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Author

*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Sun, 12 Jan 2014 20:31:31 -0500

Cite this page as follows

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2014/Jan/12/t13895767073speade3bkzg2k7.htm/, Retrieved Wed, 11 Feb 2026 01:58:57 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=233100, Retrieved Wed, 11 Feb 2026 01:58:57 +0000

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IsPrivate?

No (this computation is public)

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Estimated Impact

330

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)

-       [Exponential Smoothing] [] [2014-01-13 01:31:31] [14b1e901e86f0e99d3e5ae27817fa672] [Current]

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Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	4 seconds
R Server	'Herman Ole Andreas Wold' @ wold.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 & 4 seconds \tabularnewline
R Server & 'Herman Ole Andreas Wold' @ wold.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=233100&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]4 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Herman Ole Andreas Wold' @ wold.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=233100&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=233100&T=0

As an alternative you can also use a QR Code:

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

Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	4 seconds
R Server	'Herman Ole Andreas Wold' @ wold.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	1
beta	0.0840439979334186
gamma	FALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 1 \tabularnewline
beta & 0.0840439979334186 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=233100&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]1[/C][/ROW]
[ROW][C]beta[/C][C]0.0840439979334186[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=233100&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=233100&T=1

As an alternative you can also use a QR Code:

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

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	1
beta	0.0840439979334186
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
3	83.45	84.03	-0.579999999999998
4	83.71	84.0112544811986	-0.301254481198626
5	84.8	84.2459358502033	0.554064149796673
6	85.95	85.3825016164638	0.56749838353619
7	86.22	86.580196449437	-0.360196449436955
8	86.75	86.8199240997848	-0.069924099784842
9	87.06	87.344047398887	-0.284047398887026
10	87.17	87.630174919882	-0.460174919881979
11	87.63	87.7014999798664	-0.0714999798664167
12	87.78	88.1554908357063	-0.375490835706273
13	88.4	88.2739330846862	0.126066915313842
14	89.35	88.9045282522563	0.445471747743724
15	89.53	89.891967478903	-0.361967478903026
16	90.66	90.0415462848541	0.618453715145847
17	90.81	91.2235236076118	-0.413523607611779
18	91.55	91.3387694303882	0.211230569611757
19	91.58	92.0965220919442	-0.516522091944168
20	91.76	92.0831115103162	-0.323111510316238
21	91.78	92.235955927211	-0.455955927210965
22	91.71	92.2176355682067	-0.507635568206723
23	91.57	92.1049718455614	-0.534971845561415
24	91.95	91.9200106728786	0.0299893271213989
25	92.16	92.3025310958252	-0.142531095825234
26	92.26	92.5005522127022	-0.240552212702241
27	92.44	92.580335243035	-0.140335243035025
28	93.12	92.7485409081594	0.371459091840606
29	93.55	93.4597598153064	0.0902401846935987
30	93.63	93.8973439612023	-0.2673439612023
31	93.74	93.9548753058795	-0.214875305879502
32	94.08	94.0468163261162	0.0331836738837836
33	94.24	94.3896052147355	-0.149605214735544
34	94.66	94.5370317943775	0.122968205622527
35	94.69	94.9673665339967	-0.277366533996684
36	94.69	94.9740555415867	-0.284055541586667
37	94.69	94.9501823782366	-0.260182378236578
38	94.72	94.9283156109777	-0.208315610977749
39	95.15	94.9408079341992	0.209192065800764
40	95.28	95.3883892717451	-0.108389271745096
41	96.12	95.5092798040145	0.610720195985465
42	96.5	96.4006071709038	0.0993928290961605
43	96.67	96.788960541627	-0.11896054162699
44	96.83	96.9489626221123	-0.118962622112335
45	97.4	97.0989645277454	0.30103547225464
46	97.75	97.6942647523534	0.0557352476465667
47	97.46	98.0489489653915	-0.588948965391467
48	97.46	97.7094513397612	-0.249451339761208
49	97.56	97.6884864518778	-0.128486451877819
50	97.97	97.7776879367817	0.192312063218267
51	98.89	98.2038506114254	0.686149388574577
52	99.1	99.1815173492208	-0.0815173492208032
53	99.3	99.3846663052913	-0.0846663052913499
54	100	99.5775506105044	0.422449389495597
55	99.73	100.313054946122	-0.583054946122147
56	99.34	99.9940526774352	-0.654052677435189
57	99.78	99.5490834755645	0.230916524435514
58	99.5	100.008490623467	-0.50849062346694
59	99.6	99.6857550385591	-0.0857550385591281
60	99.52	99.7785478422757	-0.258547842275675
61	99.63	99.6768184479538	-0.0468184479537683
62	99.61	99.7828836384107	-0.172883638410696
63	99.73	99.7483538062614	-0.0183538062613735
64	100.53	99.8668112790059	0.663188720994114
65	100.87	100.722548310503	0.14745168949743
66	100.9	101.07494073999	-0.174940739989978
67	101.08	101.0902380208	-0.0102380207998038
68	102.95	101.269377576601	1.68062242339916
69	102.58	103.28062380408	-0.700623804079868
70	102.6	102.851740578538	-0.251740578537678
71	102.45	102.850583293875	-0.400583293875286
72	102.41	102.666916672353	-0.256916672352688
73	102.38	102.605324368072	-0.225324368072407
74	102.65	102.556387207348	0.0936127926522232
75	103.33	102.8342548007	0.495745199300003
76	103.68	103.555919209205	0.124080790794551
77	104.13	103.916347454931	0.213652545069408
78	104.3	104.384303668987	-0.0843036689868484
79	104.11	104.547218451605	-0.43721845160475
80	104.17	104.320472864962	-0.150472864961628
81	104.23	104.36782652381	-0.13782652380975
82	104.47	104.416243031728	0.0537569682724808
83	104.86	104.660760982258	0.199239017742087
84	104.9	105.067505825853	-0.167505825853283

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 83.45 & 84.03 & -0.579999999999998 \tabularnewline
4 & 83.71 & 84.0112544811986 & -0.301254481198626 \tabularnewline
5 & 84.8 & 84.2459358502033 & 0.554064149796673 \tabularnewline
6 & 85.95 & 85.3825016164638 & 0.56749838353619 \tabularnewline
7 & 86.22 & 86.580196449437 & -0.360196449436955 \tabularnewline
8 & 86.75 & 86.8199240997848 & -0.069924099784842 \tabularnewline
9 & 87.06 & 87.344047398887 & -0.284047398887026 \tabularnewline
10 & 87.17 & 87.630174919882 & -0.460174919881979 \tabularnewline
11 & 87.63 & 87.7014999798664 & -0.0714999798664167 \tabularnewline
12 & 87.78 & 88.1554908357063 & -0.375490835706273 \tabularnewline
13 & 88.4 & 88.2739330846862 & 0.126066915313842 \tabularnewline
14 & 89.35 & 88.9045282522563 & 0.445471747743724 \tabularnewline
15 & 89.53 & 89.891967478903 & -0.361967478903026 \tabularnewline
16 & 90.66 & 90.0415462848541 & 0.618453715145847 \tabularnewline
17 & 90.81 & 91.2235236076118 & -0.413523607611779 \tabularnewline
18 & 91.55 & 91.3387694303882 & 0.211230569611757 \tabularnewline
19 & 91.58 & 92.0965220919442 & -0.516522091944168 \tabularnewline
20 & 91.76 & 92.0831115103162 & -0.323111510316238 \tabularnewline
21 & 91.78 & 92.235955927211 & -0.455955927210965 \tabularnewline
22 & 91.71 & 92.2176355682067 & -0.507635568206723 \tabularnewline
23 & 91.57 & 92.1049718455614 & -0.534971845561415 \tabularnewline
24 & 91.95 & 91.9200106728786 & 0.0299893271213989 \tabularnewline
25 & 92.16 & 92.3025310958252 & -0.142531095825234 \tabularnewline
26 & 92.26 & 92.5005522127022 & -0.240552212702241 \tabularnewline
27 & 92.44 & 92.580335243035 & -0.140335243035025 \tabularnewline
28 & 93.12 & 92.7485409081594 & 0.371459091840606 \tabularnewline
29 & 93.55 & 93.4597598153064 & 0.0902401846935987 \tabularnewline
30 & 93.63 & 93.8973439612023 & -0.2673439612023 \tabularnewline
31 & 93.74 & 93.9548753058795 & -0.214875305879502 \tabularnewline
32 & 94.08 & 94.0468163261162 & 0.0331836738837836 \tabularnewline
33 & 94.24 & 94.3896052147355 & -0.149605214735544 \tabularnewline
34 & 94.66 & 94.5370317943775 & 0.122968205622527 \tabularnewline
35 & 94.69 & 94.9673665339967 & -0.277366533996684 \tabularnewline
36 & 94.69 & 94.9740555415867 & -0.284055541586667 \tabularnewline
37 & 94.69 & 94.9501823782366 & -0.260182378236578 \tabularnewline
38 & 94.72 & 94.9283156109777 & -0.208315610977749 \tabularnewline
39 & 95.15 & 94.9408079341992 & 0.209192065800764 \tabularnewline
40 & 95.28 & 95.3883892717451 & -0.108389271745096 \tabularnewline
41 & 96.12 & 95.5092798040145 & 0.610720195985465 \tabularnewline
42 & 96.5 & 96.4006071709038 & 0.0993928290961605 \tabularnewline
43 & 96.67 & 96.788960541627 & -0.11896054162699 \tabularnewline
44 & 96.83 & 96.9489626221123 & -0.118962622112335 \tabularnewline
45 & 97.4 & 97.0989645277454 & 0.30103547225464 \tabularnewline
46 & 97.75 & 97.6942647523534 & 0.0557352476465667 \tabularnewline
47 & 97.46 & 98.0489489653915 & -0.588948965391467 \tabularnewline
48 & 97.46 & 97.7094513397612 & -0.249451339761208 \tabularnewline
49 & 97.56 & 97.6884864518778 & -0.128486451877819 \tabularnewline
50 & 97.97 & 97.7776879367817 & 0.192312063218267 \tabularnewline
51 & 98.89 & 98.2038506114254 & 0.686149388574577 \tabularnewline
52 & 99.1 & 99.1815173492208 & -0.0815173492208032 \tabularnewline
53 & 99.3 & 99.3846663052913 & -0.0846663052913499 \tabularnewline
54 & 100 & 99.5775506105044 & 0.422449389495597 \tabularnewline
55 & 99.73 & 100.313054946122 & -0.583054946122147 \tabularnewline
56 & 99.34 & 99.9940526774352 & -0.654052677435189 \tabularnewline
57 & 99.78 & 99.5490834755645 & 0.230916524435514 \tabularnewline
58 & 99.5 & 100.008490623467 & -0.50849062346694 \tabularnewline
59 & 99.6 & 99.6857550385591 & -0.0857550385591281 \tabularnewline
60 & 99.52 & 99.7785478422757 & -0.258547842275675 \tabularnewline
61 & 99.63 & 99.6768184479538 & -0.0468184479537683 \tabularnewline
62 & 99.61 & 99.7828836384107 & -0.172883638410696 \tabularnewline
63 & 99.73 & 99.7483538062614 & -0.0183538062613735 \tabularnewline
64 & 100.53 & 99.8668112790059 & 0.663188720994114 \tabularnewline
65 & 100.87 & 100.722548310503 & 0.14745168949743 \tabularnewline
66 & 100.9 & 101.07494073999 & -0.174940739989978 \tabularnewline
67 & 101.08 & 101.0902380208 & -0.0102380207998038 \tabularnewline
68 & 102.95 & 101.269377576601 & 1.68062242339916 \tabularnewline
69 & 102.58 & 103.28062380408 & -0.700623804079868 \tabularnewline
70 & 102.6 & 102.851740578538 & -0.251740578537678 \tabularnewline
71 & 102.45 & 102.850583293875 & -0.400583293875286 \tabularnewline
72 & 102.41 & 102.666916672353 & -0.256916672352688 \tabularnewline
73 & 102.38 & 102.605324368072 & -0.225324368072407 \tabularnewline
74 & 102.65 & 102.556387207348 & 0.0936127926522232 \tabularnewline
75 & 103.33 & 102.8342548007 & 0.495745199300003 \tabularnewline
76 & 103.68 & 103.555919209205 & 0.124080790794551 \tabularnewline
77 & 104.13 & 103.916347454931 & 0.213652545069408 \tabularnewline
78 & 104.3 & 104.384303668987 & -0.0843036689868484 \tabularnewline
79 & 104.11 & 104.547218451605 & -0.43721845160475 \tabularnewline
80 & 104.17 & 104.320472864962 & -0.150472864961628 \tabularnewline
81 & 104.23 & 104.36782652381 & -0.13782652380975 \tabularnewline
82 & 104.47 & 104.416243031728 & 0.0537569682724808 \tabularnewline
83 & 104.86 & 104.660760982258 & 0.199239017742087 \tabularnewline
84 & 104.9 & 105.067505825853 & -0.167505825853283 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=233100&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]3[/C][C]83.45[/C][C]84.03[/C][C]-0.579999999999998[/C][/ROW]
[ROW][C]4[/C][C]83.71[/C][C]84.0112544811986[/C][C]-0.301254481198626[/C][/ROW]
[ROW][C]5[/C][C]84.8[/C][C]84.2459358502033[/C][C]0.554064149796673[/C][/ROW]
[ROW][C]6[/C][C]85.95[/C][C]85.3825016164638[/C][C]0.56749838353619[/C][/ROW]
[ROW][C]7[/C][C]86.22[/C][C]86.580196449437[/C][C]-0.360196449436955[/C][/ROW]
[ROW][C]8[/C][C]86.75[/C][C]86.8199240997848[/C][C]-0.069924099784842[/C][/ROW]
[ROW][C]9[/C][C]87.06[/C][C]87.344047398887[/C][C]-0.284047398887026[/C][/ROW]
[ROW][C]10[/C][C]87.17[/C][C]87.630174919882[/C][C]-0.460174919881979[/C][/ROW]
[ROW][C]11[/C][C]87.63[/C][C]87.7014999798664[/C][C]-0.0714999798664167[/C][/ROW]
[ROW][C]12[/C][C]87.78[/C][C]88.1554908357063[/C][C]-0.375490835706273[/C][/ROW]
[ROW][C]13[/C][C]88.4[/C][C]88.2739330846862[/C][C]0.126066915313842[/C][/ROW]
[ROW][C]14[/C][C]89.35[/C][C]88.9045282522563[/C][C]0.445471747743724[/C][/ROW]
[ROW][C]15[/C][C]89.53[/C][C]89.891967478903[/C][C]-0.361967478903026[/C][/ROW]
[ROW][C]16[/C][C]90.66[/C][C]90.0415462848541[/C][C]0.618453715145847[/C][/ROW]
[ROW][C]17[/C][C]90.81[/C][C]91.2235236076118[/C][C]-0.413523607611779[/C][/ROW]
[ROW][C]18[/C][C]91.55[/C][C]91.3387694303882[/C][C]0.211230569611757[/C][/ROW]
[ROW][C]19[/C][C]91.58[/C][C]92.0965220919442[/C][C]-0.516522091944168[/C][/ROW]
[ROW][C]20[/C][C]91.76[/C][C]92.0831115103162[/C][C]-0.323111510316238[/C][/ROW]
[ROW][C]21[/C][C]91.78[/C][C]92.235955927211[/C][C]-0.455955927210965[/C][/ROW]
[ROW][C]22[/C][C]91.71[/C][C]92.2176355682067[/C][C]-0.507635568206723[/C][/ROW]
[ROW][C]23[/C][C]91.57[/C][C]92.1049718455614[/C][C]-0.534971845561415[/C][/ROW]
[ROW][C]24[/C][C]91.95[/C][C]91.9200106728786[/C][C]0.0299893271213989[/C][/ROW]
[ROW][C]25[/C][C]92.16[/C][C]92.3025310958252[/C][C]-0.142531095825234[/C][/ROW]
[ROW][C]26[/C][C]92.26[/C][C]92.5005522127022[/C][C]-0.240552212702241[/C][/ROW]
[ROW][C]27[/C][C]92.44[/C][C]92.580335243035[/C][C]-0.140335243035025[/C][/ROW]
[ROW][C]28[/C][C]93.12[/C][C]92.7485409081594[/C][C]0.371459091840606[/C][/ROW]
[ROW][C]29[/C][C]93.55[/C][C]93.4597598153064[/C][C]0.0902401846935987[/C][/ROW]
[ROW][C]30[/C][C]93.63[/C][C]93.8973439612023[/C][C]-0.2673439612023[/C][/ROW]
[ROW][C]31[/C][C]93.74[/C][C]93.9548753058795[/C][C]-0.214875305879502[/C][/ROW]
[ROW][C]32[/C][C]94.08[/C][C]94.0468163261162[/C][C]0.0331836738837836[/C][/ROW]
[ROW][C]33[/C][C]94.24[/C][C]94.3896052147355[/C][C]-0.149605214735544[/C][/ROW]
[ROW][C]34[/C][C]94.66[/C][C]94.5370317943775[/C][C]0.122968205622527[/C][/ROW]
[ROW][C]35[/C][C]94.69[/C][C]94.9673665339967[/C][C]-0.277366533996684[/C][/ROW]
[ROW][C]36[/C][C]94.69[/C][C]94.9740555415867[/C][C]-0.284055541586667[/C][/ROW]
[ROW][C]37[/C][C]94.69[/C][C]94.9501823782366[/C][C]-0.260182378236578[/C][/ROW]
[ROW][C]38[/C][C]94.72[/C][C]94.9283156109777[/C][C]-0.208315610977749[/C][/ROW]
[ROW][C]39[/C][C]95.15[/C][C]94.9408079341992[/C][C]0.209192065800764[/C][/ROW]
[ROW][C]40[/C][C]95.28[/C][C]95.3883892717451[/C][C]-0.108389271745096[/C][/ROW]
[ROW][C]41[/C][C]96.12[/C][C]95.5092798040145[/C][C]0.610720195985465[/C][/ROW]
[ROW][C]42[/C][C]96.5[/C][C]96.4006071709038[/C][C]0.0993928290961605[/C][/ROW]
[ROW][C]43[/C][C]96.67[/C][C]96.788960541627[/C][C]-0.11896054162699[/C][/ROW]
[ROW][C]44[/C][C]96.83[/C][C]96.9489626221123[/C][C]-0.118962622112335[/C][/ROW]
[ROW][C]45[/C][C]97.4[/C][C]97.0989645277454[/C][C]0.30103547225464[/C][/ROW]
[ROW][C]46[/C][C]97.75[/C][C]97.6942647523534[/C][C]0.0557352476465667[/C][/ROW]
[ROW][C]47[/C][C]97.46[/C][C]98.0489489653915[/C][C]-0.588948965391467[/C][/ROW]
[ROW][C]48[/C][C]97.46[/C][C]97.7094513397612[/C][C]-0.249451339761208[/C][/ROW]
[ROW][C]49[/C][C]97.56[/C][C]97.6884864518778[/C][C]-0.128486451877819[/C][/ROW]
[ROW][C]50[/C][C]97.97[/C][C]97.7776879367817[/C][C]0.192312063218267[/C][/ROW]
[ROW][C]51[/C][C]98.89[/C][C]98.2038506114254[/C][C]0.686149388574577[/C][/ROW]
[ROW][C]52[/C][C]99.1[/C][C]99.1815173492208[/C][C]-0.0815173492208032[/C][/ROW]
[ROW][C]53[/C][C]99.3[/C][C]99.3846663052913[/C][C]-0.0846663052913499[/C][/ROW]
[ROW][C]54[/C][C]100[/C][C]99.5775506105044[/C][C]0.422449389495597[/C][/ROW]
[ROW][C]55[/C][C]99.73[/C][C]100.313054946122[/C][C]-0.583054946122147[/C][/ROW]
[ROW][C]56[/C][C]99.34[/C][C]99.9940526774352[/C][C]-0.654052677435189[/C][/ROW]
[ROW][C]57[/C][C]99.78[/C][C]99.5490834755645[/C][C]0.230916524435514[/C][/ROW]
[ROW][C]58[/C][C]99.5[/C][C]100.008490623467[/C][C]-0.50849062346694[/C][/ROW]
[ROW][C]59[/C][C]99.6[/C][C]99.6857550385591[/C][C]-0.0857550385591281[/C][/ROW]
[ROW][C]60[/C][C]99.52[/C][C]99.7785478422757[/C][C]-0.258547842275675[/C][/ROW]
[ROW][C]61[/C][C]99.63[/C][C]99.6768184479538[/C][C]-0.0468184479537683[/C][/ROW]
[ROW][C]62[/C][C]99.61[/C][C]99.7828836384107[/C][C]-0.172883638410696[/C][/ROW]
[ROW][C]63[/C][C]99.73[/C][C]99.7483538062614[/C][C]-0.0183538062613735[/C][/ROW]
[ROW][C]64[/C][C]100.53[/C][C]99.8668112790059[/C][C]0.663188720994114[/C][/ROW]
[ROW][C]65[/C][C]100.87[/C][C]100.722548310503[/C][C]0.14745168949743[/C][/ROW]
[ROW][C]66[/C][C]100.9[/C][C]101.07494073999[/C][C]-0.174940739989978[/C][/ROW]
[ROW][C]67[/C][C]101.08[/C][C]101.0902380208[/C][C]-0.0102380207998038[/C][/ROW]
[ROW][C]68[/C][C]102.95[/C][C]101.269377576601[/C][C]1.68062242339916[/C][/ROW]
[ROW][C]69[/C][C]102.58[/C][C]103.28062380408[/C][C]-0.700623804079868[/C][/ROW]
[ROW][C]70[/C][C]102.6[/C][C]102.851740578538[/C][C]-0.251740578537678[/C][/ROW]
[ROW][C]71[/C][C]102.45[/C][C]102.850583293875[/C][C]-0.400583293875286[/C][/ROW]
[ROW][C]72[/C][C]102.41[/C][C]102.666916672353[/C][C]-0.256916672352688[/C][/ROW]
[ROW][C]73[/C][C]102.38[/C][C]102.605324368072[/C][C]-0.225324368072407[/C][/ROW]
[ROW][C]74[/C][C]102.65[/C][C]102.556387207348[/C][C]0.0936127926522232[/C][/ROW]
[ROW][C]75[/C][C]103.33[/C][C]102.8342548007[/C][C]0.495745199300003[/C][/ROW]
[ROW][C]76[/C][C]103.68[/C][C]103.555919209205[/C][C]0.124080790794551[/C][/ROW]
[ROW][C]77[/C][C]104.13[/C][C]103.916347454931[/C][C]0.213652545069408[/C][/ROW]
[ROW][C]78[/C][C]104.3[/C][C]104.384303668987[/C][C]-0.0843036689868484[/C][/ROW]
[ROW][C]79[/C][C]104.11[/C][C]104.547218451605[/C][C]-0.43721845160475[/C][/ROW]
[ROW][C]80[/C][C]104.17[/C][C]104.320472864962[/C][C]-0.150472864961628[/C][/ROW]
[ROW][C]81[/C][C]104.23[/C][C]104.36782652381[/C][C]-0.13782652380975[/C][/ROW]
[ROW][C]82[/C][C]104.47[/C][C]104.416243031728[/C][C]0.0537569682724808[/C][/ROW]
[ROW][C]83[/C][C]104.86[/C][C]104.660760982258[/C][C]0.199239017742087[/C][/ROW]
[ROW][C]84[/C][C]104.9[/C][C]105.067505825853[/C][C]-0.167505825853283[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=233100&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=233100&T=2

As an alternative you can also use a QR Code:

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

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
3	83.45	84.03	-0.579999999999998
4	83.71	84.0112544811986	-0.301254481198626
5	84.8	84.2459358502033	0.554064149796673
6	85.95	85.3825016164638	0.56749838353619
7	86.22	86.580196449437	-0.360196449436955
8	86.75	86.8199240997848	-0.069924099784842
9	87.06	87.344047398887	-0.284047398887026
10	87.17	87.630174919882	-0.460174919881979
11	87.63	87.7014999798664	-0.0714999798664167
12	87.78	88.1554908357063	-0.375490835706273
13	88.4	88.2739330846862	0.126066915313842
14	89.35	88.9045282522563	0.445471747743724
15	89.53	89.891967478903	-0.361967478903026
16	90.66	90.0415462848541	0.618453715145847
17	90.81	91.2235236076118	-0.413523607611779
18	91.55	91.3387694303882	0.211230569611757
19	91.58	92.0965220919442	-0.516522091944168
20	91.76	92.0831115103162	-0.323111510316238
21	91.78	92.235955927211	-0.455955927210965
22	91.71	92.2176355682067	-0.507635568206723
23	91.57	92.1049718455614	-0.534971845561415
24	91.95	91.9200106728786	0.0299893271213989
25	92.16	92.3025310958252	-0.142531095825234
26	92.26	92.5005522127022	-0.240552212702241
27	92.44	92.580335243035	-0.140335243035025
28	93.12	92.7485409081594	0.371459091840606
29	93.55	93.4597598153064	0.0902401846935987
30	93.63	93.8973439612023	-0.2673439612023
31	93.74	93.9548753058795	-0.214875305879502
32	94.08	94.0468163261162	0.0331836738837836
33	94.24	94.3896052147355	-0.149605214735544
34	94.66	94.5370317943775	0.122968205622527
35	94.69	94.9673665339967	-0.277366533996684
36	94.69	94.9740555415867	-0.284055541586667
37	94.69	94.9501823782366	-0.260182378236578
38	94.72	94.9283156109777	-0.208315610977749
39	95.15	94.9408079341992	0.209192065800764
40	95.28	95.3883892717451	-0.108389271745096
41	96.12	95.5092798040145	0.610720195985465
42	96.5	96.4006071709038	0.0993928290961605
43	96.67	96.788960541627	-0.11896054162699
44	96.83	96.9489626221123	-0.118962622112335
45	97.4	97.0989645277454	0.30103547225464
46	97.75	97.6942647523534	0.0557352476465667
47	97.46	98.0489489653915	-0.588948965391467
48	97.46	97.7094513397612	-0.249451339761208
49	97.56	97.6884864518778	-0.128486451877819
50	97.97	97.7776879367817	0.192312063218267
51	98.89	98.2038506114254	0.686149388574577
52	99.1	99.1815173492208	-0.0815173492208032
53	99.3	99.3846663052913	-0.0846663052913499
54	100	99.5775506105044	0.422449389495597
55	99.73	100.313054946122	-0.583054946122147
56	99.34	99.9940526774352	-0.654052677435189
57	99.78	99.5490834755645	0.230916524435514
58	99.5	100.008490623467	-0.50849062346694
59	99.6	99.6857550385591	-0.0857550385591281
60	99.52	99.7785478422757	-0.258547842275675
61	99.63	99.6768184479538	-0.0468184479537683
62	99.61	99.7828836384107	-0.172883638410696
63	99.73	99.7483538062614	-0.0183538062613735
64	100.53	99.8668112790059	0.663188720994114
65	100.87	100.722548310503	0.14745168949743
66	100.9	101.07494073999	-0.174940739989978
67	101.08	101.0902380208	-0.0102380207998038
68	102.95	101.269377576601	1.68062242339916
69	102.58	103.28062380408	-0.700623804079868
70	102.6	102.851740578538	-0.251740578537678
71	102.45	102.850583293875	-0.400583293875286
72	102.41	102.666916672353	-0.256916672352688
73	102.38	102.605324368072	-0.225324368072407
74	102.65	102.556387207348	0.0936127926522232
75	103.33	102.8342548007	0.495745199300003
76	103.68	103.555919209205	0.124080790794551
77	104.13	103.916347454931	0.213652545069408
78	104.3	104.384303668987	-0.0843036689868484
79	104.11	104.547218451605	-0.43721845160475
80	104.17	104.320472864962	-0.150472864961628
81	104.23	104.36782652381	-0.13782652380975
82	104.47	104.416243031728	0.0537569682724808
83	104.86	104.660760982258	0.199239017742087
84	104.9	105.067505825853	-0.167505825853283

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
85	105.093427966571	104.348121530908	105.838734402235
86	105.286855933143	104.187648602979	106.386063263307
87	105.480283899714	104.078081208785	106.882486590643
88	105.673711866286	103.989349746714	107.358073985857
89	105.867139832857	103.910404281895	107.82387538382
90	106.060567799429	103.835880325282	108.285255273575
91	106.253995766	103.762774389841	108.74521714216
92	106.447423732572	103.689252245852	109.205595219291
93	106.640851699143	103.614127472444	109.667575925842
94	106.834279665714	103.536601710006	110.131957621423
95	107.027707632286	103.456122803824	110.599292460748
96	107.221135598857	103.372301804315	111.069969393399

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
85 & 105.093427966571 & 104.348121530908 & 105.838734402235 \tabularnewline
86 & 105.286855933143 & 104.187648602979 & 106.386063263307 \tabularnewline
87 & 105.480283899714 & 104.078081208785 & 106.882486590643 \tabularnewline
88 & 105.673711866286 & 103.989349746714 & 107.358073985857 \tabularnewline
89 & 105.867139832857 & 103.910404281895 & 107.82387538382 \tabularnewline
90 & 106.060567799429 & 103.835880325282 & 108.285255273575 \tabularnewline
91 & 106.253995766 & 103.762774389841 & 108.74521714216 \tabularnewline
92 & 106.447423732572 & 103.689252245852 & 109.205595219291 \tabularnewline
93 & 106.640851699143 & 103.614127472444 & 109.667575925842 \tabularnewline
94 & 106.834279665714 & 103.536601710006 & 110.131957621423 \tabularnewline
95 & 107.027707632286 & 103.456122803824 & 110.599292460748 \tabularnewline
96 & 107.221135598857 & 103.372301804315 & 111.069969393399 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=233100&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]85[/C][C]105.093427966571[/C][C]104.348121530908[/C][C]105.838734402235[/C][/ROW]
[ROW][C]86[/C][C]105.286855933143[/C][C]104.187648602979[/C][C]106.386063263307[/C][/ROW]
[ROW][C]87[/C][C]105.480283899714[/C][C]104.078081208785[/C][C]106.882486590643[/C][/ROW]
[ROW][C]88[/C][C]105.673711866286[/C][C]103.989349746714[/C][C]107.358073985857[/C][/ROW]
[ROW][C]89[/C][C]105.867139832857[/C][C]103.910404281895[/C][C]107.82387538382[/C][/ROW]
[ROW][C]90[/C][C]106.060567799429[/C][C]103.835880325282[/C][C]108.285255273575[/C][/ROW]
[ROW][C]91[/C][C]106.253995766[/C][C]103.762774389841[/C][C]108.74521714216[/C][/ROW]
[ROW][C]92[/C][C]106.447423732572[/C][C]103.689252245852[/C][C]109.205595219291[/C][/ROW]
[ROW][C]93[/C][C]106.640851699143[/C][C]103.614127472444[/C][C]109.667575925842[/C][/ROW]
[ROW][C]94[/C][C]106.834279665714[/C][C]103.536601710006[/C][C]110.131957621423[/C][/ROW]
[ROW][C]95[/C][C]107.027707632286[/C][C]103.456122803824[/C][C]110.599292460748[/C][/ROW]
[ROW][C]96[/C][C]107.221135598857[/C][C]103.372301804315[/C][C]111.069969393399[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=233100&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=233100&T=3

As an alternative you can also use a QR Code:

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

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
85	105.093427966571	104.348121530908	105.838734402235
86	105.286855933143	104.187648602979	106.386063263307
87	105.480283899714	104.078081208785	106.882486590643
88	105.673711866286	103.989349746714	107.358073985857
89	105.867139832857	103.910404281895	107.82387538382
90	106.060567799429	103.835880325282	108.285255273575
91	106.253995766	103.762774389841	108.74521714216
92	106.447423732572	103.689252245852	109.205595219291
93	106.640851699143	103.614127472444	109.667575925842
94	106.834279665714	103.536601710006	110.131957621423
95	107.027707632286	103.456122803824	110.599292460748
96	107.221135598857	103.372301804315	111.069969393399

Figure 1

PNG link

Postscript link

PDF link

Figure 2

PNG link

Postscript link

PDF link

Figure 3

PNG link

Postscript link

PDF link

Parameters (Session):

par1 = 12 ; par2 = Double ; par3 = additive ;

Parameters (R input):

par1 = 12 ; par2 = Double ; 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')

Free Statistics

Description of Statistical Computation

Tree of Dependent Computations

Dataset

Tables (Output of Computation)

Figures (Output of Computation)

Input Parameters & R Code