"
[211] "2008-01-13" "2008-01-20" "2008-01-27" "2008-02-03" "2008-02-10" "2008-02-17"
[217] "2008-02-24" "2008-03-02" "2008-03-09" "2008-03-16" "2008-03-23" "2008-03-30"
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[235] "2008-06-29" "2008-07-06" "2008-07-13" "2008-07-20" "2008-07-27" "2008-08-03"
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[247] "2008-09-21" "2008-09-28" "2008-10-05" "2008-10-12" "2008-10-19" "2008-10-26"
[253] "2008-11-02" "2008-11-09" "2008-11-16" "2008-11-23" "2008-11-30" "2008-12-07"
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[271] "2009-03-08" "2009-03-15" "2009-03-22" "2009-03-29" "2009-04-05" "2009-04-12"
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[343] "2010-07-25" "2010-08-01" "2010-08-08" "2010-08-15" "2010-08-22" "2010-08-29"
[349] "2010-09-05" "2010-09-12" "2010-09-19" "2010-09-26" "2010-10-03" "2010-10-10"
[355] "2010-10-17" "2010-10-24" "2010-10-31" "2010-11-07" "2010-11-14" "2010-11-21"
[361] "2010-11-28" "2010-12-05" "2010-12-12" "2010-12-19" "2010-12-26" "2011-01-02"
[367] "2011-01-09" "2011-01-16" "2011-01-23" "2011-01-30" "2011-02-06" "2011-02-13"
[373] "2011-02-20" "2011-02-27" "2011-03-06" "2011-03-13" "2011-03-20" "2011-03-27"
[379] "2011-04-03" "2011-04-10" "2011-04-17" "2011-04-24" "2011-05-01" "2011-05-08"
[385] "2011-05-15" "2011-05-22" "2011-05-29" "2011-06-05" "2011-06-12" "2011-06-19"
[391] "2011-06-26" "2011-07-03" "2011-07-10" "2011-07-17" "2011-07-24" "2011-07-31"
[397] "2011-08-07" "2011-08-14" "2011-08-21" "2011-08-28" "2011-09-04" "2011-09-11"
[403] "2011-09-18" "2011-09-25" "2011-10-02" "2011-10-09" "2011-10-16" "2011-10-23"
[409] "2011-10-30" "2011-11-06" "2011-11-13" "2011-11-20" "2011-11-27" "2011-12-04"
[415] "2011-12-11" "2011-12-18" "2011-12-25" "2012-01-01" "2012-01-08" "2012-01-15"
[421] "2012-01-22" "2012-01-29" "2012-02-05" "2012-02-12" "2012-02-19" "2012-02-26"
[427] "2012-03-04" "2012-03-11" "2012-03-18" "2012-03-25" "2012-04-01" "2012-04-08"
[433] "2012-04-15" "2012-04-22" "2012-04-29" "2012-05-06" "2012-05-13" "2012-05-20"
[439] "2012-05-27" "2012-06-03" "2012-06-10" "2012-06-17" "2012-06-24" "2012-07-01"
[445] "2012-07-08" "2012-07-15" "2012-07-22" "2012-07-29" "2012-08-05" "2012-08-12"
[451] "2012-08-19" "2012-08-26" "2012-09-02" "2012-09-09" "2012-09-16" "2012-09-23"
attr(,"class")
[1] "bsts.prediction.errors"CompareBstsModels(list("Model 1" = model1,
"Model 2" = model2,
"Model 3" = model3),
colors = c("black", "red", "blue"))
Long term forecasting
ss1 <- AddLocalLinearTrend(list(), SP500)
model1 <- bsts(SP500, state.specification = ss1, niter = 1000)=-=-=-=-= Iteration 0 Sun Sep 10 22:52:02 2017 =-=-=-=-=
=-=-=-=-= Iteration 100 Sun Sep 10 22:52:06 2017 =-=-=-=-=
=-=-=-=-= Iteration 200 Sun Sep 10 22:52:09 2017 =-=-=-=-=
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=-=-=-=-= Iteration 800 Sun Sep 10 22:52:26 2017 =-=-=-=-=
=-=-=-=-= Iteration 900 Sun Sep 10 22:52:29 2017 =-=-=-=-=pred1 <- predict(model1, horizon = 360)
ss2 <- AddSemilocalLinearTrend(list(), SP500)
model2 <- bsts(SP500, state.specification = ss2, niter = 1000)=-=-=-=-= Iteration 0 Sun Sep 10 22:52:32 2017 =-=-=-=-=
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=-=-=-=-= Iteration 800 Sun Sep 10 22:52:52 2017 =-=-=-=-=
=-=-=-=-= Iteration 900 Sun Sep 10 22:52:55 2017 =-=-=-=-=pred2 <- predict(model2, horizon = 360)plot(pred2, plot.original = 360, ylim = range(pred1))
plot(pred1, plot.original = 360, ylim = range(pred1))
The transition equation describes how the latent state evolves through time.
\[\begin{equation} \alpha_{t+1}=T_{t}\alpha_{t}+R_{t}\eta_{t} \end{equation}\]## Because 'y' is 0/1 and the state is on the logit scale the default prior
## assumed by AddLocalLevel won't work here, so we need to explicitly set the
## priors for the variance of the state innovation errors and the initial value
## of the state at time 0. The 'SdPrior' and 'NormalPrior' functions used to
## define these priors are part of the Boom package. See R help for
## documentation. Note the truncated support for the standard deviation of the
## random walk increments in the local level model.
ss <- AddLocalLevel(list(),
sigma.prior = SdPrior(sigma.guess = .1,
sample.size = 1,
upper.limit = 1),
initial.state.prior = NormalPrior(0, 5))
## Tell bsts that the observation equation should be a logistic regression by
## passing the 'family = "logit"' argument.
#ts.model <- bsts(nber ~ ., ss, data = gdp, niter = 20000,
# family = "logit", expected.model.size = 10)ss1 <- AddLocalLinearTrend(list(), SP500)
model1 <- bsts(SP500, state.specification = ss1, niter = 1000)=-=-=-=-= Iteration 0 Sun Sep 10 22:53:01 2017 =-=-=-=-=
=-=-=-=-= Iteration 100 Sun Sep 10 22:53:04 2017 =-=-=-=-=
=-=-=-=-= Iteration 200 Sun Sep 10 22:53:07 2017 =-=-=-=-=
=-=-=-=-= Iteration 300 Sun Sep 10 22:53:10 2017 =-=-=-=-=
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=-=-=-=-= Iteration 700 Sun Sep 10 22:53:22 2017 =-=-=-=-=
=-=-=-=-= Iteration 800 Sun Sep 10 22:53:25 2017 =-=-=-=-=
=-=-=-=-= Iteration 900 Sun Sep 10 22:53:28 2017 =-=-=-=-=pred1 <- predict(model1, horizon = 360)
ss2 <- AddSemilocalLinearTrend(list(), SP500)
model2 <- bsts(SP500, state.specification = ss2, niter = 1000)=-=-=-=-= Iteration 0 Sun Sep 10 22:53:31 2017 =-=-=-=-=
=-=-=-=-= Iteration 100 Sun Sep 10 22:53:34 2017 =-=-=-=-=
=-=-=-=-= Iteration 200 Sun Sep 10 22:53:36 2017 =-=-=-=-=
=-=-=-=-= Iteration 300 Sun Sep 10 22:53:38 2017 =-=-=-=-=
=-=-=-=-= Iteration 400 Sun Sep 10 22:53:41 2017 =-=-=-=-=
=-=-=-=-= Iteration 500 Sun Sep 10 22:53:43 2017 =-=-=-=-=
=-=-=-=-= Iteration 600 Sun Sep 10 22:53:45 2017 =-=-=-=-=
=-=-=-=-= Iteration 700 Sun Sep 10 22:53:48 2017 =-=-=-=-=
=-=-=-=-= Iteration 800 Sun Sep 10 22:53:50 2017 =-=-=-=-=
=-=-=-=-= Iteration 900 Sun Sep 10 22:53:53 2017 =-=-=-=-=pred2 <- predict(model2, horizon = 360)
plot(pred2, plot.original = 360, ylim = range(pred1))
plot(pred1, plot.original = 360, ylim = range(pred1))
pred1$mean
[1] -0.2556009 -0.4726404 -0.4220162 -0.2667642 -0.3170974 -0.3490364 -0.3101443
[8] -0.3557894 -0.1911030 -0.2883212 -0.3162483 -0.1676476 -0.3300988 -0.2586663
[15] -0.3001938 -0.1993543 -0.3179756 -0.2633925 -0.3279233 -0.3419511 -0.3706452
[22] -0.2813006 -0.4025930 -0.2582434 -0.2792441 -0.4354980 -0.3009352 -0.3244870
[29] -0.2677987 -0.3075440 -0.3044531 -0.2828499 -0.2765944 -0.3859214 -0.1237576
[36] -0.3804061 -0.3212224 -0.2952767 -0.3703786 -0.4031394 -0.2763961 -0.3924040
[43] -0.3178814 -0.4984948 -0.3543398 -0.2534717 -0.3161258 -0.3874293 -0.3038593
[50] -0.3403985 -0.4150325 -0.4336507 -0.4604465 -0.3856239 -0.3994486 -0.3868377
[57] -0.3612119 -0.3229177 -0.3440535 -0.4754957 -0.4180669 -0.3547644 -0.3291909
[64] -0.3128324 -0.4094765 -0.3166299 -0.4551720 -0.5102497 -0.3624292 -0.3611715
[71] -0.4612557 -0.4735380 -0.4916396 -0.3298461 -0.4160329 -0.3625403 -0.4174104
[78] -0.4857687 -0.4082052 -0.4903580 -0.3501372 -0.4528861 -0.5388649 -0.4605049
[85] -0.3842998 -0.4954819 -0.4015738 -0.3658972 -0.4530255 -0.2653239 -0.4563370
[92] -0.5626446 -0.4808652 -0.5052921 -0.4841122 -0.3953131 -0.4846965 -0.6065856
[99] -0.5560801 -0.4605891 -0.4678011 -0.5299575 -0.4956101 -0.4801398 -0.3815464
[106] -0.5155676 -0.4850492 -0.4988917 -0.4061374 -0.5052800 -0.5706830 -0.5878646
[113] -0.4200603 -0.5274138 -0.4620515 -0.7161311 -0.4654469 -0.5039324 -0.6187335
[120] -0.3389090 -0.6245469 -0.5801583 -0.5979357 -0.5749346 -0.5366157 -0.5817814
[127] -0.5157104 -0.4636360 -0.5770354 -0.5013576 -0.5750968 -0.5953769 -0.5852532
[134] -0.6021819 -0.5818276 -0.6207158 -0.5331681 -0.5705323 -0.6550389 -0.6525603
[141] -0.5441004 -0.4599245 -0.5289482 -0.6039136 -0.6016465 -0.5585580 -0.6812747
[148] -0.6167081 -0.5719553 -0.7322045 -0.6724430 -0.6384879 -0.6796741 -0.4495905
[155] -0.5455928 -0.5881620 -0.5635258 -0.6705189 -0.6340901 -0.5653046 -0.6145699
[162] -0.6973486 -0.5115888 -0.6770606 -0.5075022 -0.5894059 -0.5415249 -0.5279687
[169] -0.7092912 -0.6150194 -0.6338710 -0.6650905 -0.6093649 -0.6555860 -0.6521170
[176] -0.5897003 -0.6533317 -0.6039825 -0.6163208 -0.7200637 -0.6365002 -0.5913729
[183] -0.6463443 -0.5701451 -0.6491103 -0.6968255 -0.6395475 -0.8452485 -0.7408438
[190] -0.6621099 -0.8230722 -0.6915372 -0.7615274 -0.6655736 -0.6817409 -0.7092777
[197] -0.6353640 -0.7060133 -0.6024572 -0.7442034 -0.6620646 -0.6625303 -0.7180327
[204] -0.7367701 -0.6007341 -0.6476082 -0.6760256 -0.6557047 -0.6819957 -0.7282261
[211] -0.7980517 -0.7468444 -0.6588282 -0.6447834 -0.8165906 -0.7218301 -0.7498632
[218] -0.6289664 -0.6863808 -0.7625332 -0.8292482 -0.6968008 -0.8312630 -0.7483390
[225] -0.6829158 -0.7047419 -0.5945358 -0.8091790 -0.7608705 -0.7652975 -0.6791166
[232] -0.7992239 -0.7577969 -0.7878617 -0.6281808 -0.7297019 -0.7029118 -0.8070794
[239] -0.8896281 -0.8125424 -0.7415465 -0.7642130 -0.7303935 -0.8116643 -0.7386301
[246] -0.7602639 -0.7491017 -0.9429016 -0.8287298 -0.6120931 -0.7327052 -0.7593376
[253] -0.7054848 -0.7900960 -0.7139471 -0.7843289 -0.6980048 -0.7450880 -0.8612812
[260] -0.8816319 -0.7659895 -0.7969693 -0.7849245 -0.7193579 -0.7044749 -0.6562274
[267] -0.6736682 -0.8133440 -0.7197511 -0.6982166 -0.9229007 -0.8682694 -0.8835797
[274] -0.8628936 -0.9676803 -0.7764700 -0.7360799 -0.8289942 -0.8017648 -0.7972019
[281] -0.7767146 -0.8907595 -0.6231965 -0.8351409 -0.6952783 -0.8156239 -0.8662993
[288] -0.8325483 -0.8602648 -0.8073660 -0.9285394 -0.8324073 -0.8650151 -0.9186207
[295] -0.9248221 -0.8576073 -0.9730934 -0.9450149 -0.9149602 -0.8711743 -0.7854097
[302] -0.8384294 -0.9312366 -0.9494300 -0.8643038 -0.9980728 -0.9618233 -0.9380160
[309] -0.8316519 -0.7857023 -0.8304793 -0.9366832 -0.8568328 -0.7765663 -0.9232402
[316] -0.7983307 -0.9700090 -0.8613284 -0.9652995 -0.9163731 -0.9448452 -1.0278236
[323] -1.0056309 -0.9401244 -1.0283881 -0.9680507 -0.9348866 -0.8481019 -0.9031337
[330] -0.9010795 -0.9417035 -0.9416667 -0.8884557 -0.9580136 -0.9204205 -1.0576565
[337] -1.0690777 -0.8917485 -0.9936933 -0.7891019 -0.8931624 -0.9584151 -0.9995880
[344] -0.9774891 -0.9764564 -0.9411043 -0.8893942 -0.9619646 -1.0027302 -1.0270260
[351] -1.1194484 -1.1214368 -0.9260465 -0.8939223 -0.8968996 -1.0440923 -1.0381086
[358] -1.0880756 -1.0063487 -0.8040963
$median
[1] -0.13613913 -0.46670127 -0.39748304 -0.31221858 -0.32966688 -0.40009060 -0.45070077
[8] -0.36191848 -0.18004356 -0.25829110 -0.30490435 -0.08457108 -0.35526793 -0.22887870
[15] -0.39187435 -0.17809446 -0.36090816 -0.24283534 -0.28095077 -0.38842354 -0.37763680
[22] -0.26636046 -0.43339064 -0.21074413 -0.31577100 -0.46964649 -0.35183501 -0.35926300
[29] -0.29530517 -0.27304769 -0.35177298 -0.28253957 -0.28084682 -0.43182180 -0.09416300
[36] -0.32098830 -0.21762959 -0.29604169 -0.47246889 -0.52764525 -0.28431330 -0.45582373
[43] -0.27456851 -0.41614007 -0.37897429 -0.39715828 -0.21128934 -0.34814226 -0.27509590
[50] -0.43537445 -0.29799901 -0.37003279 -0.45991108 -0.37582636 -0.42713775 -0.35228534
[57] -0.38810593 -0.23445033 -0.20649849 -0.50176207 -0.45398961 -0.24655672 -0.29323973
[64] -0.30448841 -0.36859700 -0.37649753 -0.45345355 -0.44437216 -0.34185815 -0.27920561
[71] -0.51308673 -0.53888335 -0.58685081 -0.31363510 -0.42322164 -0.28112166 -0.36926580
[78] -0.42823998 -0.33369315 -0.41458843 -0.32227674 -0.52311836 -0.62400227 -0.42073082
[85] -0.33728216 -0.49686210 -0.43189662 -0.40116252 -0.38711254 -0.28633637 -0.51379826
[92] -0.55002679 -0.49972411 -0.49375940 -0.44725948 -0.31032734 -0.49552040 -0.61926581
[99] -0.48664890 -0.48470088 -0.41546363 -0.53098100 -0.47882942 -0.42683830 -0.39522878
[106] -0.43630748 -0.43058708 -0.45251526 -0.48220596 -0.56228397 -0.63119773 -0.51522429
[113] -0.40454907 -0.48863674 -0.33635484 -0.61296377 -0.38324007 -0.67024175 -0.70944339
[120] -0.34762481 -0.66710907 -0.47352646 -0.63295779 -0.62860651 -0.41165075 -0.45078593
[127] -0.44357336 -0.52685274 -0.54842084 -0.50240475 -0.49273033 -0.58801958 -0.53744758
[134] -0.69492552 -0.56055255 -0.57884086 -0.56288159 -0.49720727 -0.56944407 -0.52778927
[141] -0.57181145 -0.42757630 -0.46422413 -0.49363057 -0.60418319 -0.62746557 -0.62412011
[148] -0.56298747 -0.61451120 -0.61188043 -0.71337381 -0.63371481 -0.66454617 -0.45855506
[155] -0.54114415 -0.62184934 -0.59451071 -0.64520991 -0.58998395 -0.53951140 -0.62324043
[162] -0.52527165 -0.62992755 -0.56259823 -0.41896046 -0.72746600 -0.49722506 -0.48077062
[169] -0.72679690 -0.68793717 -0.51476014 -0.59148801 -0.47797170 -0.71832599 -0.69005831
[176] -0.64303051 -0.58056992 -0.61926371 -0.70417216 -0.72592096 -0.63148695 -0.63825761
[183] -0.58956868 -0.48050666 -0.72639935 -0.70557373 -0.72655378 -0.85835366 -0.77679484
[190] -0.67942842 -0.79335500 -0.73174925 -0.73541887 -0.83170117 -0.63390826 -0.68591707
[197] -0.64955360 -0.75061410 -0.62480499 -0.79929158 -0.52779287 -0.61840518 -0.74919104
[204] -0.87045797 -0.60089909 -0.60960551 -0.74982209 -0.67150422 -0.73861300 -0.61994737
[211] -0.81410733 -0.76474388 -0.62269674 -0.65053284 -0.86974468 -0.72425679 -0.79350469
[218] -0.75864823 -0.78518705 -0.87030807 -0.87460635 -0.75942279 -0.93197422 -0.79444977
[225] -0.70465749 -0.65570926 -0.66495322 -0.80377098 -0.87927930 -0.87286732 -0.67582876
[232] -0.80585122 -0.73363844 -0.75487789 -0.49175890 -0.87649686 -0.78925334 -0.81515930
[239] -0.77475946 -0.70006830 -0.68273746 -0.75336162 -0.78693060 -0.86131107 -0.75269246
[246] -0.64459031 -0.72015507 -0.96359112 -1.03455741 -0.69973009 -0.75959014 -0.78368832
[253] -0.82868409 -0.70910494 -0.54210639 -0.66531752 -0.73080364 -0.69344105 -0.85799649
[260] -0.73821469 -0.80779598 -0.99722209 -0.82225565 -0.73313111 -0.73786525 -0.75637191
[267] -0.68380657 -0.68434993 -0.79090941 -0.79835647 -0.82646948 -1.01308728 -0.83660793
[274] -0.76255010 -1.12794785 -0.78190350 -0.75553832 -0.80677899 -0.75114472 -0.79600819
[281] -0.59729231 -0.87971065 -0.55641684 -0.87905213 -0.66718899 -0.84384627 -0.79250602
[288] -0.77273492 -0.94155124 -0.75934896 -1.00673499 -0.92838930 -0.77121909 -0.90680277
[295] -0.83382369 -0.69406430 -0.98954902 -0.97809554 -0.88633879 -0.87107138 -0.78223429
[302] -0.78752227 -0.98274996 -1.06807208 -0.81631912 -0.84506344 -1.02565036 -1.05100293
[309] -0.80572258 -0.73604437 -0.82579980 -1.02569454 -0.92792469 -0.75165391 -0.80400373
[316] -0.71080081 -0.93965322 -0.84203501 -0.81933205 -0.82584868 -0.86723534 -0.97966249
[323] -1.12172705 -0.86886855 -1.08536689 -0.94277375 -0.86733958 -0.79449954 -1.02081505
[330] -1.06903388 -0.88917939 -0.74113643 -0.85682981 -0.95174986 -1.07452566 -1.25660247
[337] -1.03048270 -0.78092041 -0.89819693 -0.66412322 -0.82238119 -0.79262726 -0.81529489
[344] -0.83234717 -0.84384005 -1.17009136 -0.85935122 -1.06596426 -0.86401053 -0.98414377
[351] -1.05535930 -1.17333616 -0.90181100 -0.97227720 -0.90019830 -1.19909654 -0.97263388
[358] -0.96741564 -1.15512376 -0.81284506
$interval
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
2.5% -2.296951 -2.389824 -2.276424 -2.147877 -2.296083 -2.158859 -1.889317 -1.997337
97.5% 1.617229 1.276556 1.673753 1.347496 1.544841 1.731172 1.611061 1.507940
[,9] [,10] [,11] [,12] [,13] [,14] [,15] [,16]
2.5% -1.891083 -2.241745 -1.983490 -2.405773 -2.133396 -2.171926 -2.114811 -2.163315
97.5% 1.749398 1.457142 1.230179 1.271715 1.693461 1.632616 1.646580 1.398248
[,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24]
2.5% -2.258543 -2.074081 -2.508123 -2.250706 -2.330650 -2.186014 -2.356070 -2.134848
97.5% 1.621861 1.731049 1.629452 1.747988 2.140009 1.433887 1.354582 1.456507
[,25] [,26] [,27] [,28] [,29] [,30] [,31] [,32]
2.5% -2.082222 -2.113234 -2.163000 -2.160765 -2.135313 -2.111398 -2.230438 -2.197137
97.5% 1.573403 1.099049 1.767446 1.388689 1.594405 1.445026 1.531852 1.488889
[,33] [,34] [,35] [,36] [,37] [,38] [,39] [,40]
2.5% -2.156731 -2.365606 -2.084005 -2.455525 -2.405600 -1.950488 -1.925448 -2.452086
97.5% 1.697076 1.612265 2.037735 1.351483 1.549003 1.425425 1.465536 1.670505
[,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48]
2.5% -2.029813 -2.112231 -2.428169 -2.515544 -2.068113 -1.790876 -2.282494 -2.574797
97.5% 1.682095 1.535200 1.738251 1.154481 1.730413 1.674185 1.690552 1.390854
[,49] [,50] [,51] [,52] [,53] [,54] [,55] [,56]
2.5% -2.113605 -2.400005 -2.412863 -2.279429 -2.193487 -2.372580 -2.165342 -2.45627
97.5% 1.683719 1.586800 1.114624 1.240925 1.449768 2.029911 1.189316 1.42167
[,57] [,58] [,59] [,60] [,61] [,62] [,63] [,64]
2.5% -2.231160 -2.091454 -2.104897 -2.389119 -2.337132 -2.184746 -2.06889 -2.019065
97.5% 1.844606 1.725189 1.523562 1.418575 1.676699 1.385388 1.37313 1.586610
[,65] [,66] [,67] [,68] [,69] [,70] [,71] [,72]
2.5% -2.322495 -1.997833 -2.292249 -2.291874 -2.166163 -2.365617 -2.245343 -2.563518
97.5% 1.611247 1.508589 1.369298 1.171714 1.388552 1.378061 1.415688 1.738732
[,73] [,74] [,75] [,76] [,77] [,78] [,79] [,80]
2.5% -2.739073 -2.159579 -2.499948 -2.337985 -2.384087 -2.335901 -2.491753 -2.067118
97.5% 1.436489 1.358039 1.562405 1.844036 1.652066 1.371177 1.323823 1.531371
[,81] [,82] [,83] [,84] [,85] [,86] [,87] [,88]
2.5% -1.954670 -2.199203 -2.327172 -2.750600 -2.362949 -2.153611 -2.793066 -2.172442
97.5% 1.368036 1.824185 1.505540 1.459971 1.534556 1.637784 1.784292 1.471027
[,89] [,90] [,91] [,92] [,93] [,94] [,95] [,96]
2.5% -2.895199 -2.068962 -2.288922 -2.672865 -2.757918 -2.800870 -2.600117 -2.602289
97.5% 1.798719 1.556562 1.346709 1.315654 1.790311 1.507658 1.544152 1.587147
[,97] [,98] [,99] [,100] [,101] [,102] [,103] [,104]
2.5% -2.376235 -2.668361 -2.657447 -2.466248 -2.643423 -2.935007 -2.559258 -2.884670
97.5% 1.571542 1.624708 1.067496 1.838912 1.624774 1.441910 1.913688 1.551159
[,105] [,106] [,107] [,108] [,109] [,110] [,111] [,112]
2.5% -2.370000 -2.839148 -2.462852 -2.641073 -2.409455 -2.649164 -2.692814 -2.836755
97.5% 1.587813 1.717776 1.500298 1.287955 1.442447 1.976324 1.662962 1.393353
[,113] [,114] [,115] [,116] [,117] [,118] [,119] [,120]
2.5% -2.427913 -2.404295 -2.673549 -3.165674 -2.681005 -2.213181 -2.251483 -2.443139
97.5% 1.448316 1.280073 1.819994 1.505037 1.645357 1.873906 1.206837 1.876705
[,121] [,122] [,123] [,124] [,125] [,126] [,127] [,128]
2.5% -3.135025 -2.805695 -2.581065 -2.714693 -2.761199 -3.255686 -3.004854 -2.735171
97.5% 1.777796 1.724203 1.384799 1.465981 1.711198 1.524769 1.717411 1.772813
[,129] [,130] [,131] [,132] [,133] [,134] [,135] [,136]
2.5% -2.529800 -2.894063 -2.948735 -2.885346 -2.900774 -2.613555 -2.673191 -2.752947
97.5% 1.290102 1.754992 1.416776 1.341574 1.442758 1.685155 1.316985 2.043565
[,137] [,138] [,139] [,140] [,141] [,142] [,143] [,144]
2.5% -2.724334 -2.714582 -2.886757 -2.864954 -2.843564 -2.652145 -3.543264 -2.731892
97.5% 1.756958 1.416015 1.351477 1.521849 1.530865 1.369434 1.751739 1.490892
[,145] [,146] [,147] [,148] [,149] [,150] [,151] [,152]
2.5% -2.448832 -2.546382 -3.028771 -2.659899 -2.858268 -2.858125 -2.475036 -2.991113
97.5% 1.392580 1.906035 1.359409 1.614105 2.097159 1.129410 1.509209 1.258372
[,153] [,154] [,155] [,156] [,157] [,158] [,159] [,160]
2.5% -2.921498 -2.970309 -2.733012 -3.005615 -2.656956 -2.761294 -2.778866 -2.734282
97.5% 1.569496 1.313792 2.148207 1.581654 1.305361 1.809140 1.358400 1.525729
[,161] [,162] [,163] [,164] [,165] [,166] [,167] [,168]
2.5% -2.659510 -3.304274 -2.429458 -2.924099 -2.702165 -2.453195 -3.206374 -2.663574
97.5% 1.637367 1.630418 1.651322 1.451014 1.835708 1.702356 1.625738 1.788416
[,169] [,170] [,171] [,172] [,173] [,174] [,175] [,176]
2.5% -2.532108 -3.313519 -3.314274 -3.014612 -2.689442 -2.865689 -3.323437 -2.785513
97.5% 1.076510 1.647136 1.264863 1.394602 1.358574 2.090942 1.487223 2.175416
[,177] [,178] [,179] [,180] [,181] [,182] [,183] [,184]
2.5% -2.814799 -2.866327 -3.319886 -3.059928 -3.006270 -2.929147 -3.013504 -3.112439
97.5% 1.660533 1.765979 1.771385 1.414369 1.972942 1.452555 1.607657 1.731719
[,185] [,186] [,187] [,188] [,189] [,190] [,191] [,192]
2.5% -2.887345 -2.879336 -3.221035 -3.506657 -2.933013 -2.839807 -3.185147 -2.834244
97.5% 1.776204 1.693521 1.879732 1.632466 1.725703 1.815350 1.559153 1.587425
[,193] [,194] [,195] [,196] [,197] [,198] [,199] [,200]
2.5% -3.017553 -3.244778 -2.742322 -3.068830 -2.534517 -3.501266 -3.320164 -3.286787
97.5% 1.675979 1.955954 1.642366 1.833618 1.621085 1.624687 1.926505 1.876817
[,201] [,202] [,203] [,204] [,205] [,206] [,207] [,208]
2.5% -3.907842 -3.268725 -3.026129 -3.095119 -3.482726 -3.189993 -2.899252 -3.509318
97.5% 1.484960 1.530659 1.623681 1.612969 2.255415 2.182104 2.154368 1.863541
[,209] [,210] [,211] [,212] [,213] [,214] [,215] [,216]
2.5% -2.887721 -3.180580 -3.231899 -3.188797 -3.180648 -2.982485 -3.33006 -3.652760
97.5% 2.115452 1.381619 1.978161 1.629497 2.002516 1.955399 2.15080 2.655921
[,217] [,218] [,219] [,220] [,221] [,222] [,223] [,224]
2.5% -3.032543 -3.657159 -3.260302 -3.458813 -3.229512 -3.322962 -3.060467 -3.508118
97.5% 1.559497 1.795930 1.667615 1.772595 2.086305 1.879308 1.572429 1.841295
[,225] [,226] [,227] [,228] [,229] [,230] [,231] [,232]
2.5% -3.162500 -3.184425 -3.615646 -3.599467 -3.083437 -3.345510 -3.560051 -3.781611
97.5% 1.791507 1.969410 2.319501 1.921610 1.751247 1.896532 1.768003 1.815702
[,233] [,234] [,235] [,236] [,237] [,238] [,239] [,240]
2.5% -3.438238 -3.669535 -3.462490 -3.246156 -4.251212 -3.678972 -3.862583 -3.796666
97.5% 2.150124 1.998161 2.096892 2.076351 1.957466 1.612158 1.711108 1.923024
[,241] [,242] [,243] [,244] [,245] [,246] [,247] [,248]
2.5% -3.494328 -3.862573 -3.076948 -3.473137 -2.943298 -3.313728 -3.802964 -3.508611
97.5% 1.699478 1.874946 1.814750 1.774173 1.680087 1.641147 2.012426 1.621804
[,249] [,250] [,251] [,252] [,253] [,254] [,255] [,256]
2.5% -3.375955 -3.750931 -3.681225 -3.353925 -3.195415 -3.522450 -3.505272 -4.021303
97.5% 1.801811 2.676365 2.479328 2.436595 1.928659 2.542335 2.318070 1.914515
[,257] [,258] [,259] [,260] [,261] [,262] [,263] [,264]
2.5% -3.519930 -3.668353 -3.385777 -3.791574 -3.671649 -4.213870 -3.601885 -3.468083
97.5% 2.255953 2.050753 1.976156 1.680456 2.252309 2.825486 2.135374 2.476502
[,265] [,266] [,267] [,268] [,269] [,270] [,271] [,272]
2.5% -3.874951 -2.961018 -3.346300 -3.951392 -3.988981 -3.591272 -3.949779 -3.723123
97.5% 1.964642 1.848354 2.334566 1.583670 1.998186 2.071923 1.747085 1.955762
[,273] [,274] [,275] [,276] [,277] [,278] [,279] [,280]
2.5% -4.180170 -3.619384 -3.691367 -3.985762 -3.684176 -3.629065 -3.494013 -3.585227
97.5% 1.855781 1.810734 2.208251 2.108941 2.043633 2.565183 2.234597 2.163529
[,281] [,282] [,283] [,284] [,285] [,286] [,287] [,288]
2.5% -3.789452 -3.730953 -3.732081 -3.496243 -3.780309 -3.933094 -3.801632 -3.441345
97.5% 2.423708 1.925610 2.397261 2.001117 2.149633 1.868538 2.017391 1.801492
[,289] [,290] [,291] [,292] [,293] [,294] [,295] [,296]
2.5% -3.990535 -3.650583 -3.770018 -3.678828 -4.437306 -4.208418 -4.356425 -4.263668
97.5% 2.786984 1.998450 1.864947 2.021216 2.071416 1.991834 2.070458 1.849363
[,297] [,298] [,299] [,300] [,301] [,302] [,303] [,304]
2.5% -4.024866 -4.074366 -4.338294 -3.919870 -3.552438 -3.581350 -3.944213 -3.650550
97.5% 2.672169 2.219712 1.906414 1.567789 2.176319 2.245437 1.812348 2.331044
[,305] [,306] [,307] [,308] [,309] [,310] [,311] [,312]
2.5% -4.256078 -4.213734 -3.950571 -3.773938 -3.961477 -3.926258 -3.649230 -4.393026
97.5% 2.027528 2.137299 2.207712 2.480498 2.124059 2.663249 2.335236 2.467380
[,313] [,314] [,315] [,316] [,317] [,318] [,319] [,320]
2.5% -3.947211 -4.17358 -4.092421 -4.064271 -3.931251 -3.755325 -3.616183 -4.049439
97.5% 2.135742 2.12247 2.015311 2.533530 2.637624 2.073959 1.856677 1.987410
[,321] [,322] [,323] [,324] [,325] [,326] [,327] [,328]
2.5% -4.213979 -4.109590 -4.115586 -4.432381 -4.347394 -4.269765 -4.656187 -3.952398
97.5% 2.268832 2.029648 2.635153 2.302489 1.930560 2.378648 2.743621 2.526608
[,329] [,330] [,331] [,332] [,333] [,334] [,335] [,336]
2.5% -4.528725 -4.007317 -4.082333 -4.428298 -4.782589 -4.594700 -4.452741 -4.207426
97.5% 2.933993 2.843687 2.379352 1.855113 2.826966 2.418617 2.502431 2.649126
[,337] [,338] [,339] [,340] [,341] [,342] [,343] [,344]
2.5% -4.072772 -4.865106 -4.415988 -4.250951 -4.420700 -4.117469 -5.145838 -4.478738
97.5% 2.267495 2.447909 2.998246 2.752907 2.418887 1.877519 2.221191 2.275204
[,345] [,346] [,347] [,348] [,349] [,350] [,351] [,352]
2.5% -4.36025 -3.847235 -4.190469 -4.20591 -4.681949 -4.664822 -4.718159 -4.396947
97.5% 2.29330 2.338517 2.449445 2.56868 2.437157 2.101319 2.375465 2.342544
[,353] [,354] [,355] [,356] [,357] [,358] [,359] [,360]
2.5% -3.972375 -4.209797 -4.874081 -4.475544 -4.806167 -4.724548 -4.175987 -4.168702
97.5% 2.349024 2.372534 2.594417 2.505296 2.092920 2.217930 2.361955 2.830815
$distribution
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] -1.274777535 0.1448014708 0.899638015 1.084115868 -1.550341925 -1.402549990
[2,] 0.040413040 0.7735025773 -0.890108438 -0.338679635 -0.742220217 -0.106817853
[,7] [,8] [,9] [,10] [,11] [,12]
[1,] 0.088588656 0.0241500944 -0.481145553 0.11548802 0.13271210 0.87102812
[2,] 0.304112901 0.4108516620 -0.135489979 -0.24578108 -0.27272015 -0.47932250
[,13] [,14] [,15] [,16] [,17] [,18]
[1,] 1.95241715 -1.598551392 1.187393671 -0.065961897 -0.441672024 0.27154554
[2,] -0.69624930 0.071308717 0.753949181 -0.715197941 0.376472553 -0.61500789
[,19] [,20] [,21] [,22] [,23] [,24]
[1,] -0.681097344 0.652802822 -1.0121553763 -0.568616838 -0.7821196305 0.555275108
[2,] 0.154972548 -0.681712213 -0.0309327293 0.944127768 0.0420675252 0.689099743
[,25] [,26] [,27] [,28] [,29] [,30]
[1,] 0.016774094 0.70243164 -0.857124446 -0.39330043 1.571418912 0.799975689
[2,] -1.036423768 -0.17149638 0.314562909 -0.03260589 1.752628589 -2.002758767
[,31] [,32] [,33] [,34] [,35] [,36]
[1,] 0.24193758 -0.127852831 -0.46246151 0.1738329029 -0.508696296 0.226676459
[2,] 0.31515335 1.115351558 0.63751567 -0.1652517018 0.685829955 -0.677459888
[,37] [,38] [,39] [,40] [,41] [,42]
[1,] -0.129231749 -0.423293722 -0.605737421 -0.80258788 0.715612617 0.59044399
[2,] -1.912038873 1.143621121 0.393869501 -1.67321656 0.725415234 1.19141410
[,43] [,44] [,45] [,46] [,47] [,48]
[1,] 0.768247392 -1.07149814 1.768488898 -1.297842015 0.7057883066 -0.5394106171
[2,] 0.117029290 0.25823297 0.005413978 0.419070058 0.1973391068 -0.4941725659
[,49] [,50] [,51] [,52] [,53] [,54]
[1,] 0.47087564 -0.316346855 1.44964719 -1.12688927 -0.87886900 1.051706308
[2,] -1.29869626 -2.215202853 0.67987409 -0.82280177 1.11329677 0.551503055
[,55] [,56] [,57] [,58] [,59] [,60]
[1,] -1.05584629 0.533515461 -1.276489399 -0.6630160071 2.898081736 0.7960366280
[2,] 0.62115459 1.553394765 0.968755676 0.5962559691 0.141247006 0.4624896495
[,61] [,62] [,63] [,64] [,65] [,66]
[1,] 2.507549519 -0.4151506185 1.6706447063 -0.930294295 -0.68602904 -1.489619410
[2,] 0.088757505 -0.3293908542 1.0345185911 0.783875829 0.55335727 -0.684745779
[,67] [,68] [,69] [,70] [,71] [,72]
[1,] -3.865763e-01 0.8390863780 -0.64006866 -2.13974762 -0.003252848 -0.213596437
[2,] -5.603157e-01 -0.6475667699 0.18214205 -0.85162820 -0.278458856 0.914861138
[,73] [,74] [,75] [,76] [,77] [,78]
[1,] -1.176120679 0.301623994 -1.9083341507 -0.37887932 -1.14360322 0.678156861
[2,] -0.440229670 0.422768935 2.6077067947 2.24671269 2.12652228 -0.431692847
[,79] [,80] [,81] [,82] [,83] [,84]
[1,] -0.619490446 0.27698830 0.79044766 -1.48404349 1.027914931 0.0982040990
[2,] 0.583907139 1.44157413 2.32755136 2.09815759 -0.960032129 1.2450289326
[,85] [,86] [,87] [,88] [,89] [,90]
[1,] 0.106475925 -1.04634948 -2.029117398 0.596293077 0.109814363 -0.258278577
[2,] -1.000520931 -0.50026979 -0.461954150 -0.187970850 3.305249708 -0.156119585
[,91] [,92] [,93] [,94] [,95] [,96]
[1,] 0.81534739 0.256709634 -0.1010623091 -0.705590355 -0.36503446 -0.54742962
[2,] -2.16148823 -0.007724878 -1.4202637914 0.284852754 -0.05102175 1.09041237
[,97] [,98] [,99] [,100] [,101] [,102]
[1,] -0.43941892 -1.94266073 0.19620510 1.005355397 1.470204e+00 1.14815140
[2,] 0.67105019 0.57589809 -0.78851003 0.149351691 -9.674765e-01 0.48418349
[,103] [,104] [,105] [,106] [,107] [,108]
[1,] -0.235251123 1.563524307 -0.20191568 0.064687925 -0.899210092 -0.26797995
[2,] 0.597156162 0.810202255 1.83489886 -0.784985432 1.762315608 0.99055826
[,109] [,110] [,111] [,112] [,113] [,114]
[1,] 1.07303896 1.976480574 -1.372386048 0.017112263 -0.18672957 0.60765197
[2,] 0.33825834 -0.041772372 0.236214355 0.370266204 -0.26167335 0.33258788
[,115] [,116] [,117] [,118] [,119] [,120]
[1,] -0.33073813 0.290972245 0.930391366 -0.036056692 1.19140965 1.064974901
[2,] 1.83999699 0.374339888 0.486297067 1.531842145 1.15983969 0.635261180
[,121] [,122] [,123] [,124] [,125] [,126]
[1,] 0.21221583 -0.5338932419 0.89444189 -1.109268182 0.916907665 -0.99403721
[2,] 0.96290728 -0.3372554546 0.51910792 0.314808556 0.047638125 0.56101769
[,127] [,128] [,129] [,130] [,131] [,132]
[1,] -0.798637706 0.40715255 1.3976694514 0.365046469 -2.119399224 -0.300260764
[2,] 0.303846633 0.52173079 -0.0498695523 1.772640054 0.007131281 1.001026624
[,133] [,134] [,135] [,136] [,137] [,138]
[1,] 0.08764607 -1.141971987 -1.330736650 0.30185895 0.916574483 -0.026373912
[2,] -0.20180128 0.749481305 1.306629070 1.16470850 1.074610413 2.896022326
[,139] [,140] [,141] [,142] [,143] [,144]
[1,] -3.941791e-01 0.856722093 0.23757010 -0.64460781 -0.02592222 0.87634575
[2,] 1.416307e+00 0.437894254 0.67791045 0.33045901 1.17412693 1.22591867
[,145] [,146] [,147] [,148] [,149] [,150]
[1,] -0.96628962 -0.99286083 0.672578393 -0.65962099 0.870254887 0.19491608
[2,] -0.54629367 0.25953109 -0.308008796 -0.33078616 0.703021322 2.41601789
[,151] [,152] [,153] [,154] [,155] [,156]
[1,] 0.1107029593 -1.14243823 0.90961608 0.87282216 -0.439723921 -0.024140499
[2,] 1.1600794434 0.22829392 0.29990057 1.78195742 -0.058890789 1.217923355
[,157] [,158] [,159] [,160] [,161] [,162]
[1,] 0.60216681 -0.236214471 -0.54050448 0.85526915 -0.562003632 0.34012547
[2,] 1.02213064 -0.310092662 1.34388714 1.05209099 1.727572771 -0.43217988
[,163] [,164] [,165] [,166] [,167] [,168]
[1,] -0.7914511506 -1.34169307 0.49976158 1.59261250 0.25940471 1.38760664
[2,] 1.3146388144 1.44078035 1.83818752 1.65186223 -0.53529555 1.07886500
[,169] [,170] [,171] [,172] [,173] [,174]
[1,] -1.195665597 0.143515467 0.864858699 -0.003646311 0.581928630 -0.30684383
[2,] 1.053792581 1.193291879 -0.995376259 1.362334474 0.611875323 0.59999763
[,175] [,176] [,177] [,178] [,179] [,180]
[1,] -1.66803656 0.76338419 0.220827142 -0.53290874 0.004981508 -0.125938278
[2,] 0.22066999 0.09822331 2.160508249 1.40353782 0.359396583 0.749721129
[,181] [,182] [,183] [,184] [,185] [,186]
[1,] -0.63746321 0.191857782 -1.71359274 1.65876925 -0.19140646 -0.21988108
[2,] 0.35030059 1.186059670 0.85183144 1.10273414 -0.31246617 -0.02016989
[,187] [,188] [,189] [,190] [,191] [,192]
[1,] 1.47105762 -0.34534534 -0.016737537 -1.45859041 -0.21871528 -0.51339664
[2,] 0.58619461 0.13874879 0.002130158 0.49067719 1.91716161 1.54966246
[,193] [,194] [,195] [,196] [,197] [,198]
[1,] -0.596025666 0.618510032 0.56191185 -0.32341581 0.114075171 -0.22932103
[2,] 1.649454099 1.164076643 0.17515920 -0.30519006 0.528625624 1.05244594
[,199] [,200] [,201] [,202] [,203] [,204]
[1,] -0.15008733 0.91210029 -0.12122956 0.8885761682 -0.69176054 -0.383405806
[2,] 1.23904257 -0.17191264 1.39246499 0.9070637408 0.70695395 0.912521067
[,205] [,206] [,207] [,208] [,209] [,210]
[1,] 0.128847629 -0.49016147 -0.810986351 0.19310547 0.333006298 -0.61588900
[2,] 1.163655246 1.23287444 -0.033520360 0.84791567 1.452160224 0.18448850
[,211] [,212] [,213] [,214] [,215] [,216]
[1,] -0.671805639 -0.09840456 1.0750377 0.59099977 -0.497530402 0.009910161
[2,] 2.455510228 1.38436353 2.8312501 0.94968959 1.299520838 1.697430949
[,217] [,218] [,219] [,220] [,221] [,222]
[1,] 0.25576316 0.907534189 0.003375076 -0.23276868 1.52689779 1.688931710
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[ reached getOption("max.print") -- omitted 150 rows ]
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739 740 741 742 743 744
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823 824 825 826 827 828
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829 830 831 832 833 834
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907 908 909 910 911 912
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913 914 915 916 917 918
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919 920 921 922 923 924
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925 926 927 928 929 930
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931 932 933 934 935 936
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937 938 939 940 941 942
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943 944 945 946 947 948
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949 950 951 952 953 954
0.510754091 0.013006438 -0.030350988 -0.099789582 -0.339160252 0.245788550
955 956 957 958 959 960
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961 962 963 964 965 966
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967 968 969 970 971 972
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973 974 975 976 977 978
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979 980 981 982 983 984
0.592654989 -0.353014982 0.642674353 -0.414363751 -0.256346876 -0.220250138
985 986 987 988 989 990
-0.752935544 0.412972244 0.288069162 0.151282690 -0.250821789 -0.023817515
991 992 993 994 995 996
0.021652520 0.263783937 0.383621160 0.330713697 0.070725150 -0.100744877
997 998 999 1000
-0.453535013 -0.053872926 0.380797121 -0.568501467
[ reached getOption("max.print") -- omitted 1780 entries ]
attr(,"class")
[1] "bsts.prediction"stopImplicitCluster()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