Generalized Linear Model Theta Forecast with attention - conformal uncertainty

USAccDeaths (method=‘adj’)

(obj <- suppressWarnings(ahead::glmthetaf(USAccDeaths, h=25L, fit_func=MASS::glm.nb, attention = TRUE, type_pi = "conformal-split")))

## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo

##          Point Forecast    Lo 95     Hi 95
## Jan 1979       8268.704 6521.060 10016.349
## Feb 1979       7309.039 5561.394  9056.683
## Mar 1979       8154.793 6407.149  9902.438
## Apr 1979       8531.014 6783.369 10278.658
## May 1979       9393.218 7645.573 11140.862
## Jun 1979       9741.994 7994.350 11489.639
## Jul 1979      11003.200 9255.555 12750.844
## Aug 1979       9835.339 8087.694 11582.983
## Sep 1979       8712.499 6964.854 10460.143
## Oct 1979       9231.334 7483.690 10978.979
## Nov 1979       8576.565 6828.920 10324.209
## Dec 1979       9249.011 7501.367 10996.656
## Jan 1980       8268.724 6521.080 10016.369
## Feb 1980       7309.056 5561.412  9056.701
## Mar 1980       8154.813 6407.168  9902.457
## Apr 1980       8531.034 6783.390 10278.679
## May 1980       9393.240 7645.596 11140.885
## Jun 1980       9742.017 7994.373 11489.662
## Jul 1980      11003.226 9255.582 12750.871
## Aug 1980       9835.362 8087.718 11583.007
## Sep 1980       8712.520 6964.875 10460.164
## Oct 1980       9231.356 7483.712 10979.001
## Nov 1980       8576.585 6828.941 10324.230
## Dec 1980       9249.034 7501.389 10996.678
## Jan 1981       8268.744 6521.099 10016.388

plot(obj)

(obj <- suppressWarnings(ahead::glmthetaf(USAccDeaths, h=25L, fit_func=stats::glm, attention = TRUE, type_pi = "conformal-split")))

##          Point Forecast    Lo 95     Hi 95
## Jan 1979       8301.382 6463.642 10139.123
## Feb 1979       7350.676 5512.935  9188.417
## Mar 1979       8215.372 6377.631 10053.113
## Apr 1979       8609.165 6771.424 10446.906
## May 1979       9495.541 7657.800 11333.282
## Jun 1979       9864.997 8027.256 11702.738
## Jul 1979      11161.195 9323.454 12998.936
## Aug 1979       9993.612 8155.871 11831.352
## Sep 1979       8867.806 7030.065 10705.546
## Oct 1979       9411.894 7574.153 11249.635
## Nov 1979       8759.189 6921.448 10596.929
## Dec 1979       9461.994 7624.253 11299.734
## Jan 1980       8473.474 6635.733 10311.215
## Feb 1980       7502.722 5664.981  9340.462
## Mar 1980       8385.035 6547.294 10222.776
## Apr 1980       8786.680 6948.939 10624.421
## May 1980       9691.024 7853.283 11528.764
## Jun 1980      10067.765 8230.025 11905.506
## Jul 1980      11390.245 9552.504 13227.985
## Aug 1980      10198.378 8360.637 12036.118
## Sep 1980       9049.219 7211.478 10886.959
## Oct 1980       9604.136 7766.395 11441.877
## Nov 1980       8937.819 7100.078 10775.560
## Dec 1980       9654.655 7816.914 11492.396
## Jan 1981       8645.739 6807.998 10483.479

plot(obj)

(obj <- suppressWarnings(ahead::glmthetaf(USAccDeaths, h=25L, fit_func=MASS::rlm, attention = TRUE, type_pi = "conformal-split")))

##          Point Forecast    Lo 95     Hi 95
## Jan 1979       8305.354 6473.213 10137.494
## Feb 1979       7355.736 5523.596  9187.877
## Mar 1979       8222.735 6390.594 10054.875
## Apr 1979       8618.664 6786.523 10450.805
## May 1979       9507.979 7675.838 11340.120
## Jun 1979       9879.950 8047.809 11712.090
## Jul 1979      11180.403 9348.262 13012.544
## Aug 1979      10012.855 8180.714 11844.996
## Sep 1979       8886.690 7054.549 10718.830
## Oct 1979       9433.850 7601.710 11265.991
## Nov 1979       8781.398 6949.258 10613.539
## Dec 1979       9487.898 7655.757 11320.038
## Jan 1980       8498.379 6666.238 10330.520
## Feb 1980       7526.281 5694.140  9358.421
## Mar 1980       8413.044 6580.903 10245.185
## Apr 1980       8817.785 6985.645 10649.926
## May 1980       9727.260 7895.119 11559.401
## Jun 1980      10107.409 8275.268 11939.550
## Jul 1980      11437.350 9605.209 13269.491
## Aug 1980      10242.567 8410.426 12074.707
## Sep 1980       9090.209 7258.068 10922.350
## Oct 1980       9649.524 7817.383 11481.665
## Nov 1980       8981.806 7149.665 10813.947
## Dec 1980       9704.053 7871.912 11536.194
## Jan 1981       8691.656 6859.515 10523.796

plot(obj)

(obj <- suppressWarnings(ahead::glmthetaf(USAccDeaths, h=25L, fit_func=MASS::lqs, attention = TRUE, type_pi = "conformal-split")))

##          Point Forecast    Lo 95     Hi 95
## Jan 1979       8293.623 6466.222 10121.023
## Feb 1979       7340.789 5513.389  9168.189
## Mar 1979       8200.987 6373.586 10028.387
## Apr 1979       8590.606 6763.206 10418.006
## May 1979       9471.241 7643.841 11298.641
## Jun 1979       9835.785 8008.384 11663.185
## Jul 1979      11123.670 9296.270 12951.071
## Aug 1979       9956.019 8128.619 11783.419
## Sep 1979       8830.915 7003.515 10658.316
## Oct 1979       9369.003 7541.603 11196.403
## Nov 1979       8715.805 6888.404 10543.205
## Dec 1979       9411.394 7583.994 11238.795
## Jan 1980       8424.827 6597.427 10252.228
## Feb 1980       7456.705 5629.305  9284.106
## Mar 1980       8330.329 6502.928 10157.729
## Apr 1980       8725.928 6898.528 10553.328
## May 1980       9620.253 7792.853 11447.653
## Jun 1980       9990.343 8162.943 11817.743
## Jul 1980      11298.253 9470.852 13125.653
## Aug 1980      10112.085 8284.684 11939.485
## Sep 1980       8969.175 7141.775 10796.576
## Oct 1980       9515.508 7688.108 11342.909
## Nov 1980       8851.930 7024.529 10679.330
## Dec 1980       9558.204 7730.804 11385.604
## Jan 1981       8556.088 6728.687 10383.488

plot(obj)

(obj <- suppressWarnings(ahead::glmthetaf(USAccDeaths, h=25L, fit_func=stats::lm, attention = TRUE, type_pi = "conformal-split")))

##          Point Forecast    Lo 95     Hi 95
## Jan 1979       8301.382 6463.642 10139.123
## Feb 1979       7350.676 5512.935  9188.417
## Mar 1979       8215.372 6377.631 10053.113
## Apr 1979       8609.165 6771.424 10446.906
## May 1979       9495.541 7657.800 11333.282
## Jun 1979       9864.997 8027.256 11702.738
## Jul 1979      11161.195 9323.454 12998.936
## Aug 1979       9993.612 8155.871 11831.352
## Sep 1979       8867.806 7030.065 10705.546
## Oct 1979       9411.894 7574.153 11249.635
## Nov 1979       8759.189 6921.448 10596.929
## Dec 1979       9461.994 7624.253 11299.734
## Jan 1980       8473.474 6635.733 10311.215
## Feb 1980       7502.722 5664.981  9340.462
## Mar 1980       8385.035 6547.294 10222.776
## Apr 1980       8786.680 6948.939 10624.421
## May 1980       9691.024 7853.283 11528.764
## Jun 1980      10067.765 8230.025 11905.506
## Jul 1980      11390.245 9552.504 13227.985
## Aug 1980      10198.378 8360.637 12036.118
## Sep 1980       9049.219 7211.478 10886.959
## Oct 1980       9604.136 7766.395 11441.877
## Nov 1980       8937.819 7100.078 10775.560
## Dec 1980       9654.655 7816.914 11492.396
## Jan 1981       8645.739 6807.998 10483.479

plot(obj)

(obj <- suppressWarnings(ahead::glmthetaf(USAccDeaths, h=25L, fit_func=gam::gam, attention = TRUE, type_pi = "conformal-split")))

##          Point Forecast    Lo 95     Hi 95
## Jan 1979       8301.382 6463.642 10139.123
## Feb 1979       7350.676 5512.935  9188.417
## Mar 1979       8215.372 6377.631 10053.113
## Apr 1979       8609.165 6771.424 10446.906
## May 1979       9495.541 7657.800 11333.282
## Jun 1979       9864.997 8027.256 11702.738
## Jul 1979      11161.195 9323.454 12998.936
## Aug 1979       9993.612 8155.871 11831.352
## Sep 1979       8867.806 7030.065 10705.546
## Oct 1979       9411.894 7574.153 11249.635
## Nov 1979       8759.189 6921.448 10596.929
## Dec 1979       9461.994 7624.253 11299.734
## Jan 1980       8473.474 6635.733 10311.215
## Feb 1980       7502.722 5664.981  9340.462
## Mar 1980       8385.035 6547.294 10222.776
## Apr 1980       8786.680 6948.939 10624.421
## May 1980       9691.024 7853.283 11528.764
## Jun 1980      10067.765 8230.025 11905.506
## Jul 1980      11390.245 9552.504 13227.985
## Aug 1980      10198.378 8360.637 12036.118
## Sep 1980       9049.219 7211.478 10886.959
## Oct 1980       9604.136 7766.395 11441.877
## Nov 1980       8937.819 7100.078 10775.560
## Dec 1980       9654.655 7816.914 11492.396
## Jan 1981       8645.739 6807.998 10483.479

plot(obj)

(obj <- suppressWarnings(ahead::glmthetaf(USAccDeaths, h=25L, fit_func=quantreg::rq, attention = TRUE, type_pi = "conformal-split")))

##          Point Forecast    Lo 95     Hi 95
## Jan 1979       8302.271 6489.390 10115.152
## Feb 1979       7351.808 5538.927  9164.689
## Mar 1979       8217.019 6404.138 10029.900
## Apr 1979       8611.290 6798.409 10424.171
## May 1979       9498.323 7685.442 11311.204
## Jun 1979       9868.342 8055.461 11681.223
## Jul 1979      11165.492 9352.610 12978.373
## Aug 1979       9997.916 8185.035 11810.797
## Sep 1979       8872.030 7059.149 10684.911
## Oct 1979       9416.805 7603.924 11229.686
## Nov 1979       8764.157 6951.275 10577.038
## Dec 1979       9467.788 7654.907 11280.669
## Jan 1980       8479.045 6666.164 10291.926
## Feb 1980       7507.991 5695.110  9320.872
## Mar 1980       8391.300 6578.419 10204.181
## Apr 1980       8793.637 6980.756 10606.518
## May 1980       9699.128 7886.247 11512.009
## Jun 1980      10076.632 8263.751 11889.513
## Jul 1980      11400.780 9587.899 13213.661
## Aug 1980      10208.261 8395.380 12021.142
## Sep 1980       9058.387 7245.506 10871.268
## Oct 1980       9614.287 7801.406 11427.168
## Nov 1980       8947.657 7134.776 10760.538
## Dec 1980       9665.703 7852.822 11478.584
## Jan 1981       8656.008 6843.127 10468.889

plot(obj)

AirPassengers (method=‘adj’)

(obj <- suppressWarnings(ahead::glmthetaf(AirPassengers, h=25L, fit_func=MASS::glm.nb, attention = TRUE, type_pi = "conformal-split")))

##          Point Forecast    Lo 95    Hi 95
## Jan 1961       441.8638 312.5489 571.1787
## Feb 1961       415.3297 286.0148 544.6446
## Mar 1961       470.6905 341.3756 600.0054
## Apr 1961       466.2493 336.9344 595.5642
## May 1961       477.9140 348.5991 607.2289
## Jun 1961       552.5173 423.2024 681.8321
## Jul 1961       618.1834 488.8685 747.4982
## Aug 1961       611.8472 482.5323 741.1621
## Sep 1961       517.8358 388.5210 647.1507
## Oct 1961       448.5688 319.2539 577.8837
## Nov 1961       389.6861 260.3713 519.0010
## Dec 1961       432.9739 303.6590 562.2888
## Jan 1962       441.9548 312.6399 571.2697
## Feb 1962       415.4152 286.1003 544.7301
## Mar 1962       470.7874 341.4725 600.1023
## Apr 1962       466.3453 337.0304 595.6602
## May 1962       478.0124 348.6975 607.3273
## Jun 1962       552.6310 423.3161 681.9459
## Jul 1962       618.3107 488.9958 747.6256
## Aug 1962       611.9732 482.6583 741.2881
## Sep 1962       517.9425 388.6276 647.2574
## Oct 1962       448.6612 319.3463 577.9761
## Nov 1962       389.7664 260.4515 519.0813
## Dec 1962       433.0630 303.7481 562.3779
## Jan 1963       442.0458 312.7309 571.3607

plot(obj)

(obj <- suppressWarnings(ahead::glmthetaf(AirPassengers, h=25L, fit_func=stats::glm, attention = TRUE, type_pi = "conformal-split")))

##          Point Forecast    Lo 95    Hi 95
## Jan 1961       445.8121 385.2164 506.4079
## Feb 1961       421.7063 361.1106 482.3021
## Mar 1961       480.9357 420.3399 541.5314
## Apr 1961       479.3903 418.7946 539.9861
## May 1961       494.4538 433.8580 555.0495
## Jun 1961       575.1909 514.5952 635.7867
## Jul 1961       647.5292 586.9334 708.1249
## Aug 1961       644.8320 584.2363 705.4278
## Sep 1961       549.0895 488.4938 609.6853
## Oct 1961       478.5349 417.9391 539.1306
## Nov 1961       418.2338 357.6381 478.8296
## Dec 1961       467.4895 406.8937 528.0852
## Jan 1962       480.0432 419.4475 540.6390
## Feb 1962       453.9038 393.3081 514.4996
## Mar 1962       517.4543 456.8586 578.0501
## Apr 1962       515.5933 454.9976 576.1891
## May 1962       531.5920 470.9962 592.1877
## Jun 1962       618.1603 557.5646 678.7561
## Jul 1962       695.6431 635.0473 756.2388
## Aug 1962       692.4898 631.8941 753.0856
## Sep 1962       589.4558 528.8600 650.0515
## Oct 1962       513.5284 452.9327 574.1242
## Nov 1962       448.6570 388.0613 509.2528
## Dec 1962       501.3178 440.7220 561.9135
## Jan 1963       514.5992 454.0035 575.1950

plot(obj)

(obj <- suppressWarnings(ahead::glmthetaf(AirPassengers, h=25L, fit_func=MASS::rlm, attention = TRUE, type_pi = "conformal-split")))

##          Point Forecast    Lo 95    Hi 95
## Jan 1961       445.8309 385.5513 506.1106
## Feb 1961       421.7367 361.4570 482.0164
## Mar 1961       480.9845 420.7048 541.2641
## Apr 1961       479.4529 419.1732 539.7326
## May 1961       494.5326 434.2529 554.8122
## Jun 1961       575.2990 515.0193 635.5786
## Jul 1961       647.6690 587.3894 707.9487
## Aug 1961       644.9893 584.7096 705.2689
## Sep 1961       549.2386 488.9589 609.5182
## Oct 1961       478.6778 418.3981 538.9574
## Nov 1961       418.3700 358.0903 478.6497
## Dec 1961       467.6541 407.3745 527.9338
## Jan 1962       480.2250 419.9454 540.5047
## Feb 1962       454.0876 393.8079 514.3673
## Mar 1962       517.6772 457.3975 577.9569
## Apr 1962       515.8286 455.5489 576.1083
## May 1962       531.8481 471.5684 592.1277
## Jun 1962       618.4736 558.1940 678.7533
## Jul 1962       696.0129 635.7333 756.2926
## Aug 1962       692.8751 632.5954 753.1547
## Sep 1962       589.7981 529.5184 650.0777
## Oct 1962       513.8390 453.5594 574.1187
## Nov 1962       448.9391 388.6595 509.2188
## Dec 1962       501.6449 441.3652 561.9246
## Jan 1963       514.9471 454.6674 575.2267

plot(obj)

(obj <- suppressWarnings(ahead::glmthetaf(AirPassengers, h=25L, fit_func=stats::lm, attention = TRUE, type_pi = "conformal-split")))

##          Point Forecast    Lo 95    Hi 95
## Jan 1961       445.8121 385.2164 506.4079
## Feb 1961       421.7063 361.1106 482.3021
## Mar 1961       480.9357 420.3399 541.5314
## Apr 1961       479.3903 418.7946 539.9861
## May 1961       494.4538 433.8580 555.0495
## Jun 1961       575.1909 514.5952 635.7867
## Jul 1961       647.5292 586.9334 708.1249
## Aug 1961       644.8320 584.2363 705.4278
## Sep 1961       549.0895 488.4938 609.6853
## Oct 1961       478.5349 417.9391 539.1306
## Nov 1961       418.2338 357.6381 478.8296
## Dec 1961       467.4895 406.8937 528.0852
## Jan 1962       480.0432 419.4475 540.6390
## Feb 1962       453.9038 393.3081 514.4996
## Mar 1962       517.4543 456.8586 578.0501
## Apr 1962       515.5933 454.9976 576.1891
## May 1962       531.5920 470.9962 592.1877
## Jun 1962       618.1603 557.5646 678.7561
## Jul 1962       695.6431 635.0473 756.2388
## Aug 1962       692.4898 631.8941 753.0856
## Sep 1962       589.4558 528.8600 650.0515
## Oct 1962       513.5284 452.9327 574.1242
## Nov 1962       448.6570 388.0613 509.2528
## Dec 1962       501.3178 440.7220 561.9135
## Jan 1963       514.5992 454.0035 575.1950

plot(obj)

(obj <- suppressWarnings(ahead::glmthetaf(AirPassengers, h=25L, fit_func=MASS::lqs, attention = TRUE, type_pi = "conformal-split")))

##          Point Forecast    Lo 95    Hi 95
## Jan 1961       445.8936 386.3463 505.4410
## Feb 1961       421.8379 362.2906 481.3853
## Mar 1961       481.1472 421.5998 540.6945
## Apr 1961       479.6616 420.1143 539.2089
## May 1961       494.7952 435.2479 554.3426
## Jun 1961       575.6591 516.1118 635.2065
## Jul 1961       648.1353 588.5879 707.6826
## Aug 1961       645.5134 585.9661 705.0608
## Sep 1961       549.7353 490.1880 609.2826
## Oct 1961       479.1542 419.6069 538.7015
## Nov 1961       418.8240 359.2767 478.3713
## Dec 1961       468.2032 408.6559 527.7505
## Jan 1962       480.8311 421.2838 540.3784
## Feb 1962       454.7002 395.1529 514.2476
## Mar 1962       518.4203 458.8730 577.9676
## Apr 1962       516.6130 457.0657 576.1604
## May 1962       532.7018 473.1545 592.2491
## Jun 1962       619.5182 559.9708 679.0655
## Jul 1962       697.2461 637.6988 756.7934
## Aug 1962       694.1595 634.6122 753.7068
## Sep 1962       590.9393 531.3920 650.4867
## Oct 1962       514.8747 455.3273 574.4220
## Nov 1962       449.8797 390.3324 509.4271
## Dec 1962       502.7355 443.1882 562.2828
## Jan 1963       516.1069 456.5596 575.6542

plot(obj)

(obj <- suppressWarnings(ahead::glmthetaf(AirPassengers, h=25L, fit_func=gam::gam, attention = TRUE, type_pi = "conformal-split")))

##          Point Forecast    Lo 95    Hi 95
## Jan 1961       445.8121 385.2164 506.4079
## Feb 1961       421.7063 361.1106 482.3021
## Mar 1961       480.9357 420.3399 541.5314
## Apr 1961       479.3903 418.7946 539.9861
## May 1961       494.4538 433.8580 555.0495
## Jun 1961       575.1909 514.5952 635.7867
## Jul 1961       647.5292 586.9334 708.1249
## Aug 1961       644.8320 584.2363 705.4278
## Sep 1961       549.0895 488.4938 609.6853
## Oct 1961       478.5349 417.9391 539.1306
## Nov 1961       418.2338 357.6381 478.8296
## Dec 1961       467.4895 406.8937 528.0852
## Jan 1962       480.0432 419.4475 540.6390
## Feb 1962       453.9038 393.3081 514.4996
## Mar 1962       517.4543 456.8586 578.0501
## Apr 1962       515.5933 454.9976 576.1891
## May 1962       531.5920 470.9962 592.1877
## Jun 1962       618.1603 557.5646 678.7561
## Jul 1962       695.6431 635.0473 756.2388
## Aug 1962       692.4898 631.8941 753.0856
## Sep 1962       589.4558 528.8600 650.0515
## Oct 1962       513.5284 452.9327 574.1242
## Nov 1962       448.6570 388.0613 509.2528
## Dec 1962       501.3178 440.7220 561.9135
## Jan 1963       514.5992 454.0035 575.1950

plot(obj)

(obj <- suppressWarnings(ahead::glmthetaf(AirPassengers, h=25L, fit_func=quantreg::rq, attention = TRUE, type_pi = "conformal-split")))

##          Point Forecast    Lo 95    Hi 95
## Jan 1961       445.8959 386.0481 505.7437
## Feb 1961       421.8416 361.9938 481.6894
## Mar 1961       481.1530 421.3052 541.0009
## Apr 1961       479.6691 419.8213 539.5170
## May 1961       494.8048 434.9569 554.6526
## Jun 1961       575.6722 515.8244 635.5200
## Jul 1961       648.1522 588.3043 708.0000
## Aug 1961       645.5324 585.6846 705.3802
## Sep 1961       549.7533 489.9055 609.6011
## Oct 1961       479.1715 419.3236 539.0193
## Nov 1961       418.8404 358.9926 478.6883
## Dec 1961       468.2231 408.3753 528.0709
## Jan 1962       480.8531 421.0052 540.7009
## Feb 1962       454.7224 394.8746 514.5703
## Mar 1962       518.4472 458.5994 578.2950
## Apr 1962       516.6415 456.7937 576.4893
## May 1962       532.7327 472.8849 592.5806
## Jun 1962       619.5560 559.7082 679.4039
## Jul 1962       697.2908 637.4430 757.1386
## Aug 1962       694.2061 634.3582 754.0539
## Sep 1962       590.9807 531.1329 650.8285
## Oct 1962       514.9122 455.0644 574.7600
## Nov 1962       449.9138 390.0660 509.7617
## Dec 1962       502.7750 442.9272 562.6229
## Jan 1963       516.1489 456.3011 575.9968

plot(obj)