HOLT-WINTERS’ EXPONENTIAL SMOOTHING

Holt-Winters Additive Persamaan yang digunakan dalam Holt-Winters’ additive untuk t>s adalah sebagai berikut: Single smoothing Pemulusan tren (Trend Smoothing) Pemulusan musiman (Seasonal Smoothing)

Forecasting untuk 1 periode adalah: Forecasting m periode adalah: Dengan: s = panjang siklus musiman = konstanta pemulusan

Nilai Awal (Initial Value) Initial value untuk Holt-Winters’s additive adalah sebagai berikut:

Contoh:Quarterly sales of saws for Acme tool company The following table shows the sales of saws for the Acme tool Company. These are quarterly sales From 1994 through 2000.

Holt-Winters Additive Model alfa 0.5 beta 0.2 gamma 0.9   Year Quarter t sales L(t) b(t) S(t) F(t+1) e(i) e(i)^2 1994 1 500 125 2 350 -25 3 250 -125 4 400 375 -12.5 25 1995 5 450 343.75 -16.25 108.125 487.5 37.5 1406.25 6 351.25 -11.5 -3.625 302.5 -47.5 2256.25 7 200 332.375 -12.975 -131.638 214.75 14.75 217.5625 8 300 297.2 -17.415 5.02 344.4 44.4 1971.36 1996 9 260.83 -21.206 91.0655 387.91 37.91 1437.168 10 221.6245 -24.8059 -19.8246 235.999 35.999 1295.928 11 150 239.2281 -16.324 -93.469 65.1811 -84.8189 7194.246 12 308.942 0.883586 82.45418 227.924 -172.076 29610.14 1997 13 550 384.3801 15.79448 158.1645 400.8911 -149.109 22233.46 14 384.9995 12.75948 -33.482 380.35 30.34998 921.1212 15 370.614 7.330475 -117.9 304.29 54.29002 2947.406 16 422.7451 16.29061 122.7748 460.3987 -89.6013 8028.4 1998 17 415.4356 11.57058 136.9244 597.2003 47.20026 2227.865 18 430.2441 12.21817 -30.5679 393.5242 -6.47583 41.93636 19 455.1809 14.76189 -106.453 324.5628 -25.4372 647.0518 20 600 473.584 15.49013 126.0519 592.7176 -7.28243 53.0338 1999 21 750 551.0749 27.89028 192.7251 625.9985 -124.001 15376.37 22 554.7665 23.05056 -52.3467 548.3972 48.39723 2342.292 23 542.1349 15.91412 -138.567 471.3643 71.36433 5092.868 24 650 540.9986 12.50403 110.7065 684.1009 34.10092 1162.873 2000 850 605.3888 22.88126 239.4226 746.2277 -103.772 10768.7 26 640.3084 25.28893 -41.5122 575.9234 -24.0766 579.6841 27 627.082 17.58587 -173.23 527.0306 77.03058 5933.71 28 700 616.9807 12.04844 85.78802 755.3743 55.37433 3066.316 29 868.4517 754208.4 SSE 126812 MSE 5283.833 RMSE 72.68998

Kombinasi pemilihan konstanta pemulusan alfa beta gamma MSE RMSE 0.4 0.1 0.3 5740.124 75.76361 5874.074 76.64251 0.5 5553.978 74.52502 0.6 5492.098 74.10869 0.7 5448.754 73.81568 0.8 5425.034 73.65483 0.9 5423.918 73.64725 0.2 5295.415 72.7696 5283.833 72.68998

Plot Forecasting Holt –Winters Additive Model: menggunakan: Alfa = 0.5 beta = 0.2 gamma= 0.9 Menghasilkan: MSE = 5283.833 RMSE = 72.68998 File: Holt-Winters

Holt-Winters Multiplicative Persamaan yang digunakan dalam Holt-Winters’ Multiplicative untuk t>s adalah sebagai berikut: The single smoothing Pemulusan tren (The trend smoothing): Pemulusan musiman (The seasonal smoothing):

Holt-Winters Multiplicative (Makridakis, Spyros. 1978 Holt-Winters Multiplicative (Makridakis, Spyros. 1978. Forecasting: Methods and Applications Persamaan yang digunakan dalam Holt-Winters’ Multiplicative untuk t>s adalah sebagai berikut: The single smoothing Pemulusan tren (The trend smoothing): Pemulusan musiman (The seasonal smoothing):

Forecasting untuk 1periode adalah Forecasting untuk m periode adalah Dengan: s = panjang siklus musiman = konstanta pemulusan

Nilai Awal (Initial Value) Initial value untuk Holt-Winters’s multiplicative adalah sebagai berikut:

Holt-Winters Multiplicative Model alfa 0.93 beta 0.09 gamma 0.97   Year Quarter t sales L(t) b(t) S(t) F(t+1) e(i) e(i)^2 1994 1 500 1.333333 2 350 0.933333 3 250 0.666667 4 400 375 -12.5 1.066667 1995 5 450 339.25 -14.5925 1.326662 483.3333 -33.3333 1111.111 6 371.476 -10.3788 0.941922 303.0137 46.98633 2207.716 7 200 304.2768 -15.4927 0.657577 240.7315 -40.7315 1659.052 8 300 281.7774 -16.1233 1.06473 308.0364 -8.03641 64.58391 1996 9 263.9485 -16.2768 1.326036 352.4332 -2.43315 5.920236 10 214.8057 -19.2347 0.9314 233.2873 -33.2873 1108.046 11 150 225.8323 -16.5112 0.664011 128.603 21.39698 457.8308 12 364.0368 -2.58679 1.097768 222.8705 177.1295 31374.86 1997 13 550 411.0377 1.876101 1.337716 479.2956 70.70435 4999.106 14 378.378 -1.23212 0.925193 384.5877 -34.5877 1196.311 15 376.5452 -1.28618 0.663933 250.4289 -0.42892 0.183971 16 492.2135 9.23972 1.116812 411.9474 138.0526 19058.52 1998 17 417.47 1.681232 1.318068 670.8018 -120.802 14593.06 18 431.4189 2.785326 0.927114 387.7957 12.2043 148.9449 19 520.6544 10.56583 0.671982 288.2827 61.71731 3809.026 20 600 536.8217 11.06997 1.117663 593.2733 6.72672 45.24877 1999 21 750 567.5362 12.83798 1.321399 722.1583 27.84171 775.1607 22 542.1828 9.400758 0.922346 538.0728 -38.0728 1449.538 23 592.197 13.05597 0.675347 370.6543 29.34571 861.1706 24 650 583.2283 11.07375 1.114582 676.469 -26.469 700.6092 2000 25 850 639.8309 15.17134 1.328264 785.3099 64.69008 4184.806 26 650.8295 14.79579 0.921914 604.1385 -4.13845 17.1268 27 666.2754 14.8543 0.675395 449.5279 0.472117 0.222895 28 700 631.7547 10.41056 1.108222 759.1747 -59.1747 3501.642 29 852.9648 -852.965 727548.9 SSE 93329.8 MSE 3888.742 RMSE 62.35978

Kombinasi pemilihan konstanta pemulusan alfa beta gamma MSE RMSE 0.4 0.1 0.3 5306.15 72.84333 5309.654 72.86737 0.5 5291.192 72.74058 0.6 5254.27 72.48634 0.7 5209.442 72.17646 0.8 5169.303 71.89787 0.9 5146.042 71.73592 0.95 5144.225 71.72325 0.97 5145.742 71.73383 0.09 5171.095 71.91032 4713.89 68.65778 4503.319 67.10678 4365.369 66.07094 3912.758 62.55205 0.93 3888.742 62.35978

Holt-Winters Multiplicative Model (Makridakis, Spyros. 1978 Holt-Winters Multiplicative Model (Makridakis, Spyros. 1978. Forecasting: Methods and Applications) alfa 0.93 beta 0.09 gamma 0.97   Year Quarter t sales L(t) b(t) S(t) F(t+1) e(i) e(i)^2 1994 1 500 1.333333 2 350 0.933333 3 250 0.666667 4 400 375 -12.5 1.066667 1995 5 450 314.883 -11.3894 1.426229 483.3333 -33.3333 1111.111 6 349.9869 -10.3543 0.998036 283.2607 66.73933 4454.139 7 200 279.7932 -9.44051 0.713369 226.4217 -26.4217 698.1068 8 300 262.0839 -8.59656 1.142332 288.3763 11.62375 135.1115 1996 9 228.9916 -7.83423 1.525374 361.531 -11.531 132.9635 10 187.3093 -7.14553 1.065661 220.723 -20.723 429.4444 11 150 196.1586 -6.49818 0.763148 128.5232 21.47676 461.2512 12 326.0503 -5.85375 1.22427 216.6551 183.3449 33615.37 1997 13 550 335.9149 -5.32419 1.633961 488.4196 61.58044 3792.151 14 306.1675 -4.85298 1.14084 352.2978 -2.29776 5.279699 15 305.1266 -4.41652 0.817647 229.9475 20.05247 402.1016 16 418.1099 -3.98571 1.312709 368.1504 181.8496 33069.28 1998 17 313.5011 -3.64951 1.750767 676.6628 -126.663 16043.45 18 326.6453 -3.31728 1.222058 353.4911 46.50889 2163.077 19 398.463 -2.99894 0.876553 264.3681 85.63193 7332.828 20 600 425.3207 -2.72297 1.40776 519.1291 80.8709 6540.102 1999 21 750 398.7636 -2.48352 1.876912 739.8703 10.12971 102.6111 22 380.9633 -2.26402 1.30975 484.2773 15.72274 247.2044 23 424.6888 -2.04993 0.939907 331.9501 68.04989 4630.787 24 650 429.6061 -1.86439 1.509856 594.9743 55.02573 3027.831 2000 25 850 421.4732 -1.6983 2.012541 802.8336 47.1664 2224.67 26 426.4156 -1.5444 1.404158 549.8002 50.19976 2520.015 27 445.507 -1.40137 1.00798 399.3393 50.66067 2566.503 28 700 431.3363 -1.27811 1.619473 670.5357 29.46431 868.1458 29 865.5096 -865.51 749106.9 SSE 126573.5 MSE 5273.897 RMSE 72.6216

Plot Forecasting Holt –Winters Multiplicative Model: menggunakan: Alfa = 0.93 beta = 0.09 gamma= 0.97 Forecasting Holt-Winter MSE = 3888.742 RMSE = 62.3598 Forecasting Holt-Winters (Makridakis): MSE = 5273.897 RMSE = 72.6216 File: Holt-Winters

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