# Packages
#install.packages(c("forecast", "tseries", "ggplot2", "TSA"))
library(forecast)
## Warning: package 'forecast' was built under R version 4.3.3
## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo
library(tseries)
## Warning: package 'tseries' was built under R version 4.3.3
library(ggplot2)
library(TSA)
## Warning: package 'TSA' was built under R version 4.3.3
## Registered S3 methods overwritten by 'TSA':
##   method       from    
##   fitted.Arima forecast
##   plot.Arima   forecast
## 
## Attaching package: 'TSA'
## The following objects are masked from 'package:stats':
## 
##     acf, arima
## The following object is masked from 'package:utils':
## 
##     tar

\[ Y_t = \left(1 + \theta_1 B + \theta_2 B^2 + \theta_3 B^3 + \theta_4 B^4\right) \left(1 + \Theta_1 B^{12} + \Theta_2 B^{24} + \Theta_3 B^{36} + \Theta_4 B^{48}\right) \varepsilon_t \]

cat("Timeseries Dataset Generating Simulately")
## Timeseries Dataset Generating Simulately
# Simulating Dataset with SARIMA(0,0,4)(0,0,4)^[12] models
set.seed(258)

model <- Arima(ts(rnorm(1200), frequency=12), order = c(0,0,4), seasonal = list(order = c(0,0,4), period = 12))
sim_data <- simulate(model, nsim = 1200)
sim_data
##              Jan          Feb          Mar          Apr          May
## 101  0.771562315 -0.462772723  0.241829100 -0.400908990  0.475397395
## 102  0.612083542 -0.477170037 -0.761621709  0.277801853  1.640914518
## 103 -1.219536508  0.896536407  0.079604621 -0.310731524 -0.118424944
## 104  1.354473123 -0.229056111  0.709629935  0.601214780 -0.393988234
## 105  1.126594576  0.425566185 -0.937610680  1.064672459 -0.383145606
## 106  0.864659771 -0.900034385  0.388829658 -0.852517347  0.235365862
## 107  1.496058670 -0.252899098 -1.885121446 -0.863241402  0.711620413
## 108 -1.121823399 -1.146601401 -1.871221478  0.044489709  0.156816312
## 109 -0.523031277 -0.084169902  0.107376506 -0.491937054 -0.418758555
## 110  0.602255852  0.339925319  1.324677303  0.757940478 -0.031764930
## 111  1.229373942 -2.527403112  1.031496550 -0.754924313  0.987342095
## 112 -0.375651815  0.432536047  0.331142655  1.235131343 -0.250312033
## 113 -1.077452365 -0.109376275 -0.611704198 -0.504199346  0.672237746
## 114 -0.254389024  0.144772735 -0.312156704 -0.102649259 -1.410208830
## 115  1.197747847 -1.853868456 -0.816543838  0.253944647 -0.474718983
## 116  0.081653599 -0.450856349  1.031779263 -0.292474002 -0.036872053
## 117 -1.059830698 -1.806434636  0.413797645  0.894540215  1.015938657
## 118 -1.134867774 -0.047300648 -0.131947500 -0.348170993 -0.468565860
## 119 -1.560899701  1.447492340  0.643698629  0.291811774 -0.311806417
## 120 -1.118977494  0.228654403  0.217824787  0.262483162  0.951891243
## 121 -2.360683564  0.583501751  0.164513396 -0.523124853 -0.925721258
## 122  0.040217983  0.400349234 -1.776108772  0.648448292  0.534345412
## 123  0.887677034 -0.822412977 -0.264709359 -1.429097681  0.069713128
## 124  0.297846322  0.012227014  0.845216102 -0.032607980  0.470895265
## 125  0.380053990 -0.689586192 -0.930609977  0.256888261  0.829598864
## 126  0.604803789 -0.916189655  1.336203119 -1.719405092  0.895135950
## 127 -0.731459362 -1.097222940  1.007354904 -0.186029750  0.169323207
## 128 -0.574006319 -0.489942247  0.096376277  0.310302877  0.004637524
## 129 -1.346796239  1.157995938 -1.100054973  1.183600946 -1.109391445
## 130 -0.421666297  0.397912321  0.181979538  0.289302058  1.084307968
## 131  1.620880261 -0.246780467  0.983861775 -1.405379918  1.219486563
## 132 -0.384609137  0.576249204 -0.209620701  0.512751000  0.590988190
## 133 -0.354717764 -0.600358697 -0.740596536  0.676806173 -0.194074959
## 134  0.543750720 -0.237889337 -1.487455953 -0.373108610  0.594568512
## 135 -0.656883350 -0.051668170 -0.922990861 -1.214177937 -0.632304997
## 136  1.672318554 -0.043717115 -0.564139195  0.501205372  0.622271634
## 137 -1.020565555 -0.120949497  0.346002830  1.119660875  1.765914750
## 138  1.155637323  0.787336284  1.385342307 -1.940394149  0.172664333
## 139  1.736446299  0.173703459 -1.335657061 -2.476418912  0.492162733
## 140  1.720936571  0.032824774  0.625054817  0.805718548 -1.696992562
## 141 -1.686413029 -0.974775143  0.943988755  0.853259539  0.130724314
## 142 -0.305839397 -1.714093557  1.679690730 -1.474957209 -0.647576929
## 143  0.363076400 -1.340956092 -1.694710818  1.575765301  1.207548563
## 144  0.303244259  1.892260920 -1.228983870 -0.943748437  0.100017704
## 145 -0.164054919 -0.568594551  0.138290768  0.587078552 -0.575869546
## 146  0.594823518 -2.303674004  2.004355201 -1.069416310 -1.040037563
## 147 -1.583068712  3.029274663 -0.649745291 -0.050205990 -0.041169938
## 148  0.331992263  1.586726109 -0.269484889 -0.684784083  0.540760760
## 149 -0.738181998  0.598120588  1.280518063 -0.560595150 -0.495476280
## 150  2.441198327  1.485502927  0.459927585 -0.644546464  1.437489247
## 151  0.470634323  0.093376694  0.025658516 -0.881038075  1.359741455
## 152 -0.051326415  1.220384649 -0.796082864 -0.893300488 -0.766298563
## 153  0.777087146 -0.472713002  0.069454289 -0.857842347 -0.247861120
## 154  0.827736379  1.660104655  0.849192853 -0.880377779  0.380001004
## 155 -0.093398189  0.927133777  0.579274708 -0.558860514 -0.853222163
## 156  2.516243018  0.372587919  0.109411848  0.180112928 -0.454697204
## 157  0.762439031 -0.700815648 -0.274972121 -1.956617491  0.418738679
## 158 -0.895870948 -1.060116507 -0.009495918  0.408786559  0.671412788
## 159  0.932243126  0.443591001  1.246952116 -1.264527221  1.115610202
## 160 -1.011992253  0.130323436 -0.634287920 -0.118485634  0.681699700
## 161 -0.286764200  0.840204097 -0.086419560  0.860023195 -0.299052959
## 162 -0.695381503  0.252330019 -1.113375358 -0.279487010 -2.225938226
## 163  0.467527618  0.026607804  0.262202171 -0.336838428 -0.635571985
## 164  0.093409726 -0.229092853 -0.762410615  0.307062650  0.440203220
## 165  0.589665508 -0.944692594  0.249619428 -0.496104423 -1.121369963
## 166 -1.395552342 -0.281083476 -0.663357353  0.385725422  0.618540554
## 167 -0.847542737  0.399353244 -0.636419654  2.319722201  0.091953336
## 168 -0.890804979  0.469850728 -0.554404476 -0.299347550 -1.473342139
## 169  0.046639722 -1.874380973  0.712008780 -0.003511828  0.015093840
## 170 -0.182445123  1.177600689 -0.255044036  1.294195192 -2.086443681
## 171  0.288459038  0.784457892 -0.875131356 -0.460456497  0.027737572
## 172 -1.325244744  0.017266072 -0.599201267 -0.860634452 -0.869527058
## 173 -0.179693667 -1.656481837 -0.802661871 -1.352315878  1.588224272
## 174 -0.416613029  0.851369446 -1.307992925  0.486169435  1.362545631
## 175 -1.317614898  1.217831300 -1.054416312  0.476448222 -1.367835239
## 176  0.190292116  0.108068999  1.914994580 -0.612324484 -0.216763623
## 177 -0.804834578  0.407279164  0.411018912 -0.289101261  0.471138184
## 178  0.611523170  1.805869493  0.153005686  0.259878894 -0.076128093
## 179 -0.583890994 -0.610104725 -0.863100868  0.043042012  0.384975560
## 180  0.795835522 -1.604430137 -1.877899311  2.367102654 -0.917520875
## 181 -1.298295139  1.000167843  1.132796418  0.291845442  0.980593120
## 182 -1.214429564 -0.425741037  0.467445138 -0.228323482 -1.281225065
## 183 -2.049739827 -0.191086333  0.341006261  0.777163500 -1.069488397
## 184 -0.579299137  0.383415702 -0.753669768  1.390183431 -0.968252531
## 185 -0.678879579  1.545129873  1.409710096  0.070779759 -0.795425267
## 186 -1.502828435  1.923359949  0.756408160  0.368888612  0.508064849
## 187 -1.281975322 -0.606266258  1.244540543  0.783046177  0.920445706
## 188 -1.253088846  0.135474047 -0.462959117  2.117112879 -0.528327181
## 189 -0.980952482  0.066456923  0.562487987  1.684618838 -0.115161221
## 190 -0.894124061  0.516890346  0.704918963  0.185876427  1.821041800
## 191 -0.600607249 -0.164460655 -0.671860632  0.645445811 -0.192760578
## 192  0.832705997 -0.443225995  0.452169526 -0.751985687 -0.245621303
## 193 -0.760623519  0.448446002 -0.178927832  0.580072718  0.535818731
## 194 -0.576295998 -0.158068903  0.861116773 -0.054611573 -0.682898013
## 195 -0.765354048 -1.595427636 -0.339901333 -1.780672182 -0.920384062
## 196 -1.274006642 -0.987044664  0.505867176 -0.811500245 -0.983508864
## 197  0.601930625  0.399153359 -0.935893469  0.025412531 -0.521328207
## 198 -0.016345419 -0.440438689  0.413029371 -0.350716865  1.649297174
## 199  0.373500343 -0.635466528 -0.089989469  0.970802983 -0.320117332
## 200  0.047923090  0.190229753 -0.565159867 -0.580028913 -1.351535966
##              Jun          Jul          Aug          Sep          Oct
## 101 -0.104747253  1.075363727  0.921116942 -0.610520650 -0.323443680
## 102  0.339363114  0.676765657 -0.267981878  0.158528742 -0.257668907
## 103 -0.484652686 -1.309627006  0.013544395 -0.835022575  2.113119898
## 104  0.940653622 -1.939262802 -0.806912617 -1.050198325 -0.373192216
## 105  0.794701616  0.511788631 -0.665642905  0.928702787  1.407819605
## 106  1.414700531 -0.890036556 -0.077977143 -0.890669670 -0.613444884
## 107 -0.596719366  1.424675305 -0.211323738 -0.348367619 -1.873190084
## 108  1.059015137  0.477203958 -1.038999325 -0.825630283  0.191591558
## 109 -0.252583298 -1.480982461 -1.287226136 -0.855110450  1.296976053
## 110  0.609927119  0.109210408 -0.253713866 -0.275170295 -0.185659185
## 111 -0.960724519  2.637691971  1.302999273  0.724518217  0.770184316
## 112  0.534752451 -0.253219850 -0.233762403 -0.266500380  0.076416197
## 113  0.184334355  0.063404532  0.209363121  0.833768514 -0.523909230
## 114 -0.171274877  0.368778143 -1.727971166  0.914924509 -0.357703627
## 115 -1.845096286  0.885028472 -0.377133143 -0.101016030  1.070998881
## 116  2.509210320  0.714368619  0.066628102 -2.427844502 -0.417106749
## 117 -1.287222366 -0.584640259 -0.973361425 -0.332060121 -0.460494300
## 118 -1.132824794  1.104588897  1.267830375 -0.545508503 -1.096195030
## 119  0.261261910  0.511125121  0.616478747  0.960524817 -1.681867838
## 120  0.170077043 -0.368946173  0.950980777 -1.138106671 -0.272699231
## 121  0.161287422 -2.325980308 -0.267910485 -0.427984429  0.444073807
## 122  0.702228321 -0.464183771  0.890531936 -0.544599094 -1.054764250
## 123 -0.937244331  1.453739004 -0.846564903 -0.182681696  0.417097747
## 124 -0.141779517 -0.321382100 -0.033491204  0.708524760  0.008077054
## 125  0.310926081 -0.096435329 -1.074335598  0.233758815 -0.696268439
## 126  1.579178112 -0.032280693 -0.370299255 -1.022131060 -0.293584471
## 127 -2.193305439  0.808346132 -0.360822748  0.485065125 -0.456188952
## 128 -0.379169068 -0.411242685 -1.124425811  1.065693277  0.011297506
## 129  0.830529234  1.502850891  0.609641125  0.333599323  0.493484234
## 130  0.369226940  0.996052911  0.805100319  1.169929811 -0.475380908
## 131  0.795434386 -0.999983912 -1.366944696 -1.339926417  1.163576542
## 132 -0.380176846 -1.584218645 -0.892984389  2.555029754  0.716358801
## 133 -0.870538311  0.769454649 -0.172686544  0.479941551  0.406905045
## 134 -0.282781491 -0.817969228  0.913522467  2.360072156  1.271687578
## 135 -2.073567350  0.360811997 -0.784811272  1.924291552 -0.402596701
## 136  0.248697168 -1.045678527  1.323584365 -0.832284058  1.312098964
## 137  0.009660206 -0.090663982 -0.412794058 -0.324814933 -1.401456402
## 138  0.160102548  0.029913829 -0.721227082  0.808855798  0.208958507
## 139  0.530979776  0.399561562 -1.006408279  1.154514834 -1.428936254
## 140 -0.392418760 -0.376502261  0.835891812 -0.309538838  0.758886129
## 141 -1.353811658  0.106542982 -1.117028218  0.166376784 -0.608109606
## 142  0.644901649 -0.493944433 -0.812338632  0.586706870 -0.347209981
## 143  0.056452754 -1.139283804 -1.325647453 -0.023214741  0.567119267
## 144  0.350862821  0.013604623 -0.641697217  0.281235112  1.105023572
## 145  1.409941641  0.559507511 -0.825939266 -1.335074064  0.278599204
## 146  1.548169717  0.705118646 -0.526177126  0.651976604 -0.268013598
## 147 -0.094138864 -0.414592253  0.590059839  0.095461257  0.571099050
## 148  0.670145826 -0.796983041 -0.193144175  2.243897156  0.645358863
## 149  1.532246317 -0.340739116  1.883339862 -1.289314340  0.325085050
## 150 -0.732698308 -0.294724608 -0.676986592  0.404944618  0.834560540
## 151  0.762132182 -0.883486320 -0.081628044 -1.789614880 -0.090093914
## 152  0.264449534 -1.061488049  0.695750565  0.302473474 -1.682988040
## 153  0.484866062 -0.399538030 -1.208532787 -0.464756764  0.742071465
## 154 -1.437272388  1.831024701  0.033207934  0.489390634  0.513750909
## 155 -0.852919118  0.480895678  0.255206389  0.689366299  0.509866636
## 156  1.419549910  1.404643703  0.227508965 -2.155638583 -0.469072481
## 157 -0.864222863  0.997087568  0.540925860 -0.996112078  0.993258572
## 158 -0.248225419  0.056149104  0.763908696  0.775653683  0.008358857
## 159 -0.507021432 -0.819946073  0.740990799  0.629036429  0.477078728
## 160 -0.682843078  0.193787127  0.358570526  0.060976577 -0.619846489
## 161  2.406195660  0.180171714  0.339449576 -0.561417453  2.162753237
## 162  0.715667129 -0.495358191 -0.601225382 -0.427043046 -0.552511702
## 163  0.918840612  0.596210740  0.038845651  0.826224065  0.747705120
## 164 -0.488614177  1.108044608  0.996696627 -0.511102152 -0.485723692
## 165  0.826220067  0.104546258  0.862110948 -0.344687038 -1.739983012
## 166  2.501494904 -1.114195570  0.958541138  0.906812619  0.096619857
## 167  0.544818636  1.328343493 -0.838766119 -0.893532231  0.258279436
## 168  0.620496945  0.285710364 -0.251672500  1.567052070  0.396901032
## 169  1.614539762 -1.664579653 -0.180683732  0.946276783  0.491557605
## 170  0.558370564 -0.378317433 -1.135981489 -0.779508540  1.967779585
## 171 -0.226556708 -1.102533779  1.161807937 -0.022208986  0.441835303
## 172 -1.861410705 -0.033880038 -0.342243602  1.116345829 -0.626716251
## 173  0.322840668  1.096642316  0.227236874  0.846456800 -1.065946925
## 174  2.204287595 -0.860927136 -0.613880343 -0.909865926  0.803166474
## 175 -0.447287134  0.390654254  0.768626547  0.306729336 -0.344162042
## 176 -0.155973248 -1.029544486  0.876666277 -1.282275318  1.629192566
## 177 -0.772025841  0.085692907 -0.236339124  0.556366168 -0.906771634
## 178  0.715265292 -0.061711140 -1.212817906 -1.299379150 -0.316373020
## 179 -1.316324441 -0.731442544  1.374629060 -0.388150310 -1.109937177
## 180 -0.984029920  0.213187837  2.170339790 -0.236370380  1.253803809
## 181 -0.701027011  0.628945688  1.007428650 -1.610042093  1.563293128
## 182  0.346962188 -0.988915153  1.082802494  0.529980786  0.093444181
## 183 -0.244839754  0.270496191 -0.893747569  0.819611305 -0.177537153
## 184  0.734276018  0.414318043 -0.270059076 -0.102890882  0.636396859
## 185  0.969953142  0.055920399 -0.259264162 -0.889229869 -0.044929626
## 186  0.994285180 -0.589138787 -0.361853576  0.654549897  1.107726625
## 187 -1.479127744  0.730349952  0.052500925 -0.196528402 -0.767169628
## 188 -1.640145610 -0.115230146  1.023737453  1.360030243  0.748531796
## 189 -0.742069690 -0.112556494 -1.502956852 -0.616463047  0.887048762
## 190  1.070975255  0.777897134  0.055207003 -0.246948190  0.332232333
## 191 -2.024389905  0.674998474 -0.005336257  0.124733370 -0.868719432
## 192 -0.893540189  0.779314664 -0.826160544  0.083899890 -0.384717103
## 193 -0.008668004  0.180649053  0.213552297 -0.476685956  0.351440460
## 194 -0.500652968  1.779071458  0.619024106 -0.242499643 -0.333495040
## 195 -0.678627899  0.745377901  1.650335611  1.188821884 -3.102030247
## 196 -2.357332626 -0.513057934 -0.538631347  0.414105356  2.077908299
## 197 -0.676094098  0.844586913 -0.847054967  0.567177959  1.427929390
## 198  0.549864800 -0.118025113 -0.101289688  0.164575506 -0.262345849
## 199 -0.369476779 -0.590406674  1.905217627 -1.292119913 -0.923704237
## 200  1.132044228  0.852998966 -0.047071853  0.445788746  1.495631390
##              Nov          Dec
## 101  1.023051737  0.461014683
## 102 -1.126771420 -0.089679976
## 103 -0.238384217  0.086240934
## 104 -1.358427962  0.093466301
## 105 -0.839259954  0.163212582
## 106 -0.783607504 -1.006133958
## 107 -0.284717701  0.607242594
## 108 -0.631474708  0.182204264
## 109  1.661641446 -0.676712685
## 110  0.225411442 -0.806737488
## 111 -0.257254441 -0.660017940
## 112 -1.062958145  0.041916794
## 113 -1.225115720 -1.372935608
## 114  0.528552789  1.378584124
## 115 -0.334120968 -0.039062808
## 116  0.958219698  0.177428906
## 117 -1.092756772  0.019596010
## 118  0.321284476 -0.453569965
## 119 -0.242248898 -0.291559183
## 120 -0.307509431  1.235443347
## 121  0.174034820  0.059392147
## 122  0.405090756 -0.101834125
## 123  0.064798112 -0.101820025
## 124 -0.046139062  0.094931642
## 125  0.794582872 -1.114969878
## 126 -1.657177772 -0.594550063
## 127 -1.015289319  1.570678872
## 128 -0.203528030  0.162328917
## 129 -1.394646527 -1.226950898
## 130  0.336790655  1.330317984
## 131  0.277805008  1.938687957
## 132  0.753618506  0.966473845
## 133  0.424953233 -0.749208453
## 134  0.138557733 -1.713490669
## 135  0.254033042 -0.528544558
## 136 -1.144163663 -1.163345012
## 137  0.649207390 -0.822372924
## 138 -0.054652580  0.552046480
## 139 -0.470002442  0.800013457
## 140  0.161459430 -0.997251697
## 141  0.242242156  0.186594399
## 142 -1.151873824 -0.245559413
## 143  0.654203856 -0.379576578
## 144  0.004786473 -0.698059085
## 145  0.490773666 -1.262481895
## 146 -0.702319430  0.782874214
## 147  1.273111649  0.562486709
## 148 -1.141155097 -0.614169944
## 149 -0.721656339 -1.069007617
## 150 -1.566997226  0.712437978
## 151 -1.320426479  1.830825240
## 152  0.309299416  1.323857057
## 153 -0.275036651 -0.189651483
## 154 -0.324859301 -0.008541348
## 155 -0.259583674  0.581059598
## 156 -0.444356262  0.630898817
## 157  0.405278553 -0.732847037
## 158  0.242025812 -0.075541999
## 159  0.751025034  0.579767407
## 160  0.115784356  0.653389184
## 161  0.832224361 -0.992077728
## 162  0.267410235 -0.189493440
## 163 -0.035186530 -0.721378073
## 164 -0.173461123  1.159662406
## 165  1.379176239 -0.771372189
## 166 -1.097410823 -0.163931120
## 167  0.724174398 -0.560608300
## 168 -0.147101326 -0.156511323
## 169 -0.707566958 -0.772394199
## 170  1.641674120 -0.596230526
## 171 -0.471769025 -0.568277148
## 172  1.979679134  0.114636142
## 173 -1.343114845 -0.316052034
## 174  0.072864543 -0.291255339
## 175  1.107009864  0.600587230
## 176  1.092238443 -0.129195726
## 177 -0.339030186 -0.511472941
## 178 -1.149209176 -0.961500789
## 179  1.829792612  0.205582421
## 180  0.259451328  0.065962789
## 181 -0.348206224 -0.882158389
## 182 -0.055659941  0.940281437
## 183 -0.983288956  1.059068737
## 184 -0.391035088 -0.129223089
## 185 -0.108069546 -0.191300386
## 186  0.690161703  1.537417177
## 187 -0.345762466 -0.916599717
## 188 -0.190589984 -0.129226030
## 189  0.991849281 -0.748530502
## 190 -0.232622295  1.268910529
## 191 -1.470389310  2.655149392
## 192 -0.902783668  0.841285594
## 193  1.123553206  0.004236114
## 194  1.139118402 -1.674233092
## 195 -1.698349258 -0.183572306
## 196 -0.355592105  0.696597600
## 197  0.976438741 -0.061599360
## 198 -0.810432730  0.766231089
## 199 -0.716573116  0.278863492
## 200  0.085680422  0.029640556
sim_data_ts_m <- ts(sim_data, start = 1, frequency = 1)
plot(sim_data_ts_m, main="Simulasi Index Bulan SARIMA(0,0,4)(0,0,4)^[12]")

sim_data_ts_y <- ts(sim_data, start = 1, frequency = 12)
plot(sim_data_ts_y, main="Simulasi Index Tahun SARIMA(0,0,4)(0,0,4)^[12]")

# ACF dan PACF Plot
acf(sim_data_ts_m, lag.max=60, main="ACF")

pacf(sim_data_ts_m, , lag.max=60, main="PACF")

# Uji Stationeritas (ADF Test)

adf.test(sim_data)
## Warning in adf.test(sim_data): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  sim_data
## Dickey-Fuller = -10.616, Lag order = 10, p-value = 0.01
## alternative hypothesis: stationary
# Uji Asumsi White Noise (Ljung Box)
checkresiduals(model)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(0,0,4)(0,0,4)[12] with non-zero mean
## Q* = 8.9731, df = 16, p-value = 0.9145
## 
## Model df: 8.   Total lags used: 24
Box.test(residuals(model), lag = 35, type = "Ljung-Box")
## 
##  Box-Ljung test
## 
## data:  residuals(model)
## X-squared = 19.209, df = 35, p-value = 0.9861