An Application to HB Twofold Normal Model On sampel dataset

FIRST STEPS: Load package and load the data

library(saeHB.twofold)
data("dataTwofold")

STEPS 2: Fitting HB Model

model=NormalTF(y~x1+x2,vardir="vardir",area = "codearea",weight = "w",iter.mcmc = 100000,thin=50,burn.in = 1000,data=dataTwofold)
#> Compiling model graph
#>    Resolving undeclared variables
#>    Allocating nodes
#> Graph information:
#>    Observed stochastic nodes: 90
#>    Unobserved stochastic nodes: 125
#>    Total graph size: 953
#> 
#> Initializing model
#> 
#> Compiling model graph
#>    Resolving undeclared variables
#>    Allocating nodes
#> Graph information:
#>    Observed stochastic nodes: 90
#>    Unobserved stochastic nodes: 125
#>    Total graph size: 953
#> 
#> Initializing model
#> 
#> Compiling model graph
#>    Resolving undeclared variables
#>    Allocating nodes
#> Graph information:
#>    Observed stochastic nodes: 90
#>    Unobserved stochastic nodes: 125
#>    Total graph size: 953
#> 
#> Initializing model
#> Warning in par(opar): argument 1 does not name a graphical parameter

STEP 3 Extract mean estimation

Subarea Estimation

model$Est_sub
#>                Mean        SD        2.5%       25%       50%       75%
#> theta[1]  13.693935 2.9907207  7.79410725 11.687641 13.651013 15.756113
#> theta[2]  11.021884 1.8887456  7.24140775  9.791313 11.043135 12.224932
#> theta[3]  14.289169 2.6114699  9.00814399 12.561446 14.367107 16.043486
#> theta[4]  11.565302 1.3347049  8.96921435 10.645404 11.586031 12.497601
#> theta[5]  14.758647 0.7752779 13.23652771 14.218722 14.762563 15.278235
#> theta[6]  10.647883 1.2491440  8.26132557  9.809814 10.604563 11.507253
#> theta[7]  14.298444 0.8977892 12.54680012 13.683447 14.286063 14.938702
#> theta[8]  13.453792 1.8401307  9.95932135 12.203619 13.437748 14.701912
#> theta[9]  16.180234 0.9313269 14.34379903 15.563556 16.153791 16.791771
#> theta[10] 14.879265 0.8900639 13.06023850 14.323545 14.887421 15.503523
#> theta[11] 14.282135 0.7185156 12.85293776 13.790953 14.286779 14.758057
#> theta[12] 15.705958 1.9166356 11.87663026 14.443412 15.702365 17.029367
#> theta[13]  9.271348 0.5764731  8.14758605  8.901802  9.267310  9.662072
#> theta[14]  9.937510 0.9574809  8.00340730  9.319209  9.916980 10.569991
#> theta[15] 15.560178 0.6984295 14.16912373 15.085738 15.573015 16.018574
#> theta[16] 12.664158 1.5629059  9.52666785 11.609436 12.706893 13.690409
#> theta[17]  2.203104 0.7225219  0.80293838  1.710507  2.200799  2.711845
#> theta[18]  7.211175 1.1176978  5.00514961  6.453260  7.236005  7.961107
#> theta[19]  2.167798 0.7416359  0.76721732  1.684183  2.147463  2.667868
#> theta[20]  6.072314 1.8558269  2.39551239  4.881929  6.121745  7.345164
#> theta[21]  4.408150 0.7897354  2.90188685  3.861098  4.416221  4.949788
#> theta[22] 14.689394 1.1863310 12.34143649 13.850919 14.706597 15.528128
#> theta[23] 13.993081 0.4774903 13.07581628 13.678197 13.988196 14.304991
#> theta[24] 17.632812 1.1181235 15.43140975 16.891909 17.661932 18.405505
#> theta[25] 14.482059 1.3329031 11.91109566 13.585656 14.476319 15.396268
#> theta[26] 10.280062 1.9306761  6.52357206  8.990475 10.260903 11.579403
#> theta[27]  9.572017 2.6659067  4.53515681  7.742156  9.589982 11.379677
#> theta[28] 14.977784 0.8998436 13.20244482 14.391329 14.966302 15.566674
#> theta[29] 13.184364 1.6999774  9.87233761 12.067249 13.175719 14.329086
#> theta[30] 10.122996 1.6722072  6.85529777  9.024338 10.099267 11.243801
#> theta[31] 14.214906 2.2548685  9.80112987 12.731421 14.218126 15.673336
#> theta[32] 14.594086 0.9119349 12.81705621 13.970653 14.620481 15.188039
#> theta[33]  9.847065 3.2319295  3.49380965  7.670940  9.800527 12.056294
#> theta[34] 13.779090 1.3859364 11.08222932 12.866547 13.779015 14.696299
#> theta[35] 14.724855 1.9977525 10.88665048 13.376296 14.676986 16.087101
#> theta[36] 13.429945 1.9492077  9.72270260 12.141370 13.447926 14.715110
#> theta[37] 14.025491 0.8185450 12.38186136 13.470709 14.049907 14.570526
#> theta[38] 14.168689 0.9223053 12.33118917 13.555796 14.161017 14.786089
#> theta[39] 12.495856 1.4242153  9.67949284 11.551032 12.454718 13.463278
#> theta[40]  9.290750 2.3086690  4.74727695  7.786962  9.243464 10.821661
#> theta[41] 18.458183 0.8633562 16.68454496 17.917680 18.459179 19.006605
#> theta[42] 12.504306 1.0027508 10.53806093 11.790214 12.511577 13.166380
#> theta[43] 16.391054 3.3467816  9.73333981 14.162460 16.410557 18.564150
#> theta[44] 16.959770 1.4995306 14.07050855 15.943384 16.997048 17.963072
#> theta[45] 18.544406 1.4031244 15.67836009 17.613555 18.549013 19.484320
#> theta[46] 13.666247 1.9480836  9.76849986 12.388576 13.692576 14.982441
#> theta[47] 12.437789 2.5765002  7.35628218 10.729475 12.487263 14.122046
#> theta[48] 11.394685 0.6735983 10.08559070 10.953052 11.407259 11.851476
#> theta[49]  6.612497 1.3736153  3.95712689  5.661979  6.611700  7.571769
#> theta[50] 15.956638 1.5247427 13.01863687 14.960712 15.945552 16.977091
#> theta[51]  7.811899 1.3812439  5.01717368  6.895759  7.796244  8.726801
#> theta[52] 26.607999 0.8380639 24.93140887 26.044724 26.614751 27.178066
#> theta[53] 18.443654 1.6501583 15.22346142 17.330055 18.431570 19.533256
#> theta[54] 17.934408 1.2709243 15.46448072 17.084095 17.916619 18.790961
#> theta[55] 11.993530 0.7200001 10.62291606 11.498368 12.014689 12.495650
#> theta[56] 11.457102 0.5298765 10.43382398 11.103759 11.452049 11.805584
#> theta[57]  8.988379 0.6722448  7.67165675  8.548777  8.990322  9.452542
#> theta[58] 13.739148 1.4327947 11.02738058 12.773751 13.700317 14.655105
#> theta[59] 16.501022 0.9640695 14.57082751 15.850324 16.511746 17.138938
#> theta[60] 15.182290 0.7987066 13.58731756 14.648283 15.205642 15.703298
#> theta[61]  5.633972 1.9839501  1.74069795  4.323027  5.621226  6.974661
#> theta[62]  5.907147 0.8104232  4.32020833  5.386022  5.908321  6.443244
#> theta[63]  5.345787 0.9285898  3.49934871  4.726596  5.350993  5.965926
#> theta[64] 12.233542 1.5759335  9.19023170 11.178462 12.224014 13.254999
#> theta[65] 12.441059 2.4705346  7.66604226 10.861897 12.397437 14.121674
#> theta[66]  9.891444 1.2225462  7.49543601  9.075900  9.925679 10.736167
#> theta[67] 16.834699 1.0636311 14.74055260 16.125987 16.838004 17.536128
#> theta[68] 11.213820 1.2900793  8.65383343 10.367090 11.201986 12.084001
#> theta[69]  8.164940 0.8695323  6.46097765  7.568218  8.175364  8.769281
#> theta[70] 11.728594 0.8350003 10.11540804 11.149730 11.720960 12.313279
#> theta[71] 13.717463 0.7413192 12.29596268 13.209064 13.719799 14.195765
#> theta[72] 10.736724 0.7172862  9.32173881 10.244466 10.755119 11.238703
#> theta[73]  3.829711 1.4735561  0.77557382  2.914604  3.827939  4.822988
#> theta[74] 10.462691 1.5090432  7.44694027  9.471446 10.484507 11.435288
#> theta[75]  5.970238 1.3058257  3.37979009  5.091775  5.980230  6.838087
#> theta[76]  9.451887 0.7954995  7.88916640  8.938133  9.445954  9.972760
#> theta[77] 13.917227 0.7320858 12.45284944 13.422601 13.922239 14.407360
#> theta[78]  9.709841 2.0517371  5.50418256  8.332043  9.697690 11.089450
#> theta[79] 10.472920 1.1725138  8.21621749  9.688193 10.500675 11.242572
#> theta[80]  5.885377 0.7413730  4.44665540  5.382250  5.890962  6.399804
#> theta[81]  6.470040 0.7186338  5.10832825  5.970396  6.450965  6.948135
#> theta[82] 14.334464 2.2755902  9.96809267 12.774692 14.369945 15.938976
#> theta[83] 10.653012 1.8582612  7.05310143  9.404195 10.654906 11.997745
#> theta[84] 10.101786 2.9999949  4.09320836  8.089491 10.194831 12.104802
#> theta[85] 13.420949 1.6158732 10.36120505 12.349772 13.439530 14.449981
#> theta[86] 18.575149 1.4563524 15.82791880 17.564403 18.548614 19.581836
#> theta[87] 11.488209 0.7826168 10.02130324 10.948733 11.478634 12.001709
#> theta[88]  8.034032 1.8866459  4.31873709  6.742109  8.052728  9.304261
#> theta[89]  7.726876 1.3858310  4.97692879  6.798198  7.708597  8.632506
#> theta[90]  4.478428 2.3659624 -0.05680186  2.859148  4.536859  6.058788
#>               97.5%
#> theta[1]  19.446236
#> theta[2]  14.764375
#> theta[3]  19.287764
#> theta[4]  14.135372
#> theta[5]  16.287488
#> theta[6]  13.070641
#> theta[7]  16.029943
#> theta[8]  17.034131
#> theta[9]  18.057253
#> theta[10] 16.517110
#> theta[11] 15.677427
#> theta[12] 19.342909
#> theta[13] 10.392573
#> theta[14] 11.832711
#> theta[15] 16.882073
#> theta[16] 15.772163
#> theta[17]  3.595196
#> theta[18]  9.408534
#> theta[19]  3.649604
#> theta[20]  9.601048
#> theta[21]  5.958555
#> theta[22] 16.948818
#> theta[23] 14.923023
#> theta[24] 19.802708
#> theta[25] 17.042550
#> theta[26] 14.172464
#> theta[27] 14.821569
#> theta[28] 16.742492
#> theta[29] 16.399494
#> theta[30] 13.360624
#> theta[31] 18.826477
#> theta[32] 16.438793
#> theta[33] 16.061376
#> theta[34] 16.446325
#> theta[35] 18.678915
#> theta[36] 17.285781
#> theta[37] 15.611701
#> theta[38] 15.974672
#> theta[39] 15.272044
#> theta[40] 13.854120
#> theta[41] 20.085423
#> theta[42] 14.498339
#> theta[43] 23.197778
#> theta[44] 19.841202
#> theta[45] 21.299400
#> theta[46] 17.362929
#> theta[47] 17.621778
#> theta[48] 12.721822
#> theta[49]  9.435656
#> theta[50] 19.005955
#> theta[51] 10.564553
#> theta[52] 28.243119
#> theta[53] 21.645901
#> theta[54] 20.438322
#> theta[55] 13.386726
#> theta[56] 12.510115
#> theta[57] 10.278608
#> theta[58] 16.584997
#> theta[59] 18.340157
#> theta[60] 16.735856
#> theta[61]  9.353393
#> theta[62]  7.490230
#> theta[63]  7.087514
#> theta[64] 15.354422
#> theta[65] 17.256299
#> theta[66] 12.226799
#> theta[67] 18.950933
#> theta[68] 13.719740
#> theta[69]  9.900618
#> theta[70] 13.367590
#> theta[71] 15.237341
#> theta[72] 12.120740
#> theta[73]  6.757564
#> theta[74] 13.527221
#> theta[75]  8.576695
#> theta[76] 11.045442
#> theta[77] 15.326169
#> theta[78] 13.736792
#> theta[79] 12.734891
#> theta[80]  7.370144
#> theta[81]  7.868076
#> theta[82] 18.660188
#> theta[83] 14.079382
#> theta[84] 15.792557
#> theta[85] 16.579026
#> theta[86] 21.555367
#> theta[87] 13.061798
#> theta[88] 11.652893
#> theta[89] 10.525286
#> theta[90]  9.099498

Area Estimatio

model$Est_area
#>         Mean        SD      2.5%       25%       50%       75%     97.5%
#> 1  13.174676 1.8964022  9.465432 11.851851 13.197060 14.452074 16.847025
#> 2  12.435030 0.6541992 11.100711 12.005017 12.444293 12.882660 13.656717
#> 3  14.471977 0.7866940 12.900655 13.925519 14.473059 15.008154 15.956933
#> 4  14.855044 0.6777146 13.496261 14.409818 14.860324 15.318637 16.171839
#> 5  11.301326 0.4423385 10.441233 10.994473 11.298222 11.605755 12.165312
#> 6   7.829218 0.7416377  6.369537  7.328814  7.823200  8.338928  9.242043
#> 7   4.511784 0.8718661  2.758893  3.919648  4.534083  5.099656  6.111150
#> 8  15.870855 0.6356489 14.603152 15.431271 15.887014 16.290010 17.126473
#> 9  11.354779 1.3300468  8.796160 10.460287 11.319507 12.251505 14.110354
#> 10 12.302772 0.9540850 10.495834 11.651921 12.314185 12.942936 14.116600
#> 11 12.599379 1.6872159  9.277051 11.440858 12.612917 13.757985 15.824496
#> 12 13.990780 1.1256170 11.723329 13.238652 14.025939 14.731297 16.166025
#> 13 13.451209 0.6989780 12.065610 12.974796 13.455038 13.946018 14.788349
#> 14 13.304450 0.9323746 11.473522 12.706609 13.280167 13.937790 15.130121
#> 15 17.244118 1.6160094 14.029490 16.143443 17.282655 18.306826 20.339866
#> 16 12.627191 1.3134635 10.017830 11.763213 12.641572 13.522471 15.202562
#> 17 10.477416 0.8768515  8.762469  9.876495 10.482801 11.071932 12.215468
#> 18 20.785717 0.7753093 19.318563 20.259869 20.784291 21.325461 22.292197
#> 19 10.407083 0.3998947  9.587119 10.153623 10.414578 10.671969 11.195781
#> 20 15.245603 0.5961285 14.087379 14.861270 15.239748 15.662326 16.378580
#> 21  5.601777 0.8360844  3.891873  5.057814  5.591823  6.160392  7.211997
#> 22 11.539063 1.0943160  9.396280 10.785001 11.522628 12.272913 13.757664
#> 23 12.122676 0.6509887 10.858624 11.694505 12.132011 12.562717 13.379469
#> 24 12.106125 0.4732608 11.226162 11.776651 12.090028 12.435842 13.056795
#> 25  6.498160 0.8849327  4.757927  5.898289  6.486638  7.102867  8.202984
#> 26 11.395092 0.7355668  9.884613 10.900377 11.396302 11.893311 12.803789
#> 27  7.465617 0.5247377  6.440932  7.120691  7.455907  7.808714  8.519414
#> 28 12.074145 1.5675793  9.006168 11.037031 12.071697 13.114567 15.262457
#> 29 14.749583 0.8270243 13.158814 14.183043 14.746925 15.278746 16.427554
#> 30  6.519693 1.2785922  4.066053  5.635997  6.504943  7.373607  9.030487

Coefficient Estimation

model$coefficient
#>               Mean         SD      2.5%        25%        50%         75%
#> beta[0] -0.4393799 0.65137242 -1.725681 -0.8850581 -0.4317153 0.001575619
#> beta[1]  0.2205832 0.63298632 -0.963742 -0.2271957  0.2146501 0.653185061
#> beta[2]  1.2331151 0.06533988  1.108596  1.1893076  1.2326558 1.275856709
#>             97.5%
#> beta[0] 0.8118008
#> beta[1] 1.4332263
#> beta[2] 1.3611094

Random effect variance estimation

model$refVar
#>   var_area  var_sub
#> 1 8.040705 8.784105

STEP 3 : Extract CV and MSE Subarea

  • CV $CV(\theta \hat)=\frac{SD(\theta \hat)}{(\theta \hat)} \times 100$
  • MSE MSE = V(θ
CV=(model$Est_sub$SD)/(model$Est_sub$Mean)*100
MSE=model$Est_sub$SD^2
summary(cbind(CV,MSE))
#>        CV              MSE         
#>  Min.   : 3.150   Min.   : 0.2280  
#>  1st Qu.: 6.552   1st Qu.: 0.6985  
#>  Median :11.522   Median : 1.6847  
#>  Mean   :13.710   Mean   : 2.3755  
#>  3rd Qu.:17.622   3rd Qu.: 3.4296  
#>  Max.   :52.830   Max.   :11.2009

STEP 4 Extract CV and MSE of Area

CV2=(model$Est_area$SD)/(model$Est_area$Mean)*100
MSE2=model$Est_area$SD^2
summary(cbind(CV2,MSE2))
#>       CV2              MSE2       
#>  Min.   : 3.730   Min.   :0.1599  
#>  1st Qu.: 5.213   1st Qu.:0.4358  
#>  Median : 7.392   Median :0.6915  
#>  Mean   : 8.603   Mean   :1.0102  
#>  3rd Qu.:11.386   3rd Qu.:1.2496  
#>  Max.   :19.611   Max.   :3.5963

STEP 5 : You can also compare the CV between subarea direct estimator and HB Twofold estimator

dirCV=sqrt(dataTwofold$vardir)/(dataTwofold$y)*100
summary(cbind(dirCV,CV))
#>      dirCV               CV        
#>  Min.   :  3.187   Min.   : 3.150  
#>  1st Qu.:  6.642   1st Qu.: 6.552  
#>  Median : 12.764   Median :11.522  
#>  Mean   : 25.141   Mean   :13.710  
#>  3rd Qu.: 21.041   3rd Qu.:17.622  
#>  Max.   :680.378   Max.   :52.830
  • Look that! ,CV on our model is less than CV on direct estimator.
boxplot(cbind(dirCV,CV),ylim=c(0,50))

model$refVar
#>   var_area  var_sub
#> 1 8.040705 8.784105
model$plot
#> [[1]]
#> NULL
#> 
#> [[2]]
#> [[2]]$mar
#> [1] 5.1 4.1 4.1 2.1
#> 
#> 
#> [[3]]
#> [[1]]
#> Markov Chain Monte Carlo (MCMC) output:
#> Start = 1050 
#> End = 1e+05 
#> Thinning interval = 50 
#>              beta[0]       beta[1]  beta[2]
#>    [1,] -0.739107886  0.5426564235 1.234228
#>    [2,] -0.644374954  1.0739529923 1.226397
#>    [3,] -0.597797488 -0.6121645378 1.255090
#>    [4,] -0.145731752  0.1770570753 1.242497
#>    [5,] -1.276202120  0.1913106993 1.348151
#>    [6,] -0.301263527  0.0934289180 1.250403
#>    [7,] -1.251022363 -0.4252143296 1.318318
#>    [8,] -1.224485237 -0.5055606644 1.359902
#>    [9,]  0.130324862  0.0474149526 1.155236
#>   [10,] -0.975580329  0.7582907219 1.204156
#>   [11,] -0.998356993  0.6628774820 1.152451
#>   [12,] -1.094576691 -0.7531086786 1.327638
#>   [13,] -0.220217292 -0.5728840875 1.257189
#>   [14,] -0.731787820  1.0469429612 1.195605
#>   [15,] -1.426664916 -0.9314468225 1.358375
#>   [16,] -0.652321696 -0.3344991899 1.266324
#>   [17,] -1.183870582 -0.4400834287 1.337573
#>   [18,] -1.268203235  0.3812759820 1.377736
#>   [19,]  0.408445729 -0.2006069264 1.219338
#>   [20,] -0.997569367  0.7677607266 1.202654
#>   [21,] -0.957654953 -0.3073560134 1.284686
#>   [22,] -0.214856348 -0.5489974804 1.261799
#>   [23,] -1.318234032 -0.3117136815 1.279008
#>   [24,] -0.497476963  0.9490323137 1.151304
#>   [25,] -0.016378250  1.1160676215 1.112796
#>   [26,] -0.655081245  0.4700894969 1.117019
#>   [27,] -1.608008045  0.3801921774 1.198278
#>   [28,] -0.074268286 -0.8582805559 1.374256
#>   [29,] -1.155077081  1.5191077818 1.347713
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#>  [889,] -0.817361519  0.1944408223 1.194001
#>  [890,] -0.653566560  0.2783398406 1.306228
#>  [891,] -0.688763883 -0.4337353398 1.260395
#>  [892,] -0.518651073 -0.9494662755 1.323913
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#>  [894,] -1.219992435  0.6541247365 1.273575
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#>  [896,] -0.730870907  1.2859054650 1.188825
#>  [897,] -0.906465781  0.5370944306 1.171100
#>  [898,] -1.975186833  0.3072085850 1.319479
#>  [899,] -0.378583090  1.3917044565 1.273381
#>  [900,] -1.126179628  0.3943175094 1.207124
#>  [901,] -0.402240719 -0.0968617965 1.277154
#>  [902,] -0.434515502  0.2484971943 1.232267
#>  [903,] -0.527752895  0.1863884316 1.247730
#>  [904,]  0.760354359  0.2665479627 1.182651
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#> [1786,]  0.385845484  0.7530599767 1.100757
#> [1787,] -0.004612616  1.4701485169 1.039154
#> [1788,] -1.909045378  0.6051486861 1.297928
#> [1789,]  0.052412159 -0.0436873348 1.265263
#> [1790,]  0.004392864  0.2866311692 1.178161
#> [1791,] -0.315822312  0.6276126085 1.217045
#> [1792,] -0.978448229  0.8007846074 1.272873
#> [1793,] -0.777719073 -0.2223345634 1.354359
#> [1794,]  0.762668188  1.2848660136 1.139173
#> [1795,] -0.482682497 -0.8079133875 1.189270
#> [1796,]  0.034753510  0.4856982529 1.142782
#> [1797,] -0.873078639  0.0943257192 1.229822
#> [1798,]  0.224612465  0.1793680488 1.196708
#> [1799,] -1.705919950 -0.2305911760 1.349873
#> [1800,] -0.551338146  0.3709566377 1.228841
#> [1801,] -1.010154631  0.2655951363 1.274604
#> [1802,] -0.853030445  1.8543628088 1.211290
#> [1803,]  0.044558726 -0.3259761860 1.176695
#> [1804,] -0.578470983  1.2724016613 1.172105
#> [1805,] -1.370706268  0.3797059128 1.259693
#> [1806,] -0.296132121  0.8105770861 1.223153
#> [1807,] -0.698772596  1.1747074485 1.168274
#> [1808,] -0.857594037  0.4161468269 1.228617
#> [1809,] -0.576494865 -0.3101772857 1.267881
#> [1810,]  0.448911620  0.4890915843 1.176529
#> [1811,]  0.400714424 -0.9034148391 1.184794
#> [1812,] -0.966956006  0.9360558642 1.282424
#> [1813,]  0.077611594  0.7070861573 1.176430
#> [1814,]  0.036782777 -0.1096604542 1.194301
#> [1815,]  0.188581140 -0.0186309223 1.247393
#> [1816,]  0.398031636  0.0780368385 1.174484
#> [1817,]  0.275362254 -0.9193073030 1.239647
#> [1818,] -0.097020940  1.5532044871 1.119400
#> [1819,] -0.410661213 -0.2911812990 1.176357
#> [1820,] -1.225514481  1.1709821774 1.239072
#> [1821,] -0.520762728 -1.1428356142 1.246550
#> [1822,] -0.728182795  0.7737296482 1.236911
#> [1823,]  0.267556695 -0.4242232319 1.192967
#> [1824,] -0.536141780  0.6163826844 1.243140
#> [1825,]  0.134195350 -0.0439725372 1.185098
#> [1826,] -0.894800925 -0.0151200288 1.230215
#> [1827,] -0.579907860  0.7553479930 1.231304
#> [1828,] -0.699739918  1.2081595946 1.322774
#> [1829,]  0.420256961  0.5062152065 1.253716
#> [1830,] -0.459598695  0.2518070491 1.189488
#> [1831,] -0.216820938  0.4772697833 1.299114
#> [1832,] -0.355996356  1.2631535353 1.181997
#> [1833,] -0.426779859  0.5737764482 1.154414
#> [1834,] -1.437186731  0.1221911568 1.356035
#> [1835,] -1.503920303  0.3263920205 1.373571
#> [1836,] -0.182002384 -0.3287359224 1.370188
#> [1837,]  0.038123802  0.7099205730 1.216908
#> [1838,]  1.652548872 -0.0531271597 1.174813
#> [1839,] -0.383051703  0.3505441308 1.175266
#> [1840,]  1.352878831  0.6637443275 1.134196
#> [1841,] -0.251541486  0.3172752110 1.066201
#> [1842,] -0.964560985  0.2221917891 1.292448
#> [1843,] -0.024704988 -0.1037883460 1.260084
#> [1844,]  0.069564481  0.3744568692 1.249594
#> [1845,] -0.418657796 -0.5351888115 1.260211
#> [1846,] -0.440955042  1.3762472248 1.242220
#> [1847,] -0.737144144  0.8090910202 1.255075
#> [1848,] -0.702288876 -0.5562746225 1.280531
#> [1849,] -0.355453027 -1.5921398739 1.333453
#> [1850,]  0.083866874  0.5526965316 1.216619
#> [1851,] -1.411743236 -0.1075575511 1.317218
#> [1852,]  0.398694411  0.9231441588 1.192946
#> [1853,] -0.822577999  0.3359159578 1.279327
#> [1854,] -0.558922403 -0.0062646556 1.290027
#> [1855,] -0.811531349 -0.7273796168 1.277381
#> [1856,]  0.416954674 -0.0277641609 1.237223
#> [1857,] -0.801402878  0.1467854884 1.347535
#> [1858,] -0.633367504 -0.1204403128 1.227209
#> [1859,] -0.260026146  0.1619440847 1.315953
#> [1860,] -1.672436957  0.5209445678 1.299547
#> [1861,] -1.040672610  1.1489537059 1.255758
#> [1862,] -0.236112021  0.6107710304 1.210492
#> [1863,] -0.684008729  0.7419823189 1.124108
#> [1864,] -0.140169113 -0.3730485662 1.275386
#> [1865,] -0.651578424 -0.3474207535 1.250763
#> [1866,]  0.356382000 -0.2043920358 1.174514
#> [1867,]  0.467745982 -0.1076999764 1.137800
#> [1868,]  0.672496752  1.2731137275 1.145293
#> [1869,] -0.648381985  0.1881647164 1.212169
#> [1870,] -1.315037632  0.6424977747 1.258121
#> [1871,]  0.407784684 -0.1515225057 1.187511
#> [1872,] -1.181200415  0.4289553731 1.259130
#> [1873,] -1.439409344  1.2628099763 1.295182
#> [1874,] -0.106678395  0.9867485846 1.156718
#> [1875,] -0.104261520  0.2003281696 1.225703
#> [1876,]  0.162127451  1.4184967766 1.138434
#> [1877,]  0.499541022  0.0982547407 1.191954
#> [1878,]  0.098968585 -0.1016594953 1.257960
#> [1879,] -0.332310821  0.4230917779 1.123263
#> [1880,] -0.976638615  0.6685847741 1.196337
#> [1881,] -0.528669356  0.1101036614 1.241178
#> [1882,] -0.032094219  0.3853350094 1.166818
#> [1883,] -0.276605682  1.1165879640 1.221126
#> [1884,] -0.200455986 -0.4380666547 1.220698
#> [1885,] -0.676566901  0.8862396493 1.340224
#> [1886,]  0.035709189  1.3469374701 1.165229
#> [1887,] -0.159271741 -0.2927415018 1.233109
#> [1888,] -1.091117099 -0.6590271337 1.302627
#> [1889,] -1.760292438  0.3433777291 1.246845
#> [1890,] -0.560213605  0.1322557467 1.255001
#> [1891,]  0.503222242  1.1097134344 1.159634
#> [1892,] -0.301708168 -1.4616892100 1.220137
#> [1893,] -0.112513816  1.0776702689 1.274066
#> [1894,] -0.254887132  0.8561825446 1.222581
#> [1895,] -0.477146253  1.4995309204 1.198015
#> [1896,]  0.120260741  0.0053854607 1.176774
#> [1897,] -1.367721745  0.2277383039 1.251216
#> [1898,] -1.263399820 -0.0073706415 1.324400
#> [1899,]  0.643722750 -0.0910640730 1.297631
#> [1900,] -0.144812498 -0.7255252312 1.205565
#> [1901,] -0.652608358  0.9192313478 1.236424
#> [1902,] -0.361204773 -0.8367743219 1.288671
#> [1903,] -0.469275675 -0.0331226711 1.263261
#> [1904,] -0.445052884  0.1588767060 1.254281
#> [1905,]  0.257611559  0.3048443873 1.152133
#> [1906,]  0.186976414 -1.2362139726 1.202963
#> [1907,]  0.787031486 -0.2526966245 1.142356
#> [1908,]  0.377617553  0.3721851241 1.226069
#> [1909,] -0.767173392  0.1684221263 1.298600
#> [1910,] -1.013874307  0.4488704511 1.344895
#> [1911,] -0.628130557  1.0526664563 1.307158
#> [1912,]  0.130950214  0.4933136037 1.191277
#> [1913,] -0.800309106 -0.1056636269 1.273780
#> [1914,] -1.014986455 -0.8306048004 1.273156
#> [1915,]  0.593435047 -0.2282923156 1.236030
#> [1916,]  0.435961325  0.5577062872 1.204670
#> [1917,] -0.202801457  1.4297940751 1.140105
#> [1918,] -0.751045023 -0.0597392268 1.181685
#> [1919,]  0.689331682  0.4151554565 1.084996
#> [1920,] -0.437925291 -0.4747665396 1.189696
#> [1921,] -0.382692537  0.5187079238 1.221548
#> [1922,] -1.369922996  0.6672403006 1.345372
#> [1923,] -0.520948447  1.1131410754 1.195689
#> [1924,]  0.939248988 -0.1727345814 1.155920
#> [1925,] -0.469783671  1.1756574421 1.226128
#> [1926,]  0.237023015  0.3808897396 1.211121
#> [1927,] -0.103054554 -0.5808221485 1.200698
#> [1928,] -0.588752523  0.4969404806 1.228516
#> [1929,] -1.069463327  0.3852194657 1.266279
#> [1930,] -0.166480558  0.1987661644 1.187896
#> [1931,]  0.276524462 -0.5187687780 1.238556
#> [1932,] -0.399181665  0.6228803107 1.197207
#> [1933,]  0.176491263  0.2595247608 1.161484
#> [1934,]  0.116522624  0.3246373897 1.148021
#> [1935,] -0.360889301 -0.0536302901 1.212152
#> [1936,] -1.270073772 -0.4014100311 1.247642
#> [1937,]  0.068852850 -0.3575268561 1.275787
#> [1938,] -1.985880180  0.2723999396 1.328194
#> [1939,] -0.614656446  0.7295058505 1.256722
#> [1940,]  0.601453467  0.3144447661 1.185429
#> [1941,]  0.105126474  0.6335999934 1.165743
#> [1942,] -0.957113249  1.2713214901 1.170465
#> [1943,] -0.369802566  0.9038418016 1.209936
#> [1944,] -0.986551284  0.8481016343 1.275964
#> [1945,] -0.342969963 -0.7294425090 1.214083
#> [1946,] -0.716031729  0.0479754485 1.282623
#> [1947,]  0.343672137 -0.4304011003 1.293591
#> [1948,] -0.092828140  0.1987181907 1.219101
#> [1949,]  1.047041979  0.6862868179 1.152487
#> [1950,] -1.632733735  0.5697203273 1.245064
#> [1951,] -1.255631115  0.8491612949 1.238788
#> [1952,] -0.910146110  0.7449924085 1.291340
#> [1953,] -0.930261556 -0.1008146552 1.321975
#> [1954,] -0.799094996  0.5463347902 1.284101
#> [1955,] -0.198361093  0.2639838101 1.317133
#> [1956,]  0.343476423 -0.9437425323 1.207316
#> [1957,]  0.015159893 -1.1571603183 1.284868
#> [1958,] -0.908025719  1.2074182012 1.290830
#> [1959,] -1.316204807 -0.4387696356 1.336958
#> [1960,] -1.461825450  0.7713613117 1.199537
#> [1961,] -0.263303001 -0.1013130540 1.212374
#> [1962,] -0.879663243  0.1542651413 1.298167
#> [1963,] -0.248394559  0.3969540429 1.206682
#> [1964,] -0.581734847  0.7597395552 1.176579
#> [1965,]  0.378659199 -0.4350062532 1.117514
#> [1966,] -0.674311586 -0.3256664223 1.215391
#> [1967,] -0.016547607 -0.3944979988 1.276895
#> [1968,] -0.524605852  1.0008155147 1.192550
#> [1969,] -0.017413298  0.2536058845 1.247266
#> [1970,] -0.271191088 -0.1952943838 1.211842
#> [1971,] -1.265874537  0.2593441957 1.277978
#> [1972,] -1.070618318  1.1581405749 1.249454
#> [1973,]  0.018306239 -0.0741669132 1.210182
#> [1974,] -0.713509950 -0.1894234363 1.226268
#> [1975,] -1.944263525  0.8042338704 1.365122
#> [1976,] -0.262306234 -0.1021859960 1.255912
#> [1977,] -0.924795607 -0.2765318675 1.328291
#> [1978,] -1.304509921 -0.1301006465 1.319095
#> [1979,] -0.164059355  0.1722996669 1.209924
#> [1980,] -0.339254280  0.4626746499 1.249932
#> 
#> attr(,"class")
#> [1] "mcmc.list"
#> 
#> [[4]]
#> NULL