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Description of Statistical Computation

Author's title

Author*The author of this computation has been verified*
R Software Modulerwasp_exponentialsmoothing.wasp
Title produced by softwareExponential Smoothing
Date of computationWed, 15 Dec 2010 18:29:19 +0000
Cite this page as followsStatistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2010/Dec/15/t1292437813rh1tp5gvfp8xb2t.htm/, Retrieved Mon, 05 May 2025 14:13:04 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=110646, Retrieved Mon, 05 May 2025 14:13:04 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact227

Tree of Dependent Computations

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-     [Univariate Explorative Data Analysis] [Run Sequence gebo...] [2008-12-12 13:32:37] [76963dc1903f0f612b6153510a3818cf]
- R  D  [Univariate Explorative Data Analysis] [Run Sequence gebo...] [2008-12-17 12:14:40] [76963dc1903f0f612b6153510a3818cf]
-         [Univariate Explorative Data Analysis] [Run Sequence Plot...] [2008-12-22 18:19:51] [1ce0d16c8f4225c977b42c8fa93bc163]
- RMP       [Univariate Data Series] [Identifying Integ...] [2009-11-22 12:08:06] [b98453cac15ba1066b407e146608df68]
- RMP         [Standard Deviation-Mean Plot] [Births] [2010-11-29 10:52:49] [b98453cac15ba1066b407e146608df68]
- RMP           [ARIMA Backward Selection] [Births] [2010-11-29 17:47:06] [b98453cac15ba1066b407e146608df68]
- RMPD            [ARIMA Forecasting] [arima forecast paper] [2010-12-12 14:16:56] [7d64bf19f34ddcdf2626356c9d5bd60d]
- RMP                 [Exponential Smoothing] [additive hw] [2010-12-15 18:29:19] [5842cf9dd57f9603e676e11284d3404a] [Current]
-   PD                  [Exponential Smoothing] [Exp sm Multi] [2010-12-15 19:00:17] [7d64bf19f34ddcdf2626356c9d5bd60d]
-   PD                  [Exponential Smoothing] [additive methode] [2010-12-15 19:06:40] [7d64bf19f34ddcdf2626356c9d5bd60d]
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Dataset

Dataseries X:
Download CSV
Histogram
Boxplots 
597
593
590
580
574
573
573
620
626
620
588
566
557
561
549
532
526
511
499
555
565
542
527
510
514
517
508
493
490
469
478
528
534
518
506
502
516
528
533
536
537
524
536
587
597
581
564
558
575
580
575
563
552
537
545
601
604
586
564
549

Tables (Output of Computation)





Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time3 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135

\begin{tabular}{lllllllll}
\hline
Summary of computational transaction \tabularnewline
Raw Input & view raw input (R code)  \tabularnewline
Raw Output & view raw output of R engine  \tabularnewline
Computing time & 3 seconds \tabularnewline
R Server & 'Gwilym Jenkins' @ 72.249.127.135 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=110646&T=0

[TABLE]
[ROW][C]Summary of computational transaction[/C][/ROW]
[ROW][C]Raw Input[/C][C]view raw input (R code) [/C][/ROW]
[ROW][C]Raw Output[/C][C]view raw output of R engine [/C][/ROW]
[ROW][C]Computing time[/C][C]3 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Gwilym Jenkins' @ 72.249.127.135[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=110646&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=110646&T=0

As an alternative you can also use a QR Code:  
QR Code


The GUIDs for individual cells are displayed in the table below:

Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time3 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135






Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.902430979310268
beta0.171889535288598
gamma1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.902430979310268 \tabularnewline
beta & 0.171889535288598 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=110646&T=1

[TABLE]
[ROW][C]Estimated Parameters of Exponential Smoothing[/C][/ROW]
[ROW][C]Parameter[/C][C]Value[/C][/ROW]
[ROW][C]alpha[/C][C]0.902430979310268[/C][/ROW]
[ROW][C]beta[/C][C]0.171889535288598[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=110646&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=110646&T=1

As an alternative you can also use a QR Code:  
QR Code


The GUIDs for individual cells are displayed in the table below:

Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.902430979310268
beta0.171889535288598
gamma1






Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
13557582.066773504274-25.0667735042737
14561560.5137154054380.486284594562449
15549545.5542935929533.4457064070466
16532529.341704979862.65829502014026
17526523.8308825187672.16911748123323
18511508.2984145489892.70158545101117
19499514.290527895811-15.2905278958111
20555541.17415791578613.8258420842142
21565555.5196119562539.48038804374698
22542556.080843337813-14.0808433378133
23527507.48715780075119.512842199249
24510502.8195898092517.18041019074894
25514499.77425995300814.2257400469916
26517523.357456043095-6.35745604309477
27508508.633474020973-0.633474020973267
28493494.152819018376-1.15281901837568
29490490.053765896008-0.0537658960076328
30469477.121206562276-8.12120656227648
31478474.466164683443.53383531656021
32528526.9734852730041.02651472699631
33534533.1541827187740.845817281225663
34518526.0948171399-8.09481713989953
35506489.5797066024416.4202933975604
36502483.8372486190218.1627513809797
37516496.01287842949919.9871215705006
38528528.303486745401-0.30348674540096
39533526.0568051095436.9431948904571
40536525.99370756823010.0062924317696
41537541.434008533813-4.43400853381263
42524532.443786526393-8.44378652639284
43536539.267107680346-3.2671076803457
44587592.969756290473-5.96975629047301
45597599.311267383891-2.31126738389071
46581594.532895626929-13.5328956269292
47564560.6610374343743.33896256562559
48558546.4132630353911.5867369646098
49575554.94210647903220.0578935209677
50580587.437435597709-7.43743559770917
51575580.473890899085-5.47389089908484
52563568.591956654152-5.59195665415223
53552565.215275065266-13.2152750652664
54537543.215486277513-6.21548627751315
55545548.206578842009-3.20657884200909
56601597.3613456612283.63865433877197
57604609.882370920297-5.88237092029669
58586597.38412898002-11.3841289800203
59564564.028555948256-0.0285559482556437
60549543.9551924560235.04480754397673

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 557 & 582.066773504274 & -25.0667735042737 \tabularnewline
14 & 561 & 560.513715405438 & 0.486284594562449 \tabularnewline
15 & 549 & 545.554293592953 & 3.4457064070466 \tabularnewline
16 & 532 & 529.34170497986 & 2.65829502014026 \tabularnewline
17 & 526 & 523.830882518767 & 2.16911748123323 \tabularnewline
18 & 511 & 508.298414548989 & 2.70158545101117 \tabularnewline
19 & 499 & 514.290527895811 & -15.2905278958111 \tabularnewline
20 & 555 & 541.174157915786 & 13.8258420842142 \tabularnewline
21 & 565 & 555.519611956253 & 9.48038804374698 \tabularnewline
22 & 542 & 556.080843337813 & -14.0808433378133 \tabularnewline
23 & 527 & 507.487157800751 & 19.512842199249 \tabularnewline
24 & 510 & 502.819589809251 & 7.18041019074894 \tabularnewline
25 & 514 & 499.774259953008 & 14.2257400469916 \tabularnewline
26 & 517 & 523.357456043095 & -6.35745604309477 \tabularnewline
27 & 508 & 508.633474020973 & -0.633474020973267 \tabularnewline
28 & 493 & 494.152819018376 & -1.15281901837568 \tabularnewline
29 & 490 & 490.053765896008 & -0.0537658960076328 \tabularnewline
30 & 469 & 477.121206562276 & -8.12120656227648 \tabularnewline
31 & 478 & 474.46616468344 & 3.53383531656021 \tabularnewline
32 & 528 & 526.973485273004 & 1.02651472699631 \tabularnewline
33 & 534 & 533.154182718774 & 0.845817281225663 \tabularnewline
34 & 518 & 526.0948171399 & -8.09481713989953 \tabularnewline
35 & 506 & 489.57970660244 & 16.4202933975604 \tabularnewline
36 & 502 & 483.83724861902 & 18.1627513809797 \tabularnewline
37 & 516 & 496.012878429499 & 19.9871215705006 \tabularnewline
38 & 528 & 528.303486745401 & -0.30348674540096 \tabularnewline
39 & 533 & 526.056805109543 & 6.9431948904571 \tabularnewline
40 & 536 & 525.993707568230 & 10.0062924317696 \tabularnewline
41 & 537 & 541.434008533813 & -4.43400853381263 \tabularnewline
42 & 524 & 532.443786526393 & -8.44378652639284 \tabularnewline
43 & 536 & 539.267107680346 & -3.2671076803457 \tabularnewline
44 & 587 & 592.969756290473 & -5.96975629047301 \tabularnewline
45 & 597 & 599.311267383891 & -2.31126738389071 \tabularnewline
46 & 581 & 594.532895626929 & -13.5328956269292 \tabularnewline
47 & 564 & 560.661037434374 & 3.33896256562559 \tabularnewline
48 & 558 & 546.41326303539 & 11.5867369646098 \tabularnewline
49 & 575 & 554.942106479032 & 20.0578935209677 \tabularnewline
50 & 580 & 587.437435597709 & -7.43743559770917 \tabularnewline
51 & 575 & 580.473890899085 & -5.47389089908484 \tabularnewline
52 & 563 & 568.591956654152 & -5.59195665415223 \tabularnewline
53 & 552 & 565.215275065266 & -13.2152750652664 \tabularnewline
54 & 537 & 543.215486277513 & -6.21548627751315 \tabularnewline
55 & 545 & 548.206578842009 & -3.20657884200909 \tabularnewline
56 & 601 & 597.361345661228 & 3.63865433877197 \tabularnewline
57 & 604 & 609.882370920297 & -5.88237092029669 \tabularnewline
58 & 586 & 597.38412898002 & -11.3841289800203 \tabularnewline
59 & 564 & 564.028555948256 & -0.0285559482556437 \tabularnewline
60 & 549 & 543.955192456023 & 5.04480754397673 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=110646&T=2

[TABLE]
[ROW][C]Interpolation Forecasts of Exponential Smoothing[/C][/ROW]
[ROW][C]t[/C][C]Observed[/C][C]Fitted[/C][C]Residuals[/C][/ROW]
[ROW][C]13[/C][C]557[/C][C]582.066773504274[/C][C]-25.0667735042737[/C][/ROW]
[ROW][C]14[/C][C]561[/C][C]560.513715405438[/C][C]0.486284594562449[/C][/ROW]
[ROW][C]15[/C][C]549[/C][C]545.554293592953[/C][C]3.4457064070466[/C][/ROW]
[ROW][C]16[/C][C]532[/C][C]529.34170497986[/C][C]2.65829502014026[/C][/ROW]
[ROW][C]17[/C][C]526[/C][C]523.830882518767[/C][C]2.16911748123323[/C][/ROW]
[ROW][C]18[/C][C]511[/C][C]508.298414548989[/C][C]2.70158545101117[/C][/ROW]
[ROW][C]19[/C][C]499[/C][C]514.290527895811[/C][C]-15.2905278958111[/C][/ROW]
[ROW][C]20[/C][C]555[/C][C]541.174157915786[/C][C]13.8258420842142[/C][/ROW]
[ROW][C]21[/C][C]565[/C][C]555.519611956253[/C][C]9.48038804374698[/C][/ROW]
[ROW][C]22[/C][C]542[/C][C]556.080843337813[/C][C]-14.0808433378133[/C][/ROW]
[ROW][C]23[/C][C]527[/C][C]507.487157800751[/C][C]19.512842199249[/C][/ROW]
[ROW][C]24[/C][C]510[/C][C]502.819589809251[/C][C]7.18041019074894[/C][/ROW]
[ROW][C]25[/C][C]514[/C][C]499.774259953008[/C][C]14.2257400469916[/C][/ROW]
[ROW][C]26[/C][C]517[/C][C]523.357456043095[/C][C]-6.35745604309477[/C][/ROW]
[ROW][C]27[/C][C]508[/C][C]508.633474020973[/C][C]-0.633474020973267[/C][/ROW]
[ROW][C]28[/C][C]493[/C][C]494.152819018376[/C][C]-1.15281901837568[/C][/ROW]
[ROW][C]29[/C][C]490[/C][C]490.053765896008[/C][C]-0.0537658960076328[/C][/ROW]
[ROW][C]30[/C][C]469[/C][C]477.121206562276[/C][C]-8.12120656227648[/C][/ROW]
[ROW][C]31[/C][C]478[/C][C]474.46616468344[/C][C]3.53383531656021[/C][/ROW]
[ROW][C]32[/C][C]528[/C][C]526.973485273004[/C][C]1.02651472699631[/C][/ROW]
[ROW][C]33[/C][C]534[/C][C]533.154182718774[/C][C]0.845817281225663[/C][/ROW]
[ROW][C]34[/C][C]518[/C][C]526.0948171399[/C][C]-8.09481713989953[/C][/ROW]
[ROW][C]35[/C][C]506[/C][C]489.57970660244[/C][C]16.4202933975604[/C][/ROW]
[ROW][C]36[/C][C]502[/C][C]483.83724861902[/C][C]18.1627513809797[/C][/ROW]
[ROW][C]37[/C][C]516[/C][C]496.012878429499[/C][C]19.9871215705006[/C][/ROW]
[ROW][C]38[/C][C]528[/C][C]528.303486745401[/C][C]-0.30348674540096[/C][/ROW]
[ROW][C]39[/C][C]533[/C][C]526.056805109543[/C][C]6.9431948904571[/C][/ROW]
[ROW][C]40[/C][C]536[/C][C]525.993707568230[/C][C]10.0062924317696[/C][/ROW]
[ROW][C]41[/C][C]537[/C][C]541.434008533813[/C][C]-4.43400853381263[/C][/ROW]
[ROW][C]42[/C][C]524[/C][C]532.443786526393[/C][C]-8.44378652639284[/C][/ROW]
[ROW][C]43[/C][C]536[/C][C]539.267107680346[/C][C]-3.2671076803457[/C][/ROW]
[ROW][C]44[/C][C]587[/C][C]592.969756290473[/C][C]-5.96975629047301[/C][/ROW]
[ROW][C]45[/C][C]597[/C][C]599.311267383891[/C][C]-2.31126738389071[/C][/ROW]
[ROW][C]46[/C][C]581[/C][C]594.532895626929[/C][C]-13.5328956269292[/C][/ROW]
[ROW][C]47[/C][C]564[/C][C]560.661037434374[/C][C]3.33896256562559[/C][/ROW]
[ROW][C]48[/C][C]558[/C][C]546.41326303539[/C][C]11.5867369646098[/C][/ROW]
[ROW][C]49[/C][C]575[/C][C]554.942106479032[/C][C]20.0578935209677[/C][/ROW]
[ROW][C]50[/C][C]580[/C][C]587.437435597709[/C][C]-7.43743559770917[/C][/ROW]
[ROW][C]51[/C][C]575[/C][C]580.473890899085[/C][C]-5.47389089908484[/C][/ROW]
[ROW][C]52[/C][C]563[/C][C]568.591956654152[/C][C]-5.59195665415223[/C][/ROW]
[ROW][C]53[/C][C]552[/C][C]565.215275065266[/C][C]-13.2152750652664[/C][/ROW]
[ROW][C]54[/C][C]537[/C][C]543.215486277513[/C][C]-6.21548627751315[/C][/ROW]
[ROW][C]55[/C][C]545[/C][C]548.206578842009[/C][C]-3.20657884200909[/C][/ROW]
[ROW][C]56[/C][C]601[/C][C]597.361345661228[/C][C]3.63865433877197[/C][/ROW]
[ROW][C]57[/C][C]604[/C][C]609.882370920297[/C][C]-5.88237092029669[/C][/ROW]
[ROW][C]58[/C][C]586[/C][C]597.38412898002[/C][C]-11.3841289800203[/C][/ROW]
[ROW][C]59[/C][C]564[/C][C]564.028555948256[/C][C]-0.0285559482556437[/C][/ROW]
[ROW][C]60[/C][C]549[/C][C]543.955192456023[/C][C]5.04480754397673[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=110646&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=110646&T=2

As an alternative you can also use a QR Code:  
QR Code


The GUIDs for individual cells are displayed in the table below:

Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
13557582.066773504274-25.0667735042737
14561560.5137154054380.486284594562449
15549545.5542935929533.4457064070466
16532529.341704979862.65829502014026
17526523.8308825187672.16911748123323
18511508.2984145489892.70158545101117
19499514.290527895811-15.2905278958111
20555541.17415791578613.8258420842142
21565555.5196119562539.48038804374698
22542556.080843337813-14.0808433378133
23527507.48715780075119.512842199249
24510502.8195898092517.18041019074894
25514499.77425995300814.2257400469916
26517523.357456043095-6.35745604309477
27508508.633474020973-0.633474020973267
28493494.152819018376-1.15281901837568
29490490.053765896008-0.0537658960076328
30469477.121206562276-8.12120656227648
31478474.466164683443.53383531656021
32528526.9734852730041.02651472699631
33534533.1541827187740.845817281225663
34518526.0948171399-8.09481713989953
35506489.5797066024416.4202933975604
36502483.8372486190218.1627513809797
37516496.01287842949919.9871215705006
38528528.303486745401-0.30348674540096
39533526.0568051095436.9431948904571
40536525.99370756823010.0062924317696
41537541.434008533813-4.43400853381263
42524532.443786526393-8.44378652639284
43536539.267107680346-3.2671076803457
44587592.969756290473-5.96975629047301
45597599.311267383891-2.31126738389071
46581594.532895626929-13.5328956269292
47564560.6610374343743.33896256562559
48558546.4132630353911.5867369646098
49575554.94210647903220.0578935209677
50580587.437435597709-7.43743559770917
51575580.473890899085-5.47389089908484
52563568.591956654152-5.59195665415223
53552565.215275065266-13.2152750652664
54537543.215486277513-6.21548627751315
55545548.206578842009-3.20657884200909
56601597.3613456612283.63865433877197
57604609.882370920297-5.88237092029669
58586597.38412898002-11.3841289800203
59564564.028555948256-0.0285559482556437
60549543.9551924560235.04480754397673






Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
61542.800781344236523.096887611713562.504675076759
62546.795067217103518.116520134701575.473614299506
63540.171072944562502.842858183237577.499287705887
64527.502726293093481.462127800306573.54332478588
65523.581313943002468.625277354667578.537350531336
66511.393008216494447.257472209831575.528544223158
67520.453507751861446.846283979418594.060731524304
68571.834056317333488.450491910470655.217620724195
69578.2422506406484.773342110523671.711159170677
70569.527866088077465.664681296064673.391050880089
71548.331748990947433.767900805657662.895597176237
72529.561701063319403.994735075952655.128667050687

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
61 & 542.800781344236 & 523.096887611713 & 562.504675076759 \tabularnewline
62 & 546.795067217103 & 518.116520134701 & 575.473614299506 \tabularnewline
63 & 540.171072944562 & 502.842858183237 & 577.499287705887 \tabularnewline
64 & 527.502726293093 & 481.462127800306 & 573.54332478588 \tabularnewline
65 & 523.581313943002 & 468.625277354667 & 578.537350531336 \tabularnewline
66 & 511.393008216494 & 447.257472209831 & 575.528544223158 \tabularnewline
67 & 520.453507751861 & 446.846283979418 & 594.060731524304 \tabularnewline
68 & 571.834056317333 & 488.450491910470 & 655.217620724195 \tabularnewline
69 & 578.2422506406 & 484.773342110523 & 671.711159170677 \tabularnewline
70 & 569.527866088077 & 465.664681296064 & 673.391050880089 \tabularnewline
71 & 548.331748990947 & 433.767900805657 & 662.895597176237 \tabularnewline
72 & 529.561701063319 & 403.994735075952 & 655.128667050687 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=110646&T=3

[TABLE]
[ROW][C]Extrapolation Forecasts of Exponential Smoothing[/C][/ROW]
[ROW][C]t[/C][C]Forecast[/C][C]95% Lower Bound[/C][C]95% Upper Bound[/C][/ROW]
[ROW][C]61[/C][C]542.800781344236[/C][C]523.096887611713[/C][C]562.504675076759[/C][/ROW]
[ROW][C]62[/C][C]546.795067217103[/C][C]518.116520134701[/C][C]575.473614299506[/C][/ROW]
[ROW][C]63[/C][C]540.171072944562[/C][C]502.842858183237[/C][C]577.499287705887[/C][/ROW]
[ROW][C]64[/C][C]527.502726293093[/C][C]481.462127800306[/C][C]573.54332478588[/C][/ROW]
[ROW][C]65[/C][C]523.581313943002[/C][C]468.625277354667[/C][C]578.537350531336[/C][/ROW]
[ROW][C]66[/C][C]511.393008216494[/C][C]447.257472209831[/C][C]575.528544223158[/C][/ROW]
[ROW][C]67[/C][C]520.453507751861[/C][C]446.846283979418[/C][C]594.060731524304[/C][/ROW]
[ROW][C]68[/C][C]571.834056317333[/C][C]488.450491910470[/C][C]655.217620724195[/C][/ROW]
[ROW][C]69[/C][C]578.2422506406[/C][C]484.773342110523[/C][C]671.711159170677[/C][/ROW]
[ROW][C]70[/C][C]569.527866088077[/C][C]465.664681296064[/C][C]673.391050880089[/C][/ROW]
[ROW][C]71[/C][C]548.331748990947[/C][C]433.767900805657[/C][C]662.895597176237[/C][/ROW]
[ROW][C]72[/C][C]529.561701063319[/C][C]403.994735075952[/C][C]655.128667050687[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=110646&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=110646&T=3

As an alternative you can also use a QR Code:  
QR Code


The GUIDs for individual cells are displayed in the table below:

Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
61542.800781344236523.096887611713562.504675076759
62546.795067217103518.116520134701575.473614299506
63540.171072944562502.842858183237577.499287705887
64527.502726293093481.462127800306573.54332478588
65523.581313943002468.625277354667578.537350531336
66511.393008216494447.257472209831575.528544223158
67520.453507751861446.846283979418594.060731524304
68571.834056317333488.450491910470655.217620724195
69578.2422506406484.773342110523671.711159170677
70569.527866088077465.664681296064673.391050880089
71548.331748990947433.767900805657662.895597176237
72529.561701063319403.994735075952655.128667050687


Figures (Output of Computation)

Input Parameters & R Code

Parameters (Session):
par1 = 12 ; par2 = Triple ; par3 = additive ;
Parameters (R input):
par1 = 12 ; par2 = Triple ; par3 = additive ;
R code (references can be found in the software module):
par1 <- as.numeric(par1)
if (par2 == 'Single') K <- 1
if (par2 == 'Double') K <- 2
if (par2 == 'Triple') K <- par1
nx <- length(x)
nxmK <- nx - K
x <- ts(x, frequency = par1)
if (par2 == 'Single') fit <- HoltWinters(x, gamma=F, beta=F)
if (par2 == 'Double') fit <- HoltWinters(x, gamma=F)
if (par2 == 'Triple') fit <- HoltWinters(x, seasonal=par3)
fit
myresid <- x - fit$fitted[,'xhat']
bitmap(file='test1.png')
op <- par(mfrow=c(2,1))
plot(fit,ylab='Observed (black) / Fitted (red)',main='Interpolation Fit of Exponential Smoothing')
plot(myresid,ylab='Residuals',main='Interpolation Prediction Errors')
par(op)
dev.off()
bitmap(file='test2.png')
p <- predict(fit, par1, prediction.interval=TRUE)
np <- length(p[,1])
plot(fit,p,ylab='Observed (black) / Fitted (red)',main='Extrapolation Fit of Exponential Smoothing')
dev.off()
bitmap(file='test3.png')
op <- par(mfrow = c(2,2))
acf(as.numeric(myresid),lag.max = nx/2,main='Residual ACF')
spectrum(myresid,main='Residals Periodogram')
cpgram(myresid,main='Residal Cumulative Periodogram')
qqnorm(myresid,main='Residual Normal QQ Plot')
qqline(myresid)
par(op)
dev.off()
load(file='createtable')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Estimated Parameters of Exponential Smoothing',2,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Parameter',header=TRUE)
a<-table.element(a,'Value',header=TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'alpha',header=TRUE)
a<-table.element(a,fit$alpha)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'beta',header=TRUE)
a<-table.element(a,fit$beta)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'gamma',header=TRUE)
a<-table.element(a,fit$gamma)
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file='mytable.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Interpolation Forecasts of Exponential Smoothing',4,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'t',header=TRUE)
a<-table.element(a,'Observed',header=TRUE)
a<-table.element(a,'Fitted',header=TRUE)
a<-table.element(a,'Residuals',header=TRUE)
a<-table.row.end(a)
for (i in 1:nxmK) {
a<-table.row.start(a)
a<-table.element(a,i+K,header=TRUE)
a<-table.element(a,x[i+K])
a<-table.element(a,fit$fitted[i,'xhat'])
a<-table.element(a,myresid[i])
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable1.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Extrapolation Forecasts of Exponential Smoothing',4,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'t',header=TRUE)
a<-table.element(a,'Forecast',header=TRUE)
a<-table.element(a,'95% Lower Bound',header=TRUE)
a<-table.element(a,'95% Upper Bound',header=TRUE)
a<-table.row.end(a)
for (i in 1:np) {
a<-table.row.start(a)
a<-table.element(a,nx+i,header=TRUE)
a<-table.element(a,p[i,'fit'])
a<-table.element(a,p[i,'lwr'])
a<-table.element(a,p[i,'upr'])
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable2.tab')