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POLITECNICO DI TORINO
ESERCITAZIONI DI LOGISTICA
Laurea in Ingegneria Logistica e della Produzione
Corso di Logistica e di Distribuzione 1
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Problem 16.32 - Number of Airline Tickets Sold by a Local Travel Agenc
Month Year Tickets
January 1995 605
February 1995 647
March 1995 636
April 1995 612
May 1995 714
June 1995 765
July 1995 698
August 1995 615
September 1995 588
October 1995 685November 1995 711
December 1995 664
January 1996 630
February 1996 696
March 1996 670
April 1996 671
May 1996 724
June 1996 787
July 1996 724August 1996 651
September 1996 589
October 1996 697
November 1996 750
December 1996 705
January 1997 664
February 1997 704
March 1997 691
April 1997 672
May 1997 753
June 1997 787
July 1997 751
August 1997 695
September 1997 643
October 1997 724
November 1997 803
December 1997 705
January 1998 720
February 1998 757March 1998 707
April 1998 692
May 1998 828
June 1998 827
July 1998 763
August 1998 710
Part (e): As the time series chart indicates, there is
seasonality, so that Winters' model does a significantly
better job than Holt's and simple models. However, it
doesn't help much in any of the models to optimize over the
smoothing constants - only marginal improvements are
8/3/2019 Esempio Time Series
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Time series plot of Tickets
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Janu
ary
March
M
ay July
Septe
mbe
r
Novembe
r
Janu
ary
March
M
ay July
Septe
mbe
r
Novembe
r
Janu
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March
M
ay July
Septe
mbe
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Novembe
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Janu
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March
M
ay July
Septe
mbe
r
Novembe
r
Month
Tickets
There is some upward trend and
some seasonality, so Winters' method
looks like a good choice.
8/3/2019 Esempio Time Series
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Forecasting results for Tickets Date Observation SmLevel Forecast Error
gen-95 605,000 605,000
Simple exponential smoothing feb-95 647,000 609,200 605,000 42,000
mar-95 636,000 611,880 609,200 26,800
Smoothing constant(s) apr-95 612,000 611,892 611,880 0,120
Level 0,100 alfa mag-95 714,000 622,103 611,892 102,108
giu-95 765,000 636,393 622,103 142,897
Estimation period lug-95 698,000 642,553 636,393 61,607
ago-95 615,000 639,798 642,553 -27,553
MAE 49,3871 set-95 588,000 634,618 639,798 -51,798
RMSE 61,2139 ott-95 685,000 639,656 634,618 50,382
MAPE 6,79% nov-95 711,000 646,791 639,656 71,344
dic-95 664,000 648,512 646,791 17,209gen-96 630,000 646,660 648,512 -18,512
feb-96 696,000 651,594 646,660 49,340
mar-96 670,000 653,435 651,594 18,406
apr-96 671,000 655,191 653,435 17,565
mag-96 724,000 662,072 655,191 68,809
giu-96 787,000 674,565 662,072 124,928
lug-96 724,000 679,509 674,565 49,435
ago-96 651,000 676,658 679,509 -28,509
set-96 589,000 667,892 676,658 -87,658
ott-96 697,000 670,803 667,892 29,108nov-96 750,000 678,722 670,803 79,197
dic-96 705,000 681,350 678,722 26,278
gen-97 664,000 679,615 681,350 -17,350
feb-97 704,000 682,054 679,615 24,385
mar-97 691,000 682,948 682,054 8,946
apr-97 672,000 681,853 682,948 -10,948
mag-97 753,000 688,968 681,853 71,147
giu-97 787,000 698,771 688,968 98,032
lug-97 751,000 703,994 698,771 52,229
ago-97 695,000 703,095 703,994 -8,994set-97 643,000 697,085 703,095 -60,095
ott-97 724,000 699,777 697,085 26,915
nov-97 803,000 710,099 699,777 103,223
dic-97 705,000 709,589 710,099 -5,099
gen-98 720,000 710,630 709,589 10,411
feb-98 757,000 715,267 710,630 46,370
mar-98 707,000 714,441 715,267 -8,267
apr-98 692,000 712,196 714,441 -22,441
mag-98 828,000 723,777 712,196 115,804
giu-98 827,000 734,099 723,777 103,223
lug-98 763,000 736,989 734,099 28,901
ago-98 710,000 734,290 736,989 -26,989
set-98 673,000 728,161 734,290 -61,290
ott-98 793,000 734,645 728,161 64,839
nov-98 852,000 746,381 734,645 117,355
dic-98 710,000 742,743 746,381 -36,381
8/3/2019 Esempio Time Series
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Time series of Tickets with forecasts superimposed
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700
750
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900
gen-95
mar
-95
mag
-95
lug-95
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nov-9
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nov-9
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Date
Observation
Forecast
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Forecasting results for Tickets Date Observation SmLevel Forecast Error
gen-95 605,000 605,000
Simple exponential smoothing feb-95 647,000 613,408 605,000 42,000
mar-95 636,000 617,931 613,408 22,592
Smoothing constant(s) apr-95 612,000 616,743 617,931 -5,931
Level 0,200 mag-95 714,000 636,213 616,743 97,257
giu-95 765,000 661,995 636,213 128,787
Estimation period lug-95 698,000 669,202 661,995 36,005
ago-95 615,000 658,352 669,202 -54,202
MAE 47,4880 set-95 588,000 644,268 658,352 -70,352
RMSE 58,4296 ott-95 685,000 652,422 644,268 40,732
MAPE 6,62% nov-95 711,000 664,149 652,422 58,578
dic-95 664,000 664,119 664,149 -0,149gen-96 630,000 657,289 664,119 -34,119
feb-96 696,000 665,038 657,289 38,711
mar-96 670,000 666,032 665,038 4,962
apr-96 671,000 667,026 666,032 4,968
mag-96 724,000 678,432 667,026 56,974
giu-96 787,000 700,166 678,432 108,568
lug-96 724,000 704,937 700,166 23,834
ago-96 651,000 694,140 704,937 -53,937
set-96 589,000 673,092 694,140 -105,140
ott-96 697,000 677,878 673,092 23,908nov-96 750,000 692,316 677,878 72,122
dic-96 705,000 694,855 692,316 12,684
gen-97 664,000 688,678 694,855 -30,855
feb-97 704,000 691,746 688,678 15,322
mar-97 691,000 691,596 691,746 -0,746
apr-97 672,000 687,673 691,596 -19,596
mag-97 753,000 700,751 687,673 65,327
giu-97 787,000 718,017 700,751 86,249
lug-97 751,000 724,620 718,017 32,983
ago-97 695,000 718,690 724,620 -29,620set-97 643,000 703,538 718,690 -75,690
ott-97 724,000 707,634 703,538 20,462
nov-97 803,000 726,725 707,634 95,366
dic-97 705,000 722,376 726,725 -21,725
gen-98 720,000 721,900 722,376 -2,376
feb-98 757,000 728,927 721,900 35,100
mar-98 707,000 724,537 728,927 -21,927
apr-98 692,000 718,024 724,537 -32,537
mag-98 828,000 740,040 718,024 109,976
giu-98 827,000 757,448 740,040 86,960
lug-98 763,000 758,560 757,448 5,552
ago-98 710,000 748,839 758,560 -48,560
set-98 673,000 733,657 748,839 -75,839
ott-98 793,000 745,536 733,657 59,343
nov-98 852,000 766,849 745,536 106,464
dic-98 710,000 755,469 766,849 -56,849
These summary measures are
marginally better than in part (b),
but nothing much to get excited
8/3/2019 Esempio Time Series
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Time series of Tickets with forecasts superimposed
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gen-95
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mag
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set-9
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Date
Observation
Forecast
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Forecasting results for Tickets Date Observation SmLevel SmTrend Forecast Error Esercizio con aggiunta del trend
gen-95 605,000 605,000 2,188 inizializzazione del trend senza la retta di regressione
Holt's exponential smoothing feb-95 647,000 611,169 2,586 607,188 39,813
mar-95 636,000 615,979 2,808 613,754 22,246
Smoothing constant(s) apr-95 612,000 618,108 2,740 618,787 -6,787
Level 0,100 mag-95 714,000 630,164 3,672 620,849 93,151
Trend 0,100 giu-95 765,000 646,952 4,983 633,835 131,165
lug-95 698,000 656,542 5,444 651,935 46,065
Estimation period ago-95 615,000 657,287 4,974 661,986 -46,986
set-95 588,000 654,835 4,232 662,261 -74,261
MAE 46,3444 ott-95 685,000 661,660 4,491 659,067 25,933
RMSE 56,1873 nov-95 711,000 670,636 4,939 666,151 44,849
MAPE 6,57% dic-95 664,000 674,418 4,824 675,575 -11,575
gen-96 630,000 674,317 4,331 679,241 -49,241
feb-96 696,000 680,384 4,505 678,648 17,352
mar-96 670,000 683,399 4,356 684,888 -14,888
apr-96 671,000 686,080 4,188 687,755 -16,755
mag-96 724,000 693,641 4,526 690,268 33,732
giu-96 787,000 707,050 5,414 698,167 88,833
lug-96 724,000 713,618 5,529 712,464 11,536
ago-96 651,000 712,332 4,848 719,147 -68,147
set-96 589,000 704,362 3,566 717,180 -128,180
ott-96 697,000 706,835 3,457 707,928 -10,928
nov-96 750,000 714,263 3,854 710,292 39,708
dic-96 705,000 716,805 3,723 718,117 -13,117
gen-97 664,000 714,875 3,157 720,528 -56,528
feb-97 704,000 716,629 3,017 718,032 -14,032
mar-97 691,000 716,782 2,731 719,646 -28,646
apr-97 672,000 714,761 2,255 719,512 -47,512
mag-97 753,000 720,615 2,615 717,016 35,984
giu-97 787,000 729,607 3,253 723,230 63,770
lug-97 751,000 734,674 3,434 732,860 18,140
ago-97 695,000 733,798 3,003 738,108 -43,108
set-97 643,000 727,421 2,065 736,801 -93,801
ott-97 724,000 728,938 2,010 729,486 -5,486
nov-97 803,000 738,153 2,731 730,948 72,052
dic-97 705,000 737,296 2,372 740,884 -35,884
gen-98 720,000 737,701 2,175 739,668 -19,668
feb-98 757,000 741,589 2,347 739,877 17,123
mar-98 707,000 740,242 1,977 743,936 -36,936
apr-98 692,000 737,197 1,475 742,219 -50,219
mag-98 828,000 747,605 2,368 738,673 89,327giu-98 827,000 757,676 3,139 749,974 77,026
lug-98 763,000 761,034 3,161 760,815 2,185
ago-98 710,000 758,775 2,619 764,194 -54,194
set-98 673,000 752,554 1,735 761,393 -88,393
ott-98 793,000 758,160 2,122 754,289 38,711
nov-98 852,000 769,453 3,039 760,281 91,719
dic-98 710,000 766,243 2,414 772,492 -62,492
gen-99 768,657
feb-99 771,071
mar-99 773,485
apr-99 775,899
mag-99 778,313
giu-99 780,727
lug-99 783,141
ago-99 785,555
set-99 787,969
ott-99 790,383
nov-99 792,797dic-99 795,211
8/3/2019 Esempio Time Series
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Time series of Tickets with forecasts superimposed
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600
650
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850
900
gen-95
mar
-95
mag
-95
lug-95
set-9
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Date
Observation
Forecast
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Forecasting results for Tickets Date Observation SmLevel SmTrend Forecast Error
gen-95 605,000 605,000 2,188
Holt's exponential smoothing feb-95 647,000 610,234 2,272 607,188 39,813
mar-95 636,000 614,303 2,322 612,506 23,494Smoothing constant(s) apr-95 612,000 616,271 2,312 616,625 -4,625
Level 0,077 mag-95 714,000 625,883 2,514 618,583 95,417
Trend 0,028 giu-95 765,000 638,849 2,804 628,397 136,603
lug-95 698,000 645,964 2,923 641,653 56,347
Estimation period ago-95 615,000 646,294 2,851 648,887 -33,887
set-95 588,000 644,467 2,722 649,146 -61,146
MAE 44,9023 ott-95 685,000 650,082 2,802 647,189 37,811
RMSE 55,4748 nov-95 711,000 657,330 2,925 652,884 58,116
MAPE 6,31% dic-95 664,000 660,542 2,933 660,255 3,745
gen-96 630,000 660,913 2,862 663,474 -33,474
feb-96 696,000 666,241 2,930 663,775 32,225mar-96 670,000 669,234 2,932 669,171 0,829
apr-96 671,000 672,077 2,930 672,166 -1,166
mag-96 724,000 678,755 3,033 675,007 48,993
giu-96 787,000 689,838 3,256 681,789 105,211
lug-96 724,000 695,459 3,322 693,095 30,905
ago-96 651,000 695,125 3,221 698,781 -47,781
set-96 589,000 689,980 2,989 698,346 -109,346
ott-96 697,000 693,277 2,997 692,969 4,031
nov-96 750,000 700,385 3,111 696,274 53,726
dic-96 705,000 703,611 3,114 703,496 1,504
gen-97 664,000 703,457 3,024 706,726 -42,726feb-97 704,000 706,291 3,019 706,480 -2,480
mar-97 691,000 707,908 2,980 709,309 -18,309
apr-97 672,000 707,913 2,897 710,888 -38,888
mag-97 753,000 714,038 2,987 710,810 42,190
giu-97 787,000 722,379 3,135 717,025 69,975
lug-97 751,000 727,464 3,189 725,514 25,486
ago-97 695,000 727,925 3,114 730,653 -35,653
set-97 643,000 724,303 2,927 731,039 -88,039
ott-97 724,000 726,983 2,920 727,230 -3,230
nov-97 803,000 735,496 3,075 729,903 73,097
dic-97 705,000 736,002 3,004 738,571 -33,571gen-98 720,000 737,552 2,964 739,006 -19,006
feb-98 757,000 741,777 2,999 740,515 16,485
mar-98 707,000 741,885 2,918 744,775 -37,775
apr-98 692,000 740,763 2,807 744,803 -52,803
mag-98 828,000 750,030 2,986 743,570 84,430
giu-98 827,000 758,676 3,142 753,015 73,985
lug-98 763,000 761,909 3,145 761,818 1,182
ago-98 710,000 760,841 3,028 765,054 -55,054
set-98 673,000 756,917 2,836 763,869 -90,869
ott-98 793,000 762,296 2,906 759,752 33,248
nov-98 852,000 771,843 3,090 765,202 86,798dic-98 710,000 769,965 2,952 774,933 -64,933
gen-99 772,917
feb-99 775,870
mar-99 778,822
apr-99 781,774
mag-99 784,727
giu-99 787,679
Again, this is only a marginal
improvement over the non-
optimized Holt's model, and it's only
a little better than the simple
exponential smoothing models.
8/3/2019 Esempio Time Series
11/15
Time series of Tickets with forecasts superimposed
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900
gen-95
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Date
Observation
Forecast
8/3/2019 Esempio Time Series
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Forecasting results for Tickets Date Observation SmLevel SmTrend SmSeason Forecast Error
gen-95 605,000 636,231 1,796 0,951
Winters' exponential smoothing feb-95 647,000 637,914 1,784 1,016 648,146 -1,146
mar-95 636,000 641,229 1,938 0,973 621,136 14,864
Smoothing constant(s) apr-95 612,000 643,117 1,933 0,952 612,469 -0,469Level 0,100 mag-95 714,000 647,160 2,144 1,075 691,378 22,622
Trend 0,100 giu-95 765,000 653,057 2,519 1,120 723,196 41,804
Seasonality 0,100 lug-95 698,000 656,586 2,620 1,050 687,414 10,586
ago-95 615,000 658,614 2,561 0,941 620,572 -5,572
Estimation period set-95 588,000 662,573 2,701 0,873 575,823 12,177
ott-95 685,000 666,894 2,863 1,007 668,708 16,292
MAE 14,3915 nov-95 711,000 668,889 2,776 1,074 720,335 -9,335
RMSE 17,5676 dic-95 664,000 672,060 2,815 0,983 660,113 3,887
MAPE 2,00% gen-96 630,000 673,640 2,692 0,949 641,748 -11,748
feb-96 696,000 677,223 2,781 1,017 686,950 9,050
mar-96 670,000 680,858 2,866 0,974 661,691 8,309
apr-96 671,000 685,820 3,076 0,955 651,046 19,954mag-96 724,000 687,357 2,922 1,073 740,540 -16,540
giu-96 787,000 691,546 3,049 1,121 772,811 14,189
lug-96 724,000 694,087 2,998 1,049 729,336 -5,336
ago-96 651,000 696,585 2,948 0,940 655,701 -4,701
set-96 589,000 697,082 2,703 0,870 610,387 -21,387
ott-96 697,000 698,997 2,624 1,006 704,937 -7,937
nov-96 750,000 701,275 2,589 1,074 753,724 -3,724
dic-96 705,000 705,173 2,720 0,985 692,125 12,875
gen-97 664,000 707,047 2,636 0,948 672,034 -8,034
feb-97 704,000 707,945 2,462 1,015 721,678 -17,678
mar-97 691,000 710,298 2,451 0,974 692,055 -1,055
apr-97 672,000 711,854 2,362 0,954 680,550 -8,550mag-97 753,000 712,984 2,238 1,071 766,211 -13,211
giu-97 787,000 713,880 2,104 1,120 802,057 -15,057
lug-97 751,000 715,955 2,101 1,049 751,299 -0,299
ago-97 695,000 720,185 2,314 0,943 674,991 20,009
set-97 643,000 724,174 2,482 0,872 628,432 14,568
ott-97 724,000 725,934 2,409 1,005 731,262 -7,262
nov-97 803,000 730,291 2,604 1,076 782,083 20,917
dic-97 705,000 731,182 2,433 0,983 721,876 -16,876
gen-98 720,000 736,177 2,689 0,951 695,702 24,298
feb-98 757,000 739,586 2,761 1,016 749,694 7,306
mar-98 707,000 740,697 2,596 0,972 723,072 -16,072
apr-98 692,000 741,520 2,419 0,952 708,911 -16,911mag-98 828,000 746,847 2,710 1,075 796,858 31,142
giu-98 827,000 748,472 2,601 1,118 839,137 -12,137
lug-98 763,000 748,682 2,362 1,046 788,091 -25,091
ago-98 710,000 751,269 2,385 0,943 707,878 2,122
set-98 673,000 755,501 2,569 0,874 656,894 16,106
ott-98 793,000 761,135 2,876 1,009 762,194 30,806
nov-98 852,000 766,765 3,151 1,080 822,351 29,649
dic-98 710,000 765,160 2,676 0,977 756,741 -46,741
gen-99 730,436
feb-99 782,488
mar-99 751,601
apr-99 738,382mag-99 836,840
giu-99 873,438
lug-99 820,160
ago-99 741,558
set-99 689,427
ott-99 799,109
nov-99 858,033
dic-99 779,241
8/3/2019 Esempio Time Series
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Time series of Tickets with forecasts superimposed
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gen-95
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-95
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-95
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nov-9
8
gen-99
mar
-99
mag-99
lug-99
set-9
9
nov-9
9
Date
Observation
Forecast
8/3/2019 Esempio Time Series
14/15
Forecasting results for Tickets Date Observation SmLevel SmTrend SmSeason Forecast Error
gen-95 605,000 636,231 1,796 0,951
Winters' exponential smoothing feb-95 647,000 637,898 1,788 1,016 648,146 -1,146
mar-95 636,000 641,429 1,894 0,971 621,124 14,876
Smoothing constant(s) apr-95 612,000 643,249 1,889 0,952 612,618 -0,618Level 0,114 mag-95 714,000 647,529 2,035 1,072 691,473 22,527
Trend 0,061 giu-95 765,000 653,803 2,293 1,114 723,485 41,515
Seasonality 0,000 lug-95 698,000 657,185 2,359 1,049 687,959 10,041
ago-95 615,000 658,832 2,316 0,941 620,890 -5,890
Estimation period set-95 588,000 662,741 2,412 0,871 575,800 12,200
ott-95 685,000 667,011 2,525 1,005 668,588 16,412
MAE 13,8179 nov-95 711,000 668,574 2,467 1,076 720,097 -9,097
RMSE 17,1263 dic-95 664,000 671,562 2,499 0,983 659,499 4,501
MAPE 1,93% gen-96 630,000 672,748 2,419 0,951 640,973 -10,973
feb-96 696,000 676,300 2,488 1,016 685,875 10,125
mar-96 670,000 680,066 2,565 0,971 659,091 10,909
apr-96 671,000 685,133 2,718 0,952 650,050 20,950mag-96 724,000 686,444 2,632 1,072 737,253 -13,253
giu-96 787,000 691,068 2,753 1,114 767,495 19,505
lug-96 724,000 693,440 2,730 1,049 727,517 -3,517
ago-96 651,000 695,642 2,698 0,941 655,370 -4,370
set-96 589,000 695,833 2,545 0,871 608,191 -19,191
ott-96 697,000 697,815 2,511 1,005 701,984 -4,984
nov-96 750,000 699,986 2,490 1,076 753,212 -3,212
dic-96 705,000 704,167 2,593 0,983 690,395 14,605
gen-97 664,000 705,795 2,535 0,951 672,067 -8,067
feb-97 704,000 706,587 2,429 1,016 719,565 -15,565
mar-97 691,000 709,315 2,447 0,971 688,442 2,558
apr-97 672,000 711,070 2,405 0,952 677,791 -5,791mag-97 753,000 712,232 2,329 1,072 764,718 -11,718
giu-97 787,000 713,654 2,274 1,114 795,879 -8,879
lug-97 751,000 715,961 2,276 1,049 750,697 0,303
ago-97 695,000 720,515 2,415 0,941 676,144 18,856
set-97 643,000 724,679 2,521 0,871 629,607 13,393
ott-97 724,000 726,413 2,473 1,005 730,954 -6,954
nov-97 803,000 730,903 2,596 1,076 783,929 19,071
dic-97 705,000 731,660 2,484 0,983 720,883 -15,883
gen-98 720,000 736,763 2,643 0,951 698,107 21,893
feb-98 757,000 740,063 2,683 1,016 751,134 5,866
mar-98 707,000 741,083 2,582 0,971 721,194 -14,194
apr-98 692,000 741,734 2,465 0,952 708,171 -16,171mag-98 828,000 747,419 2,660 1,072 797,647 30,353
giu-98 827,000 749,218 2,608 1,114 835,440 -8,440
lug-98 763,000 749,077 2,441 1,049 788,338 -25,338
ago-98 710,000 751,823 2,459 0,941 707,474 2,526
set-98 673,000 756,384 2,587 0,871 656,912 16,088
ott-98 793,000 762,378 2,794 1,005 762,890 30,110
nov-98 852,000 768,244 2,981 1,076 822,956 29,044
dic-98 710,000 765,675 2,644 0,983 757,961 -47,961
gen-99 730,604
feb-99 783,190
mar-99 751,158
apr-99 739,200
mag-99 834,834
giu-99 870,477
lug-99 822,264
ago-99 740,711
set-99 687,555
ott-99 796,201
nov-99 854,772
dic-99 783,684
Slightly better than the Winters' nonoptimized model, but much better
than Holt's and simple models.
8/3/2019 Esempio Time Series
15/15
Time series of Tickets with forecasts superimposed
550
600
650
700
750
800
850
900
gen-95
mar
-95
mag
-95
lug-95
set-9
5
nov-9
5
gen-96
mar
-96
mag
-96
lug-96
set-9
6
nov-9
6
gen-97
mar
-97
mag
-97
lug-97
set-9
7
nov-9
7
gen-98
mar
-98
mag-98
lug-98
set-9
8
nov-9
8
gen-99
mar
-99
mag-99
lug-99
set-9
9
nov-9
9
Date
Observation
Forecast