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Author's title

Author

*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Mon, 26 Dec 2011 08:07:48 -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/2011/Dec/26/t1324904975obfkmubmp6r0hzh.htm/, Retrieved Wed, 11 Feb 2026 18:05:47 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=160805, Retrieved Wed, 11 Feb 2026 18:05:47 +0000

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Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords

KDGP2W102

Estimated Impact

344

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

-       [Exponential Smoothing] [] [2011-12-26 13:07:48] [de3755f9a1b164103f4acd66d3337cdc] [Current]

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Dataseries X:

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4,23
4,38
4,43
4,44
4,44
4,44
4,44
4,44
4,45
4,45
4,45
4,45
4,45
4,45
4,45
4,45
4,46
4,46
4,46
4,48
4,58
4,67
4,68
4,68
4,69
4,69
4,69
4,69
4,69
4,69
4,69
4,73
4,78
4,79
4,79
4,8
4,8
4,81
5,16
5,26
5,29
5,29
5,29
5,3
5,3
5,3
5,3
5,3
5,3
5,3
5,3
5,35
5,44
5,47
5,47
5,48
5,48
5,48
5,48
5,48
5,48
5,48
5,5
5,55
5,55
5,57
5,58
5,58
5,58
5,59
5,59
5,59
5,61
5,61
5,61
5,63
5,69
5,7
5,7
5,7
5,7
5,7
5,7
5,7

Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	2 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 & 2 seconds \tabularnewline
R Server & 'Herman Ole Andreas Wold' @ wold.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=160805&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]'Herman Ole Andreas Wold' @ wold.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=160805&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=160805&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	2 seconds
R Server	'Herman Ole Andreas Wold' @ wold.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	1
beta	0.483752617647451
gamma	FALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 1 \tabularnewline
beta & 0.483752617647451 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=160805&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]0.483752617647451[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=160805&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=160805&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	0.483752617647451
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
3	4.43	4.53	-0.0999999999999996
4	4.44	4.53162473823525	-0.0916247382352537
5	4.44	4.49730103127269	-0.0573010312726883
6	4.44	4.46958150740063	-0.0295815074006267
7	4.44	4.45527137576162	-0.0152713757616159
8	4.44	4.44788380776186	-0.00788380776185615
9	4.45	4.44406999512003	0.00593000487997042
10	4.45	4.45693865050338	-0.00693865050337727
11	4.45	4.45358206015943	-0.00358206015942741
12	4.45	4.45184922918073	-0.00184922918073394
13	4.45	4.45095465972392	-0.000954659723923612
14	4.45	4.45049284058351	-0.000492840583513399
15	4.45	4.45025442766116	-0.000254427661155354
16	4.45	4.45013134761407	-0.000131347614069632
17	4.46	4.45006780786194	0.0099321921380584
18	4.46	4.46487253180771	-0.00487253180770519
19	4.46	4.46251543179116	-0.00251543179115732
20	4.48	4.46129858507767	0.0187014149223295
21	4.58	4.49034544350006	0.0896545564999407
22	4.67	4.63371606989093	0.0362839301090734
23	4.68	4.74126851605973	-0.0612685160597284
24	4.68	4.72162971103646	-0.0416297110364594
25	4.69	4.70149122935067	-0.011491229350665
26	4.69	4.70593231707229	-0.015932317072294
27	4.69	4.69822501698338	-0.00822501698338307
28	4.69	4.69424614348748	-0.00424614348747632
29	4.69	4.6921920604605	-0.00219206046050324
30	4.69	4.69113164547469	-0.00113164547469324
31	4.69	4.69058420901406	-0.000584209014061088
32	4.73	4.69030159637426	0.0396984036257439
33	4.78	4.74950580304464	0.0304941969553649
34	4.79	4.81425745064485	-0.0242574506448499
35	4.79	4.81252284539795	-0.0225228453979494
36	4.8	4.80162735997982	-0.00162735997982288
37	4.8	4.81084012032973	-0.0108401203297284
38	4.81	4.80559618374461	0.00440381625539032
39	5.16	4.81772654138579	0.342273458614208
40	5.26	5.33330222294166	-0.0733022229416624
41	5.29	5.39784208071426	-0.107842080714255
42	5.29	5.37567319187619	-0.0856731918761868
43	5.29	5.33422856104387	-0.0442285610438695
44	5.3	5.31283287886412	-0.0128328788641179
45	5.3	5.31662494012165	-0.0166249401216483
46	5.3	5.30858258181957	-0.00858258181956817
47	5.3	5.30443073539818	-0.00443073539817895
48	5.3	5.30228735555121	-0.00228735555120618
49	5.3	5.30118084131582	-0.0011808413158203
50	5.3	5.30060960623827	-0.00060960623826567
51	5.3	5.30031470762477	-0.000314707624770705
52	5.35	5.30016246698749	0.0498375330125054
53	5.44	5.37427150403938	0.0657284959606157
54	5.47	5.49606783601436	-0.0260678360143638
55	5.47	5.51345745210601	-0.0434574521060096
56	5.48	5.49243479589344	-0.0124347958934381
57	5.48	5.49641943083008	-0.0164194308300765
58	5.48	5.48847648818575	-0.00847648818574598
59	5.48	5.48437596483743	-0.00437596483743352
60	5.48	5.48225908039259	-0.00225908039259171
61	5.48	5.4811662443392	-0.0011662443391991
62	5.48	5.4806020705873	-0.000602070587294889
63	5.5	5.48031081736468	0.0196891826353172
64	5.55	5.50983551100386	0.0401644889961439
65	5.55	5.57926518769221	-0.0292651876922125
66	5.57	5.56510807654016	0.00489192345983991
67	5.58	5.58747455731919	-0.00747455731918922
68	5.58	5.59385872065028	-0.0138587206502754
69	5.58	5.58715452825846	-0.00715452825845997
70	5.59	5.5836935064854	0.00630649351460288
71	5.59	5.59674428923126	-0.00674428923126325
72	5.59	5.59348172166147	-0.00348172166146821
73	5.61	5.59179742969381	0.0182025703061877
74	5.61	5.62060297072734	-0.0106029707273425
75	5.61	5.61547375588315	-0.00547375588315191
76	5.63	5.61282581214631	0.0171741878536862
77	5.69	5.6411338704765	0.0488661295234971
78	5.7	5.72477298854779	-0.0247729885477943
79	5.7	5.72278899049085	-0.0227889904908487
80	5.7	5.71176475668736	-0.0117647566873575
81	5.7	5.70607352484386	-0.00607352484386325
82	5.7	5.7031354413023	-0.00313544130229726
83	5.7	5.70161866336483	-0.00161866336483119
84	5.7	5.700835630725	-0.000835630725004144

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 4.43 & 4.53 & -0.0999999999999996 \tabularnewline
4 & 4.44 & 4.53162473823525 & -0.0916247382352537 \tabularnewline
5 & 4.44 & 4.49730103127269 & -0.0573010312726883 \tabularnewline
6 & 4.44 & 4.46958150740063 & -0.0295815074006267 \tabularnewline
7 & 4.44 & 4.45527137576162 & -0.0152713757616159 \tabularnewline
8 & 4.44 & 4.44788380776186 & -0.00788380776185615 \tabularnewline
9 & 4.45 & 4.44406999512003 & 0.00593000487997042 \tabularnewline
10 & 4.45 & 4.45693865050338 & -0.00693865050337727 \tabularnewline
11 & 4.45 & 4.45358206015943 & -0.00358206015942741 \tabularnewline
12 & 4.45 & 4.45184922918073 & -0.00184922918073394 \tabularnewline
13 & 4.45 & 4.45095465972392 & -0.000954659723923612 \tabularnewline
14 & 4.45 & 4.45049284058351 & -0.000492840583513399 \tabularnewline
15 & 4.45 & 4.45025442766116 & -0.000254427661155354 \tabularnewline
16 & 4.45 & 4.45013134761407 & -0.000131347614069632 \tabularnewline
17 & 4.46 & 4.45006780786194 & 0.0099321921380584 \tabularnewline
18 & 4.46 & 4.46487253180771 & -0.00487253180770519 \tabularnewline
19 & 4.46 & 4.46251543179116 & -0.00251543179115732 \tabularnewline
20 & 4.48 & 4.46129858507767 & 0.0187014149223295 \tabularnewline
21 & 4.58 & 4.49034544350006 & 0.0896545564999407 \tabularnewline
22 & 4.67 & 4.63371606989093 & 0.0362839301090734 \tabularnewline
23 & 4.68 & 4.74126851605973 & -0.0612685160597284 \tabularnewline
24 & 4.68 & 4.72162971103646 & -0.0416297110364594 \tabularnewline
25 & 4.69 & 4.70149122935067 & -0.011491229350665 \tabularnewline
26 & 4.69 & 4.70593231707229 & -0.015932317072294 \tabularnewline
27 & 4.69 & 4.69822501698338 & -0.00822501698338307 \tabularnewline
28 & 4.69 & 4.69424614348748 & -0.00424614348747632 \tabularnewline
29 & 4.69 & 4.6921920604605 & -0.00219206046050324 \tabularnewline
30 & 4.69 & 4.69113164547469 & -0.00113164547469324 \tabularnewline
31 & 4.69 & 4.69058420901406 & -0.000584209014061088 \tabularnewline
32 & 4.73 & 4.69030159637426 & 0.0396984036257439 \tabularnewline
33 & 4.78 & 4.74950580304464 & 0.0304941969553649 \tabularnewline
34 & 4.79 & 4.81425745064485 & -0.0242574506448499 \tabularnewline
35 & 4.79 & 4.81252284539795 & -0.0225228453979494 \tabularnewline
36 & 4.8 & 4.80162735997982 & -0.00162735997982288 \tabularnewline
37 & 4.8 & 4.81084012032973 & -0.0108401203297284 \tabularnewline
38 & 4.81 & 4.80559618374461 & 0.00440381625539032 \tabularnewline
39 & 5.16 & 4.81772654138579 & 0.342273458614208 \tabularnewline
40 & 5.26 & 5.33330222294166 & -0.0733022229416624 \tabularnewline
41 & 5.29 & 5.39784208071426 & -0.107842080714255 \tabularnewline
42 & 5.29 & 5.37567319187619 & -0.0856731918761868 \tabularnewline
43 & 5.29 & 5.33422856104387 & -0.0442285610438695 \tabularnewline
44 & 5.3 & 5.31283287886412 & -0.0128328788641179 \tabularnewline
45 & 5.3 & 5.31662494012165 & -0.0166249401216483 \tabularnewline
46 & 5.3 & 5.30858258181957 & -0.00858258181956817 \tabularnewline
47 & 5.3 & 5.30443073539818 & -0.00443073539817895 \tabularnewline
48 & 5.3 & 5.30228735555121 & -0.00228735555120618 \tabularnewline
49 & 5.3 & 5.30118084131582 & -0.0011808413158203 \tabularnewline
50 & 5.3 & 5.30060960623827 & -0.00060960623826567 \tabularnewline
51 & 5.3 & 5.30031470762477 & -0.000314707624770705 \tabularnewline
52 & 5.35 & 5.30016246698749 & 0.0498375330125054 \tabularnewline
53 & 5.44 & 5.37427150403938 & 0.0657284959606157 \tabularnewline
54 & 5.47 & 5.49606783601436 & -0.0260678360143638 \tabularnewline
55 & 5.47 & 5.51345745210601 & -0.0434574521060096 \tabularnewline
56 & 5.48 & 5.49243479589344 & -0.0124347958934381 \tabularnewline
57 & 5.48 & 5.49641943083008 & -0.0164194308300765 \tabularnewline
58 & 5.48 & 5.48847648818575 & -0.00847648818574598 \tabularnewline
59 & 5.48 & 5.48437596483743 & -0.00437596483743352 \tabularnewline
60 & 5.48 & 5.48225908039259 & -0.00225908039259171 \tabularnewline
61 & 5.48 & 5.4811662443392 & -0.0011662443391991 \tabularnewline
62 & 5.48 & 5.4806020705873 & -0.000602070587294889 \tabularnewline
63 & 5.5 & 5.48031081736468 & 0.0196891826353172 \tabularnewline
64 & 5.55 & 5.50983551100386 & 0.0401644889961439 \tabularnewline
65 & 5.55 & 5.57926518769221 & -0.0292651876922125 \tabularnewline
66 & 5.57 & 5.56510807654016 & 0.00489192345983991 \tabularnewline
67 & 5.58 & 5.58747455731919 & -0.00747455731918922 \tabularnewline
68 & 5.58 & 5.59385872065028 & -0.0138587206502754 \tabularnewline
69 & 5.58 & 5.58715452825846 & -0.00715452825845997 \tabularnewline
70 & 5.59 & 5.5836935064854 & 0.00630649351460288 \tabularnewline
71 & 5.59 & 5.59674428923126 & -0.00674428923126325 \tabularnewline
72 & 5.59 & 5.59348172166147 & -0.00348172166146821 \tabularnewline
73 & 5.61 & 5.59179742969381 & 0.0182025703061877 \tabularnewline
74 & 5.61 & 5.62060297072734 & -0.0106029707273425 \tabularnewline
75 & 5.61 & 5.61547375588315 & -0.00547375588315191 \tabularnewline
76 & 5.63 & 5.61282581214631 & 0.0171741878536862 \tabularnewline
77 & 5.69 & 5.6411338704765 & 0.0488661295234971 \tabularnewline
78 & 5.7 & 5.72477298854779 & -0.0247729885477943 \tabularnewline
79 & 5.7 & 5.72278899049085 & -0.0227889904908487 \tabularnewline
80 & 5.7 & 5.71176475668736 & -0.0117647566873575 \tabularnewline
81 & 5.7 & 5.70607352484386 & -0.00607352484386325 \tabularnewline
82 & 5.7 & 5.7031354413023 & -0.00313544130229726 \tabularnewline
83 & 5.7 & 5.70161866336483 & -0.00161866336483119 \tabularnewline
84 & 5.7 & 5.700835630725 & -0.000835630725004144 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=160805&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]4.43[/C][C]4.53[/C][C]-0.0999999999999996[/C][/ROW]
[ROW][C]4[/C][C]4.44[/C][C]4.53162473823525[/C][C]-0.0916247382352537[/C][/ROW]
[ROW][C]5[/C][C]4.44[/C][C]4.49730103127269[/C][C]-0.0573010312726883[/C][/ROW]
[ROW][C]6[/C][C]4.44[/C][C]4.46958150740063[/C][C]-0.0295815074006267[/C][/ROW]
[ROW][C]7[/C][C]4.44[/C][C]4.45527137576162[/C][C]-0.0152713757616159[/C][/ROW]
[ROW][C]8[/C][C]4.44[/C][C]4.44788380776186[/C][C]-0.00788380776185615[/C][/ROW]
[ROW][C]9[/C][C]4.45[/C][C]4.44406999512003[/C][C]0.00593000487997042[/C][/ROW]
[ROW][C]10[/C][C]4.45[/C][C]4.45693865050338[/C][C]-0.00693865050337727[/C][/ROW]
[ROW][C]11[/C][C]4.45[/C][C]4.45358206015943[/C][C]-0.00358206015942741[/C][/ROW]
[ROW][C]12[/C][C]4.45[/C][C]4.45184922918073[/C][C]-0.00184922918073394[/C][/ROW]
[ROW][C]13[/C][C]4.45[/C][C]4.45095465972392[/C][C]-0.000954659723923612[/C][/ROW]
[ROW][C]14[/C][C]4.45[/C][C]4.45049284058351[/C][C]-0.000492840583513399[/C][/ROW]
[ROW][C]15[/C][C]4.45[/C][C]4.45025442766116[/C][C]-0.000254427661155354[/C][/ROW]
[ROW][C]16[/C][C]4.45[/C][C]4.45013134761407[/C][C]-0.000131347614069632[/C][/ROW]
[ROW][C]17[/C][C]4.46[/C][C]4.45006780786194[/C][C]0.0099321921380584[/C][/ROW]
[ROW][C]18[/C][C]4.46[/C][C]4.46487253180771[/C][C]-0.00487253180770519[/C][/ROW]
[ROW][C]19[/C][C]4.46[/C][C]4.46251543179116[/C][C]-0.00251543179115732[/C][/ROW]
[ROW][C]20[/C][C]4.48[/C][C]4.46129858507767[/C][C]0.0187014149223295[/C][/ROW]
[ROW][C]21[/C][C]4.58[/C][C]4.49034544350006[/C][C]0.0896545564999407[/C][/ROW]
[ROW][C]22[/C][C]4.67[/C][C]4.63371606989093[/C][C]0.0362839301090734[/C][/ROW]
[ROW][C]23[/C][C]4.68[/C][C]4.74126851605973[/C][C]-0.0612685160597284[/C][/ROW]
[ROW][C]24[/C][C]4.68[/C][C]4.72162971103646[/C][C]-0.0416297110364594[/C][/ROW]
[ROW][C]25[/C][C]4.69[/C][C]4.70149122935067[/C][C]-0.011491229350665[/C][/ROW]
[ROW][C]26[/C][C]4.69[/C][C]4.70593231707229[/C][C]-0.015932317072294[/C][/ROW]
[ROW][C]27[/C][C]4.69[/C][C]4.69822501698338[/C][C]-0.00822501698338307[/C][/ROW]
[ROW][C]28[/C][C]4.69[/C][C]4.69424614348748[/C][C]-0.00424614348747632[/C][/ROW]
[ROW][C]29[/C][C]4.69[/C][C]4.6921920604605[/C][C]-0.00219206046050324[/C][/ROW]
[ROW][C]30[/C][C]4.69[/C][C]4.69113164547469[/C][C]-0.00113164547469324[/C][/ROW]
[ROW][C]31[/C][C]4.69[/C][C]4.69058420901406[/C][C]-0.000584209014061088[/C][/ROW]
[ROW][C]32[/C][C]4.73[/C][C]4.69030159637426[/C][C]0.0396984036257439[/C][/ROW]
[ROW][C]33[/C][C]4.78[/C][C]4.74950580304464[/C][C]0.0304941969553649[/C][/ROW]
[ROW][C]34[/C][C]4.79[/C][C]4.81425745064485[/C][C]-0.0242574506448499[/C][/ROW]
[ROW][C]35[/C][C]4.79[/C][C]4.81252284539795[/C][C]-0.0225228453979494[/C][/ROW]
[ROW][C]36[/C][C]4.8[/C][C]4.80162735997982[/C][C]-0.00162735997982288[/C][/ROW]
[ROW][C]37[/C][C]4.8[/C][C]4.81084012032973[/C][C]-0.0108401203297284[/C][/ROW]
[ROW][C]38[/C][C]4.81[/C][C]4.80559618374461[/C][C]0.00440381625539032[/C][/ROW]
[ROW][C]39[/C][C]5.16[/C][C]4.81772654138579[/C][C]0.342273458614208[/C][/ROW]
[ROW][C]40[/C][C]5.26[/C][C]5.33330222294166[/C][C]-0.0733022229416624[/C][/ROW]
[ROW][C]41[/C][C]5.29[/C][C]5.39784208071426[/C][C]-0.107842080714255[/C][/ROW]
[ROW][C]42[/C][C]5.29[/C][C]5.37567319187619[/C][C]-0.0856731918761868[/C][/ROW]
[ROW][C]43[/C][C]5.29[/C][C]5.33422856104387[/C][C]-0.0442285610438695[/C][/ROW]
[ROW][C]44[/C][C]5.3[/C][C]5.31283287886412[/C][C]-0.0128328788641179[/C][/ROW]
[ROW][C]45[/C][C]5.3[/C][C]5.31662494012165[/C][C]-0.0166249401216483[/C][/ROW]
[ROW][C]46[/C][C]5.3[/C][C]5.30858258181957[/C][C]-0.00858258181956817[/C][/ROW]
[ROW][C]47[/C][C]5.3[/C][C]5.30443073539818[/C][C]-0.00443073539817895[/C][/ROW]
[ROW][C]48[/C][C]5.3[/C][C]5.30228735555121[/C][C]-0.00228735555120618[/C][/ROW]
[ROW][C]49[/C][C]5.3[/C][C]5.30118084131582[/C][C]-0.0011808413158203[/C][/ROW]
[ROW][C]50[/C][C]5.3[/C][C]5.30060960623827[/C][C]-0.00060960623826567[/C][/ROW]
[ROW][C]51[/C][C]5.3[/C][C]5.30031470762477[/C][C]-0.000314707624770705[/C][/ROW]
[ROW][C]52[/C][C]5.35[/C][C]5.30016246698749[/C][C]0.0498375330125054[/C][/ROW]
[ROW][C]53[/C][C]5.44[/C][C]5.37427150403938[/C][C]0.0657284959606157[/C][/ROW]
[ROW][C]54[/C][C]5.47[/C][C]5.49606783601436[/C][C]-0.0260678360143638[/C][/ROW]
[ROW][C]55[/C][C]5.47[/C][C]5.51345745210601[/C][C]-0.0434574521060096[/C][/ROW]
[ROW][C]56[/C][C]5.48[/C][C]5.49243479589344[/C][C]-0.0124347958934381[/C][/ROW]
[ROW][C]57[/C][C]5.48[/C][C]5.49641943083008[/C][C]-0.0164194308300765[/C][/ROW]
[ROW][C]58[/C][C]5.48[/C][C]5.48847648818575[/C][C]-0.00847648818574598[/C][/ROW]
[ROW][C]59[/C][C]5.48[/C][C]5.48437596483743[/C][C]-0.00437596483743352[/C][/ROW]
[ROW][C]60[/C][C]5.48[/C][C]5.48225908039259[/C][C]-0.00225908039259171[/C][/ROW]
[ROW][C]61[/C][C]5.48[/C][C]5.4811662443392[/C][C]-0.0011662443391991[/C][/ROW]
[ROW][C]62[/C][C]5.48[/C][C]5.4806020705873[/C][C]-0.000602070587294889[/C][/ROW]
[ROW][C]63[/C][C]5.5[/C][C]5.48031081736468[/C][C]0.0196891826353172[/C][/ROW]
[ROW][C]64[/C][C]5.55[/C][C]5.50983551100386[/C][C]0.0401644889961439[/C][/ROW]
[ROW][C]65[/C][C]5.55[/C][C]5.57926518769221[/C][C]-0.0292651876922125[/C][/ROW]
[ROW][C]66[/C][C]5.57[/C][C]5.56510807654016[/C][C]0.00489192345983991[/C][/ROW]
[ROW][C]67[/C][C]5.58[/C][C]5.58747455731919[/C][C]-0.00747455731918922[/C][/ROW]
[ROW][C]68[/C][C]5.58[/C][C]5.59385872065028[/C][C]-0.0138587206502754[/C][/ROW]
[ROW][C]69[/C][C]5.58[/C][C]5.58715452825846[/C][C]-0.00715452825845997[/C][/ROW]
[ROW][C]70[/C][C]5.59[/C][C]5.5836935064854[/C][C]0.00630649351460288[/C][/ROW]
[ROW][C]71[/C][C]5.59[/C][C]5.59674428923126[/C][C]-0.00674428923126325[/C][/ROW]
[ROW][C]72[/C][C]5.59[/C][C]5.59348172166147[/C][C]-0.00348172166146821[/C][/ROW]
[ROW][C]73[/C][C]5.61[/C][C]5.59179742969381[/C][C]0.0182025703061877[/C][/ROW]
[ROW][C]74[/C][C]5.61[/C][C]5.62060297072734[/C][C]-0.0106029707273425[/C][/ROW]
[ROW][C]75[/C][C]5.61[/C][C]5.61547375588315[/C][C]-0.00547375588315191[/C][/ROW]
[ROW][C]76[/C][C]5.63[/C][C]5.61282581214631[/C][C]0.0171741878536862[/C][/ROW]
[ROW][C]77[/C][C]5.69[/C][C]5.6411338704765[/C][C]0.0488661295234971[/C][/ROW]
[ROW][C]78[/C][C]5.7[/C][C]5.72477298854779[/C][C]-0.0247729885477943[/C][/ROW]
[ROW][C]79[/C][C]5.7[/C][C]5.72278899049085[/C][C]-0.0227889904908487[/C][/ROW]
[ROW][C]80[/C][C]5.7[/C][C]5.71176475668736[/C][C]-0.0117647566873575[/C][/ROW]
[ROW][C]81[/C][C]5.7[/C][C]5.70607352484386[/C][C]-0.00607352484386325[/C][/ROW]
[ROW][C]82[/C][C]5.7[/C][C]5.7031354413023[/C][C]-0.00313544130229726[/C][/ROW]
[ROW][C]83[/C][C]5.7[/C][C]5.70161866336483[/C][C]-0.00161866336483119[/C][/ROW]
[ROW][C]84[/C][C]5.7[/C][C]5.700835630725[/C][C]-0.000835630725004144[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=160805&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=160805&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	4.43	4.53	-0.0999999999999996
4	4.44	4.53162473823525	-0.0916247382352537
5	4.44	4.49730103127269	-0.0573010312726883
6	4.44	4.46958150740063	-0.0295815074006267
7	4.44	4.45527137576162	-0.0152713757616159
8	4.44	4.44788380776186	-0.00788380776185615
9	4.45	4.44406999512003	0.00593000487997042
10	4.45	4.45693865050338	-0.00693865050337727
11	4.45	4.45358206015943	-0.00358206015942741
12	4.45	4.45184922918073	-0.00184922918073394
13	4.45	4.45095465972392	-0.000954659723923612
14	4.45	4.45049284058351	-0.000492840583513399
15	4.45	4.45025442766116	-0.000254427661155354
16	4.45	4.45013134761407	-0.000131347614069632
17	4.46	4.45006780786194	0.0099321921380584
18	4.46	4.46487253180771	-0.00487253180770519
19	4.46	4.46251543179116	-0.00251543179115732
20	4.48	4.46129858507767	0.0187014149223295
21	4.58	4.49034544350006	0.0896545564999407
22	4.67	4.63371606989093	0.0362839301090734
23	4.68	4.74126851605973	-0.0612685160597284
24	4.68	4.72162971103646	-0.0416297110364594
25	4.69	4.70149122935067	-0.011491229350665
26	4.69	4.70593231707229	-0.015932317072294
27	4.69	4.69822501698338	-0.00822501698338307
28	4.69	4.69424614348748	-0.00424614348747632
29	4.69	4.6921920604605	-0.00219206046050324
30	4.69	4.69113164547469	-0.00113164547469324
31	4.69	4.69058420901406	-0.000584209014061088
32	4.73	4.69030159637426	0.0396984036257439
33	4.78	4.74950580304464	0.0304941969553649
34	4.79	4.81425745064485	-0.0242574506448499
35	4.79	4.81252284539795	-0.0225228453979494
36	4.8	4.80162735997982	-0.00162735997982288
37	4.8	4.81084012032973	-0.0108401203297284
38	4.81	4.80559618374461	0.00440381625539032
39	5.16	4.81772654138579	0.342273458614208
40	5.26	5.33330222294166	-0.0733022229416624
41	5.29	5.39784208071426	-0.107842080714255
42	5.29	5.37567319187619	-0.0856731918761868
43	5.29	5.33422856104387	-0.0442285610438695
44	5.3	5.31283287886412	-0.0128328788641179
45	5.3	5.31662494012165	-0.0166249401216483
46	5.3	5.30858258181957	-0.00858258181956817
47	5.3	5.30443073539818	-0.00443073539817895
48	5.3	5.30228735555121	-0.00228735555120618
49	5.3	5.30118084131582	-0.0011808413158203
50	5.3	5.30060960623827	-0.00060960623826567
51	5.3	5.30031470762477	-0.000314707624770705
52	5.35	5.30016246698749	0.0498375330125054
53	5.44	5.37427150403938	0.0657284959606157
54	5.47	5.49606783601436	-0.0260678360143638
55	5.47	5.51345745210601	-0.0434574521060096
56	5.48	5.49243479589344	-0.0124347958934381
57	5.48	5.49641943083008	-0.0164194308300765
58	5.48	5.48847648818575	-0.00847648818574598
59	5.48	5.48437596483743	-0.00437596483743352
60	5.48	5.48225908039259	-0.00225908039259171
61	5.48	5.4811662443392	-0.0011662443391991
62	5.48	5.4806020705873	-0.000602070587294889
63	5.5	5.48031081736468	0.0196891826353172
64	5.55	5.50983551100386	0.0401644889961439
65	5.55	5.57926518769221	-0.0292651876922125
66	5.57	5.56510807654016	0.00489192345983991
67	5.58	5.58747455731919	-0.00747455731918922
68	5.58	5.59385872065028	-0.0138587206502754
69	5.58	5.58715452825846	-0.00715452825845997
70	5.59	5.5836935064854	0.00630649351460288
71	5.59	5.59674428923126	-0.00674428923126325
72	5.59	5.59348172166147	-0.00348172166146821
73	5.61	5.59179742969381	0.0182025703061877
74	5.61	5.62060297072734	-0.0106029707273425
75	5.61	5.61547375588315	-0.00547375588315191
76	5.63	5.61282581214631	0.0171741878536862
77	5.69	5.6411338704765	0.0488661295234971
78	5.7	5.72477298854779	-0.0247729885477943
79	5.7	5.72278899049085	-0.0227889904908487
80	5.7	5.71176475668736	-0.0117647566873575
81	5.7	5.70607352484386	-0.00607352484386325
82	5.7	5.7031354413023	-0.00313544130229726
83	5.7	5.70161866336483	-0.00161866336483119
84	5.7	5.700835630725	-0.000835630725004144

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
85	5.7004313921744	5.60232085173317	5.79854193261562
86	5.70086278434879	5.52531558637762	5.87640998231997
87	5.70129417652319	5.44037549463251	5.96221285841387
88	5.70172556869759	5.34687907438241	6.05657206301276
89	5.70215696087198	5.2451729529788	6.15914096876517
90	5.70258835304638	5.13572087096913	6.26945583512363
91	5.70301974522078	5.01896254901771	6.38707694142385
92	5.70345113739517	4.89528900227505	6.5116132725153
93	5.70388252956957	4.76504322171081	6.64272183742833
94	5.70431392174397	4.62852629746881	6.78010154601913
95	5.70474531391836	4.48600396597758	6.92348666185915
96	5.70517670609276	4.33771235016186	7.07264106202366

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
85 & 5.7004313921744 & 5.60232085173317 & 5.79854193261562 \tabularnewline
86 & 5.70086278434879 & 5.52531558637762 & 5.87640998231997 \tabularnewline
87 & 5.70129417652319 & 5.44037549463251 & 5.96221285841387 \tabularnewline
88 & 5.70172556869759 & 5.34687907438241 & 6.05657206301276 \tabularnewline
89 & 5.70215696087198 & 5.2451729529788 & 6.15914096876517 \tabularnewline
90 & 5.70258835304638 & 5.13572087096913 & 6.26945583512363 \tabularnewline
91 & 5.70301974522078 & 5.01896254901771 & 6.38707694142385 \tabularnewline
92 & 5.70345113739517 & 4.89528900227505 & 6.5116132725153 \tabularnewline
93 & 5.70388252956957 & 4.76504322171081 & 6.64272183742833 \tabularnewline
94 & 5.70431392174397 & 4.62852629746881 & 6.78010154601913 \tabularnewline
95 & 5.70474531391836 & 4.48600396597758 & 6.92348666185915 \tabularnewline
96 & 5.70517670609276 & 4.33771235016186 & 7.07264106202366 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=160805&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]5.7004313921744[/C][C]5.60232085173317[/C][C]5.79854193261562[/C][/ROW]
[ROW][C]86[/C][C]5.70086278434879[/C][C]5.52531558637762[/C][C]5.87640998231997[/C][/ROW]
[ROW][C]87[/C][C]5.70129417652319[/C][C]5.44037549463251[/C][C]5.96221285841387[/C][/ROW]
[ROW][C]88[/C][C]5.70172556869759[/C][C]5.34687907438241[/C][C]6.05657206301276[/C][/ROW]
[ROW][C]89[/C][C]5.70215696087198[/C][C]5.2451729529788[/C][C]6.15914096876517[/C][/ROW]
[ROW][C]90[/C][C]5.70258835304638[/C][C]5.13572087096913[/C][C]6.26945583512363[/C][/ROW]
[ROW][C]91[/C][C]5.70301974522078[/C][C]5.01896254901771[/C][C]6.38707694142385[/C][/ROW]
[ROW][C]92[/C][C]5.70345113739517[/C][C]4.89528900227505[/C][C]6.5116132725153[/C][/ROW]
[ROW][C]93[/C][C]5.70388252956957[/C][C]4.76504322171081[/C][C]6.64272183742833[/C][/ROW]
[ROW][C]94[/C][C]5.70431392174397[/C][C]4.62852629746881[/C][C]6.78010154601913[/C][/ROW]
[ROW][C]95[/C][C]5.70474531391836[/C][C]4.48600396597758[/C][C]6.92348666185915[/C][/ROW]
[ROW][C]96[/C][C]5.70517670609276[/C][C]4.33771235016186[/C][C]7.07264106202366[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=160805&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=160805&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	5.7004313921744	5.60232085173317	5.79854193261562
86	5.70086278434879	5.52531558637762	5.87640998231997
87	5.70129417652319	5.44037549463251	5.96221285841387
88	5.70172556869759	5.34687907438241	6.05657206301276
89	5.70215696087198	5.2451729529788	6.15914096876517
90	5.70258835304638	5.13572087096913	6.26945583512363
91	5.70301974522078	5.01896254901771	6.38707694142385
92	5.70345113739517	4.89528900227505	6.5116132725153
93	5.70388252956957	4.76504322171081	6.64272183742833
94	5.70431392174397	4.62852629746881	6.78010154601913
95	5.70474531391836	4.48600396597758	6.92348666185915
96	5.70517670609276	4.33771235016186	7.07264106202366

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