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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 computationThu, 01 Dec 2011 09:49:47 -0500
Cite this page as followsStatistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2011/Dec/01/t1322751010ww217ffgjsusglg.htm/, Retrieved Tue, 23 Sep 2025 01:16:38 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=149750, Retrieved Tue, 23 Sep 2025 01:16:38 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact442

Tree of Dependent Computations

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
F     [Univariate Data Series] [HPC Retail Sales] [2008-03-02 15:42:48] [74be16979710d4c4e7c6647856088456]
- RMPD  [Structural Time Series Models] [HPC Retail Sales] [2008-03-06 16:52:55] [74be16979710d4c4e7c6647856088456]
- R  D    [Structural Time Series Models] [HPC Retail Sales] [2008-03-08 11:33:35] [74be16979710d4c4e7c6647856088456]
- RM D      [Structural Time Series Models] [] [2010-11-26 10:41:09] [d39e5c40c631ed6c22677d2e41dbfc7d]
- RMP           [Exponential Smoothing] [Exponential smoot...] [2011-12-01 14:49:47] [b625935f05df8270d3a5abfea0142dde] [Current]
Feedback Forum

Post a new message

Dataset

Dataseries X:
Download CSV
Histogram
Boxplots 
9700
9081
9084
9743
8587
9731
9563
9998
9437
10038
9918
9252
9737
9035
9133
9487
8700
9627
8947
9283
8829
9947
9628
9318
9605
8640
9214
9567
8547
9185
9470
9123
9278
10170
9434
9655
9429
8739
9552
9687
9019
9672
9206
9069
9788
10312
10105
9863
9656
9295
9946
9701
9049
10190
9706
9765
9893
9994
10433
10073
10112
9266
9820
10097
9115
10411
9678
10408
10153
10368
10581
10597
10680
9738
9556

Tables (Output of Computation)





Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time2 seconds
R Server'George Udny Yule' @ yule.wessa.net

\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 & 2 seconds \tabularnewline
R Server & 'George Udny Yule' @ yule.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=149750&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]2 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'George Udny Yule' @ yule.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=149750&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=149750&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 time2 seconds
R Server'George Udny Yule' @ yule.wessa.net






Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.158806071446012
betaFALSE
gammaFALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.158806071446012 \tabularnewline
beta & FALSE \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=149750&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]0.158806071446012[/C][/ROW]
[ROW][C]beta[/C][C]FALSE[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=149750&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=149750&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
alpha0.158806071446012
betaFALSE
gammaFALSE






Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
290819700-619
390849601.69904177492-517.699041774918
497439519.48529075928223.514709240721
585879554.9807836442-967.980783644196
697319401.25955815843329.740441841572
795639453.62434232416109.375657675839
899989470.99386083148527.006139168516
994379554.68563542077-117.685635420767
10100389535.99644199397502.003558006032
1199189615.71765489283302.282345107174
1292529663.72192658678-411.721926586784
1397379598.33798489735138.662015102645
1490359620.3583547746-585.358354774593
1591339527.39989406474-394.39989406474
1694879464.766796309622.2332036904045
1787009468.29756404333-768.297564043327
1896279346.28724619607280.712753803935
1989479390.86613583246-443.866135832459
2092839320.37749855298-37.3774985529835
2188299314.4417248473-485.441724847306
2299479237.35063160833709.64936839167
2396289350.04725990675277.952740093246
2493189394.18784260862-76.1878426086168
2596059382.088750632222.911249368004
2686409417.48841042525-777.48841042525
2792149294.01853037081-80.0185303708113
2895679281.31110191974285.688898080261
2985479326.6802334796-779.680233479605
3091859202.8622786166-17.8622786165997
3194709200.02564032242269.974359677577
3291239242.89920777397-119.899207773971
3392789223.858485617954.1415143821014
34101709232.45648681906937.543513180943
3594349381.3440889570252.6559110429844
3696559389.70616732816265.293832671838
3794299431.83643867363-2.83643867363207
3887399431.38599499097-692.385994990975
3995529321.43089520222230.56910479778
4096879358.04666893198328.95333106802
4190199410.28645512797-391.286455127971
4296729348.14779037906323.852209620938
4392069399.57748751807-193.577487518074
4490699368.83620720494-299.836207204939
4597889321.22039706145466.77960293855
46103129395.34783203525916.65216796475
47101059540.9177617122564.082238287798
4898639630.49744594716232.502554052839
4996569667.42026315746-11.4202631574572
5092959665.60665603054-370.606656030543
5199469606.75206893459339.247931065411
5297019660.6267001132740.3732998867254
5390499667.0382252596-618.038225259597
54101909568.89000270266621.109997297344
5597069667.5260413092938.4739586907108
5697659673.6359395419491.3640604580632
5798939688.14510705464204.854892945363
5899949720.67730781978273.322692180216
59104339764.08261080197668.917389198028
60100739870.31075350243202.689246497566
61101129902.49903646306209.500963536935
6292669935.76906144652-669.769061446519
6398209829.40566802211-9.4056680221147
64100979827.9119908342269.088009165802
6591159870.64480044305-755.644800443048
66104119750.64381827608660.356181723919
6796789855.51238925075-177.512389250745
68104089827.32234408084580.67765591916
69101539919.53748139384233.462518606158
70103689956.61274680358411.387253196423
711058110021.9435403267559.056459673333
721059710110.7251004039486.274899596096
731068010187.9485068516492.051493148436
74973810266.0892714276-528.089271427612
75955610182.2254888594-626.225488859407

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
2 & 9081 & 9700 & -619 \tabularnewline
3 & 9084 & 9601.69904177492 & -517.699041774918 \tabularnewline
4 & 9743 & 9519.48529075928 & 223.514709240721 \tabularnewline
5 & 8587 & 9554.9807836442 & -967.980783644196 \tabularnewline
6 & 9731 & 9401.25955815843 & 329.740441841572 \tabularnewline
7 & 9563 & 9453.62434232416 & 109.375657675839 \tabularnewline
8 & 9998 & 9470.99386083148 & 527.006139168516 \tabularnewline
9 & 9437 & 9554.68563542077 & -117.685635420767 \tabularnewline
10 & 10038 & 9535.99644199397 & 502.003558006032 \tabularnewline
11 & 9918 & 9615.71765489283 & 302.282345107174 \tabularnewline
12 & 9252 & 9663.72192658678 & -411.721926586784 \tabularnewline
13 & 9737 & 9598.33798489735 & 138.662015102645 \tabularnewline
14 & 9035 & 9620.3583547746 & -585.358354774593 \tabularnewline
15 & 9133 & 9527.39989406474 & -394.39989406474 \tabularnewline
16 & 9487 & 9464.7667963096 & 22.2332036904045 \tabularnewline
17 & 8700 & 9468.29756404333 & -768.297564043327 \tabularnewline
18 & 9627 & 9346.28724619607 & 280.712753803935 \tabularnewline
19 & 8947 & 9390.86613583246 & -443.866135832459 \tabularnewline
20 & 9283 & 9320.37749855298 & -37.3774985529835 \tabularnewline
21 & 8829 & 9314.4417248473 & -485.441724847306 \tabularnewline
22 & 9947 & 9237.35063160833 & 709.64936839167 \tabularnewline
23 & 9628 & 9350.04725990675 & 277.952740093246 \tabularnewline
24 & 9318 & 9394.18784260862 & -76.1878426086168 \tabularnewline
25 & 9605 & 9382.088750632 & 222.911249368004 \tabularnewline
26 & 8640 & 9417.48841042525 & -777.48841042525 \tabularnewline
27 & 9214 & 9294.01853037081 & -80.0185303708113 \tabularnewline
28 & 9567 & 9281.31110191974 & 285.688898080261 \tabularnewline
29 & 8547 & 9326.6802334796 & -779.680233479605 \tabularnewline
30 & 9185 & 9202.8622786166 & -17.8622786165997 \tabularnewline
31 & 9470 & 9200.02564032242 & 269.974359677577 \tabularnewline
32 & 9123 & 9242.89920777397 & -119.899207773971 \tabularnewline
33 & 9278 & 9223.8584856179 & 54.1415143821014 \tabularnewline
34 & 10170 & 9232.45648681906 & 937.543513180943 \tabularnewline
35 & 9434 & 9381.34408895702 & 52.6559110429844 \tabularnewline
36 & 9655 & 9389.70616732816 & 265.293832671838 \tabularnewline
37 & 9429 & 9431.83643867363 & -2.83643867363207 \tabularnewline
38 & 8739 & 9431.38599499097 & -692.385994990975 \tabularnewline
39 & 9552 & 9321.43089520222 & 230.56910479778 \tabularnewline
40 & 9687 & 9358.04666893198 & 328.95333106802 \tabularnewline
41 & 9019 & 9410.28645512797 & -391.286455127971 \tabularnewline
42 & 9672 & 9348.14779037906 & 323.852209620938 \tabularnewline
43 & 9206 & 9399.57748751807 & -193.577487518074 \tabularnewline
44 & 9069 & 9368.83620720494 & -299.836207204939 \tabularnewline
45 & 9788 & 9321.22039706145 & 466.77960293855 \tabularnewline
46 & 10312 & 9395.34783203525 & 916.65216796475 \tabularnewline
47 & 10105 & 9540.9177617122 & 564.082238287798 \tabularnewline
48 & 9863 & 9630.49744594716 & 232.502554052839 \tabularnewline
49 & 9656 & 9667.42026315746 & -11.4202631574572 \tabularnewline
50 & 9295 & 9665.60665603054 & -370.606656030543 \tabularnewline
51 & 9946 & 9606.75206893459 & 339.247931065411 \tabularnewline
52 & 9701 & 9660.62670011327 & 40.3732998867254 \tabularnewline
53 & 9049 & 9667.0382252596 & -618.038225259597 \tabularnewline
54 & 10190 & 9568.89000270266 & 621.109997297344 \tabularnewline
55 & 9706 & 9667.52604130929 & 38.4739586907108 \tabularnewline
56 & 9765 & 9673.63593954194 & 91.3640604580632 \tabularnewline
57 & 9893 & 9688.14510705464 & 204.854892945363 \tabularnewline
58 & 9994 & 9720.67730781978 & 273.322692180216 \tabularnewline
59 & 10433 & 9764.08261080197 & 668.917389198028 \tabularnewline
60 & 10073 & 9870.31075350243 & 202.689246497566 \tabularnewline
61 & 10112 & 9902.49903646306 & 209.500963536935 \tabularnewline
62 & 9266 & 9935.76906144652 & -669.769061446519 \tabularnewline
63 & 9820 & 9829.40566802211 & -9.4056680221147 \tabularnewline
64 & 10097 & 9827.9119908342 & 269.088009165802 \tabularnewline
65 & 9115 & 9870.64480044305 & -755.644800443048 \tabularnewline
66 & 10411 & 9750.64381827608 & 660.356181723919 \tabularnewline
67 & 9678 & 9855.51238925075 & -177.512389250745 \tabularnewline
68 & 10408 & 9827.32234408084 & 580.67765591916 \tabularnewline
69 & 10153 & 9919.53748139384 & 233.462518606158 \tabularnewline
70 & 10368 & 9956.61274680358 & 411.387253196423 \tabularnewline
71 & 10581 & 10021.9435403267 & 559.056459673333 \tabularnewline
72 & 10597 & 10110.7251004039 & 486.274899596096 \tabularnewline
73 & 10680 & 10187.9485068516 & 492.051493148436 \tabularnewline
74 & 9738 & 10266.0892714276 & -528.089271427612 \tabularnewline
75 & 9556 & 10182.2254888594 & -626.225488859407 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=149750&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]2[/C][C]9081[/C][C]9700[/C][C]-619[/C][/ROW]
[ROW][C]3[/C][C]9084[/C][C]9601.69904177492[/C][C]-517.699041774918[/C][/ROW]
[ROW][C]4[/C][C]9743[/C][C]9519.48529075928[/C][C]223.514709240721[/C][/ROW]
[ROW][C]5[/C][C]8587[/C][C]9554.9807836442[/C][C]-967.980783644196[/C][/ROW]
[ROW][C]6[/C][C]9731[/C][C]9401.25955815843[/C][C]329.740441841572[/C][/ROW]
[ROW][C]7[/C][C]9563[/C][C]9453.62434232416[/C][C]109.375657675839[/C][/ROW]
[ROW][C]8[/C][C]9998[/C][C]9470.99386083148[/C][C]527.006139168516[/C][/ROW]
[ROW][C]9[/C][C]9437[/C][C]9554.68563542077[/C][C]-117.685635420767[/C][/ROW]
[ROW][C]10[/C][C]10038[/C][C]9535.99644199397[/C][C]502.003558006032[/C][/ROW]
[ROW][C]11[/C][C]9918[/C][C]9615.71765489283[/C][C]302.282345107174[/C][/ROW]
[ROW][C]12[/C][C]9252[/C][C]9663.72192658678[/C][C]-411.721926586784[/C][/ROW]
[ROW][C]13[/C][C]9737[/C][C]9598.33798489735[/C][C]138.662015102645[/C][/ROW]
[ROW][C]14[/C][C]9035[/C][C]9620.3583547746[/C][C]-585.358354774593[/C][/ROW]
[ROW][C]15[/C][C]9133[/C][C]9527.39989406474[/C][C]-394.39989406474[/C][/ROW]
[ROW][C]16[/C][C]9487[/C][C]9464.7667963096[/C][C]22.2332036904045[/C][/ROW]
[ROW][C]17[/C][C]8700[/C][C]9468.29756404333[/C][C]-768.297564043327[/C][/ROW]
[ROW][C]18[/C][C]9627[/C][C]9346.28724619607[/C][C]280.712753803935[/C][/ROW]
[ROW][C]19[/C][C]8947[/C][C]9390.86613583246[/C][C]-443.866135832459[/C][/ROW]
[ROW][C]20[/C][C]9283[/C][C]9320.37749855298[/C][C]-37.3774985529835[/C][/ROW]
[ROW][C]21[/C][C]8829[/C][C]9314.4417248473[/C][C]-485.441724847306[/C][/ROW]
[ROW][C]22[/C][C]9947[/C][C]9237.35063160833[/C][C]709.64936839167[/C][/ROW]
[ROW][C]23[/C][C]9628[/C][C]9350.04725990675[/C][C]277.952740093246[/C][/ROW]
[ROW][C]24[/C][C]9318[/C][C]9394.18784260862[/C][C]-76.1878426086168[/C][/ROW]
[ROW][C]25[/C][C]9605[/C][C]9382.088750632[/C][C]222.911249368004[/C][/ROW]
[ROW][C]26[/C][C]8640[/C][C]9417.48841042525[/C][C]-777.48841042525[/C][/ROW]
[ROW][C]27[/C][C]9214[/C][C]9294.01853037081[/C][C]-80.0185303708113[/C][/ROW]
[ROW][C]28[/C][C]9567[/C][C]9281.31110191974[/C][C]285.688898080261[/C][/ROW]
[ROW][C]29[/C][C]8547[/C][C]9326.6802334796[/C][C]-779.680233479605[/C][/ROW]
[ROW][C]30[/C][C]9185[/C][C]9202.8622786166[/C][C]-17.8622786165997[/C][/ROW]
[ROW][C]31[/C][C]9470[/C][C]9200.02564032242[/C][C]269.974359677577[/C][/ROW]
[ROW][C]32[/C][C]9123[/C][C]9242.89920777397[/C][C]-119.899207773971[/C][/ROW]
[ROW][C]33[/C][C]9278[/C][C]9223.8584856179[/C][C]54.1415143821014[/C][/ROW]
[ROW][C]34[/C][C]10170[/C][C]9232.45648681906[/C][C]937.543513180943[/C][/ROW]
[ROW][C]35[/C][C]9434[/C][C]9381.34408895702[/C][C]52.6559110429844[/C][/ROW]
[ROW][C]36[/C][C]9655[/C][C]9389.70616732816[/C][C]265.293832671838[/C][/ROW]
[ROW][C]37[/C][C]9429[/C][C]9431.83643867363[/C][C]-2.83643867363207[/C][/ROW]
[ROW][C]38[/C][C]8739[/C][C]9431.38599499097[/C][C]-692.385994990975[/C][/ROW]
[ROW][C]39[/C][C]9552[/C][C]9321.43089520222[/C][C]230.56910479778[/C][/ROW]
[ROW][C]40[/C][C]9687[/C][C]9358.04666893198[/C][C]328.95333106802[/C][/ROW]
[ROW][C]41[/C][C]9019[/C][C]9410.28645512797[/C][C]-391.286455127971[/C][/ROW]
[ROW][C]42[/C][C]9672[/C][C]9348.14779037906[/C][C]323.852209620938[/C][/ROW]
[ROW][C]43[/C][C]9206[/C][C]9399.57748751807[/C][C]-193.577487518074[/C][/ROW]
[ROW][C]44[/C][C]9069[/C][C]9368.83620720494[/C][C]-299.836207204939[/C][/ROW]
[ROW][C]45[/C][C]9788[/C][C]9321.22039706145[/C][C]466.77960293855[/C][/ROW]
[ROW][C]46[/C][C]10312[/C][C]9395.34783203525[/C][C]916.65216796475[/C][/ROW]
[ROW][C]47[/C][C]10105[/C][C]9540.9177617122[/C][C]564.082238287798[/C][/ROW]
[ROW][C]48[/C][C]9863[/C][C]9630.49744594716[/C][C]232.502554052839[/C][/ROW]
[ROW][C]49[/C][C]9656[/C][C]9667.42026315746[/C][C]-11.4202631574572[/C][/ROW]
[ROW][C]50[/C][C]9295[/C][C]9665.60665603054[/C][C]-370.606656030543[/C][/ROW]
[ROW][C]51[/C][C]9946[/C][C]9606.75206893459[/C][C]339.247931065411[/C][/ROW]
[ROW][C]52[/C][C]9701[/C][C]9660.62670011327[/C][C]40.3732998867254[/C][/ROW]
[ROW][C]53[/C][C]9049[/C][C]9667.0382252596[/C][C]-618.038225259597[/C][/ROW]
[ROW][C]54[/C][C]10190[/C][C]9568.89000270266[/C][C]621.109997297344[/C][/ROW]
[ROW][C]55[/C][C]9706[/C][C]9667.52604130929[/C][C]38.4739586907108[/C][/ROW]
[ROW][C]56[/C][C]9765[/C][C]9673.63593954194[/C][C]91.3640604580632[/C][/ROW]
[ROW][C]57[/C][C]9893[/C][C]9688.14510705464[/C][C]204.854892945363[/C][/ROW]
[ROW][C]58[/C][C]9994[/C][C]9720.67730781978[/C][C]273.322692180216[/C][/ROW]
[ROW][C]59[/C][C]10433[/C][C]9764.08261080197[/C][C]668.917389198028[/C][/ROW]
[ROW][C]60[/C][C]10073[/C][C]9870.31075350243[/C][C]202.689246497566[/C][/ROW]
[ROW][C]61[/C][C]10112[/C][C]9902.49903646306[/C][C]209.500963536935[/C][/ROW]
[ROW][C]62[/C][C]9266[/C][C]9935.76906144652[/C][C]-669.769061446519[/C][/ROW]
[ROW][C]63[/C][C]9820[/C][C]9829.40566802211[/C][C]-9.4056680221147[/C][/ROW]
[ROW][C]64[/C][C]10097[/C][C]9827.9119908342[/C][C]269.088009165802[/C][/ROW]
[ROW][C]65[/C][C]9115[/C][C]9870.64480044305[/C][C]-755.644800443048[/C][/ROW]
[ROW][C]66[/C][C]10411[/C][C]9750.64381827608[/C][C]660.356181723919[/C][/ROW]
[ROW][C]67[/C][C]9678[/C][C]9855.51238925075[/C][C]-177.512389250745[/C][/ROW]
[ROW][C]68[/C][C]10408[/C][C]9827.32234408084[/C][C]580.67765591916[/C][/ROW]
[ROW][C]69[/C][C]10153[/C][C]9919.53748139384[/C][C]233.462518606158[/C][/ROW]
[ROW][C]70[/C][C]10368[/C][C]9956.61274680358[/C][C]411.387253196423[/C][/ROW]
[ROW][C]71[/C][C]10581[/C][C]10021.9435403267[/C][C]559.056459673333[/C][/ROW]
[ROW][C]72[/C][C]10597[/C][C]10110.7251004039[/C][C]486.274899596096[/C][/ROW]
[ROW][C]73[/C][C]10680[/C][C]10187.9485068516[/C][C]492.051493148436[/C][/ROW]
[ROW][C]74[/C][C]9738[/C][C]10266.0892714276[/C][C]-528.089271427612[/C][/ROW]
[ROW][C]75[/C][C]9556[/C][C]10182.2254888594[/C][C]-626.225488859407[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=149750&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=149750&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
290819700-619
390849601.69904177492-517.699041774918
497439519.48529075928223.514709240721
585879554.9807836442-967.980783644196
697319401.25955815843329.740441841572
795639453.62434232416109.375657675839
899989470.99386083148527.006139168516
994379554.68563542077-117.685635420767
10100389535.99644199397502.003558006032
1199189615.71765489283302.282345107174
1292529663.72192658678-411.721926586784
1397379598.33798489735138.662015102645
1490359620.3583547746-585.358354774593
1591339527.39989406474-394.39989406474
1694879464.766796309622.2332036904045
1787009468.29756404333-768.297564043327
1896279346.28724619607280.712753803935
1989479390.86613583246-443.866135832459
2092839320.37749855298-37.3774985529835
2188299314.4417248473-485.441724847306
2299479237.35063160833709.64936839167
2396289350.04725990675277.952740093246
2493189394.18784260862-76.1878426086168
2596059382.088750632222.911249368004
2686409417.48841042525-777.48841042525
2792149294.01853037081-80.0185303708113
2895679281.31110191974285.688898080261
2985479326.6802334796-779.680233479605
3091859202.8622786166-17.8622786165997
3194709200.02564032242269.974359677577
3291239242.89920777397-119.899207773971
3392789223.858485617954.1415143821014
34101709232.45648681906937.543513180943
3594349381.3440889570252.6559110429844
3696559389.70616732816265.293832671838
3794299431.83643867363-2.83643867363207
3887399431.38599499097-692.385994990975
3995529321.43089520222230.56910479778
4096879358.04666893198328.95333106802
4190199410.28645512797-391.286455127971
4296729348.14779037906323.852209620938
4392069399.57748751807-193.577487518074
4490699368.83620720494-299.836207204939
4597889321.22039706145466.77960293855
46103129395.34783203525916.65216796475
47101059540.9177617122564.082238287798
4898639630.49744594716232.502554052839
4996569667.42026315746-11.4202631574572
5092959665.60665603054-370.606656030543
5199469606.75206893459339.247931065411
5297019660.6267001132740.3732998867254
5390499667.0382252596-618.038225259597
54101909568.89000270266621.109997297344
5597069667.5260413092938.4739586907108
5697659673.6359395419491.3640604580632
5798939688.14510705464204.854892945363
5899949720.67730781978273.322692180216
59104339764.08261080197668.917389198028
60100739870.31075350243202.689246497566
61101129902.49903646306209.500963536935
6292669935.76906144652-669.769061446519
6398209829.40566802211-9.4056680221147
64100979827.9119908342269.088009165802
6591159870.64480044305-755.644800443048
66104119750.64381827608660.356181723919
6796789855.51238925075-177.512389250745
68104089827.32234408084580.67765591916
69101539919.53748139384233.462518606158
70103689956.61274680358411.387253196423
711058110021.9435403267559.056459673333
721059710110.7251004039486.274899596096
731068010187.9485068516492.051493148436
74973810266.0892714276-528.089271427612
75955610182.2254888594-626.225488859407






Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
7610082.77707913439200.1100086341310965.4441496344
7710082.77707913439189.049158345610976.504999923
7810082.77707913439178.1235347045810987.430623564
7910082.77707913439167.3282960223210998.2258622463
8010082.77707913439156.658882815311008.8952754533
8110082.77707913439146.110995300211019.4431629684
8210082.77707913439135.6805731450411029.8735851235
8310082.77707913439125.3637772061211040.1903810625
8410082.77707913439115.1569730177711050.3971852508
8510082.77707913439105.0567158335611060.497442435
8610082.77707913439095.0597370445311070.494421224
8710082.77707913439085.162931822411080.3912264462

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
76 & 10082.7770791343 & 9200.11000863413 & 10965.4441496344 \tabularnewline
77 & 10082.7770791343 & 9189.0491583456 & 10976.504999923 \tabularnewline
78 & 10082.7770791343 & 9178.12353470458 & 10987.430623564 \tabularnewline
79 & 10082.7770791343 & 9167.32829602232 & 10998.2258622463 \tabularnewline
80 & 10082.7770791343 & 9156.6588828153 & 11008.8952754533 \tabularnewline
81 & 10082.7770791343 & 9146.1109953002 & 11019.4431629684 \tabularnewline
82 & 10082.7770791343 & 9135.68057314504 & 11029.8735851235 \tabularnewline
83 & 10082.7770791343 & 9125.36377720612 & 11040.1903810625 \tabularnewline
84 & 10082.7770791343 & 9115.15697301777 & 11050.3971852508 \tabularnewline
85 & 10082.7770791343 & 9105.05671583356 & 11060.497442435 \tabularnewline
86 & 10082.7770791343 & 9095.05973704453 & 11070.494421224 \tabularnewline
87 & 10082.7770791343 & 9085.1629318224 & 11080.3912264462 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=149750&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]76[/C][C]10082.7770791343[/C][C]9200.11000863413[/C][C]10965.4441496344[/C][/ROW]
[ROW][C]77[/C][C]10082.7770791343[/C][C]9189.0491583456[/C][C]10976.504999923[/C][/ROW]
[ROW][C]78[/C][C]10082.7770791343[/C][C]9178.12353470458[/C][C]10987.430623564[/C][/ROW]
[ROW][C]79[/C][C]10082.7770791343[/C][C]9167.32829602232[/C][C]10998.2258622463[/C][/ROW]
[ROW][C]80[/C][C]10082.7770791343[/C][C]9156.6588828153[/C][C]11008.8952754533[/C][/ROW]
[ROW][C]81[/C][C]10082.7770791343[/C][C]9146.1109953002[/C][C]11019.4431629684[/C][/ROW]
[ROW][C]82[/C][C]10082.7770791343[/C][C]9135.68057314504[/C][C]11029.8735851235[/C][/ROW]
[ROW][C]83[/C][C]10082.7770791343[/C][C]9125.36377720612[/C][C]11040.1903810625[/C][/ROW]
[ROW][C]84[/C][C]10082.7770791343[/C][C]9115.15697301777[/C][C]11050.3971852508[/C][/ROW]
[ROW][C]85[/C][C]10082.7770791343[/C][C]9105.05671583356[/C][C]11060.497442435[/C][/ROW]
[ROW][C]86[/C][C]10082.7770791343[/C][C]9095.05973704453[/C][C]11070.494421224[/C][/ROW]
[ROW][C]87[/C][C]10082.7770791343[/C][C]9085.1629318224[/C][C]11080.3912264462[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=149750&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=149750&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
7610082.77707913439200.1100086341310965.4441496344
7710082.77707913439189.049158345610976.504999923
7810082.77707913439178.1235347045810987.430623564
7910082.77707913439167.3282960223210998.2258622463
8010082.77707913439156.658882815311008.8952754533
8110082.77707913439146.110995300211019.4431629684
8210082.77707913439135.6805731450411029.8735851235
8310082.77707913439125.3637772061211040.1903810625
8410082.77707913439115.1569730177711050.3971852508
8510082.77707913439105.0567158335611060.497442435
8610082.77707913439095.0597370445311070.494421224
8710082.77707913439085.162931822411080.3912264462


Figures (Output of Computation)

Input Parameters & R Code

Parameters (Session):
par1 = 12 ; par2 = Single ; par3 = additive ;
Parameters (R input):
par1 = 12 ; par2 = Single ; 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')