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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 computationSat, 10 Nov 2012 09:54:36 -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/2012/Nov/10/t1352559292b25xgrwva6rd2sv.htm/, Retrieved Sat, 02 May 2026 09:58:22 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=187362, Retrieved Sat, 02 May 2026 09:58:22 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact454

Tree of Dependent Computations

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-     [Exponential Smoothing] [HPC Retail Sales] [2008-03-10 17:43:04] [74be16979710d4c4e7c6647856088456]
- RM D    [Exponential Smoothing] [Single smoothing] [2012-11-10 14:54:36] [c63d55528b56cf8bb48e0b5d1a959d8e] [Current]
- R         [Exponential Smoothing] [double smoothing] [2012-11-10 14:57:27] [86dcce9422b96d4554cb918e531c1d5d]
-             [Exponential Smoothing] [Triple smoothing] [2012-11-10 14:59:32] [86dcce9422b96d4554cb918e531c1d5d]
-   P           [Exponential Smoothing] [Triple Smoothing ...] [2012-11-11 17:39:21] [74be16979710d4c4e7c6647856088456]
-  M              [Exponential Smoothing] [Triple Smoothing ...] [2012-11-11 17:40:47] [74be16979710d4c4e7c6647856088456]
-   P         [Exponential Smoothing] [Double Smoothing ...] [2012-11-11 17:31:29] [74be16979710d4c4e7c6647856088456]
- R  D      [Exponential Smoothing] [Single smoothing] [2012-11-11 17:23:55] [74be16979710d4c4e7c6647856088456]
- R         [Exponential Smoothing] [Single Smoothing] [2012-11-11 17:35:30] [74be16979710d4c4e7c6647856088456]
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Dataset

Dataseries X:
Download CSV
Histogram
Boxplots 
236422
250580
279515
264417
283706
281288
271146
283944
269155
270899
276507
319957
250746
247772
280449
274925
296013
287881
279098
294763
261924
291596
287537
326201
255598
253086
285261
284747
300402
288854
295433
307256
273189
287540
290705
337006
268335
259060
293703
294262
312404
301014
309942
317079
293912
304060
301299
357634
281493
282478
319111
315223
328445
321081
328040
326362
313566
319768
324315
387243
293308
295109
339190
335678
345401
351002
351889
355773
333363
336214
343910
405788

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'Gertrude Mary Cox' @ cox.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 & 'Gertrude Mary Cox' @ cox.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=187362&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]'Gertrude Mary Cox' @ cox.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=187362&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=187362&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'Gertrude Mary Cox' @ cox.wessa.net






Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.245575870418928
betaFALSE
gammaFALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.245575870418928 \tabularnewline
beta & FALSE \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=187362&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]0.245575870418928[/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=187362&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=187362&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
alpha0.245575870418928
betaFALSE
gammaFALSE






Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
225058023642214158
3279515239898.86317339139616.1368266088
4264417249627.63045722114789.369542779
5283706253259.54275563630446.4572443639
6281288260736.45799459320551.5420054065
7271146265783.4208110225362.57918897766
8283944267100.34086304616843.6591369541
9269155271236.737116643-2081.73711664317
10270899270725.51271224173.487287759839
11276507270768.1170039385738.88299606158
12319957272177.44819092947779.5518090714
13250746283910.953214668-33164.9532146676
14247772275766.440961573-27994.4409615726
15280449268891.68175554311557.3182444569
16274925271729.8802431343195.11975686584
17296013272514.52455851923498.4754414808
18287881278285.1831185799595.81688142137
19279098280641.684201614-1543.68420161435
20294763280262.59261015114500.407389849
21261924283823.542776342-21899.5427763422
22291596278445.54349726513150.4565027346
23287537281674.9782993315862.02170066925
24326201283114.54938088743086.4506191128
25255598293695.541994938-38097.541994938
26253086284339.704958709-31253.7049587094
27285261276664.5491596588596.45084034197
28284747278775.6300572895971.36994271149
29300402280242.05442856320159.9455714367
30288854285192.8506098673661.14939013292
31295433286091.9405580839341.05944191728
32307256288385.87936116718870.1206388335
33273189293019.925661958-19830.9256619582
34287540288149.92883131-609.928831309779
35290705288000.1450276672704.85497233272
36337006288664.39214185548341.6078581451
37268335300535.924569069-32200.9245690693
38259060292628.154489726-33568.1544897259
39293703284384.6257325549318.37426744559
40294262286672.9936041727589.00639582827
41312404288536.67045544223867.329544558
42301014294397.9106829226616.08931707777
43309942296022.66257573313919.337424267
44317079299440.91597935217638.0840206479
45293912303772.403815245-9860.40381524485
46304060301350.9265656342709.07343436603
47301299302016.209632307-717.209632307175
48357634301840.08025258155793.9197474195
49281493315541.720658637-34048.7206586368
50282478307180.176446241-24702.1764462412
51319111301113.91796421417997.0820357865
52315223305533.5670501539689.43294984737
53328445307913.05798067720531.9420193228
54321081312955.2075134638125.79248653661
55328040314950.70607618813089.2939238118
56326362318165.1208246978196.87917530257
57313566320178.076562891-6612.07656289113
58319768318554.3101056831213.68989431747
59324315318852.3630578985462.63694210182
60387243320193.85487973767049.1451202626
61293308336659.507053491-43351.5070534909
62295109326013.422974858-30904.4229748576
63339190318424.04240301220765.9575969877
64335678323523.66051497512154.3394850249
65345401326508.47301337718892.5269866228
66351002331148.0217725319853.9782274698
67351889336023.6797570215865.3202429804
68355773339919.81958516515853.1804148355
69333363343812.978164446-10449.9781644461
70336214341246.715680853-5032.71568085346
71343910340010.8021469573899.19785304309
72405788340968.35105365464819.6489463464

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
2 & 250580 & 236422 & 14158 \tabularnewline
3 & 279515 & 239898.863173391 & 39616.1368266088 \tabularnewline
4 & 264417 & 249627.630457221 & 14789.369542779 \tabularnewline
5 & 283706 & 253259.542755636 & 30446.4572443639 \tabularnewline
6 & 281288 & 260736.457994593 & 20551.5420054065 \tabularnewline
7 & 271146 & 265783.420811022 & 5362.57918897766 \tabularnewline
8 & 283944 & 267100.340863046 & 16843.6591369541 \tabularnewline
9 & 269155 & 271236.737116643 & -2081.73711664317 \tabularnewline
10 & 270899 & 270725.51271224 & 173.487287759839 \tabularnewline
11 & 276507 & 270768.117003938 & 5738.88299606158 \tabularnewline
12 & 319957 & 272177.448190929 & 47779.5518090714 \tabularnewline
13 & 250746 & 283910.953214668 & -33164.9532146676 \tabularnewline
14 & 247772 & 275766.440961573 & -27994.4409615726 \tabularnewline
15 & 280449 & 268891.681755543 & 11557.3182444569 \tabularnewline
16 & 274925 & 271729.880243134 & 3195.11975686584 \tabularnewline
17 & 296013 & 272514.524558519 & 23498.4754414808 \tabularnewline
18 & 287881 & 278285.183118579 & 9595.81688142137 \tabularnewline
19 & 279098 & 280641.684201614 & -1543.68420161435 \tabularnewline
20 & 294763 & 280262.592610151 & 14500.407389849 \tabularnewline
21 & 261924 & 283823.542776342 & -21899.5427763422 \tabularnewline
22 & 291596 & 278445.543497265 & 13150.4565027346 \tabularnewline
23 & 287537 & 281674.978299331 & 5862.02170066925 \tabularnewline
24 & 326201 & 283114.549380887 & 43086.4506191128 \tabularnewline
25 & 255598 & 293695.541994938 & -38097.541994938 \tabularnewline
26 & 253086 & 284339.704958709 & -31253.7049587094 \tabularnewline
27 & 285261 & 276664.549159658 & 8596.45084034197 \tabularnewline
28 & 284747 & 278775.630057289 & 5971.36994271149 \tabularnewline
29 & 300402 & 280242.054428563 & 20159.9455714367 \tabularnewline
30 & 288854 & 285192.850609867 & 3661.14939013292 \tabularnewline
31 & 295433 & 286091.940558083 & 9341.05944191728 \tabularnewline
32 & 307256 & 288385.879361167 & 18870.1206388335 \tabularnewline
33 & 273189 & 293019.925661958 & -19830.9256619582 \tabularnewline
34 & 287540 & 288149.92883131 & -609.928831309779 \tabularnewline
35 & 290705 & 288000.145027667 & 2704.85497233272 \tabularnewline
36 & 337006 & 288664.392141855 & 48341.6078581451 \tabularnewline
37 & 268335 & 300535.924569069 & -32200.9245690693 \tabularnewline
38 & 259060 & 292628.154489726 & -33568.1544897259 \tabularnewline
39 & 293703 & 284384.625732554 & 9318.37426744559 \tabularnewline
40 & 294262 & 286672.993604172 & 7589.00639582827 \tabularnewline
41 & 312404 & 288536.670455442 & 23867.329544558 \tabularnewline
42 & 301014 & 294397.910682922 & 6616.08931707777 \tabularnewline
43 & 309942 & 296022.662575733 & 13919.337424267 \tabularnewline
44 & 317079 & 299440.915979352 & 17638.0840206479 \tabularnewline
45 & 293912 & 303772.403815245 & -9860.40381524485 \tabularnewline
46 & 304060 & 301350.926565634 & 2709.07343436603 \tabularnewline
47 & 301299 & 302016.209632307 & -717.209632307175 \tabularnewline
48 & 357634 & 301840.080252581 & 55793.9197474195 \tabularnewline
49 & 281493 & 315541.720658637 & -34048.7206586368 \tabularnewline
50 & 282478 & 307180.176446241 & -24702.1764462412 \tabularnewline
51 & 319111 & 301113.917964214 & 17997.0820357865 \tabularnewline
52 & 315223 & 305533.567050153 & 9689.43294984737 \tabularnewline
53 & 328445 & 307913.057980677 & 20531.9420193228 \tabularnewline
54 & 321081 & 312955.207513463 & 8125.79248653661 \tabularnewline
55 & 328040 & 314950.706076188 & 13089.2939238118 \tabularnewline
56 & 326362 & 318165.120824697 & 8196.87917530257 \tabularnewline
57 & 313566 & 320178.076562891 & -6612.07656289113 \tabularnewline
58 & 319768 & 318554.310105683 & 1213.68989431747 \tabularnewline
59 & 324315 & 318852.363057898 & 5462.63694210182 \tabularnewline
60 & 387243 & 320193.854879737 & 67049.1451202626 \tabularnewline
61 & 293308 & 336659.507053491 & -43351.5070534909 \tabularnewline
62 & 295109 & 326013.422974858 & -30904.4229748576 \tabularnewline
63 & 339190 & 318424.042403012 & 20765.9575969877 \tabularnewline
64 & 335678 & 323523.660514975 & 12154.3394850249 \tabularnewline
65 & 345401 & 326508.473013377 & 18892.5269866228 \tabularnewline
66 & 351002 & 331148.02177253 & 19853.9782274698 \tabularnewline
67 & 351889 & 336023.67975702 & 15865.3202429804 \tabularnewline
68 & 355773 & 339919.819585165 & 15853.1804148355 \tabularnewline
69 & 333363 & 343812.978164446 & -10449.9781644461 \tabularnewline
70 & 336214 & 341246.715680853 & -5032.71568085346 \tabularnewline
71 & 343910 & 340010.802146957 & 3899.19785304309 \tabularnewline
72 & 405788 & 340968.351053654 & 64819.6489463464 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=187362&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]250580[/C][C]236422[/C][C]14158[/C][/ROW]
[ROW][C]3[/C][C]279515[/C][C]239898.863173391[/C][C]39616.1368266088[/C][/ROW]
[ROW][C]4[/C][C]264417[/C][C]249627.630457221[/C][C]14789.369542779[/C][/ROW]
[ROW][C]5[/C][C]283706[/C][C]253259.542755636[/C][C]30446.4572443639[/C][/ROW]
[ROW][C]6[/C][C]281288[/C][C]260736.457994593[/C][C]20551.5420054065[/C][/ROW]
[ROW][C]7[/C][C]271146[/C][C]265783.420811022[/C][C]5362.57918897766[/C][/ROW]
[ROW][C]8[/C][C]283944[/C][C]267100.340863046[/C][C]16843.6591369541[/C][/ROW]
[ROW][C]9[/C][C]269155[/C][C]271236.737116643[/C][C]-2081.73711664317[/C][/ROW]
[ROW][C]10[/C][C]270899[/C][C]270725.51271224[/C][C]173.487287759839[/C][/ROW]
[ROW][C]11[/C][C]276507[/C][C]270768.117003938[/C][C]5738.88299606158[/C][/ROW]
[ROW][C]12[/C][C]319957[/C][C]272177.448190929[/C][C]47779.5518090714[/C][/ROW]
[ROW][C]13[/C][C]250746[/C][C]283910.953214668[/C][C]-33164.9532146676[/C][/ROW]
[ROW][C]14[/C][C]247772[/C][C]275766.440961573[/C][C]-27994.4409615726[/C][/ROW]
[ROW][C]15[/C][C]280449[/C][C]268891.681755543[/C][C]11557.3182444569[/C][/ROW]
[ROW][C]16[/C][C]274925[/C][C]271729.880243134[/C][C]3195.11975686584[/C][/ROW]
[ROW][C]17[/C][C]296013[/C][C]272514.524558519[/C][C]23498.4754414808[/C][/ROW]
[ROW][C]18[/C][C]287881[/C][C]278285.183118579[/C][C]9595.81688142137[/C][/ROW]
[ROW][C]19[/C][C]279098[/C][C]280641.684201614[/C][C]-1543.68420161435[/C][/ROW]
[ROW][C]20[/C][C]294763[/C][C]280262.592610151[/C][C]14500.407389849[/C][/ROW]
[ROW][C]21[/C][C]261924[/C][C]283823.542776342[/C][C]-21899.5427763422[/C][/ROW]
[ROW][C]22[/C][C]291596[/C][C]278445.543497265[/C][C]13150.4565027346[/C][/ROW]
[ROW][C]23[/C][C]287537[/C][C]281674.978299331[/C][C]5862.02170066925[/C][/ROW]
[ROW][C]24[/C][C]326201[/C][C]283114.549380887[/C][C]43086.4506191128[/C][/ROW]
[ROW][C]25[/C][C]255598[/C][C]293695.541994938[/C][C]-38097.541994938[/C][/ROW]
[ROW][C]26[/C][C]253086[/C][C]284339.704958709[/C][C]-31253.7049587094[/C][/ROW]
[ROW][C]27[/C][C]285261[/C][C]276664.549159658[/C][C]8596.45084034197[/C][/ROW]
[ROW][C]28[/C][C]284747[/C][C]278775.630057289[/C][C]5971.36994271149[/C][/ROW]
[ROW][C]29[/C][C]300402[/C][C]280242.054428563[/C][C]20159.9455714367[/C][/ROW]
[ROW][C]30[/C][C]288854[/C][C]285192.850609867[/C][C]3661.14939013292[/C][/ROW]
[ROW][C]31[/C][C]295433[/C][C]286091.940558083[/C][C]9341.05944191728[/C][/ROW]
[ROW][C]32[/C][C]307256[/C][C]288385.879361167[/C][C]18870.1206388335[/C][/ROW]
[ROW][C]33[/C][C]273189[/C][C]293019.925661958[/C][C]-19830.9256619582[/C][/ROW]
[ROW][C]34[/C][C]287540[/C][C]288149.92883131[/C][C]-609.928831309779[/C][/ROW]
[ROW][C]35[/C][C]290705[/C][C]288000.145027667[/C][C]2704.85497233272[/C][/ROW]
[ROW][C]36[/C][C]337006[/C][C]288664.392141855[/C][C]48341.6078581451[/C][/ROW]
[ROW][C]37[/C][C]268335[/C][C]300535.924569069[/C][C]-32200.9245690693[/C][/ROW]
[ROW][C]38[/C][C]259060[/C][C]292628.154489726[/C][C]-33568.1544897259[/C][/ROW]
[ROW][C]39[/C][C]293703[/C][C]284384.625732554[/C][C]9318.37426744559[/C][/ROW]
[ROW][C]40[/C][C]294262[/C][C]286672.993604172[/C][C]7589.00639582827[/C][/ROW]
[ROW][C]41[/C][C]312404[/C][C]288536.670455442[/C][C]23867.329544558[/C][/ROW]
[ROW][C]42[/C][C]301014[/C][C]294397.910682922[/C][C]6616.08931707777[/C][/ROW]
[ROW][C]43[/C][C]309942[/C][C]296022.662575733[/C][C]13919.337424267[/C][/ROW]
[ROW][C]44[/C][C]317079[/C][C]299440.915979352[/C][C]17638.0840206479[/C][/ROW]
[ROW][C]45[/C][C]293912[/C][C]303772.403815245[/C][C]-9860.40381524485[/C][/ROW]
[ROW][C]46[/C][C]304060[/C][C]301350.926565634[/C][C]2709.07343436603[/C][/ROW]
[ROW][C]47[/C][C]301299[/C][C]302016.209632307[/C][C]-717.209632307175[/C][/ROW]
[ROW][C]48[/C][C]357634[/C][C]301840.080252581[/C][C]55793.9197474195[/C][/ROW]
[ROW][C]49[/C][C]281493[/C][C]315541.720658637[/C][C]-34048.7206586368[/C][/ROW]
[ROW][C]50[/C][C]282478[/C][C]307180.176446241[/C][C]-24702.1764462412[/C][/ROW]
[ROW][C]51[/C][C]319111[/C][C]301113.917964214[/C][C]17997.0820357865[/C][/ROW]
[ROW][C]52[/C][C]315223[/C][C]305533.567050153[/C][C]9689.43294984737[/C][/ROW]
[ROW][C]53[/C][C]328445[/C][C]307913.057980677[/C][C]20531.9420193228[/C][/ROW]
[ROW][C]54[/C][C]321081[/C][C]312955.207513463[/C][C]8125.79248653661[/C][/ROW]
[ROW][C]55[/C][C]328040[/C][C]314950.706076188[/C][C]13089.2939238118[/C][/ROW]
[ROW][C]56[/C][C]326362[/C][C]318165.120824697[/C][C]8196.87917530257[/C][/ROW]
[ROW][C]57[/C][C]313566[/C][C]320178.076562891[/C][C]-6612.07656289113[/C][/ROW]
[ROW][C]58[/C][C]319768[/C][C]318554.310105683[/C][C]1213.68989431747[/C][/ROW]
[ROW][C]59[/C][C]324315[/C][C]318852.363057898[/C][C]5462.63694210182[/C][/ROW]
[ROW][C]60[/C][C]387243[/C][C]320193.854879737[/C][C]67049.1451202626[/C][/ROW]
[ROW][C]61[/C][C]293308[/C][C]336659.507053491[/C][C]-43351.5070534909[/C][/ROW]
[ROW][C]62[/C][C]295109[/C][C]326013.422974858[/C][C]-30904.4229748576[/C][/ROW]
[ROW][C]63[/C][C]339190[/C][C]318424.042403012[/C][C]20765.9575969877[/C][/ROW]
[ROW][C]64[/C][C]335678[/C][C]323523.660514975[/C][C]12154.3394850249[/C][/ROW]
[ROW][C]65[/C][C]345401[/C][C]326508.473013377[/C][C]18892.5269866228[/C][/ROW]
[ROW][C]66[/C][C]351002[/C][C]331148.02177253[/C][C]19853.9782274698[/C][/ROW]
[ROW][C]67[/C][C]351889[/C][C]336023.67975702[/C][C]15865.3202429804[/C][/ROW]
[ROW][C]68[/C][C]355773[/C][C]339919.819585165[/C][C]15853.1804148355[/C][/ROW]
[ROW][C]69[/C][C]333363[/C][C]343812.978164446[/C][C]-10449.9781644461[/C][/ROW]
[ROW][C]70[/C][C]336214[/C][C]341246.715680853[/C][C]-5032.71568085346[/C][/ROW]
[ROW][C]71[/C][C]343910[/C][C]340010.802146957[/C][C]3899.19785304309[/C][/ROW]
[ROW][C]72[/C][C]405788[/C][C]340968.351053654[/C][C]64819.6489463464[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=187362&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=187362&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
225058023642214158
3279515239898.86317339139616.1368266088
4264417249627.63045722114789.369542779
5283706253259.54275563630446.4572443639
6281288260736.45799459320551.5420054065
7271146265783.4208110225362.57918897766
8283944267100.34086304616843.6591369541
9269155271236.737116643-2081.73711664317
10270899270725.51271224173.487287759839
11276507270768.1170039385738.88299606158
12319957272177.44819092947779.5518090714
13250746283910.953214668-33164.9532146676
14247772275766.440961573-27994.4409615726
15280449268891.68175554311557.3182444569
16274925271729.8802431343195.11975686584
17296013272514.52455851923498.4754414808
18287881278285.1831185799595.81688142137
19279098280641.684201614-1543.68420161435
20294763280262.59261015114500.407389849
21261924283823.542776342-21899.5427763422
22291596278445.54349726513150.4565027346
23287537281674.9782993315862.02170066925
24326201283114.54938088743086.4506191128
25255598293695.541994938-38097.541994938
26253086284339.704958709-31253.7049587094
27285261276664.5491596588596.45084034197
28284747278775.6300572895971.36994271149
29300402280242.05442856320159.9455714367
30288854285192.8506098673661.14939013292
31295433286091.9405580839341.05944191728
32307256288385.87936116718870.1206388335
33273189293019.925661958-19830.9256619582
34287540288149.92883131-609.928831309779
35290705288000.1450276672704.85497233272
36337006288664.39214185548341.6078581451
37268335300535.924569069-32200.9245690693
38259060292628.154489726-33568.1544897259
39293703284384.6257325549318.37426744559
40294262286672.9936041727589.00639582827
41312404288536.67045544223867.329544558
42301014294397.9106829226616.08931707777
43309942296022.66257573313919.337424267
44317079299440.91597935217638.0840206479
45293912303772.403815245-9860.40381524485
46304060301350.9265656342709.07343436603
47301299302016.209632307-717.209632307175
48357634301840.08025258155793.9197474195
49281493315541.720658637-34048.7206586368
50282478307180.176446241-24702.1764462412
51319111301113.91796421417997.0820357865
52315223305533.5670501539689.43294984737
53328445307913.05798067720531.9420193228
54321081312955.2075134638125.79248653661
55328040314950.70607618813089.2939238118
56326362318165.1208246978196.87917530257
57313566320178.076562891-6612.07656289113
58319768318554.3101056831213.68989431747
59324315318852.3630578985462.63694210182
60387243320193.85487973767049.1451202626
61293308336659.507053491-43351.5070534909
62295109326013.422974858-30904.4229748576
63339190318424.04240301220765.9575969877
64335678323523.66051497512154.3394850249
65345401326508.47301337718892.5269866228
66351002331148.0217725319853.9782274698
67351889336023.6797570215865.3202429804
68355773339919.81958516515853.1804148355
69333363343812.978164446-10449.9781644461
70336214341246.715680853-5032.71568085346
71343910340010.8021469573899.19785304309
72405788340968.35105365464819.6489463464






Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
73356886.492763902311646.165149274402126.82037853
74356886.492763902310301.969043997403471.016483807
75356886.492763902308995.4867425404777.498785304
76356886.492763902307723.711462663406049.274065141
77356886.492763902306484.015912015407288.969615789
78356886.492763902305274.08843765408498.897090154
79356886.492763902304091.882358406409681.103169398
80356886.492763902302935.57529812410837.410229684
81356886.492763902301803.536208074411969.44931973
82356886.492763902300694.298373019413078.687154785
83356886.492763902299606.537125003414166.448402801
84356886.492763902298539.05129878415233.934229024

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
73 & 356886.492763902 & 311646.165149274 & 402126.82037853 \tabularnewline
74 & 356886.492763902 & 310301.969043997 & 403471.016483807 \tabularnewline
75 & 356886.492763902 & 308995.4867425 & 404777.498785304 \tabularnewline
76 & 356886.492763902 & 307723.711462663 & 406049.274065141 \tabularnewline
77 & 356886.492763902 & 306484.015912015 & 407288.969615789 \tabularnewline
78 & 356886.492763902 & 305274.08843765 & 408498.897090154 \tabularnewline
79 & 356886.492763902 & 304091.882358406 & 409681.103169398 \tabularnewline
80 & 356886.492763902 & 302935.57529812 & 410837.410229684 \tabularnewline
81 & 356886.492763902 & 301803.536208074 & 411969.44931973 \tabularnewline
82 & 356886.492763902 & 300694.298373019 & 413078.687154785 \tabularnewline
83 & 356886.492763902 & 299606.537125003 & 414166.448402801 \tabularnewline
84 & 356886.492763902 & 298539.05129878 & 415233.934229024 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=187362&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]73[/C][C]356886.492763902[/C][C]311646.165149274[/C][C]402126.82037853[/C][/ROW]
[ROW][C]74[/C][C]356886.492763902[/C][C]310301.969043997[/C][C]403471.016483807[/C][/ROW]
[ROW][C]75[/C][C]356886.492763902[/C][C]308995.4867425[/C][C]404777.498785304[/C][/ROW]
[ROW][C]76[/C][C]356886.492763902[/C][C]307723.711462663[/C][C]406049.274065141[/C][/ROW]
[ROW][C]77[/C][C]356886.492763902[/C][C]306484.015912015[/C][C]407288.969615789[/C][/ROW]
[ROW][C]78[/C][C]356886.492763902[/C][C]305274.08843765[/C][C]408498.897090154[/C][/ROW]
[ROW][C]79[/C][C]356886.492763902[/C][C]304091.882358406[/C][C]409681.103169398[/C][/ROW]
[ROW][C]80[/C][C]356886.492763902[/C][C]302935.57529812[/C][C]410837.410229684[/C][/ROW]
[ROW][C]81[/C][C]356886.492763902[/C][C]301803.536208074[/C][C]411969.44931973[/C][/ROW]
[ROW][C]82[/C][C]356886.492763902[/C][C]300694.298373019[/C][C]413078.687154785[/C][/ROW]
[ROW][C]83[/C][C]356886.492763902[/C][C]299606.537125003[/C][C]414166.448402801[/C][/ROW]
[ROW][C]84[/C][C]356886.492763902[/C][C]298539.05129878[/C][C]415233.934229024[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=187362&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=187362&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
73356886.492763902311646.165149274402126.82037853
74356886.492763902310301.969043997403471.016483807
75356886.492763902308995.4867425404777.498785304
76356886.492763902307723.711462663406049.274065141
77356886.492763902306484.015912015407288.969615789
78356886.492763902305274.08843765408498.897090154
79356886.492763902304091.882358406409681.103169398
80356886.492763902302935.57529812410837.410229684
81356886.492763902301803.536208074411969.44931973
82356886.492763902300694.298373019413078.687154785
83356886.492763902299606.537125003414166.448402801
84356886.492763902298539.05129878415233.934229024


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')