library(crossvalidation)
## Loading required package: doSNOW
## Loading required package: foreach
## Loading required package: iterators
## Loading required package: snow
## Loading required package: forecast
## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo
data("AirPassengers")

# Example 1 -----

(res1 <- crossval_ts(y=AirPassengers, initial_window = 10,
horizon = 3, fcast_func = forecast::thetaf))
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##                     ME       RMSE        MAE          MPE       MAPE
## result.1    -28.794660  29.300287  28.794660  0.259750743 0.25975074
## result.2     16.198526  16.894302  16.198526 -0.134036396 0.13403640
## result.3     11.201494  15.993359  12.578276 -0.081116675 0.09308869
## result.4     21.430125  22.483895  21.430125 -0.157908838 0.15790884
## result.5     10.055765  11.527746  10.055765 -0.073295759 0.07329576
## result.6     -2.640822  10.676714   9.999466  0.024882775 0.07426965
## result.7     14.296434  23.709132  20.531135 -0.081722428 0.13160003
## result.8     38.247497  39.529998  38.247497 -0.231651791 0.23165179
## result.9     23.043159  23.947630  23.043159 -0.137610422 0.13761042
## result.10   -21.689067  27.907560  21.689067  0.154732863 0.15473286
## result.11   -41.782157  46.664199  41.782157  0.335689202 0.33568920
## result.12   -34.934831  36.512081  34.934831  0.280155112 0.28015511
## result.13    -4.002700  12.728771   9.999100  0.040488081 0.08233473
## result.14    30.349582  30.588761  30.349582 -0.208731397 0.20873140
## result.15    21.192349  25.806712  21.192349 -0.127304850 0.12730485
## result.16    23.193143  25.914875  23.193143 -0.137455344 0.13745534
## result.17    30.081542  30.679960  30.081542 -0.174862066 0.17486207
## result.18    -6.530509   9.111376   6.999059  0.039593797 0.04222610
## result.19    19.907586  23.010762  19.907586 -0.105338252 0.10533825
## result.20    17.631089  19.829355  17.631089 -0.089550147 0.08955015
## result.21    11.738022  14.718185  12.229846 -0.058884295 0.06155725
## result.22   -21.787490  28.489509  21.787490  0.129489104 0.12948910
## result.23   -43.557571  47.527244  43.557571  0.278950791 0.27895079
## result.24   -34.473558  35.514155  34.473558  0.221812109 0.22181211
## result.25    -4.699360  10.498595   7.201224  0.033422378 0.04808847
## result.26    25.974138  26.581272  25.974138 -0.149797833 0.14979783
## result.27    16.905109  19.474600  16.905109 -0.090819469 0.09081947
## result.28    15.218760  16.352917  15.218760 -0.081484738 0.08148474
## result.29     7.625241   8.933828   7.625241 -0.040408146 0.04040815
## result.30     2.261970  17.595326  15.666212 -0.004267259 0.07797211
## result.31    30.404914  36.592516  30.404914 -0.136176156 0.13617616
## result.32    46.412292  47.386236  46.412292 -0.200384649 0.20038465
## result.33     4.651769  15.532402  15.001095 -0.016552192 0.06607050
## result.34   -23.538570  33.729923  29.025988  0.121976444 0.14465172
## result.35   -60.690177  63.572981  60.690177  0.328287148 0.32828715
## result.36   -30.676401  32.211315  30.676401  0.168456030 0.16845603
## result.37    -8.577804  12.511784   8.577804  0.048889552 0.04888955
## result.38    21.858632  21.864774  21.858632 -0.111896409 0.11189641
## result.39    15.511300  24.464353  15.511300 -0.066956116 0.06695612
## result.40    27.351766  33.292899  27.351766 -0.116287717 0.11628772
## result.41    40.030438  40.086293  40.030438 -0.171470545 0.17147055
## result.42     1.913204   6.585371   5.082179 -0.007480486 0.02131880
## result.43    10.484756  17.849104  14.434313 -0.039461169 0.05670814
## result.44    28.459896  30.643222  28.459896 -0.107735883 0.10773588
## result.45     9.494149  19.388080  18.988959 -0.033029178 0.07309167
## result.46   -31.346158  42.472223  34.784073  0.144434512 0.15707390
## result.47   -73.108813  78.130452  73.108813  0.368808906 0.36880891
## result.48   -47.391791  49.482768  47.391791  0.245955653 0.24595565
## result.49   -20.263420  22.215913  20.263420  0.106896888 0.10689689
## result.50    18.627251  19.778162  18.627251 -0.093146961 0.09314696
## result.51    12.850828  24.491415  18.283609 -0.052964282 0.08186206
## result.52    18.630902  29.191109  27.309290 -0.076261222 0.12242286
## result.53    51.629084  51.820741  51.629084 -0.222377386 0.22237739
## result.54    13.235345  22.822393  16.377438 -0.050065336 0.06390716
## result.55    44.587147  53.679665  44.587147 -0.157309556 0.15730956
## result.56    54.072225  56.591957  54.072225 -0.186048489 0.18604849
## result.57    23.241489  30.809955  26.322921 -0.077182075 0.08907949
## result.58   -48.442256  57.148030  48.442256  0.199756472 0.19975647
## result.59   -76.172134  81.227076  76.172134  0.346233997 0.34623400
## result.60   -49.727782  51.411167  49.727782  0.229654227 0.22965423
## result.61   -12.015506  17.811915  13.001128  0.058148711 0.06222153
## result.62    28.827561  29.294075  28.827561 -0.122413712 0.12241371
## result.63    19.978440  24.903081  19.978440 -0.077636903 0.07763690
## result.64    17.110530  24.500048  22.184318 -0.062408624 0.08418454
## result.65    40.684200  40.816355  40.684200 -0.151376898 0.15137690
## result.66    22.682439  32.492379  22.682439 -0.074237521 0.07423752
## result.67    50.630668  64.359435  50.630668 -0.146962154 0.14696215
## result.68    71.054598  73.834184  71.054598 -0.204938760 0.20493876
## result.69    18.573839  30.848787  28.001381 -0.049971189 0.08018767
## result.70   -67.050346  75.890258  67.050346  0.228774453 0.22877445
## result.71   -89.147978  96.654776  89.147978  0.344537272 0.34453727
## result.72   -62.834495  65.591765  62.834495  0.245236494 0.24523649
## result.73   -17.192156  24.448634  17.192156  0.070396960 0.07039696
## result.74    39.147639  39.310613  39.147639 -0.139867698 0.13986770
## result.75    15.464746  23.499340  15.599360 -0.049536142 0.05002211
## result.76    21.388632  28.622347  25.036615 -0.067092418 0.08026203
## result.77    44.769285  44.904644  44.769285 -0.141625136 0.14162514
## result.78    24.337654  38.632254  24.890329 -0.066081607 0.06784734
## result.79    60.018016  72.625037  60.018016 -0.152620633 0.15262063
## result.80    78.806534  80.547853  78.806534 -0.196880257 0.19688026
## result.81    11.811108  31.117683  30.676430 -0.025278321 0.07842007
## result.82   -73.039284  86.689647  73.039284  0.223481827 0.22348183
## result.83  -112.674445 120.216380 112.674445  0.382042857 0.38204286
## result.84   -75.193608  77.210395  75.193608  0.259592270 0.25959227
## result.85   -18.005367  23.707491  18.005367  0.064266651 0.06426665
## result.86    34.032748  34.590333  34.032748 -0.110400738 0.11040074
## result.87    21.143051  32.190728  22.381017 -0.060361477 0.06447432
## result.88    25.770886  36.725269  33.180194 -0.071385272 0.09600091
## result.89    61.405131  61.602493  61.405131 -0.173880072 0.17388007
## result.90    28.569615  46.690860  30.711084 -0.068438510 0.07459216
## result.91    72.884393  87.299176  72.884393 -0.164968096 0.16496810
## result.92    97.243170  99.503251  97.243170 -0.213704240 0.21370424
## result.93    17.748567  36.930435  36.435381 -0.035014060 0.08126855
## result.94   -76.493543  94.890278  76.493543  0.208139134 0.20813913
## result.95  -136.613003 145.258940 136.613003  0.409412213 0.40941221
## result.96   -92.679937  95.004482  92.679937  0.285555972 0.28555597
## result.97   -32.018423  33.966942  32.018423  0.099842072 0.09984207
## result.98    22.498638  24.963399  22.498638 -0.067003620 0.06700362
## result.99     7.476001  20.150283  17.158667 -0.019143446 0.04959208
## result.100    9.982584  22.773032  22.210356 -0.025942187 0.06439430
## result.101   52.362071  53.101568  52.362071 -0.146065760 0.14606576
## result.102   34.853629  55.893897  39.234038 -0.081274141 0.09386152
## result.103   95.047754 111.264404  95.047754 -0.207545291 0.20754529
## result.104  121.335201 125.765542 121.335201 -0.250852140 0.25085214
## result.105   27.661546  53.669523  52.335667 -0.049684215 0.11075877
## result.106  -82.928463 106.536749  87.398376  0.222291120 0.23114243
## result.107 -168.429957 174.864018 168.429957  0.490890905 0.49089091
## result.108  -86.047368  89.349691  86.047368  0.261674036 0.26167404
## result.109  -35.392983  38.646195  35.392983  0.108684894 0.10868489
## result.110   32.273683  33.691668  32.273683 -0.092489738 0.09248974
## result.111   35.911969  45.528574  35.911969 -0.092371900 0.09237190
## result.112   28.584481  41.791438  38.166544 -0.069168689 0.09718642
## result.113   78.144295  79.313101  78.144295 -0.191198246 0.19119825
## result.114   37.152546  52.614035  39.210445 -0.080727044 0.08592376
## result.115   95.078342 110.886020  95.078342 -0.187505147 0.18750515
## result.116  109.166178 116.176118 109.166178 -0.202725580 0.20272558
## result.117   41.289554  62.020845  57.334902 -0.071737094 0.10639227
## result.118  -92.399494 116.617773  92.824074  0.216693887 0.21745342
## result.119 -175.618445 183.279552 175.618445  0.444857203 0.44485720
## result.120  -94.580461  97.360392  94.580461  0.245373593 0.24537359
## result.121  -27.751828  32.935586  27.751828  0.073365613 0.07336561
## result.122   36.177008  38.166461  36.177008 -0.088751649 0.08875165
## result.123    5.992278  14.161847  13.997426 -0.013666896 0.03414042
## result.124   12.637863  33.652691  27.980298 -0.025012667 0.06425163
## result.125   71.834372  76.950725  71.834372 -0.156620714 0.15662071
## result.126   85.518711  93.750937  85.518711 -0.170478555 0.17047855
## result.127   94.429064 115.523968  94.429064 -0.162279474 0.16227947
## result.128  173.325805 177.666522 173.325805 -0.291744683 0.29174468
## result.129   33.890665  63.841907  61.668605 -0.050341919 0.10502290
## result.130 -119.059067 137.736851 119.059067  0.244484820 0.24448482
## result.131 -180.821172 190.452409 180.821172  0.418676479 0.41867648
## result.132 -103.156396 108.618814 103.156396  0.247654263 0.24765426
##           ME         RMSE          MAE          MPE         MAPE 
##  2.657082195 51.427170382 46.511874693  0.003423843  0.155428590
(res1_parallel <- crossval_ts(y=AirPassengers, initial_window = 10,
horizon = 3, fcast_func = forecast::thetaf, cl=2L))
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##                     ME       RMSE        MAE          MPE       MAPE
## result.1    -28.794660  29.300287  28.794660  0.259750743 0.25975074
## result.2     16.198526  16.894302  16.198526 -0.134036396 0.13403640
## result.3     11.201494  15.993359  12.578276 -0.081116675 0.09308869
## result.4     21.430125  22.483895  21.430125 -0.157908838 0.15790884
## result.5     10.055765  11.527746  10.055765 -0.073295759 0.07329576
## result.6     -2.640822  10.676714   9.999466  0.024882775 0.07426965
## result.7     14.296434  23.709132  20.531135 -0.081722428 0.13160003
## result.8     38.247497  39.529998  38.247497 -0.231651791 0.23165179
## result.9     23.043159  23.947630  23.043159 -0.137610422 0.13761042
## result.10   -21.689067  27.907560  21.689067  0.154732863 0.15473286
## result.11   -41.782157  46.664199  41.782157  0.335689202 0.33568920
## result.12   -34.934831  36.512081  34.934831  0.280155112 0.28015511
## result.13    -4.002700  12.728771   9.999100  0.040488081 0.08233473
## result.14    30.349582  30.588761  30.349582 -0.208731397 0.20873140
## result.15    21.192349  25.806712  21.192349 -0.127304850 0.12730485
## result.16    23.193143  25.914875  23.193143 -0.137455344 0.13745534
## result.17    30.081542  30.679960  30.081542 -0.174862066 0.17486207
## result.18    -6.530509   9.111376   6.999059  0.039593797 0.04222610
## result.19    19.907586  23.010762  19.907586 -0.105338252 0.10533825
## result.20    17.631089  19.829355  17.631089 -0.089550147 0.08955015
## result.21    11.738022  14.718185  12.229846 -0.058884295 0.06155725
## result.22   -21.787490  28.489509  21.787490  0.129489104 0.12948910
## result.23   -43.557571  47.527244  43.557571  0.278950791 0.27895079
## result.24   -34.473558  35.514155  34.473558  0.221812109 0.22181211
## result.25    -4.699360  10.498595   7.201224  0.033422378 0.04808847
## result.26    25.974138  26.581272  25.974138 -0.149797833 0.14979783
## result.27    16.905109  19.474600  16.905109 -0.090819469 0.09081947
## result.28    15.218760  16.352917  15.218760 -0.081484738 0.08148474
## result.29     7.625241   8.933828   7.625241 -0.040408146 0.04040815
## result.30     2.261970  17.595326  15.666212 -0.004267259 0.07797211
## result.31    30.404914  36.592516  30.404914 -0.136176156 0.13617616
## result.32    46.412292  47.386236  46.412292 -0.200384649 0.20038465
## result.33     4.651769  15.532402  15.001095 -0.016552192 0.06607050
## result.34   -23.538570  33.729923  29.025988  0.121976444 0.14465172
## result.35   -60.690177  63.572981  60.690177  0.328287148 0.32828715
## result.36   -30.676401  32.211315  30.676401  0.168456030 0.16845603
## result.37    -8.577804  12.511784   8.577804  0.048889552 0.04888955
## result.38    21.858632  21.864774  21.858632 -0.111896409 0.11189641
## result.39    15.511300  24.464353  15.511300 -0.066956116 0.06695612
## result.40    27.351766  33.292899  27.351766 -0.116287717 0.11628772
## result.41    40.030438  40.086293  40.030438 -0.171470545 0.17147055
## result.42     1.913204   6.585371   5.082179 -0.007480486 0.02131880
## result.43    10.484756  17.849104  14.434313 -0.039461169 0.05670814
## result.44    28.459896  30.643222  28.459896 -0.107735883 0.10773588
## result.45     9.494149  19.388080  18.988959 -0.033029178 0.07309167
## result.46   -31.346158  42.472223  34.784073  0.144434512 0.15707390
## result.47   -73.108813  78.130452  73.108813  0.368808906 0.36880891
## result.48   -47.391791  49.482768  47.391791  0.245955653 0.24595565
## result.49   -20.263420  22.215913  20.263420  0.106896888 0.10689689
## result.50    18.627251  19.778162  18.627251 -0.093146961 0.09314696
## result.51    12.850828  24.491415  18.283609 -0.052964282 0.08186206
## result.52    18.630902  29.191109  27.309290 -0.076261222 0.12242286
## result.53    51.629084  51.820741  51.629084 -0.222377386 0.22237739
## result.54    13.235345  22.822393  16.377438 -0.050065336 0.06390716
## result.55    44.587147  53.679665  44.587147 -0.157309556 0.15730956
## result.56    54.072225  56.591957  54.072225 -0.186048489 0.18604849
## result.57    23.241489  30.809955  26.322921 -0.077182075 0.08907949
## result.58   -48.442256  57.148030  48.442256  0.199756472 0.19975647
## result.59   -76.172134  81.227076  76.172134  0.346233997 0.34623400
## result.60   -49.727782  51.411167  49.727782  0.229654227 0.22965423
## result.61   -12.015506  17.811915  13.001128  0.058148711 0.06222153
## result.62    28.827561  29.294075  28.827561 -0.122413712 0.12241371
## result.63    19.978440  24.903081  19.978440 -0.077636903 0.07763690
## result.64    17.110530  24.500048  22.184318 -0.062408624 0.08418454
## result.65    40.684200  40.816355  40.684200 -0.151376898 0.15137690
## result.66    22.682439  32.492379  22.682439 -0.074237521 0.07423752
## result.67    50.630668  64.359435  50.630668 -0.146962154 0.14696215
## result.68    71.054598  73.834184  71.054598 -0.204938760 0.20493876
## result.69    18.573839  30.848787  28.001381 -0.049971189 0.08018767
## result.70   -67.050346  75.890258  67.050346  0.228774453 0.22877445
## result.71   -89.147978  96.654776  89.147978  0.344537272 0.34453727
## result.72   -62.834495  65.591765  62.834495  0.245236494 0.24523649
## result.73   -17.192156  24.448634  17.192156  0.070396960 0.07039696
## result.74    39.147639  39.310613  39.147639 -0.139867698 0.13986770
## result.75    15.464746  23.499340  15.599360 -0.049536142 0.05002211
## result.76    21.388632  28.622347  25.036615 -0.067092418 0.08026203
## result.77    44.769285  44.904644  44.769285 -0.141625136 0.14162514
## result.78    24.337654  38.632254  24.890329 -0.066081607 0.06784734
## result.79    60.018016  72.625037  60.018016 -0.152620633 0.15262063
## result.80    78.806534  80.547853  78.806534 -0.196880257 0.19688026
## result.81    11.811108  31.117683  30.676430 -0.025278321 0.07842007
## result.82   -73.039284  86.689647  73.039284  0.223481827 0.22348183
## result.83  -112.674445 120.216380 112.674445  0.382042857 0.38204286
## result.84   -75.193608  77.210395  75.193608  0.259592270 0.25959227
## result.85   -18.005367  23.707491  18.005367  0.064266651 0.06426665
## result.86    34.032748  34.590333  34.032748 -0.110400738 0.11040074
## result.87    21.143051  32.190728  22.381017 -0.060361477 0.06447432
## result.88    25.770886  36.725269  33.180194 -0.071385272 0.09600091
## result.89    61.405131  61.602493  61.405131 -0.173880072 0.17388007
## result.90    28.569615  46.690860  30.711084 -0.068438510 0.07459216
## result.91    72.884393  87.299176  72.884393 -0.164968096 0.16496810
## result.92    97.243170  99.503251  97.243170 -0.213704240 0.21370424
## result.93    17.748567  36.930435  36.435381 -0.035014060 0.08126855
## result.94   -76.493543  94.890278  76.493543  0.208139134 0.20813913
## result.95  -136.613003 145.258940 136.613003  0.409412213 0.40941221
## result.96   -92.679937  95.004482  92.679937  0.285555972 0.28555597
## result.97   -32.018423  33.966942  32.018423  0.099842072 0.09984207
## result.98    22.498638  24.963399  22.498638 -0.067003620 0.06700362
## result.99     7.476001  20.150283  17.158667 -0.019143446 0.04959208
## result.100    9.982584  22.773032  22.210356 -0.025942187 0.06439430
## result.101   52.362071  53.101568  52.362071 -0.146065760 0.14606576
## result.102   34.853629  55.893897  39.234038 -0.081274141 0.09386152
## result.103   95.047754 111.264404  95.047754 -0.207545291 0.20754529
## result.104  121.335201 125.765542 121.335201 -0.250852140 0.25085214
## result.105   27.661546  53.669523  52.335667 -0.049684215 0.11075877
## result.106  -82.928463 106.536749  87.398376  0.222291120 0.23114243
## result.107 -168.429957 174.864018 168.429957  0.490890905 0.49089091
## result.108  -86.047368  89.349691  86.047368  0.261674036 0.26167404
## result.109  -35.392983  38.646195  35.392983  0.108684894 0.10868489
## result.110   32.273683  33.691668  32.273683 -0.092489738 0.09248974
## result.111   35.911969  45.528574  35.911969 -0.092371900 0.09237190
## result.112   28.584481  41.791438  38.166544 -0.069168689 0.09718642
## result.113   78.144295  79.313101  78.144295 -0.191198246 0.19119825
## result.114   37.152546  52.614035  39.210445 -0.080727044 0.08592376
## result.115   95.078342 110.886020  95.078342 -0.187505147 0.18750515
## result.116  109.166178 116.176118 109.166178 -0.202725580 0.20272558
## result.117   41.289554  62.020845  57.334902 -0.071737094 0.10639227
## result.118  -92.399494 116.617773  92.824074  0.216693887 0.21745342
## result.119 -175.618445 183.279552 175.618445  0.444857203 0.44485720
## result.120  -94.580461  97.360392  94.580461  0.245373593 0.24537359
## result.121  -27.751828  32.935586  27.751828  0.073365613 0.07336561
## result.122   36.177008  38.166461  36.177008 -0.088751649 0.08875165
## result.123    5.992278  14.161847  13.997426 -0.013666896 0.03414042
## result.124   12.637863  33.652691  27.980298 -0.025012667 0.06425163
## result.125   71.834372  76.950725  71.834372 -0.156620714 0.15662071
## result.126   85.518711  93.750937  85.518711 -0.170478555 0.17047855
## result.127   94.429064 115.523968  94.429064 -0.162279474 0.16227947
## result.128  173.325805 177.666522 173.325805 -0.291744683 0.29174468
## result.129   33.890665  63.841907  61.668605 -0.050341919 0.10502290
## result.130 -119.059067 137.736851 119.059067  0.244484820 0.24448482
## result.131 -180.821172 190.452409 180.821172  0.418676479 0.41867648
## result.132 -103.156396 108.618814 103.156396  0.247654263 0.24765426
print(colMeans(res1_parallel))
##           ME         RMSE          MAE          MPE         MAPE 
##  2.657082195 51.427170382 46.511874693  0.003423843  0.155428590
# Example 2 -----

fcast_func <- function (y, h, ...)
{
      forecast::forecast(forecast::auto.arima(y, ...),
      h=h, ...)
}

(res2 <- crossval_ts(y=AirPassengers, initial_window = 10, horizon = 3,
fcast_func = fcast_func))
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##                      ME       RMSE        MAE          MPE       MAPE
## result.1    -17.4666667  18.474487  17.466667  0.158924192 0.15892419
## result.2     -9.3333333  10.424331   9.333333  0.079589664 0.07958966
## result.3     -2.9321548  12.406309  10.872257  0.031080911 0.08739369
## result.4     10.3308133  10.682605  10.330813 -0.076563916 0.07656392
## result.5      5.0881248   7.171584   6.433807 -0.036313796 0.04707925
## result.6      5.5253837  11.598542   8.508769 -0.035515033 0.05938212
## result.7     18.0932850  26.110632  21.832590 -0.107845328 0.13775977
## result.8     36.3000000  37.625656  36.300000 -0.219692065 0.21969206
## result.9     39.2000000  39.606060  39.200000 -0.235234550 0.23523455
## result.10   -16.3333333  22.457367  16.333333  0.118048285 0.11804829
## result.11   -35.0000000  39.365806  35.000000  0.281790975 0.28179098
## result.12   -29.0000000  31.010751  29.000000  0.234168755 0.23416876
## result.13     0.0000000  13.589211  12.666667  0.011302682 0.09980843
## result.14     2.9000000   5.007661   4.300000 -0.019222222 0.02922222
## result.15     9.8521041  20.833322  16.825631 -0.052991094 0.10108438
## result.16    22.4654809  24.115632  22.465481 -0.134305527 0.13430553
## result.17    23.5102600  23.830725  23.510260 -0.136968702 0.13696870
## result.18    20.3000000  21.215325  20.300000 -0.117556040 0.11755604
## result.19    30.9000000  32.997121  30.900000 -0.165627228 0.16562723
## result.20    39.7000000  40.915645  39.700000 -0.204576252 0.20457625
## result.21    16.0000000  17.492856  16.000000 -0.081221324 0.08122132
## result.22   -17.3333333  23.050669  17.333333  0.103305600 0.10330560
## result.23   -53.8888889  58.697558  53.888889  0.344973669 0.34497367
## result.24   -26.0000000  27.398297  26.000000  0.168170059 0.16817006
## result.25    -1.0000000  10.847427   9.666667  0.010953692 0.06210567
## result.26    -0.7666667   5.843230   5.366667  0.005572818 0.03112837
## result.27     4.6462332  10.873135   9.350221 -0.023004506 0.05051321
## result.28    10.6666667  12.192894  10.666667 -0.056817654 0.05681765
## result.29     6.1200918   6.233263   6.120092 -0.032842259 0.03284226
## result.30    16.2000000  23.475661  16.200000 -0.076832843 0.07683284
## result.31    29.2783498  35.414530  29.278350 -0.130975527 0.13097553
## result.32    53.5000000  54.389797  53.500000 -0.231212275 0.23121228
## result.33     9.0000000  16.340135  15.000000 -0.036095089 0.06480322
## result.34   -16.0000000  26.495283  24.000000  0.085026725 0.11808458
## result.35   -72.6666667  76.469796  72.666667  0.393554409 0.39355441
## result.36   -23.3333333  25.285042  23.333333  0.128892235 0.12889223
## result.37    -3.6666667  11.474610   9.000000  0.023163665 0.05047975
## result.38    -4.5666667   4.662975   4.566667  0.023402763 0.02340276
## result.39     8.0333333  20.496097  15.100000 -0.030984089 0.06703851
## result.40    20.7333333  27.870414  24.466667 -0.086439648 0.10548727
## result.41    30.2333333  30.390953  30.233333 -0.129417299 0.12941730
## result.42    27.2666667  27.863237  27.266667 -0.115178218 0.11517822
## result.43    33.7555478  37.384187  33.755548 -0.134253573 0.13425357
## result.44    49.6666667  51.150106  49.666667 -0.189429700 0.18942970
## result.45    14.6666667  20.960280  18.666667 -0.053615549 0.07049319
## result.46   -24.0000000  34.650637  29.333333  0.111899040 0.13150688
## result.47   -84.8888889  90.848991  84.888889  0.428425354 0.42842535
## result.48   -39.6666667  41.717303  39.666667  0.206331298 0.20633130
## result.49   -36.9000000  38.413669  36.900000  0.192943127 0.19294313
## result.50   -32.6333333  33.364003  32.633333  0.166564238 0.16656424
## result.51   -22.0691733  29.513574  24.898987  0.115042825 0.12708458
## result.52    12.6666667  24.124676  23.333333 -0.049376699 0.10611429
## result.53    44.0000000  44.143705  44.000000 -0.189462455 0.18946245
## result.54     6.6666667  17.378147  12.666667 -0.023444230 0.04978809
## result.55    39.2166799  48.016281  39.216680 -0.137811607 0.13781161
## result.56    67.4333333  69.355197  67.433333 -0.232966640 0.23296664
## result.57    66.5666667  69.094211  66.566667 -0.230453676 0.23045368
## result.58   -41.6666667  49.190107  41.666667  0.171839061 0.17183906
## result.59   -62.6666667  66.713317  62.666667  0.284699956 0.28469996
## result.60   -38.6666667  40.562709  38.666667  0.179290268 0.17929027
## result.61    -4.3333333  16.782928  13.000000  0.024786603 0.06059928
## result.62     2.5573555   5.472031   5.335920 -0.010893798 0.02281897
## result.63     9.6790764  16.292424  13.076944 -0.036274151 0.05085727
## result.64    -4.2059672  14.820698  11.833293  0.020213878 0.04862415
## result.65    27.1640237  27.565240  27.164024 -0.101187906 0.10118791
## result.66    25.2149388  34.124180  25.214939 -0.083252932 0.08325293
## result.67    49.8465053  63.382919  49.846505 -0.144671266 0.14467127
## result.68    92.6000000  94.802039  92.600000 -0.268119048 0.26811905
## result.69    26.0000000  33.832923  28.000000 -0.072406340 0.07881660
## result.70   -53.0000000  60.808442  53.000000  0.181375058 0.18137506
## result.71  -104.6666667 113.416636 104.666667  0.404446628 0.40444663
## result.72   -49.0000000  52.360927  49.000000  0.192481329 0.19248133
## result.73    -7.6666667  22.248595  17.000000  0.035506129 0.06857263
## result.74    27.3210530  27.919842  27.321053 -0.097574099 0.09757410
## result.75   -42.7784559  49.307489  42.778456  0.151316841 0.15131684
## result.76     7.3333333  19.425070  19.333333 -0.020642223 0.06396352
## result.77    20.2000000  20.315183  20.200000 -0.063880123 0.06388012
## result.78    34.5000000  44.214440  34.500000 -0.097163871 0.09716387
## result.79    55.6464085  68.000349  55.646408 -0.141150461 0.14115046
## result.80   101.6333333 103.015646 101.633333 -0.254418067 0.25441807
## result.81    17.0000000  30.784195  29.666667 -0.039151025 0.07483178
## result.82   -57.6666667  70.420168  57.666667  0.177602190 0.17760219
## result.83  -131.6666667 140.714710 131.666667  0.446689337 0.44668934
## result.84   -60.6666667  62.870237  60.666667  0.210074846 0.21007485
## result.85    -8.6666667  20.864643  14.666667  0.033526621 0.05257424
## result.86    10.7588165  28.062649  27.322081 -0.034055867 0.08908332
## result.87     4.3133849  21.588922  17.212592 -0.008841490 0.05169600
## result.88     7.6442215  15.089531  14.421224 -0.020003432 0.04251839
## result.89    52.0000000  52.121653  52.000000 -0.147221510 0.14722151
## result.90    19.0000000  38.388366  25.000000 -0.043530899 0.06073450
## result.91    64.1495383  78.185931  64.149538 -0.144525734 0.14452573
## result.92   119.5333333 121.322326 119.533333 -0.263234394 0.26323439
## result.93    23.3333333  37.407664  35.333333 -0.048092802 0.07779577
## result.94   -59.0000000  76.700717  60.333333  0.162255027 0.16511013
## result.95  -158.5555556 168.936019 158.555556  0.475514278 0.47551428
## result.96   -74.6666667  76.755022  74.666667  0.230412082 0.23041208
## result.97   -20.0000000  25.390287  20.000000  0.063677083 0.06367708
## result.98     4.7179135  15.737887  15.573390 -0.012981051 0.04711777
## result.99   -40.5685047  44.425409  40.568505  0.122443324 0.12244332
## result.100  -23.4647665  24.390453  23.464767  0.069463121 0.06946312
## result.101   39.6666667  40.253364  39.666667 -0.110573600 0.11057360
## result.102   20.0000000  42.918527  29.333333 -0.043447009 0.07026693
## result.103   81.6666667  97.027488  81.666667 -0.177521559 0.17752156
## result.104  118.0000000 121.814066 118.000000 -0.244220219 0.24422022
## result.105  110.8666667 119.530917 110.866667 -0.230036297 0.23003630
## result.106  -68.3333333  91.631508  77.666667  0.185103928 0.20358578
## result.107 -191.7777778 199.935195 191.777778  0.559859147 0.55985915
## result.108  -68.6666667  71.530879  68.666667  0.209129017 0.20912902
## result.109  -23.3333333  31.016125  24.000000  0.073522879 0.07537473
## result.110  -13.3528520  26.710767  25.131944  0.037819912 0.07277271
## result.111  -20.4938723  33.828711  31.209336  0.060949242 0.08734201
## result.112   -1.1369689  19.588567  17.651912  0.006977388 0.04796086
## result.113   65.3333333  66.070669  65.333333 -0.159904463 0.15990446
## result.114   23.3333333  39.378505  30.000000 -0.049303772 0.06613879
## result.115   78.1547246  93.226203  78.154725 -0.153254906 0.15325491
## result.116  142.4333333 147.592265 142.433333 -0.266447674 0.26644767
## result.117   51.3333333  66.897932  57.333333 -0.091627583 0.10458655
## result.118  -71.6666667  95.266293  79.000000  0.170114888 0.18323355
## result.119 -203.6666667 213.448320 203.666667  0.516665935 0.51666593
## result.120  -71.6666667  74.612331  71.666667  0.186602513 0.18660251
## result.121  -12.3333333  26.639570  19.000000  0.035088950 0.05107616
## result.122   19.6712652  24.761271  20.710301 -0.047781679 0.05043906
## result.123 -100.9369837 105.441362 100.936984  0.249034227 0.24903423
## result.124   -6.9640723  20.706650  20.699766  0.019259081 0.04905451
## result.125   59.6666667  63.887923  59.666667 -0.130093254 0.13009325
## result.126   70.3333333  77.522040  70.333333 -0.140072285 0.14007229
## result.127   82.0000000 102.498780  82.000000 -0.140155095 0.14015510
## result.128  152.0666667 156.696607 152.066667 -0.255553656 0.25555366
## result.129  -82.3333333 129.430805  98.333333  0.159123572 0.18484705
## result.130 -271.0000000 300.908070 271.000000  0.550765811 0.55076581
## result.131 -207.2222222 218.898298 207.222222  0.480275734 0.48027573
## result.132  -80.3333333  85.457592  80.333333  0.193480769 0.19348077
print(res2)
##                      ME       RMSE        MAE          MPE       MAPE
## result.1    -17.4666667  18.474487  17.466667  0.158924192 0.15892419
## result.2     -9.3333333  10.424331   9.333333  0.079589664 0.07958966
## result.3     -2.9321548  12.406309  10.872257  0.031080911 0.08739369
## result.4     10.3308133  10.682605  10.330813 -0.076563916 0.07656392
## result.5      5.0881248   7.171584   6.433807 -0.036313796 0.04707925
## result.6      5.5253837  11.598542   8.508769 -0.035515033 0.05938212
## result.7     18.0932850  26.110632  21.832590 -0.107845328 0.13775977
## result.8     36.3000000  37.625656  36.300000 -0.219692065 0.21969206
## result.9     39.2000000  39.606060  39.200000 -0.235234550 0.23523455
## result.10   -16.3333333  22.457367  16.333333  0.118048285 0.11804829
## result.11   -35.0000000  39.365806  35.000000  0.281790975 0.28179098
## result.12   -29.0000000  31.010751  29.000000  0.234168755 0.23416876
## result.13     0.0000000  13.589211  12.666667  0.011302682 0.09980843
## result.14     2.9000000   5.007661   4.300000 -0.019222222 0.02922222
## result.15     9.8521041  20.833322  16.825631 -0.052991094 0.10108438
## result.16    22.4654809  24.115632  22.465481 -0.134305527 0.13430553
## result.17    23.5102600  23.830725  23.510260 -0.136968702 0.13696870
## result.18    20.3000000  21.215325  20.300000 -0.117556040 0.11755604
## result.19    30.9000000  32.997121  30.900000 -0.165627228 0.16562723
## result.20    39.7000000  40.915645  39.700000 -0.204576252 0.20457625
## result.21    16.0000000  17.492856  16.000000 -0.081221324 0.08122132
## result.22   -17.3333333  23.050669  17.333333  0.103305600 0.10330560
## result.23   -53.8888889  58.697558  53.888889  0.344973669 0.34497367
## result.24   -26.0000000  27.398297  26.000000  0.168170059 0.16817006
## result.25    -1.0000000  10.847427   9.666667  0.010953692 0.06210567
## result.26    -0.7666667   5.843230   5.366667  0.005572818 0.03112837
## result.27     4.6462332  10.873135   9.350221 -0.023004506 0.05051321
## result.28    10.6666667  12.192894  10.666667 -0.056817654 0.05681765
## result.29     6.1200918   6.233263   6.120092 -0.032842259 0.03284226
## result.30    16.2000000  23.475661  16.200000 -0.076832843 0.07683284
## result.31    29.2783498  35.414530  29.278350 -0.130975527 0.13097553
## result.32    53.5000000  54.389797  53.500000 -0.231212275 0.23121228
## result.33     9.0000000  16.340135  15.000000 -0.036095089 0.06480322
## result.34   -16.0000000  26.495283  24.000000  0.085026725 0.11808458
## result.35   -72.6666667  76.469796  72.666667  0.393554409 0.39355441
## result.36   -23.3333333  25.285042  23.333333  0.128892235 0.12889223
## result.37    -3.6666667  11.474610   9.000000  0.023163665 0.05047975
## result.38    -4.5666667   4.662975   4.566667  0.023402763 0.02340276
## result.39     8.0333333  20.496097  15.100000 -0.030984089 0.06703851
## result.40    20.7333333  27.870414  24.466667 -0.086439648 0.10548727
## result.41    30.2333333  30.390953  30.233333 -0.129417299 0.12941730
## result.42    27.2666667  27.863237  27.266667 -0.115178218 0.11517822
## result.43    33.7555478  37.384187  33.755548 -0.134253573 0.13425357
## result.44    49.6666667  51.150106  49.666667 -0.189429700 0.18942970
## result.45    14.6666667  20.960280  18.666667 -0.053615549 0.07049319
## result.46   -24.0000000  34.650637  29.333333  0.111899040 0.13150688
## result.47   -84.8888889  90.848991  84.888889  0.428425354 0.42842535
## result.48   -39.6666667  41.717303  39.666667  0.206331298 0.20633130
## result.49   -36.9000000  38.413669  36.900000  0.192943127 0.19294313
## result.50   -32.6333333  33.364003  32.633333  0.166564238 0.16656424
## result.51   -22.0691733  29.513574  24.898987  0.115042825 0.12708458
## result.52    12.6666667  24.124676  23.333333 -0.049376699 0.10611429
## result.53    44.0000000  44.143705  44.000000 -0.189462455 0.18946245
## result.54     6.6666667  17.378147  12.666667 -0.023444230 0.04978809
## result.55    39.2166799  48.016281  39.216680 -0.137811607 0.13781161
## result.56    67.4333333  69.355197  67.433333 -0.232966640 0.23296664
## result.57    66.5666667  69.094211  66.566667 -0.230453676 0.23045368
## result.58   -41.6666667  49.190107  41.666667  0.171839061 0.17183906
## result.59   -62.6666667  66.713317  62.666667  0.284699956 0.28469996
## result.60   -38.6666667  40.562709  38.666667  0.179290268 0.17929027
## result.61    -4.3333333  16.782928  13.000000  0.024786603 0.06059928
## result.62     2.5573555   5.472031   5.335920 -0.010893798 0.02281897
## result.63     9.6790764  16.292424  13.076944 -0.036274151 0.05085727
## result.64    -4.2059672  14.820698  11.833293  0.020213878 0.04862415
## result.65    27.1640237  27.565240  27.164024 -0.101187906 0.10118791
## result.66    25.2149388  34.124180  25.214939 -0.083252932 0.08325293
## result.67    49.8465053  63.382919  49.846505 -0.144671266 0.14467127
## result.68    92.6000000  94.802039  92.600000 -0.268119048 0.26811905
## result.69    26.0000000  33.832923  28.000000 -0.072406340 0.07881660
## result.70   -53.0000000  60.808442  53.000000  0.181375058 0.18137506
## result.71  -104.6666667 113.416636 104.666667  0.404446628 0.40444663
## result.72   -49.0000000  52.360927  49.000000  0.192481329 0.19248133
## result.73    -7.6666667  22.248595  17.000000  0.035506129 0.06857263
## result.74    27.3210530  27.919842  27.321053 -0.097574099 0.09757410
## result.75   -42.7784559  49.307489  42.778456  0.151316841 0.15131684
## result.76     7.3333333  19.425070  19.333333 -0.020642223 0.06396352
## result.77    20.2000000  20.315183  20.200000 -0.063880123 0.06388012
## result.78    34.5000000  44.214440  34.500000 -0.097163871 0.09716387
## result.79    55.6464085  68.000349  55.646408 -0.141150461 0.14115046
## result.80   101.6333333 103.015646 101.633333 -0.254418067 0.25441807
## result.81    17.0000000  30.784195  29.666667 -0.039151025 0.07483178
## result.82   -57.6666667  70.420168  57.666667  0.177602190 0.17760219
## result.83  -131.6666667 140.714710 131.666667  0.446689337 0.44668934
## result.84   -60.6666667  62.870237  60.666667  0.210074846 0.21007485
## result.85    -8.6666667  20.864643  14.666667  0.033526621 0.05257424
## result.86    10.7588165  28.062649  27.322081 -0.034055867 0.08908332
## result.87     4.3133849  21.588922  17.212592 -0.008841490 0.05169600
## result.88     7.6442215  15.089531  14.421224 -0.020003432 0.04251839
## result.89    52.0000000  52.121653  52.000000 -0.147221510 0.14722151
## result.90    19.0000000  38.388366  25.000000 -0.043530899 0.06073450
## result.91    64.1495383  78.185931  64.149538 -0.144525734 0.14452573
## result.92   119.5333333 121.322326 119.533333 -0.263234394 0.26323439
## result.93    23.3333333  37.407664  35.333333 -0.048092802 0.07779577
## result.94   -59.0000000  76.700717  60.333333  0.162255027 0.16511013
## result.95  -158.5555556 168.936019 158.555556  0.475514278 0.47551428
## result.96   -74.6666667  76.755022  74.666667  0.230412082 0.23041208
## result.97   -20.0000000  25.390287  20.000000  0.063677083 0.06367708
## result.98     4.7179135  15.737887  15.573390 -0.012981051 0.04711777
## result.99   -40.5685047  44.425409  40.568505  0.122443324 0.12244332
## result.100  -23.4647665  24.390453  23.464767  0.069463121 0.06946312
## result.101   39.6666667  40.253364  39.666667 -0.110573600 0.11057360
## result.102   20.0000000  42.918527  29.333333 -0.043447009 0.07026693
## result.103   81.6666667  97.027488  81.666667 -0.177521559 0.17752156
## result.104  118.0000000 121.814066 118.000000 -0.244220219 0.24422022
## result.105  110.8666667 119.530917 110.866667 -0.230036297 0.23003630
## result.106  -68.3333333  91.631508  77.666667  0.185103928 0.20358578
## result.107 -191.7777778 199.935195 191.777778  0.559859147 0.55985915
## result.108  -68.6666667  71.530879  68.666667  0.209129017 0.20912902
## result.109  -23.3333333  31.016125  24.000000  0.073522879 0.07537473
## result.110  -13.3528520  26.710767  25.131944  0.037819912 0.07277271
## result.111  -20.4938723  33.828711  31.209336  0.060949242 0.08734201
## result.112   -1.1369689  19.588567  17.651912  0.006977388 0.04796086
## result.113   65.3333333  66.070669  65.333333 -0.159904463 0.15990446
## result.114   23.3333333  39.378505  30.000000 -0.049303772 0.06613879
## result.115   78.1547246  93.226203  78.154725 -0.153254906 0.15325491
## result.116  142.4333333 147.592265 142.433333 -0.266447674 0.26644767
## result.117   51.3333333  66.897932  57.333333 -0.091627583 0.10458655
## result.118  -71.6666667  95.266293  79.000000  0.170114888 0.18323355
## result.119 -203.6666667 213.448320 203.666667  0.516665935 0.51666593
## result.120  -71.6666667  74.612331  71.666667  0.186602513 0.18660251
## result.121  -12.3333333  26.639570  19.000000  0.035088950 0.05107616
## result.122   19.6712652  24.761271  20.710301 -0.047781679 0.05043906
## result.123 -100.9369837 105.441362 100.936984  0.249034227 0.24903423
## result.124   -6.9640723  20.706650  20.699766  0.019259081 0.04905451
## result.125   59.6666667  63.887923  59.666667 -0.130093254 0.13009325
## result.126   70.3333333  77.522040  70.333333 -0.140072285 0.14007229
## result.127   82.0000000 102.498780  82.000000 -0.140155095 0.14015510
## result.128  152.0666667 156.696607 152.066667 -0.255553656 0.25555366
## result.129  -82.3333333 129.430805  98.333333  0.159123572 0.18484705
## result.130 -271.0000000 300.908070 271.000000  0.550765811 0.55076581
## result.131 -207.2222222 218.898298 207.222222  0.480275734 0.48027573
## result.132  -80.3333333  85.457592  80.333333  0.193480769 0.19348077
##          ME        RMSE         MAE         MPE        MAPE 
## -2.73547115 51.91416756 47.09134413  0.01929547  0.15385307
(res2_parallel <- crossval_ts(y=AirPassengers, initial_window = 10, horizon = 3,
fcast_func = fcast_func, cl=2L))
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##                      ME       RMSE        MAE          MPE       MAPE
## result.1    -17.4666667  18.474487  17.466667  0.158924192 0.15892419
## result.2     -9.3333333  10.424331   9.333333  0.079589664 0.07958966
## result.3     -2.9321548  12.406309  10.872257  0.031080911 0.08739369
## result.4     10.3308133  10.682605  10.330813 -0.076563916 0.07656392
## result.5      5.0881248   7.171584   6.433807 -0.036313796 0.04707925
## result.6      5.5253837  11.598542   8.508769 -0.035515033 0.05938212
## result.7     18.0932850  26.110632  21.832590 -0.107845328 0.13775977
## result.8     36.3000000  37.625656  36.300000 -0.219692065 0.21969206
## result.9     39.2000000  39.606060  39.200000 -0.235234550 0.23523455
## result.10   -16.3333333  22.457367  16.333333  0.118048285 0.11804829
## result.11   -35.0000000  39.365806  35.000000  0.281790975 0.28179098
## result.12   -29.0000000  31.010751  29.000000  0.234168755 0.23416876
## result.13     0.0000000  13.589211  12.666667  0.011302682 0.09980843
## result.14     2.9000000   5.007661   4.300000 -0.019222222 0.02922222
## result.15     9.8521041  20.833322  16.825631 -0.052991094 0.10108438
## result.16    22.4654809  24.115632  22.465481 -0.134305527 0.13430553
## result.17    23.5102600  23.830725  23.510260 -0.136968702 0.13696870
## result.18    20.3000000  21.215325  20.300000 -0.117556040 0.11755604
## result.19    30.9000000  32.997121  30.900000 -0.165627228 0.16562723
## result.20    39.7000000  40.915645  39.700000 -0.204576252 0.20457625
## result.21    16.0000000  17.492856  16.000000 -0.081221324 0.08122132
## result.22   -17.3333333  23.050669  17.333333  0.103305600 0.10330560
## result.23   -53.8888889  58.697558  53.888889  0.344973669 0.34497367
## result.24   -26.0000000  27.398297  26.000000  0.168170059 0.16817006
## result.25    -1.0000000  10.847427   9.666667  0.010953692 0.06210567
## result.26    -0.7666667   5.843230   5.366667  0.005572818 0.03112837
## result.27     4.6462332  10.873135   9.350221 -0.023004506 0.05051321
## result.28    10.6666667  12.192894  10.666667 -0.056817654 0.05681765
## result.29     6.1200918   6.233263   6.120092 -0.032842259 0.03284226
## result.30    16.2000000  23.475661  16.200000 -0.076832843 0.07683284
## result.31    29.2783498  35.414530  29.278350 -0.130975527 0.13097553
## result.32    53.5000000  54.389797  53.500000 -0.231212275 0.23121228
## result.33     9.0000000  16.340135  15.000000 -0.036095089 0.06480322
## result.34   -16.0000000  26.495283  24.000000  0.085026725 0.11808458
## result.35   -72.6666667  76.469796  72.666667  0.393554409 0.39355441
## result.36   -23.3333333  25.285042  23.333333  0.128892235 0.12889223
## result.37    -3.6666667  11.474610   9.000000  0.023163665 0.05047975
## result.38    -4.5666667   4.662975   4.566667  0.023402763 0.02340276
## result.39     8.0333333  20.496097  15.100000 -0.030984089 0.06703851
## result.40    20.7333333  27.870414  24.466667 -0.086439648 0.10548727
## result.41    30.2333333  30.390953  30.233333 -0.129417299 0.12941730
## result.42    27.2666667  27.863237  27.266667 -0.115178218 0.11517822
## result.43    33.7555478  37.384187  33.755548 -0.134253573 0.13425357
## result.44    49.6666667  51.150106  49.666667 -0.189429700 0.18942970
## result.45    14.6666667  20.960280  18.666667 -0.053615549 0.07049319
## result.46   -24.0000000  34.650637  29.333333  0.111899040 0.13150688
## result.47   -84.8888889  90.848991  84.888889  0.428425354 0.42842535
## result.48   -39.6666667  41.717303  39.666667  0.206331298 0.20633130
## result.49   -36.9000000  38.413669  36.900000  0.192943127 0.19294313
## result.50   -32.6333333  33.364003  32.633333  0.166564238 0.16656424
## result.51   -22.0691733  29.513574  24.898987  0.115042825 0.12708458
## result.52    12.6666667  24.124676  23.333333 -0.049376699 0.10611429
## result.53    44.0000000  44.143705  44.000000 -0.189462455 0.18946245
## result.54     6.6666667  17.378147  12.666667 -0.023444230 0.04978809
## result.55    39.2166799  48.016281  39.216680 -0.137811607 0.13781161
## result.56    67.4333333  69.355197  67.433333 -0.232966640 0.23296664
## result.57    66.5666667  69.094211  66.566667 -0.230453676 0.23045368
## result.58   -41.6666667  49.190107  41.666667  0.171839061 0.17183906
## result.59   -62.6666667  66.713317  62.666667  0.284699956 0.28469996
## result.60   -38.6666667  40.562709  38.666667  0.179290268 0.17929027
## result.61    -4.3333333  16.782928  13.000000  0.024786603 0.06059928
## result.62     2.5573555   5.472031   5.335920 -0.010893798 0.02281897
## result.63     9.6790764  16.292424  13.076944 -0.036274151 0.05085727
## result.64    -4.2059672  14.820698  11.833293  0.020213878 0.04862415
## result.65    27.1640237  27.565240  27.164024 -0.101187906 0.10118791
## result.66    25.2149388  34.124180  25.214939 -0.083252932 0.08325293
## result.67    49.8465053  63.382919  49.846505 -0.144671266 0.14467127
## result.68    92.6000000  94.802039  92.600000 -0.268119048 0.26811905
## result.69    26.0000000  33.832923  28.000000 -0.072406340 0.07881660
## result.70   -53.0000000  60.808442  53.000000  0.181375058 0.18137506
## result.71  -104.6666667 113.416636 104.666667  0.404446628 0.40444663
## result.72   -49.0000000  52.360927  49.000000  0.192481329 0.19248133
## result.73    -7.6666667  22.248595  17.000000  0.035506129 0.06857263
## result.74    27.3210530  27.919842  27.321053 -0.097574099 0.09757410
## result.75   -42.7784559  49.307489  42.778456  0.151316841 0.15131684
## result.76     7.3333333  19.425070  19.333333 -0.020642223 0.06396352
## result.77    20.2000000  20.315183  20.200000 -0.063880123 0.06388012
## result.78    34.5000000  44.214440  34.500000 -0.097163871 0.09716387
## result.79    55.6464085  68.000349  55.646408 -0.141150461 0.14115046
## result.80   101.6333333 103.015646 101.633333 -0.254418067 0.25441807
## result.81    17.0000000  30.784195  29.666667 -0.039151025 0.07483178
## result.82   -57.6666667  70.420168  57.666667  0.177602190 0.17760219
## result.83  -131.6666667 140.714710 131.666667  0.446689337 0.44668934
## result.84   -60.6666667  62.870237  60.666667  0.210074846 0.21007485
## result.85    -8.6666667  20.864643  14.666667  0.033526621 0.05257424
## result.86    10.7588165  28.062649  27.322081 -0.034055867 0.08908332
## result.87     4.3133849  21.588922  17.212592 -0.008841490 0.05169600
## result.88     7.6442215  15.089531  14.421224 -0.020003432 0.04251839
## result.89    52.0000000  52.121653  52.000000 -0.147221510 0.14722151
## result.90    19.0000000  38.388366  25.000000 -0.043530899 0.06073450
## result.91    64.1495383  78.185931  64.149538 -0.144525734 0.14452573
## result.92   119.5333333 121.322326 119.533333 -0.263234394 0.26323439
## result.93    23.3333333  37.407664  35.333333 -0.048092802 0.07779577
## result.94   -59.0000000  76.700717  60.333333  0.162255027 0.16511013
## result.95  -158.5555556 168.936019 158.555556  0.475514278 0.47551428
## result.96   -74.6666667  76.755022  74.666667  0.230412082 0.23041208
## result.97   -20.0000000  25.390287  20.000000  0.063677083 0.06367708
## result.98     4.7179135  15.737887  15.573390 -0.012981051 0.04711777
## result.99   -40.5685047  44.425409  40.568505  0.122443324 0.12244332
## result.100  -23.4647665  24.390453  23.464767  0.069463121 0.06946312
## result.101   39.6666667  40.253364  39.666667 -0.110573600 0.11057360
## result.102   20.0000000  42.918527  29.333333 -0.043447009 0.07026693
## result.103   81.6666667  97.027488  81.666667 -0.177521559 0.17752156
## result.104  118.0000000 121.814066 118.000000 -0.244220219 0.24422022
## result.105  110.8666667 119.530917 110.866667 -0.230036297 0.23003630
## result.106  -68.3333333  91.631508  77.666667  0.185103928 0.20358578
## result.107 -191.7777778 199.935195 191.777778  0.559859147 0.55985915
## result.108  -68.6666667  71.530879  68.666667  0.209129017 0.20912902
## result.109  -23.3333333  31.016125  24.000000  0.073522879 0.07537473
## result.110  -13.3528520  26.710767  25.131944  0.037819912 0.07277271
## result.111  -20.4938723  33.828711  31.209336  0.060949242 0.08734201
## result.112   -1.1369689  19.588567  17.651912  0.006977388 0.04796086
## result.113   65.3333333  66.070669  65.333333 -0.159904463 0.15990446
## result.114   23.3333333  39.378505  30.000000 -0.049303772 0.06613879
## result.115   78.1547246  93.226203  78.154725 -0.153254906 0.15325491
## result.116  142.4333333 147.592265 142.433333 -0.266447674 0.26644767
## result.117   51.3333333  66.897932  57.333333 -0.091627583 0.10458655
## result.118  -71.6666667  95.266293  79.000000  0.170114888 0.18323355
## result.119 -203.6666667 213.448320 203.666667  0.516665935 0.51666593
## result.120  -71.6666667  74.612331  71.666667  0.186602513 0.18660251
## result.121  -12.3333333  26.639570  19.000000  0.035088950 0.05107616
## result.122   19.6712652  24.761271  20.710301 -0.047781679 0.05043906
## result.123 -100.9369837 105.441362 100.936984  0.249034227 0.24903423
## result.124   -6.9640723  20.706650  20.699766  0.019259081 0.04905451
## result.125   59.6666667  63.887923  59.666667 -0.130093254 0.13009325
## result.126   70.3333333  77.522040  70.333333 -0.140072285 0.14007229
## result.127   82.0000000 102.498780  82.000000 -0.140155095 0.14015510
## result.128  152.0666667 156.696607 152.066667 -0.255553656 0.25555366
## result.129  -82.3333333 129.430805  98.333333  0.159123572 0.18484705
## result.130 -271.0000000 300.908070 271.000000  0.550765811 0.55076581
## result.131 -207.2222222 218.898298 207.222222  0.480275734 0.48027573
## result.132  -80.3333333  85.457592  80.333333  0.193480769 0.19348077
print(res2_parallel)
##                      ME       RMSE        MAE          MPE       MAPE
## result.1    -17.4666667  18.474487  17.466667  0.158924192 0.15892419
## result.2     -9.3333333  10.424331   9.333333  0.079589664 0.07958966
## result.3     -2.9321548  12.406309  10.872257  0.031080911 0.08739369
## result.4     10.3308133  10.682605  10.330813 -0.076563916 0.07656392
## result.5      5.0881248   7.171584   6.433807 -0.036313796 0.04707925
## result.6      5.5253837  11.598542   8.508769 -0.035515033 0.05938212
## result.7     18.0932850  26.110632  21.832590 -0.107845328 0.13775977
## result.8     36.3000000  37.625656  36.300000 -0.219692065 0.21969206
## result.9     39.2000000  39.606060  39.200000 -0.235234550 0.23523455
## result.10   -16.3333333  22.457367  16.333333  0.118048285 0.11804829
## result.11   -35.0000000  39.365806  35.000000  0.281790975 0.28179098
## result.12   -29.0000000  31.010751  29.000000  0.234168755 0.23416876
## result.13     0.0000000  13.589211  12.666667  0.011302682 0.09980843
## result.14     2.9000000   5.007661   4.300000 -0.019222222 0.02922222
## result.15     9.8521041  20.833322  16.825631 -0.052991094 0.10108438
## result.16    22.4654809  24.115632  22.465481 -0.134305527 0.13430553
## result.17    23.5102600  23.830725  23.510260 -0.136968702 0.13696870
## result.18    20.3000000  21.215325  20.300000 -0.117556040 0.11755604
## result.19    30.9000000  32.997121  30.900000 -0.165627228 0.16562723
## result.20    39.7000000  40.915645  39.700000 -0.204576252 0.20457625
## result.21    16.0000000  17.492856  16.000000 -0.081221324 0.08122132
## result.22   -17.3333333  23.050669  17.333333  0.103305600 0.10330560
## result.23   -53.8888889  58.697558  53.888889  0.344973669 0.34497367
## result.24   -26.0000000  27.398297  26.000000  0.168170059 0.16817006
## result.25    -1.0000000  10.847427   9.666667  0.010953692 0.06210567
## result.26    -0.7666667   5.843230   5.366667  0.005572818 0.03112837
## result.27     4.6462332  10.873135   9.350221 -0.023004506 0.05051321
## result.28    10.6666667  12.192894  10.666667 -0.056817654 0.05681765
## result.29     6.1200918   6.233263   6.120092 -0.032842259 0.03284226
## result.30    16.2000000  23.475661  16.200000 -0.076832843 0.07683284
## result.31    29.2783498  35.414530  29.278350 -0.130975527 0.13097553
## result.32    53.5000000  54.389797  53.500000 -0.231212275 0.23121228
## result.33     9.0000000  16.340135  15.000000 -0.036095089 0.06480322
## result.34   -16.0000000  26.495283  24.000000  0.085026725 0.11808458
## result.35   -72.6666667  76.469796  72.666667  0.393554409 0.39355441
## result.36   -23.3333333  25.285042  23.333333  0.128892235 0.12889223
## result.37    -3.6666667  11.474610   9.000000  0.023163665 0.05047975
## result.38    -4.5666667   4.662975   4.566667  0.023402763 0.02340276
## result.39     8.0333333  20.496097  15.100000 -0.030984089 0.06703851
## result.40    20.7333333  27.870414  24.466667 -0.086439648 0.10548727
## result.41    30.2333333  30.390953  30.233333 -0.129417299 0.12941730
## result.42    27.2666667  27.863237  27.266667 -0.115178218 0.11517822
## result.43    33.7555478  37.384187  33.755548 -0.134253573 0.13425357
## result.44    49.6666667  51.150106  49.666667 -0.189429700 0.18942970
## result.45    14.6666667  20.960280  18.666667 -0.053615549 0.07049319
## result.46   -24.0000000  34.650637  29.333333  0.111899040 0.13150688
## result.47   -84.8888889  90.848991  84.888889  0.428425354 0.42842535
## result.48   -39.6666667  41.717303  39.666667  0.206331298 0.20633130
## result.49   -36.9000000  38.413669  36.900000  0.192943127 0.19294313
## result.50   -32.6333333  33.364003  32.633333  0.166564238 0.16656424
## result.51   -22.0691733  29.513574  24.898987  0.115042825 0.12708458
## result.52    12.6666667  24.124676  23.333333 -0.049376699 0.10611429
## result.53    44.0000000  44.143705  44.000000 -0.189462455 0.18946245
## result.54     6.6666667  17.378147  12.666667 -0.023444230 0.04978809
## result.55    39.2166799  48.016281  39.216680 -0.137811607 0.13781161
## result.56    67.4333333  69.355197  67.433333 -0.232966640 0.23296664
## result.57    66.5666667  69.094211  66.566667 -0.230453676 0.23045368
## result.58   -41.6666667  49.190107  41.666667  0.171839061 0.17183906
## result.59   -62.6666667  66.713317  62.666667  0.284699956 0.28469996
## result.60   -38.6666667  40.562709  38.666667  0.179290268 0.17929027
## result.61    -4.3333333  16.782928  13.000000  0.024786603 0.06059928
## result.62     2.5573555   5.472031   5.335920 -0.010893798 0.02281897
## result.63     9.6790764  16.292424  13.076944 -0.036274151 0.05085727
## result.64    -4.2059672  14.820698  11.833293  0.020213878 0.04862415
## result.65    27.1640237  27.565240  27.164024 -0.101187906 0.10118791
## result.66    25.2149388  34.124180  25.214939 -0.083252932 0.08325293
## result.67    49.8465053  63.382919  49.846505 -0.144671266 0.14467127
## result.68    92.6000000  94.802039  92.600000 -0.268119048 0.26811905
## result.69    26.0000000  33.832923  28.000000 -0.072406340 0.07881660
## result.70   -53.0000000  60.808442  53.000000  0.181375058 0.18137506
## result.71  -104.6666667 113.416636 104.666667  0.404446628 0.40444663
## result.72   -49.0000000  52.360927  49.000000  0.192481329 0.19248133
## result.73    -7.6666667  22.248595  17.000000  0.035506129 0.06857263
## result.74    27.3210530  27.919842  27.321053 -0.097574099 0.09757410
## result.75   -42.7784559  49.307489  42.778456  0.151316841 0.15131684
## result.76     7.3333333  19.425070  19.333333 -0.020642223 0.06396352
## result.77    20.2000000  20.315183  20.200000 -0.063880123 0.06388012
## result.78    34.5000000  44.214440  34.500000 -0.097163871 0.09716387
## result.79    55.6464085  68.000349  55.646408 -0.141150461 0.14115046
## result.80   101.6333333 103.015646 101.633333 -0.254418067 0.25441807
## result.81    17.0000000  30.784195  29.666667 -0.039151025 0.07483178
## result.82   -57.6666667  70.420168  57.666667  0.177602190 0.17760219
## result.83  -131.6666667 140.714710 131.666667  0.446689337 0.44668934
## result.84   -60.6666667  62.870237  60.666667  0.210074846 0.21007485
## result.85    -8.6666667  20.864643  14.666667  0.033526621 0.05257424
## result.86    10.7588165  28.062649  27.322081 -0.034055867 0.08908332
## result.87     4.3133849  21.588922  17.212592 -0.008841490 0.05169600
## result.88     7.6442215  15.089531  14.421224 -0.020003432 0.04251839
## result.89    52.0000000  52.121653  52.000000 -0.147221510 0.14722151
## result.90    19.0000000  38.388366  25.000000 -0.043530899 0.06073450
## result.91    64.1495383  78.185931  64.149538 -0.144525734 0.14452573
## result.92   119.5333333 121.322326 119.533333 -0.263234394 0.26323439
## result.93    23.3333333  37.407664  35.333333 -0.048092802 0.07779577
## result.94   -59.0000000  76.700717  60.333333  0.162255027 0.16511013
## result.95  -158.5555556 168.936019 158.555556  0.475514278 0.47551428
## result.96   -74.6666667  76.755022  74.666667  0.230412082 0.23041208
## result.97   -20.0000000  25.390287  20.000000  0.063677083 0.06367708
## result.98     4.7179135  15.737887  15.573390 -0.012981051 0.04711777
## result.99   -40.5685047  44.425409  40.568505  0.122443324 0.12244332
## result.100  -23.4647665  24.390453  23.464767  0.069463121 0.06946312
## result.101   39.6666667  40.253364  39.666667 -0.110573600 0.11057360
## result.102   20.0000000  42.918527  29.333333 -0.043447009 0.07026693
## result.103   81.6666667  97.027488  81.666667 -0.177521559 0.17752156
## result.104  118.0000000 121.814066 118.000000 -0.244220219 0.24422022
## result.105  110.8666667 119.530917 110.866667 -0.230036297 0.23003630
## result.106  -68.3333333  91.631508  77.666667  0.185103928 0.20358578
## result.107 -191.7777778 199.935195 191.777778  0.559859147 0.55985915
## result.108  -68.6666667  71.530879  68.666667  0.209129017 0.20912902
## result.109  -23.3333333  31.016125  24.000000  0.073522879 0.07537473
## result.110  -13.3528520  26.710767  25.131944  0.037819912 0.07277271
## result.111  -20.4938723  33.828711  31.209336  0.060949242 0.08734201
## result.112   -1.1369689  19.588567  17.651912  0.006977388 0.04796086
## result.113   65.3333333  66.070669  65.333333 -0.159904463 0.15990446
## result.114   23.3333333  39.378505  30.000000 -0.049303772 0.06613879
## result.115   78.1547246  93.226203  78.154725 -0.153254906 0.15325491
## result.116  142.4333333 147.592265 142.433333 -0.266447674 0.26644767
## result.117   51.3333333  66.897932  57.333333 -0.091627583 0.10458655
## result.118  -71.6666667  95.266293  79.000000  0.170114888 0.18323355
## result.119 -203.6666667 213.448320 203.666667  0.516665935 0.51666593
## result.120  -71.6666667  74.612331  71.666667  0.186602513 0.18660251
## result.121  -12.3333333  26.639570  19.000000  0.035088950 0.05107616
## result.122   19.6712652  24.761271  20.710301 -0.047781679 0.05043906
## result.123 -100.9369837 105.441362 100.936984  0.249034227 0.24903423
## result.124   -6.9640723  20.706650  20.699766  0.019259081 0.04905451
## result.125   59.6666667  63.887923  59.666667 -0.130093254 0.13009325
## result.126   70.3333333  77.522040  70.333333 -0.140072285 0.14007229
## result.127   82.0000000 102.498780  82.000000 -0.140155095 0.14015510
## result.128  152.0666667 156.696607 152.066667 -0.255553656 0.25555366
## result.129  -82.3333333 129.430805  98.333333  0.159123572 0.18484705
## result.130 -271.0000000 300.908070 271.000000  0.550765811 0.55076581
## result.131 -207.2222222 218.898298 207.222222  0.480275734 0.48027573
## result.132  -80.3333333  85.457592  80.333333  0.193480769 0.19348077
print(colMeans(res2_parallel))
##          ME        RMSE         MAE         MPE        MAPE 
## -2.73547115 51.91416756 47.09134413  0.01929547  0.15385307
# Example 3 -----

fcast_func <- function (y, h, ...)
{
      forecast::forecast(forecast::ets(y, ...),
      h=h, ...)
}

(res3 <- crossval_ts(y=AirPassengers,
initial_window = 10, horizon = 3, fcast_func = fcast_func))
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##                       ME       RMSE        MAE          MPE       MAPE
## result.1   -1.746854e+01  18.476258  17.468540  0.158940920 0.15894092
## result.2    1.566517e+01  16.338696  15.665166 -0.129620578 0.12962058
## result.3    9.334734e+00  14.166609  11.333800 -0.066852965 0.08423615
## result.4    1.899970e+01  19.974699  18.999700 -0.139946721 0.13994672
## result.5    7.667768e+00  10.116828   8.333700 -0.055024805 0.06035227
## result.6   -4.665166e+00  10.892780   9.999500  0.039573326 0.07537422
## result.7    1.299940e+01  22.516314  19.666467 -0.073276577 0.12661311
## result.8    3.799900e+01  39.267339  37.998999 -0.230155699 0.23015570
## result.9    2.638065e+01  26.980335  26.380646 -0.157917523 0.15791752
## result.10  -1.194882e+01  19.502045  14.871827  0.089212388 0.10640658
## result.11  -3.500000e+01  39.365806  35.000000  0.281790974 0.28179097
## result.12  -2.900120e+01  31.011874  29.001201  0.234178134 0.23417813
## result.13  -2.503146e-03  13.589212  12.665832  0.011321715 0.09980403
## result.14   3.099810e+01  31.265776  30.998098 -0.213156000 0.21315600
## result.15   2.061362e+01  25.215278  20.613620 -0.123719557 0.12371956
## result.16   1.976075e+01  22.833664  19.760752 -0.116436221 0.11643622
## result.17   2.559638e+01  26.328207  25.596376 -0.148569678 0.14856968
## result.18  -6.997195e+00   9.325274   6.999065  0.042286329 0.04229683
## result.19   1.999849e+01  23.107133  19.998490 -0.105824971 0.10582497
## result.20   2.000094e+01  22.316759  20.000943 -0.101693143 0.10169314
## result.21   1.716873e+01  18.567860  17.168734 -0.087253955 0.08725395
## result.22  -1.309391e+01  20.058398  15.920192  0.079801175 0.09400360
## result.23  -3.498229e+01  38.293959  34.982290  0.224201208 0.22420121
## result.24  -2.780590e+01  29.117601  27.805899  0.179635262 0.17963526
## result.25  -1.002243e+00  10.847634   9.665919  0.010967687 0.06210191
## result.26   2.633173e+01  26.961372  26.331730 -0.151847236 0.15184724
## result.27   1.533533e+01  17.796854  15.335334 -0.082315697 0.08231570
## result.28   1.366717e+01  14.888934  13.667167 -0.073082132 0.07308213
## result.29   5.667569e+00   7.725082   5.667569 -0.029763467 0.02976347
## result.30   1.001302e+00  17.019673  15.666233  0.002081324 0.07853791
## result.31   2.933213e+01  35.467303  29.332133 -0.131226644 0.13122664
## result.32   4.700020e+01  48.010611  47.000200 -0.202900850 0.20290085
## result.33   9.003504e+00  16.342065  15.001168 -0.036110584 0.06480754
## result.34  -1.599880e+01  26.494558  23.999600  0.085021061 0.11808222
## result.35  -5.133213e+01  53.508971  51.332132  0.277289389 0.27728939
## result.36  -2.333664e+01  25.288089  23.336635  0.128910070 0.12891007
## result.37  -3.668468e+00  11.475185   8.999400  0.023173314 0.05047708
## result.38   2.333143e+01  23.350474  23.331432 -0.119423608 0.11942361
## result.39   1.533553e+01  24.305024  15.335535 -0.066135351 0.06613535
## result.40   2.633353e+01  32.254362  26.333534 -0.111817232 0.11181723
## result.41   3.733333e+01  37.461091  37.333333 -0.159851259 0.15985126
## result.42  -3.293244e-01   5.744331   4.998664  0.001988452 0.02120384
## result.43   1.033323e+01  17.710579  14.333300 -0.038856064 0.05632360
## result.44   3.066607e+01  33.014590  30.666065 -0.116090067 0.11609007
## result.45   1.596020e+01  21.884930  19.097844 -0.058653322 0.07189233
## result.46  -2.186468e+01  33.207394  28.621561  0.102905641 0.12774711
## result.47  -6.266586e+01  66.857306  62.665864  0.315959437 0.31595944
## result.48  -3.967020e+01  41.720658  39.670195  0.206349258 0.20634926
## result.49  -1.600260e+01  19.237548  16.002602  0.085442448 0.08544245
## result.50   1.766356e+01  18.979561  17.663564 -0.088210238 0.08821024
## result.51   8.002104e+00  21.087919  16.667368 -0.030089418 0.07618125
## result.52   1.266697e+01  24.124834  23.333434 -0.049378101 0.10611462
## result.53   4.399840e+01  44.142108  43.998398 -0.189455546 0.18945555
## result.54   6.671416e+00  17.379970  12.665083 -0.023463967 0.04978035
## result.55   3.937407e+01  48.213476  39.374066 -0.138360925 0.13836092
## result.56   5.245156e+01  54.900407  52.451557 -0.180469964 0.18046996
## result.57   2.833573e+01  33.849702  28.335728 -0.095559229 0.09555923
## result.58  -3.035671e+01  40.063520  31.896682  0.127953483 0.13320935
## result.59  -6.290693e+01  66.939055  62.906928  0.285753416 0.28575342
## result.60  -3.872977e+01  40.622870  38.729773  0.179577606 0.17957761
## result.61  -4.336335e+00  16.783703  13.001001  0.024800038 0.06060444
## result.62   3.166407e+01  32.127382  31.664066 -0.134472082 0.13447208
## result.63   1.833594e+01  23.304408  18.335936 -0.071080054 0.07108005
## result.64   1.433463e+01  21.871685  20.333767 -0.051797796 0.07754515
## result.65   3.566577e+01  35.687567  35.665766 -0.132732266 0.13273227
## result.66   1.767007e+01  27.793052  17.670073 -0.056987701 0.05698770
## result.67   4.733353e+01  60.942752  47.333534 -0.136908797 0.13690880
## result.68   7.200010e+01  74.810969  72.000100 -0.207667267 0.20766727
## result.69   2.600458e+01  33.836441  28.001526 -0.072419822 0.07882030
## result.70  -5.299496e+01  60.804046  52.994956  0.181358689 0.18135869
## result.71  -7.266837e+01  78.855862  72.668366  0.280918896 0.28091890
## result.72  -4.900354e+01  52.364240  49.003541  0.192494862 0.19249486
## result.73  -7.670468e+00  22.249906  16.998733  0.035520495 0.06856896
## result.74   4.266297e+01  42.774809  42.662966 -0.152446535 0.15244653
## result.75   1.467077e+01  22.791526  15.334700 -0.046862407 0.04925928
## result.76   1.833393e+01  25.684406  23.000201 -0.057162646 0.07400837
## result.77   3.899930e+01  39.059081  38.999297 -0.123374360 0.12337436
## result.78   1.800402e+01  32.997142  20.668006 -0.047602582 0.05611372
## result.79   5.533293e+01  67.690145  55.332933 -0.140317443 0.14031744
## result.80   7.933383e+01  81.097140  79.333835 -0.198191813 0.19819181
## result.81   3.492602e+01  43.341592  35.642007 -0.085205032 0.08722189
## result.82  -5.039041e+01  64.596821  50.390413  0.156855140 0.15685514
## result.83  -9.433413e+01 100.428171  94.334133  0.319615748 0.31961575
## result.84  -6.067167e+01  62.875067  60.671673  0.210091910 0.21009191
## result.85  -8.671568e+00  20.866679  14.668300  0.033543175 0.05258042
## result.86   3.632983e+01  36.788750  36.329831 -0.117898398 0.11789840
## result.87   1.800350e+01  29.475290  21.334501 -0.050814052 0.06188049
## result.88   2.000090e+01  31.443643  29.333633 -0.054497535 0.08550329
## result.89   5.199860e+01  52.120255  51.998599 -0.147217542 0.14721754
## result.90   1.900550e+01  38.391090  24.998166 -0.043545680 0.06072841
## result.91   6.599920e+01  80.028501  65.999199 -0.148893596 0.14889360
## result.92   9.633403e+01  98.545092  96.334034 -0.211720092 0.21172009
## result.93   4.001969e+01  49.562736  40.895453 -0.085732296 0.08790002
## result.94  -5.320270e+01  72.336208  58.400899  0.147764823 0.15889588
## result.95  -1.149998e+02 121.946520 114.999799  0.344302575 0.34430258
## result.96  -7.467297e+01  76.761154  74.672971  0.230431282 0.23043128
## result.97  -2.000570e+01  25.394778  20.005701  0.063694559 0.06369456
## result.98   2.632913e+01  28.013900  26.329129 -0.078681829 0.07868183
## result.99   4.003100e+00  18.403572  16.001033 -0.009003855 0.04673321
## result.100  2.667068e+00  18.547295  17.333467 -0.004861041 0.05098179
## result.101  3.966447e+01  40.251196  39.664466 -0.110567445 0.11056744
## result.102  2.000441e+01  42.920584  29.334804 -0.043458671 0.07027014
## result.103  8.166527e+01  97.026309  81.665266 -0.177518248 0.17751825
## result.104  1.140015e+02 117.944939 114.001500 -0.235802433 0.23580243
## result.105  4.548324e+01  63.756939  56.938858 -0.088544264 0.11689975
## result.106 -6.733087e+01  90.886381  77.332511  0.182684317 0.20248956
## result.107 -1.473319e+02 152.250626 147.331929  0.428568527 0.42856853
## result.108 -6.867677e+01  71.540578  68.676770  0.209159255 0.20915926
## result.109 -2.333783e+01  31.019511  24.001500  0.073536339 0.07537986
## result.110  3.632843e+01  37.647183  36.328430 -0.104177443 0.10417744
## result.111  3.233603e+01  42.093245  32.336034 -0.082827186 0.08282719
## result.112  2.133563e+01  35.290671  33.334100 -0.050532069 0.08561530
## result.113  6.533153e+01  66.068888  65.331532 -0.159900038 0.15990004
## result.114  2.333975e+01  39.382309  30.002140 -0.049318805 0.06614301
## result.115  8.399900e+01  99.088337  83.998999 -0.165175246 0.16517525
## result.116  1.063358e+02 113.152617 106.335753 -0.197472775 0.19747278
## result.117  5.133863e+01  66.901999  57.335100 -0.091637783 0.10458911
## result.118 -6.860590e+01  92.985616  77.979744  0.163579394 0.18034835
## result.119 -1.483322e+02 153.978372 148.332229  0.374999511 0.37499951
## result.120 -7.167639e+01  74.621670  71.676389  0.186627429 0.18662743
## result.121 -1.233895e+01  26.642171  19.001872  0.035103235 0.05108146
## result.122  4.232883e+01  43.641938  42.328830 -0.104067563 0.10406756
## result.123  4.004304e+00  13.367914  13.334768 -0.008805279 0.03266836
## result.124  6.667869e+00  29.529918  24.000401 -0.011243449 0.05557217
## result.125  5.966407e+01  63.885493  59.664065 -0.130087467 0.13008747
## result.126  7.033614e+01  77.524589  70.336143 -0.140078053 0.14007805
## result.127  8.200427e+01 102.502194  82.004267 -0.140163054 0.14016305
## result.128  1.156678e+02 121.690542 115.667768 -0.193347441 0.19334744
## result.129  4.367297e+01  66.684958  61.668767 -0.067972141 0.10339694
## result.130 -9.699129e+01 114.262754  96.991290  0.200000819 0.20000082
## result.131 -1.530016e+02 160.505938 153.001600  0.353767961 0.35376796
## result.132 -8.034313e+01  85.466805  80.343135  0.193503795 0.19350380
##           ME         RMSE          MAE          MPE         MAPE 
##  5.606310555 45.344040192 40.688899803 -0.008682302  0.136229866
(res3_parallel <- crossval_ts(y=AirPassengers,
initial_window = 10, horizon = 3, fcast_func = fcast_func, cl=2L))
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##                       ME       RMSE        MAE          MPE       MAPE
## result.1   -1.746854e+01  18.476258  17.468540  0.158940920 0.15894092
## result.2    1.566517e+01  16.338696  15.665166 -0.129620578 0.12962058
## result.3    9.334734e+00  14.166609  11.333800 -0.066852965 0.08423615
## result.4    1.899970e+01  19.974699  18.999700 -0.139946721 0.13994672
## result.5    7.667768e+00  10.116828   8.333700 -0.055024805 0.06035227
## result.6   -4.665166e+00  10.892780   9.999500  0.039573326 0.07537422
## result.7    1.299940e+01  22.516314  19.666467 -0.073276577 0.12661311
## result.8    3.799900e+01  39.267339  37.998999 -0.230155699 0.23015570
## result.9    2.638065e+01  26.980335  26.380646 -0.157917523 0.15791752
## result.10  -1.194882e+01  19.502045  14.871827  0.089212388 0.10640658
## result.11  -3.500000e+01  39.365806  35.000000  0.281790974 0.28179097
## result.12  -2.900120e+01  31.011874  29.001201  0.234178134 0.23417813
## result.13  -2.503146e-03  13.589212  12.665832  0.011321715 0.09980403
## result.14   3.099810e+01  31.265776  30.998098 -0.213156000 0.21315600
## result.15   2.061362e+01  25.215278  20.613620 -0.123719557 0.12371956
## result.16   1.976075e+01  22.833664  19.760752 -0.116436221 0.11643622
## result.17   2.559638e+01  26.328207  25.596376 -0.148569678 0.14856968
## result.18  -6.997195e+00   9.325274   6.999065  0.042286329 0.04229683
## result.19   1.999849e+01  23.107133  19.998490 -0.105824971 0.10582497
## result.20   2.000094e+01  22.316759  20.000943 -0.101693143 0.10169314
## result.21   1.716873e+01  18.567860  17.168734 -0.087253955 0.08725395
## result.22  -1.309391e+01  20.058398  15.920192  0.079801175 0.09400360
## result.23  -3.498229e+01  38.293959  34.982290  0.224201208 0.22420121
## result.24  -2.780590e+01  29.117601  27.805899  0.179635262 0.17963526
## result.25  -1.002243e+00  10.847634   9.665919  0.010967687 0.06210191
## result.26   2.633173e+01  26.961372  26.331730 -0.151847236 0.15184724
## result.27   1.533533e+01  17.796854  15.335334 -0.082315697 0.08231570
## result.28   1.366717e+01  14.888934  13.667167 -0.073082132 0.07308213
## result.29   5.667569e+00   7.725082   5.667569 -0.029763467 0.02976347
## result.30   1.001302e+00  17.019673  15.666233  0.002081324 0.07853791
## result.31   2.933213e+01  35.467303  29.332133 -0.131226644 0.13122664
## result.32   4.700020e+01  48.010611  47.000200 -0.202900850 0.20290085
## result.33   9.003504e+00  16.342065  15.001168 -0.036110584 0.06480754
## result.34  -1.599880e+01  26.494558  23.999600  0.085021061 0.11808222
## result.35  -5.133213e+01  53.508971  51.332132  0.277289389 0.27728939
## result.36  -2.333664e+01  25.288089  23.336635  0.128910070 0.12891007
## result.37  -3.668468e+00  11.475185   8.999400  0.023173314 0.05047708
## result.38   2.333143e+01  23.350474  23.331432 -0.119423608 0.11942361
## result.39   1.533553e+01  24.305024  15.335535 -0.066135351 0.06613535
## result.40   2.633353e+01  32.254362  26.333534 -0.111817232 0.11181723
## result.41   3.733333e+01  37.461091  37.333333 -0.159851259 0.15985126
## result.42  -3.293244e-01   5.744331   4.998664  0.001988452 0.02120384
## result.43   1.033323e+01  17.710579  14.333300 -0.038856064 0.05632360
## result.44   3.066607e+01  33.014590  30.666065 -0.116090067 0.11609007
## result.45   1.596020e+01  21.884930  19.097844 -0.058653322 0.07189233
## result.46  -2.186468e+01  33.207394  28.621561  0.102905641 0.12774711
## result.47  -6.266586e+01  66.857306  62.665864  0.315959437 0.31595944
## result.48  -3.967020e+01  41.720658  39.670195  0.206349258 0.20634926
## result.49  -1.600260e+01  19.237548  16.002602  0.085442448 0.08544245
## result.50   1.766356e+01  18.979561  17.663564 -0.088210238 0.08821024
## result.51   8.002104e+00  21.087919  16.667368 -0.030089418 0.07618125
## result.52   1.266697e+01  24.124834  23.333434 -0.049378101 0.10611462
## result.53   4.399840e+01  44.142108  43.998398 -0.189455546 0.18945555
## result.54   6.671416e+00  17.379970  12.665083 -0.023463967 0.04978035
## result.55   3.937407e+01  48.213476  39.374066 -0.138360925 0.13836092
## result.56   5.245156e+01  54.900407  52.451557 -0.180469964 0.18046996
## result.57   2.833573e+01  33.849702  28.335728 -0.095559229 0.09555923
## result.58  -3.035671e+01  40.063520  31.896682  0.127953483 0.13320935
## result.59  -6.290693e+01  66.939055  62.906928  0.285753416 0.28575342
## result.60  -3.872977e+01  40.622870  38.729773  0.179577606 0.17957761
## result.61  -4.336335e+00  16.783703  13.001001  0.024800038 0.06060444
## result.62   3.166407e+01  32.127382  31.664066 -0.134472082 0.13447208
## result.63   1.833594e+01  23.304408  18.335936 -0.071080054 0.07108005
## result.64   1.433463e+01  21.871685  20.333767 -0.051797796 0.07754515
## result.65   3.566577e+01  35.687567  35.665766 -0.132732266 0.13273227
## result.66   1.767007e+01  27.793052  17.670073 -0.056987701 0.05698770
## result.67   4.733353e+01  60.942752  47.333534 -0.136908797 0.13690880
## result.68   7.200010e+01  74.810969  72.000100 -0.207667267 0.20766727
## result.69   2.600458e+01  33.836441  28.001526 -0.072419822 0.07882030
## result.70  -5.299496e+01  60.804046  52.994956  0.181358689 0.18135869
## result.71  -7.266837e+01  78.855862  72.668366  0.280918896 0.28091890
## result.72  -4.900354e+01  52.364240  49.003541  0.192494862 0.19249486
## result.73  -7.670468e+00  22.249906  16.998733  0.035520495 0.06856896
## result.74   4.266297e+01  42.774809  42.662966 -0.152446535 0.15244653
## result.75   1.467077e+01  22.791526  15.334700 -0.046862407 0.04925928
## result.76   1.833393e+01  25.684406  23.000201 -0.057162646 0.07400837
## result.77   3.899930e+01  39.059081  38.999297 -0.123374360 0.12337436
## result.78   1.800402e+01  32.997142  20.668006 -0.047602582 0.05611372
## result.79   5.533293e+01  67.690145  55.332933 -0.140317443 0.14031744
## result.80   7.933383e+01  81.097140  79.333835 -0.198191813 0.19819181
## result.81   3.492602e+01  43.341592  35.642007 -0.085205032 0.08722189
## result.82  -5.039041e+01  64.596821  50.390413  0.156855140 0.15685514
## result.83  -9.433413e+01 100.428171  94.334133  0.319615748 0.31961575
## result.84  -6.067167e+01  62.875067  60.671673  0.210091910 0.21009191
## result.85  -8.671568e+00  20.866679  14.668300  0.033543175 0.05258042
## result.86   3.632983e+01  36.788750  36.329831 -0.117898398 0.11789840
## result.87   1.800350e+01  29.475290  21.334501 -0.050814052 0.06188049
## result.88   2.000090e+01  31.443643  29.333633 -0.054497535 0.08550329
## result.89   5.199860e+01  52.120255  51.998599 -0.147217542 0.14721754
## result.90   1.900550e+01  38.391090  24.998166 -0.043545680 0.06072841
## result.91   6.599920e+01  80.028501  65.999199 -0.148893596 0.14889360
## result.92   9.633403e+01  98.545092  96.334034 -0.211720092 0.21172009
## result.93   4.001969e+01  49.562736  40.895453 -0.085732296 0.08790002
## result.94  -5.320270e+01  72.336208  58.400899  0.147764823 0.15889588
## result.95  -1.149998e+02 121.946520 114.999799  0.344302575 0.34430258
## result.96  -7.467297e+01  76.761154  74.672971  0.230431282 0.23043128
## result.97  -2.000570e+01  25.394778  20.005701  0.063694559 0.06369456
## result.98   2.632913e+01  28.013900  26.329129 -0.078681829 0.07868183
## result.99   4.003100e+00  18.403572  16.001033 -0.009003855 0.04673321
## result.100  2.667068e+00  18.547295  17.333467 -0.004861041 0.05098179
## result.101  3.966447e+01  40.251196  39.664466 -0.110567445 0.11056744
## result.102  2.000441e+01  42.920584  29.334804 -0.043458671 0.07027014
## result.103  8.166527e+01  97.026309  81.665266 -0.177518248 0.17751825
## result.104  1.140015e+02 117.944939 114.001500 -0.235802433 0.23580243
## result.105  4.548324e+01  63.756939  56.938858 -0.088544264 0.11689975
## result.106 -6.733087e+01  90.886381  77.332511  0.182684317 0.20248956
## result.107 -1.473319e+02 152.250626 147.331929  0.428568527 0.42856853
## result.108 -6.867677e+01  71.540578  68.676770  0.209159255 0.20915926
## result.109 -2.333783e+01  31.019511  24.001500  0.073536339 0.07537986
## result.110  3.632843e+01  37.647183  36.328430 -0.104177443 0.10417744
## result.111  3.233603e+01  42.093245  32.336034 -0.082827186 0.08282719
## result.112  2.133563e+01  35.290671  33.334100 -0.050532069 0.08561530
## result.113  6.533153e+01  66.068888  65.331532 -0.159900038 0.15990004
## result.114  2.333975e+01  39.382309  30.002140 -0.049318805 0.06614301
## result.115  8.399900e+01  99.088337  83.998999 -0.165175246 0.16517525
## result.116  1.063358e+02 113.152617 106.335753 -0.197472775 0.19747278
## result.117  5.133863e+01  66.901999  57.335100 -0.091637783 0.10458911
## result.118 -6.860590e+01  92.985616  77.979744  0.163579394 0.18034835
## result.119 -1.483322e+02 153.978372 148.332229  0.374999511 0.37499951
## result.120 -7.167639e+01  74.621670  71.676389  0.186627429 0.18662743
## result.121 -1.233895e+01  26.642171  19.001872  0.035103235 0.05108146
## result.122  4.232883e+01  43.641938  42.328830 -0.104067563 0.10406756
## result.123  4.004304e+00  13.367914  13.334768 -0.008805279 0.03266836
## result.124  6.667869e+00  29.529918  24.000401 -0.011243449 0.05557217
## result.125  5.966407e+01  63.885493  59.664065 -0.130087467 0.13008747
## result.126  7.033614e+01  77.524589  70.336143 -0.140078053 0.14007805
## result.127  8.200427e+01 102.502194  82.004267 -0.140163054 0.14016305
## result.128  1.156678e+02 121.690542 115.667768 -0.193347441 0.19334744
## result.129  4.367297e+01  66.684958  61.668767 -0.067972141 0.10339694
## result.130 -9.699129e+01 114.262754  96.991290  0.200000819 0.20000082
## result.131 -1.530016e+02 160.505938 153.001600  0.353767961 0.35376796
## result.132 -8.034313e+01  85.466805  80.343135  0.193503795 0.19350380
print(colMeans(res3_parallel))
##           ME         RMSE          MAE          MPE         MAPE 
##  5.606310555 45.344040192 40.688899803 -0.008682302  0.136229866
# Example 4 -----

(res4 <- crossval_ts(y=AirPassengers, fcast_func = forecast::thetaf,
initial_window = 10,
horizon = 3,
fixed_window = TRUE))
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##                     ME       RMSE        MAE          MPE       MAPE
## result.1    -28.794660  29.300287  28.794660  0.259750743 0.25975074
## result.2     16.198526  16.894302  16.198526 -0.134036396 0.13403640
## result.3     11.201494  15.993359  12.578276 -0.081116675 0.09308869
## result.4     21.430125  22.483895  21.430125 -0.157908838 0.15790884
## result.5     10.055765  11.527746  10.055765 -0.073295759 0.07329576
## result.6     -2.640822  10.676714   9.999466  0.024882775 0.07426965
## result.7     14.296434  23.709132  20.531135 -0.081722428 0.13160003
## result.8     38.247497  39.529998  38.247497 -0.231651791 0.23165179
## result.9     23.043159  23.947630  23.043159 -0.137610422 0.13761042
## result.10   -21.689067  27.907560  21.689067  0.154732863 0.15473286
## result.11   -41.782157  46.664199  41.782157  0.335689202 0.33568920
## result.12   -34.934831  36.512081  34.934831  0.280155112 0.28015511
## result.13    -4.002700  12.728771   9.999100  0.040488081 0.08233473
## result.14    30.349582  30.588761  30.349582 -0.208731397 0.20873140
## result.15    21.192349  25.806712  21.192349 -0.127304850 0.12730485
## result.16    23.193143  25.914875  23.193143 -0.137455344 0.13745534
## result.17    30.081542  30.679960  30.081542 -0.174862066 0.17486207
## result.18    -6.530509   9.111376   6.999059  0.039593797 0.04222610
## result.19    19.907586  23.010762  19.907586 -0.105338252 0.10533825
## result.20    17.631089  19.829355  17.631089 -0.089550147 0.08955015
## result.21    11.738022  14.718185  12.229846 -0.058884295 0.06155725
## result.22   -21.787490  28.489509  21.787490  0.129489104 0.12948910
## result.23   -43.557571  47.527244  43.557571  0.278950791 0.27895079
## result.24   -34.473558  35.514155  34.473558  0.221812109 0.22181211
## result.25    -4.699360  10.498595   7.201224  0.033422378 0.04808847
## result.26    25.974138  26.581272  25.974138 -0.149797833 0.14979783
## result.27    16.905109  19.474600  16.905109 -0.090819469 0.09081947
## result.28    15.218760  16.352917  15.218760 -0.081484738 0.08148474
## result.29     7.625241   8.933828   7.625241 -0.040408146 0.04040815
## result.30     2.261970  17.595326  15.666212 -0.004267259 0.07797211
## result.31    30.404914  36.592516  30.404914 -0.136176156 0.13617616
## result.32    46.412292  47.386236  46.412292 -0.200384649 0.20038465
## result.33     4.651769  15.532402  15.001095 -0.016552192 0.06607050
## result.34   -23.538570  33.729923  29.025988  0.121976444 0.14465172
## result.35   -60.690177  63.572981  60.690177  0.328287148 0.32828715
## result.36   -30.676401  32.211315  30.676401  0.168456030 0.16845603
## result.37    -8.577804  12.511784   8.577804  0.048889552 0.04888955
## result.38    21.858632  21.864774  21.858632 -0.111896409 0.11189641
## result.39    15.511300  24.464353  15.511300 -0.066956116 0.06695612
## result.40    27.351766  33.292899  27.351766 -0.116287717 0.11628772
## result.41    40.030438  40.086293  40.030438 -0.171470545 0.17147055
## result.42     1.913204   6.585371   5.082179 -0.007480486 0.02131880
## result.43    10.484756  17.849104  14.434313 -0.039461169 0.05670814
## result.44    28.459896  30.643222  28.459896 -0.107735883 0.10773588
## result.45     9.494149  19.388080  18.988959 -0.033029178 0.07309167
## result.46   -31.346158  42.472223  34.784073  0.144434512 0.15707390
## result.47   -73.108813  78.130452  73.108813  0.368808906 0.36880891
## result.48   -47.391791  49.482768  47.391791  0.245955653 0.24595565
## result.49   -20.263420  22.215913  20.263420  0.106896888 0.10689689
## result.50    18.627251  19.778162  18.627251 -0.093146961 0.09314696
## result.51    12.850828  24.491415  18.283609 -0.052964282 0.08186206
## result.52    18.630902  29.191109  27.309290 -0.076261222 0.12242286
## result.53    51.629084  51.820741  51.629084 -0.222377386 0.22237739
## result.54    13.235345  22.822393  16.377438 -0.050065336 0.06390716
## result.55    44.587147  53.679665  44.587147 -0.157309556 0.15730956
## result.56    54.072225  56.591957  54.072225 -0.186048489 0.18604849
## result.57    23.241489  30.809955  26.322921 -0.077182075 0.08907949
## result.58   -48.442256  57.148030  48.442256  0.199756472 0.19975647
## result.59   -76.172134  81.227076  76.172134  0.346233997 0.34623400
## result.60   -49.727782  51.411167  49.727782  0.229654227 0.22965423
## result.61   -12.015506  17.811915  13.001128  0.058148711 0.06222153
## result.62    28.827561  29.294075  28.827561 -0.122413712 0.12241371
## result.63    19.978440  24.903081  19.978440 -0.077636903 0.07763690
## result.64    17.110530  24.500048  22.184318 -0.062408624 0.08418454
## result.65    40.684200  40.816355  40.684200 -0.151376898 0.15137690
## result.66    22.682439  32.492379  22.682439 -0.074237521 0.07423752
## result.67    50.630668  64.359435  50.630668 -0.146962154 0.14696215
## result.68    71.054598  73.834184  71.054598 -0.204938760 0.20493876
## result.69    18.573839  30.848787  28.001381 -0.049971189 0.08018767
## result.70   -67.050346  75.890258  67.050346  0.228774453 0.22877445
## result.71   -89.147978  96.654776  89.147978  0.344537272 0.34453727
## result.72   -62.834495  65.591765  62.834495  0.245236494 0.24523649
## result.73   -17.192156  24.448634  17.192156  0.070396960 0.07039696
## result.74    39.147639  39.310613  39.147639 -0.139867698 0.13986770
## result.75    15.464746  23.499340  15.599360 -0.049536142 0.05002211
## result.76    21.388632  28.622347  25.036615 -0.067092418 0.08026203
## result.77    44.769285  44.904644  44.769285 -0.141625136 0.14162514
## result.78    24.337654  38.632254  24.890329 -0.066081607 0.06784734
## result.79    60.018016  72.625037  60.018016 -0.152620633 0.15262063
## result.80    78.806534  80.547853  78.806534 -0.196880257 0.19688026
## result.81    11.811108  31.117683  30.676430 -0.025278321 0.07842007
## result.82   -73.039284  86.689647  73.039284  0.223481827 0.22348183
## result.83  -112.674445 120.216380 112.674445  0.382042857 0.38204286
## result.84   -75.193608  77.210395  75.193608  0.259592270 0.25959227
## result.85   -18.005367  23.707491  18.005367  0.064266651 0.06426665
## result.86    34.032748  34.590333  34.032748 -0.110400738 0.11040074
## result.87    21.143051  32.190728  22.381017 -0.060361477 0.06447432
## result.88    25.770886  36.725269  33.180194 -0.071385272 0.09600091
## result.89    61.405131  61.602493  61.405131 -0.173880072 0.17388007
## result.90    28.569615  46.690860  30.711084 -0.068438510 0.07459216
## result.91    72.884393  87.299176  72.884393 -0.164968096 0.16496810
## result.92    97.243170  99.503251  97.243170 -0.213704240 0.21370424
## result.93    17.748567  36.930435  36.435381 -0.035014060 0.08126855
## result.94   -76.493543  94.890278  76.493543  0.208139134 0.20813913
## result.95  -136.613003 145.258940 136.613003  0.409412213 0.40941221
## result.96   -92.679937  95.004482  92.679937  0.285555972 0.28555597
## result.97   -32.018423  33.966942  32.018423  0.099842072 0.09984207
## result.98    22.498638  24.963399  22.498638 -0.067003620 0.06700362
## result.99     7.476001  20.150283  17.158667 -0.019143446 0.04959208
## result.100    9.982584  22.773032  22.210356 -0.025942187 0.06439430
## result.101   52.362071  53.101568  52.362071 -0.146065760 0.14606576
## result.102   34.853629  55.893897  39.234038 -0.081274141 0.09386152
## result.103   95.047754 111.264404  95.047754 -0.207545291 0.20754529
## result.104  121.335201 125.765542 121.335201 -0.250852140 0.25085214
## result.105   27.661546  53.669523  52.335667 -0.049684215 0.11075877
## result.106  -82.928463 106.536749  87.398376  0.222291120 0.23114243
## result.107 -168.429957 174.864018 168.429957  0.490890905 0.49089091
## result.108  -86.047368  89.349691  86.047368  0.261674036 0.26167404
## result.109  -35.392983  38.646195  35.392983  0.108684894 0.10868489
## result.110   32.273683  33.691668  32.273683 -0.092489738 0.09248974
## result.111   35.911969  45.528574  35.911969 -0.092371900 0.09237190
## result.112   28.584481  41.791438  38.166544 -0.069168689 0.09718642
## result.113   78.144295  79.313101  78.144295 -0.191198246 0.19119825
## result.114   37.152546  52.614035  39.210445 -0.080727044 0.08592376
## result.115   95.078342 110.886020  95.078342 -0.187505147 0.18750515
## result.116  109.166178 116.176118 109.166178 -0.202725580 0.20272558
## result.117   41.289554  62.020845  57.334902 -0.071737094 0.10639227
## result.118  -92.399494 116.617773  92.824074  0.216693887 0.21745342
## result.119 -175.618445 183.279552 175.618445  0.444857203 0.44485720
## result.120  -94.580461  97.360392  94.580461  0.245373593 0.24537359
## result.121  -27.751828  32.935586  27.751828  0.073365613 0.07336561
## result.122   36.177008  38.166461  36.177008 -0.088751649 0.08875165
## result.123    5.992278  14.161847  13.997426 -0.013666896 0.03414042
## result.124   12.637863  33.652691  27.980298 -0.025012667 0.06425163
## result.125   71.834372  76.950725  71.834372 -0.156620714 0.15662071
## result.126   85.518711  93.750937  85.518711 -0.170478555 0.17047855
## result.127   94.429064 115.523968  94.429064 -0.162279474 0.16227947
## result.128  173.325805 177.666522 173.325805 -0.291744683 0.29174468
## result.129   33.890665  63.841907  61.668605 -0.050341919 0.10502290
## result.130 -119.059067 137.736851 119.059067  0.244484820 0.24448482
## result.131 -180.821172 190.452409 180.821172  0.418676479 0.41867648
## result.132 -103.156396 108.618814 103.156396  0.247654263 0.24765426
##           ME         RMSE          MAE          MPE         MAPE 
##  2.657082195 51.427170382 46.511874693  0.003423843  0.155428590
(res4_parallel <- crossval_ts(y=AirPassengers, fcast_func = forecast::thetaf,
initial_window = 10,
horizon = 3,
fixed_window = TRUE, cl=2L))
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##                     ME       RMSE        MAE          MPE       MAPE
## result.1    -28.794660  29.300287  28.794660  0.259750743 0.25975074
## result.2     16.198526  16.894302  16.198526 -0.134036396 0.13403640
## result.3     11.201494  15.993359  12.578276 -0.081116675 0.09308869
## result.4     21.430125  22.483895  21.430125 -0.157908838 0.15790884
## result.5     10.055765  11.527746  10.055765 -0.073295759 0.07329576
## result.6     -2.640822  10.676714   9.999466  0.024882775 0.07426965
## result.7     14.296434  23.709132  20.531135 -0.081722428 0.13160003
## result.8     38.247497  39.529998  38.247497 -0.231651791 0.23165179
## result.9     23.043159  23.947630  23.043159 -0.137610422 0.13761042
## result.10   -21.689067  27.907560  21.689067  0.154732863 0.15473286
## result.11   -41.782157  46.664199  41.782157  0.335689202 0.33568920
## result.12   -34.934831  36.512081  34.934831  0.280155112 0.28015511
## result.13    -4.002700  12.728771   9.999100  0.040488081 0.08233473
## result.14    30.349582  30.588761  30.349582 -0.208731397 0.20873140
## result.15    21.192349  25.806712  21.192349 -0.127304850 0.12730485
## result.16    23.193143  25.914875  23.193143 -0.137455344 0.13745534
## result.17    30.081542  30.679960  30.081542 -0.174862066 0.17486207
## result.18    -6.530509   9.111376   6.999059  0.039593797 0.04222610
## result.19    19.907586  23.010762  19.907586 -0.105338252 0.10533825
## result.20    17.631089  19.829355  17.631089 -0.089550147 0.08955015
## result.21    11.738022  14.718185  12.229846 -0.058884295 0.06155725
## result.22   -21.787490  28.489509  21.787490  0.129489104 0.12948910
## result.23   -43.557571  47.527244  43.557571  0.278950791 0.27895079
## result.24   -34.473558  35.514155  34.473558  0.221812109 0.22181211
## result.25    -4.699360  10.498595   7.201224  0.033422378 0.04808847
## result.26    25.974138  26.581272  25.974138 -0.149797833 0.14979783
## result.27    16.905109  19.474600  16.905109 -0.090819469 0.09081947
## result.28    15.218760  16.352917  15.218760 -0.081484738 0.08148474
## result.29     7.625241   8.933828   7.625241 -0.040408146 0.04040815
## result.30     2.261970  17.595326  15.666212 -0.004267259 0.07797211
## result.31    30.404914  36.592516  30.404914 -0.136176156 0.13617616
## result.32    46.412292  47.386236  46.412292 -0.200384649 0.20038465
## result.33     4.651769  15.532402  15.001095 -0.016552192 0.06607050
## result.34   -23.538570  33.729923  29.025988  0.121976444 0.14465172
## result.35   -60.690177  63.572981  60.690177  0.328287148 0.32828715
## result.36   -30.676401  32.211315  30.676401  0.168456030 0.16845603
## result.37    -8.577804  12.511784   8.577804  0.048889552 0.04888955
## result.38    21.858632  21.864774  21.858632 -0.111896409 0.11189641
## result.39    15.511300  24.464353  15.511300 -0.066956116 0.06695612
## result.40    27.351766  33.292899  27.351766 -0.116287717 0.11628772
## result.41    40.030438  40.086293  40.030438 -0.171470545 0.17147055
## result.42     1.913204   6.585371   5.082179 -0.007480486 0.02131880
## result.43    10.484756  17.849104  14.434313 -0.039461169 0.05670814
## result.44    28.459896  30.643222  28.459896 -0.107735883 0.10773588
## result.45     9.494149  19.388080  18.988959 -0.033029178 0.07309167
## result.46   -31.346158  42.472223  34.784073  0.144434512 0.15707390
## result.47   -73.108813  78.130452  73.108813  0.368808906 0.36880891
## result.48   -47.391791  49.482768  47.391791  0.245955653 0.24595565
## result.49   -20.263420  22.215913  20.263420  0.106896888 0.10689689
## result.50    18.627251  19.778162  18.627251 -0.093146961 0.09314696
## result.51    12.850828  24.491415  18.283609 -0.052964282 0.08186206
## result.52    18.630902  29.191109  27.309290 -0.076261222 0.12242286
## result.53    51.629084  51.820741  51.629084 -0.222377386 0.22237739
## result.54    13.235345  22.822393  16.377438 -0.050065336 0.06390716
## result.55    44.587147  53.679665  44.587147 -0.157309556 0.15730956
## result.56    54.072225  56.591957  54.072225 -0.186048489 0.18604849
## result.57    23.241489  30.809955  26.322921 -0.077182075 0.08907949
## result.58   -48.442256  57.148030  48.442256  0.199756472 0.19975647
## result.59   -76.172134  81.227076  76.172134  0.346233997 0.34623400
## result.60   -49.727782  51.411167  49.727782  0.229654227 0.22965423
## result.61   -12.015506  17.811915  13.001128  0.058148711 0.06222153
## result.62    28.827561  29.294075  28.827561 -0.122413712 0.12241371
## result.63    19.978440  24.903081  19.978440 -0.077636903 0.07763690
## result.64    17.110530  24.500048  22.184318 -0.062408624 0.08418454
## result.65    40.684200  40.816355  40.684200 -0.151376898 0.15137690
## result.66    22.682439  32.492379  22.682439 -0.074237521 0.07423752
## result.67    50.630668  64.359435  50.630668 -0.146962154 0.14696215
## result.68    71.054598  73.834184  71.054598 -0.204938760 0.20493876
## result.69    18.573839  30.848787  28.001381 -0.049971189 0.08018767
## result.70   -67.050346  75.890258  67.050346  0.228774453 0.22877445
## result.71   -89.147978  96.654776  89.147978  0.344537272 0.34453727
## result.72   -62.834495  65.591765  62.834495  0.245236494 0.24523649
## result.73   -17.192156  24.448634  17.192156  0.070396960 0.07039696
## result.74    39.147639  39.310613  39.147639 -0.139867698 0.13986770
## result.75    15.464746  23.499340  15.599360 -0.049536142 0.05002211
## result.76    21.388632  28.622347  25.036615 -0.067092418 0.08026203
## result.77    44.769285  44.904644  44.769285 -0.141625136 0.14162514
## result.78    24.337654  38.632254  24.890329 -0.066081607 0.06784734
## result.79    60.018016  72.625037  60.018016 -0.152620633 0.15262063
## result.80    78.806534  80.547853  78.806534 -0.196880257 0.19688026
## result.81    11.811108  31.117683  30.676430 -0.025278321 0.07842007
## result.82   -73.039284  86.689647  73.039284  0.223481827 0.22348183
## result.83  -112.674445 120.216380 112.674445  0.382042857 0.38204286
## result.84   -75.193608  77.210395  75.193608  0.259592270 0.25959227
## result.85   -18.005367  23.707491  18.005367  0.064266651 0.06426665
## result.86    34.032748  34.590333  34.032748 -0.110400738 0.11040074
## result.87    21.143051  32.190728  22.381017 -0.060361477 0.06447432
## result.88    25.770886  36.725269  33.180194 -0.071385272 0.09600091
## result.89    61.405131  61.602493  61.405131 -0.173880072 0.17388007
## result.90    28.569615  46.690860  30.711084 -0.068438510 0.07459216
## result.91    72.884393  87.299176  72.884393 -0.164968096 0.16496810
## result.92    97.243170  99.503251  97.243170 -0.213704240 0.21370424
## result.93    17.748567  36.930435  36.435381 -0.035014060 0.08126855
## result.94   -76.493543  94.890278  76.493543  0.208139134 0.20813913
## result.95  -136.613003 145.258940 136.613003  0.409412213 0.40941221
## result.96   -92.679937  95.004482  92.679937  0.285555972 0.28555597
## result.97   -32.018423  33.966942  32.018423  0.099842072 0.09984207
## result.98    22.498638  24.963399  22.498638 -0.067003620 0.06700362
## result.99     7.476001  20.150283  17.158667 -0.019143446 0.04959208
## result.100    9.982584  22.773032  22.210356 -0.025942187 0.06439430
## result.101   52.362071  53.101568  52.362071 -0.146065760 0.14606576
## result.102   34.853629  55.893897  39.234038 -0.081274141 0.09386152
## result.103   95.047754 111.264404  95.047754 -0.207545291 0.20754529
## result.104  121.335201 125.765542 121.335201 -0.250852140 0.25085214
## result.105   27.661546  53.669523  52.335667 -0.049684215 0.11075877
## result.106  -82.928463 106.536749  87.398376  0.222291120 0.23114243
## result.107 -168.429957 174.864018 168.429957  0.490890905 0.49089091
## result.108  -86.047368  89.349691  86.047368  0.261674036 0.26167404
## result.109  -35.392983  38.646195  35.392983  0.108684894 0.10868489
## result.110   32.273683  33.691668  32.273683 -0.092489738 0.09248974
## result.111   35.911969  45.528574  35.911969 -0.092371900 0.09237190
## result.112   28.584481  41.791438  38.166544 -0.069168689 0.09718642
## result.113   78.144295  79.313101  78.144295 -0.191198246 0.19119825
## result.114   37.152546  52.614035  39.210445 -0.080727044 0.08592376
## result.115   95.078342 110.886020  95.078342 -0.187505147 0.18750515
## result.116  109.166178 116.176118 109.166178 -0.202725580 0.20272558
## result.117   41.289554  62.020845  57.334902 -0.071737094 0.10639227
## result.118  -92.399494 116.617773  92.824074  0.216693887 0.21745342
## result.119 -175.618445 183.279552 175.618445  0.444857203 0.44485720
## result.120  -94.580461  97.360392  94.580461  0.245373593 0.24537359
## result.121  -27.751828  32.935586  27.751828  0.073365613 0.07336561
## result.122   36.177008  38.166461  36.177008 -0.088751649 0.08875165
## result.123    5.992278  14.161847  13.997426 -0.013666896 0.03414042
## result.124   12.637863  33.652691  27.980298 -0.025012667 0.06425163
## result.125   71.834372  76.950725  71.834372 -0.156620714 0.15662071
## result.126   85.518711  93.750937  85.518711 -0.170478555 0.17047855
## result.127   94.429064 115.523968  94.429064 -0.162279474 0.16227947
## result.128  173.325805 177.666522 173.325805 -0.291744683 0.29174468
## result.129   33.890665  63.841907  61.668605 -0.050341919 0.10502290
## result.130 -119.059067 137.736851 119.059067  0.244484820 0.24448482
## result.131 -180.821172 190.452409 180.821172  0.418676479 0.41867648
## result.132 -103.156396 108.618814 103.156396  0.247654263 0.24765426
print(colMeans(res4_parallel))
##           ME         RMSE          MAE          MPE         MAPE 
##  2.657082195 51.427170382 46.511874693  0.003423843  0.155428590
# Example 5 -----

x <- ts(matrix(rnorm(50), nrow = 25))

fcast_func <- function(y, h = 5, type_forecast=c("mean", "median"))
{
 type_forecast <- match.arg(type_forecast)

 if (type_forecast == "mean")
 {
  means <- colMeans(y)
  return(list(mean = t(replicate(n = h, expr = means))))
 } else {
  medians <- apply(y, 2, median)
  return(list(mean = t(replicate(n = h, expr = medians))))
 }
}

print(fcast_func(x))
## $mean
##      Series 1   Series 2
## [1,] 0.102532 -0.2203926
## [2,] 0.102532 -0.2203926
## [3,] 0.102532 -0.2203926
## [4,] 0.102532 -0.2203926
## [5,] 0.102532 -0.2203926
(res5 <- crossval_ts(y = x, fcast_func = fcast_func, fit_params = list(type_forecast = "median")))
##   |                                                                              |                                                                      |   0%  |                                                                              |====                                                                  |   6%  |                                                                              |========                                                              |  11%  |                                                                              |============                                                          |  17%  |                                                                              |================                                                      |  22%  |                                                                              |===================                                                   |  28%  |                                                                              |=======================                                               |  33%  |                                                                              |===========================                                           |  39%  |                                                                              |===============================                                       |  44%  |                                                                              |===================================                                   |  50%  |                                                                              |=======================================                               |  56%  |                                                                              |===========================================                           |  61%  |                                                                              |===============================================                       |  67%  |                                                                              |===================================================                   |  72%  |                                                                              |======================================================                |  78%  |                                                                              |==========================================================            |  83%  |                                                                              |==============================================================        |  89%  |                                                                              |==================================================================    |  94%  |                                                                              |======================================================================| 100%
##                     ME      RMSE       MAE          MPE       MAPE
## result.1  -0.690181348 0.7174868 0.6901813  -1.00666702  1.0066670
## result.2  -0.435640782 0.6057156 0.4739418  -0.55630440  0.6187708
## result.3   0.126111488 1.1351391 0.8461277  -1.30437537  1.3043754
## result.4   0.505731988 1.2015685 0.8835714  -1.51229401  1.6102254
## result.5   0.736719142 1.2834659 0.7909988  -0.04985672  1.4712553
## result.6   0.353676707 1.0388520 0.8421400   0.13018660  2.0926560
## result.7   0.092177436 0.8930188 0.7647644  -1.66143029  2.9844934
## result.8   0.428849725 1.1656455 1.0036998  -3.54201382  3.5818273
## result.9  -0.079844260 0.6057046 0.4548511  -0.31729076  2.0604926
## result.10 -0.151638101 0.5606515 0.3671110  -0.65323588  0.8020641
## result.11  0.003433341 0.9261484 0.7451840  -1.47103754  1.5800523
## result.12  0.315744215 0.9573762 0.7906209  -0.99652990  0.9965299
## result.13 -0.074613231 1.1513017 1.0680000  -1.32135851  1.3213585
## result.14 -0.539434135 1.1850508 1.0579015  -5.44631065  5.4463106
## result.15 -0.354664009 0.9410527 0.8136701 -21.00656982 21.0065698
## result.16  0.430718876 0.8067862 0.6427804   8.86186135 10.4858752
## result.17  1.196308130 1.4542929 1.1963081  -1.07940107  1.1237354
## result.18  0.897734735 1.2789986 0.9905868   0.77932927  2.3833998
colMeans(res5)
##         ME       RMSE        MAE        MPE       MAPE 
##  0.1533994  0.9949031  0.8012466 -1.7862944  3.4375922
(res6 <- crossval_ts(y = x, fcast_func = fcast_func, fit_params = list(type_forecast = "mean")))
##   |                                                                              |                                                                      |   0%  |                                                                              |====                                                                  |   6%  |                                                                              |========                                                              |  11%  |                                                                              |============                                                          |  17%  |                                                                              |================                                                      |  22%  |                                                                              |===================                                                   |  28%  |                                                                              |=======================                                               |  33%  |                                                                              |===========================                                           |  39%  |                                                                              |===============================                                       |  44%  |                                                                              |===================================                                   |  50%  |                                                                              |=======================================                               |  56%  |                                                                              |===========================================                           |  61%  |                                                                              |===============================================                       |  67%  |                                                                              |===================================================                   |  72%  |                                                                              |======================================================                |  78%  |                                                                              |==========================================================            |  83%  |                                                                              |==============================================================        |  89%  |                                                                              |==================================================================    |  94%  |                                                                              |======================================================================| 100%
##                    ME      RMSE       MAE          MPE       MAPE
## result.1  -0.75589737 0.7806978 0.7558974  -1.10683248  1.1068325
## result.2  -0.53356883 0.6801790 0.5399884  -0.67230817  0.6723082
## result.3  -0.15829870 1.1552093 0.9409311  -0.95776218  0.9577622
## result.4   0.39400864 1.1504684 0.8463303  -1.40153985  1.4156069
## result.5   0.61616617 1.1548871 0.7504685   0.03685634  1.4276698
## result.6   0.05078208 0.8763353 0.7408291   0.05584046  1.6183760
## result.7  -0.10115737 0.8424740 0.7003194  -0.79743904  2.1078490
## result.8   0.09398145 0.8447896 0.6603921  -1.61103270  1.6796623
## result.9   0.21243646 0.6156534 0.5316482  -1.37702971  1.9963495
## result.10 -0.02381856 0.6147361 0.4177251  -0.60036097  1.0499744
## result.11  0.26709553 0.9403369 0.6954727  -1.36395217  1.3639522
## result.12  0.55286331 1.0461495 0.7631215  -0.80998406  0.8099841
## result.13  0.12163144 1.1557368 1.0025851  -1.07785082  1.0778508
## result.14 -0.52185272 1.2158927 1.0661078  -1.00186961  1.0018696
## result.15 -0.48320067 0.9908049 0.8565157 -18.76920371 18.7692037
## result.16  0.16687224 0.7097811 0.5995854  -4.05305261  4.0530526
## result.17  0.94982075 1.2590424 1.0138025  -0.90542703  0.9054270
## result.18  0.48750897 1.0572565 0.8227185  -0.62628466  1.1412910
colMeans(res6)
##          ME        RMSE         MAE         MPE        MAPE 
##  0.07418738  0.94946838  0.76135771 -2.05773517  2.39750121
# Example 6 -----

eval_metric <- function(predicted, observed)
{
  error <- observed - predicted$mean

  res7 <- apply(error, 2, function(x) sqrt(mean(x ^ 2, na.rm = FALSE)))

  return(res7)
}

(res8 <- crossval_ts(y = x, fcast_func = fcast_func, 
                    fit_params = list(type_forecast = "mean"),
eval_metric = eval_metric))
##   |                                                                              |                                                                      |   0%  |                                                                              |====                                                                  |   6%  |                                                                              |========                                                              |  11%  |                                                                              |============                                                          |  17%  |                                                                              |================                                                      |  22%  |                                                                              |===================                                                   |  28%  |                                                                              |=======================                                               |  33%  |                                                                              |===========================                                           |  39%  |                                                                              |===============================                                       |  44%  |                                                                              |===================================                                   |  50%  |                                                                              |=======================================                               |  56%  |                                                                              |===========================================                           |  61%  |                                                                              |===============================================                       |  67%  |                                                                              |===================================================                   |  72%  |                                                                              |======================================================                |  78%  |                                                                              |==========================================================            |  83%  |                                                                              |==============================================================        |  89%  |                                                                              |==================================================================    |  94%  |                                                                              |======================================================================| 100%
##            Series 1  Series 2
## result.1  0.7945859 0.7665582
## result.2  0.7286174 0.6280157
## result.3  1.5193724 0.6004367
## result.4  1.6000053 0.2951919
## result.5  1.6172965 0.2277724
## result.6  0.8056898 0.9416959
## result.7  0.6211421 1.0167141
## result.8  0.6156449 1.0238751
## result.9  0.5556410 0.6703143
## result.10 0.7038364 0.5103089
## result.11 0.5742717 1.1994495
## result.12 1.0191007 1.0725163
## result.13 1.0266029 1.2718262
## result.14 1.0450063 1.3655592
## result.15 0.6586148 1.2367761
## result.16 0.4399057 0.9022535
## result.17 1.2822554 1.2353933
## result.18 1.2999875 0.7386575
colMeans(res8)
##  Series 1  Series 2 
## 0.9393098 0.8724064