Esempio Time Series

8/3/2019 Esempio Time Series

1/15

POLITECNICO DI TORINO

ESERCITAZIONI DI LOGISTICA

Laurea in Ingegneria Logistica e della Produzione

Corso di Logistica e di Distribuzione 1


2/15

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


3/15

Time series plot of Tickets

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Tickets

There is some upward trend and

some seasonality, so Winters' method

looks like a good choice.


4/15

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


5/15

Time series of Tickets with forecasts superimposed

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6/15

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


7/15


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8/15

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


9/15


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10/15

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.


11/15


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12/15

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-95 647,000 637,914 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


13/15


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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.


15/15


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Date

Observation

Forecast

Esempio Time Series

Documents

Transcript of Esempio Time Series