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Author

*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Sat, 11 Jan 2014 16:44:36 -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/11/t1389476716kt6kqty62md1pjs.htm/, Retrieved Wed, 11 Feb 2026 13:40:02 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=232944, Retrieved Wed, 11 Feb 2026 13:40:02 +0000

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IsPrivate?

No (this computation is public)

User-defined keywords

Estimated Impact

358

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

-       [Exponential Smoothing] [] [2014-01-11 21:44:36] [c13b0833c91505664fff70cc44050808] [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	3 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 & 3 seconds \tabularnewline
R Server & 'Herman Ole Andreas Wold' @ wold.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232944&T=0

[TABLE]
[ROW][C]Summary of computational transaction[/C][/ROW]
[ROW][C]Raw Input[/C][C]view raw input (R code) [/C][/ROW]
[ROW][C]Raw Output[/C][C]view raw output of R engine [/C][/ROW]
[ROW][C]Computing time[/C][C]3 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Herman Ole Andreas Wold' @ wold.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232944&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232944&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	3 seconds
R Server	'Herman Ole Andreas Wold' @ wold.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.682288509201968
beta	0.0363764273498533
gamma	1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.682288509201968 \tabularnewline
beta & 0.0363764273498533 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232944&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]0.682288509201968[/C][/ROW]
[ROW][C]beta[/C][C]0.0363764273498533[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232944&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232944&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	0.682288509201968
beta	0.0363764273498533
gamma	1

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	48.4	48.0974755456824	0.302524454317584
14	48.5	48.4180446782984	0.0819553217016207
15	48.41	48.4011353845685	0.00886461543149863
16	48.35	48.3599012742241	-0.00990127422406317
17	48.53	48.543152152768	-0.0131521527679865
18	48.52	48.5442555140545	-0.0242555140545235
19	48.52	48.6573947588819	-0.137394758881932
20	48.49	48.488833468595	0.00116653140504752
21	48.45	48.4482302517551	0.00176974824494636
22	48.65	48.5073140957999	0.142685904200142
23	48.74	48.7925170891644	-0.0525170891644109
24	48.74	48.9398375998561	-0.199837599856124
25	48.74	48.9100054396662	-0.170005439666213
26	48.79	48.8287909475206	-0.0387909475205603
27	48.82	48.6931755897683	0.126824410231741
28	48.82	48.7164227720463	0.103577227953664
29	49.2	48.9708624947968	0.229137505203241
30	49.3	49.1327526449417	0.167247355058294
31	49.3	49.3454665919582	-0.0454665919581743
32	49.34	49.2889343812242	0.051065618775759
33	49.47	49.2888314721187	0.181168527881297
34	49.65	49.5283322243154	0.121667775684571
35	49.7	49.7503745307362	-0.0503745307361783
36	49.75	49.865707304539	-0.115707304538986
37	49.75	49.9180481877385	-0.168048187738535
38	49.7	49.8945805029529	-0.194580502952867
39	50.09	49.7133706044651	0.376629395534884
40	50.19	49.9131638397459	0.276836160254085
41	50.53	50.3507240938616	0.179275906138422
42	50.55	50.4764202523037	0.0735797476963356
43	50.55	50.5740006291667	-0.0240006291666717
44	50.55	50.579004159375	-0.0290041593749777
45	50.58	50.5797288727598	0.000271127240225155
46	50.61	50.6886693232595	-0.0786693232595397
47	50.94	50.7258177103463	0.214182289653714
48	51.01	51.0150286579794	-0.00502865797945162
49	51.01	51.1429017810961	-0.132901781096102
50	51.04	51.1517793206041	-0.111779320604136
51	51.15	51.2285118747268	-0.0785118747268214
52	51.31	51.0895029767421	0.220497023257877
53	51.31	51.4662502404456	-0.15625024044558
54	51.34	51.3250489005762	0.0149510994238184
55	51.34	51.3465935400721	-0.00659354007210311
56	51.34	51.3573432260545	-0.0173432260544928
57	51.47	51.3712466662668	0.098753333733228
58	51.95	51.5214969306433	0.428503069356672
59	51.97	52.0118469840796	-0.0418469840795694
60	51.92	52.0619487888161	-0.141948788816116
61	51.92	52.0578582263177	-0.137858226317697
62	51.91	52.0723966639187	-0.162396663918699
63	51.97	52.1273273709331	-0.157327370933139
64	52.14	52.0268880238223	0.113111976177706
65	52.33	52.2069557358363	0.123044264163681
66	52.4	52.3125480299527	0.0874519700472618
67	52.4	52.3799850904706	0.0200149095293582
68	52.41	52.4095254326108	0.000474567389154856
69	52.71	52.4779713904202	0.232028609579785
70	53.17	52.8347592809833	0.33524071901671
71	53.33	53.1179653548523	0.21203464514771
72	53.32	53.3214810443782	-0.00148104437816698
73	53.32	53.4313092527852	-0.111309252785155
74	53.3	53.4738667828506	-0.173866782850602
75	53.31	53.5419980313716	-0.231998031371624
76	53.72	53.4921683873874	0.227831612612562
77	53.87	53.7724861698642	0.0975138301357887
78	53.91	53.8647003772588	0.0452996227411688
79	53.91	53.8955326320105	0.014467367989468
80	53.96	53.9291458030961	0.0308541969039027
81	54.02	54.1103310232845	-0.0903310232845342
82	54.33	54.2920518970097	0.0379481029902848
83	54.48	54.3328424799963	0.147157520003667
84	54.54	54.4218558771116	0.11814412288836

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 48.4 & 48.0974755456824 & 0.302524454317584 \tabularnewline
14 & 48.5 & 48.4180446782984 & 0.0819553217016207 \tabularnewline
15 & 48.41 & 48.4011353845685 & 0.00886461543149863 \tabularnewline
16 & 48.35 & 48.3599012742241 & -0.00990127422406317 \tabularnewline
17 & 48.53 & 48.543152152768 & -0.0131521527679865 \tabularnewline
18 & 48.52 & 48.5442555140545 & -0.0242555140545235 \tabularnewline
19 & 48.52 & 48.6573947588819 & -0.137394758881932 \tabularnewline
20 & 48.49 & 48.488833468595 & 0.00116653140504752 \tabularnewline
21 & 48.45 & 48.4482302517551 & 0.00176974824494636 \tabularnewline
22 & 48.65 & 48.5073140957999 & 0.142685904200142 \tabularnewline
23 & 48.74 & 48.7925170891644 & -0.0525170891644109 \tabularnewline
24 & 48.74 & 48.9398375998561 & -0.199837599856124 \tabularnewline
25 & 48.74 & 48.9100054396662 & -0.170005439666213 \tabularnewline
26 & 48.79 & 48.8287909475206 & -0.0387909475205603 \tabularnewline
27 & 48.82 & 48.6931755897683 & 0.126824410231741 \tabularnewline
28 & 48.82 & 48.7164227720463 & 0.103577227953664 \tabularnewline
29 & 49.2 & 48.9708624947968 & 0.229137505203241 \tabularnewline
30 & 49.3 & 49.1327526449417 & 0.167247355058294 \tabularnewline
31 & 49.3 & 49.3454665919582 & -0.0454665919581743 \tabularnewline
32 & 49.34 & 49.2889343812242 & 0.051065618775759 \tabularnewline
33 & 49.47 & 49.2888314721187 & 0.181168527881297 \tabularnewline
34 & 49.65 & 49.5283322243154 & 0.121667775684571 \tabularnewline
35 & 49.7 & 49.7503745307362 & -0.0503745307361783 \tabularnewline
36 & 49.75 & 49.865707304539 & -0.115707304538986 \tabularnewline
37 & 49.75 & 49.9180481877385 & -0.168048187738535 \tabularnewline
38 & 49.7 & 49.8945805029529 & -0.194580502952867 \tabularnewline
39 & 50.09 & 49.7133706044651 & 0.376629395534884 \tabularnewline
40 & 50.19 & 49.9131638397459 & 0.276836160254085 \tabularnewline
41 & 50.53 & 50.3507240938616 & 0.179275906138422 \tabularnewline
42 & 50.55 & 50.4764202523037 & 0.0735797476963356 \tabularnewline
43 & 50.55 & 50.5740006291667 & -0.0240006291666717 \tabularnewline
44 & 50.55 & 50.579004159375 & -0.0290041593749777 \tabularnewline
45 & 50.58 & 50.5797288727598 & 0.000271127240225155 \tabularnewline
46 & 50.61 & 50.6886693232595 & -0.0786693232595397 \tabularnewline
47 & 50.94 & 50.7258177103463 & 0.214182289653714 \tabularnewline
48 & 51.01 & 51.0150286579794 & -0.00502865797945162 \tabularnewline
49 & 51.01 & 51.1429017810961 & -0.132901781096102 \tabularnewline
50 & 51.04 & 51.1517793206041 & -0.111779320604136 \tabularnewline
51 & 51.15 & 51.2285118747268 & -0.0785118747268214 \tabularnewline
52 & 51.31 & 51.0895029767421 & 0.220497023257877 \tabularnewline
53 & 51.31 & 51.4662502404456 & -0.15625024044558 \tabularnewline
54 & 51.34 & 51.3250489005762 & 0.0149510994238184 \tabularnewline
55 & 51.34 & 51.3465935400721 & -0.00659354007210311 \tabularnewline
56 & 51.34 & 51.3573432260545 & -0.0173432260544928 \tabularnewline
57 & 51.47 & 51.3712466662668 & 0.098753333733228 \tabularnewline
58 & 51.95 & 51.5214969306433 & 0.428503069356672 \tabularnewline
59 & 51.97 & 52.0118469840796 & -0.0418469840795694 \tabularnewline
60 & 51.92 & 52.0619487888161 & -0.141948788816116 \tabularnewline
61 & 51.92 & 52.0578582263177 & -0.137858226317697 \tabularnewline
62 & 51.91 & 52.0723966639187 & -0.162396663918699 \tabularnewline
63 & 51.97 & 52.1273273709331 & -0.157327370933139 \tabularnewline
64 & 52.14 & 52.0268880238223 & 0.113111976177706 \tabularnewline
65 & 52.33 & 52.2069557358363 & 0.123044264163681 \tabularnewline
66 & 52.4 & 52.3125480299527 & 0.0874519700472618 \tabularnewline
67 & 52.4 & 52.3799850904706 & 0.0200149095293582 \tabularnewline
68 & 52.41 & 52.4095254326108 & 0.000474567389154856 \tabularnewline
69 & 52.71 & 52.4779713904202 & 0.232028609579785 \tabularnewline
70 & 53.17 & 52.8347592809833 & 0.33524071901671 \tabularnewline
71 & 53.33 & 53.1179653548523 & 0.21203464514771 \tabularnewline
72 & 53.32 & 53.3214810443782 & -0.00148104437816698 \tabularnewline
73 & 53.32 & 53.4313092527852 & -0.111309252785155 \tabularnewline
74 & 53.3 & 53.4738667828506 & -0.173866782850602 \tabularnewline
75 & 53.31 & 53.5419980313716 & -0.231998031371624 \tabularnewline
76 & 53.72 & 53.4921683873874 & 0.227831612612562 \tabularnewline
77 & 53.87 & 53.7724861698642 & 0.0975138301357887 \tabularnewline
78 & 53.91 & 53.8647003772588 & 0.0452996227411688 \tabularnewline
79 & 53.91 & 53.8955326320105 & 0.014467367989468 \tabularnewline
80 & 53.96 & 53.9291458030961 & 0.0308541969039027 \tabularnewline
81 & 54.02 & 54.1103310232845 & -0.0903310232845342 \tabularnewline
82 & 54.33 & 54.2920518970097 & 0.0379481029902848 \tabularnewline
83 & 54.48 & 54.3328424799963 & 0.147157520003667 \tabularnewline
84 & 54.54 & 54.4218558771116 & 0.11814412288836 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232944&T=2

[TABLE]
[ROW][C]Interpolation Forecasts of Exponential Smoothing[/C][/ROW]
[ROW][C]t[/C][C]Observed[/C][C]Fitted[/C][C]Residuals[/C][/ROW]
[ROW][C]13[/C][C]48.4[/C][C]48.0974755456824[/C][C]0.302524454317584[/C][/ROW]
[ROW][C]14[/C][C]48.5[/C][C]48.4180446782984[/C][C]0.0819553217016207[/C][/ROW]
[ROW][C]15[/C][C]48.41[/C][C]48.4011353845685[/C][C]0.00886461543149863[/C][/ROW]
[ROW][C]16[/C][C]48.35[/C][C]48.3599012742241[/C][C]-0.00990127422406317[/C][/ROW]
[ROW][C]17[/C][C]48.53[/C][C]48.543152152768[/C][C]-0.0131521527679865[/C][/ROW]
[ROW][C]18[/C][C]48.52[/C][C]48.5442555140545[/C][C]-0.0242555140545235[/C][/ROW]
[ROW][C]19[/C][C]48.52[/C][C]48.6573947588819[/C][C]-0.137394758881932[/C][/ROW]
[ROW][C]20[/C][C]48.49[/C][C]48.488833468595[/C][C]0.00116653140504752[/C][/ROW]
[ROW][C]21[/C][C]48.45[/C][C]48.4482302517551[/C][C]0.00176974824494636[/C][/ROW]
[ROW][C]22[/C][C]48.65[/C][C]48.5073140957999[/C][C]0.142685904200142[/C][/ROW]
[ROW][C]23[/C][C]48.74[/C][C]48.7925170891644[/C][C]-0.0525170891644109[/C][/ROW]
[ROW][C]24[/C][C]48.74[/C][C]48.9398375998561[/C][C]-0.199837599856124[/C][/ROW]
[ROW][C]25[/C][C]48.74[/C][C]48.9100054396662[/C][C]-0.170005439666213[/C][/ROW]
[ROW][C]26[/C][C]48.79[/C][C]48.8287909475206[/C][C]-0.0387909475205603[/C][/ROW]
[ROW][C]27[/C][C]48.82[/C][C]48.6931755897683[/C][C]0.126824410231741[/C][/ROW]
[ROW][C]28[/C][C]48.82[/C][C]48.7164227720463[/C][C]0.103577227953664[/C][/ROW]
[ROW][C]29[/C][C]49.2[/C][C]48.9708624947968[/C][C]0.229137505203241[/C][/ROW]
[ROW][C]30[/C][C]49.3[/C][C]49.1327526449417[/C][C]0.167247355058294[/C][/ROW]
[ROW][C]31[/C][C]49.3[/C][C]49.3454665919582[/C][C]-0.0454665919581743[/C][/ROW]
[ROW][C]32[/C][C]49.34[/C][C]49.2889343812242[/C][C]0.051065618775759[/C][/ROW]
[ROW][C]33[/C][C]49.47[/C][C]49.2888314721187[/C][C]0.181168527881297[/C][/ROW]
[ROW][C]34[/C][C]49.65[/C][C]49.5283322243154[/C][C]0.121667775684571[/C][/ROW]
[ROW][C]35[/C][C]49.7[/C][C]49.7503745307362[/C][C]-0.0503745307361783[/C][/ROW]
[ROW][C]36[/C][C]49.75[/C][C]49.865707304539[/C][C]-0.115707304538986[/C][/ROW]
[ROW][C]37[/C][C]49.75[/C][C]49.9180481877385[/C][C]-0.168048187738535[/C][/ROW]
[ROW][C]38[/C][C]49.7[/C][C]49.8945805029529[/C][C]-0.194580502952867[/C][/ROW]
[ROW][C]39[/C][C]50.09[/C][C]49.7133706044651[/C][C]0.376629395534884[/C][/ROW]
[ROW][C]40[/C][C]50.19[/C][C]49.9131638397459[/C][C]0.276836160254085[/C][/ROW]
[ROW][C]41[/C][C]50.53[/C][C]50.3507240938616[/C][C]0.179275906138422[/C][/ROW]
[ROW][C]42[/C][C]50.55[/C][C]50.4764202523037[/C][C]0.0735797476963356[/C][/ROW]
[ROW][C]43[/C][C]50.55[/C][C]50.5740006291667[/C][C]-0.0240006291666717[/C][/ROW]
[ROW][C]44[/C][C]50.55[/C][C]50.579004159375[/C][C]-0.0290041593749777[/C][/ROW]
[ROW][C]45[/C][C]50.58[/C][C]50.5797288727598[/C][C]0.000271127240225155[/C][/ROW]
[ROW][C]46[/C][C]50.61[/C][C]50.6886693232595[/C][C]-0.0786693232595397[/C][/ROW]
[ROW][C]47[/C][C]50.94[/C][C]50.7258177103463[/C][C]0.214182289653714[/C][/ROW]
[ROW][C]48[/C][C]51.01[/C][C]51.0150286579794[/C][C]-0.00502865797945162[/C][/ROW]
[ROW][C]49[/C][C]51.01[/C][C]51.1429017810961[/C][C]-0.132901781096102[/C][/ROW]
[ROW][C]50[/C][C]51.04[/C][C]51.1517793206041[/C][C]-0.111779320604136[/C][/ROW]
[ROW][C]51[/C][C]51.15[/C][C]51.2285118747268[/C][C]-0.0785118747268214[/C][/ROW]
[ROW][C]52[/C][C]51.31[/C][C]51.0895029767421[/C][C]0.220497023257877[/C][/ROW]
[ROW][C]53[/C][C]51.31[/C][C]51.4662502404456[/C][C]-0.15625024044558[/C][/ROW]
[ROW][C]54[/C][C]51.34[/C][C]51.3250489005762[/C][C]0.0149510994238184[/C][/ROW]
[ROW][C]55[/C][C]51.34[/C][C]51.3465935400721[/C][C]-0.00659354007210311[/C][/ROW]
[ROW][C]56[/C][C]51.34[/C][C]51.3573432260545[/C][C]-0.0173432260544928[/C][/ROW]
[ROW][C]57[/C][C]51.47[/C][C]51.3712466662668[/C][C]0.098753333733228[/C][/ROW]
[ROW][C]58[/C][C]51.95[/C][C]51.5214969306433[/C][C]0.428503069356672[/C][/ROW]
[ROW][C]59[/C][C]51.97[/C][C]52.0118469840796[/C][C]-0.0418469840795694[/C][/ROW]
[ROW][C]60[/C][C]51.92[/C][C]52.0619487888161[/C][C]-0.141948788816116[/C][/ROW]
[ROW][C]61[/C][C]51.92[/C][C]52.0578582263177[/C][C]-0.137858226317697[/C][/ROW]
[ROW][C]62[/C][C]51.91[/C][C]52.0723966639187[/C][C]-0.162396663918699[/C][/ROW]
[ROW][C]63[/C][C]51.97[/C][C]52.1273273709331[/C][C]-0.157327370933139[/C][/ROW]
[ROW][C]64[/C][C]52.14[/C][C]52.0268880238223[/C][C]0.113111976177706[/C][/ROW]
[ROW][C]65[/C][C]52.33[/C][C]52.2069557358363[/C][C]0.123044264163681[/C][/ROW]
[ROW][C]66[/C][C]52.4[/C][C]52.3125480299527[/C][C]0.0874519700472618[/C][/ROW]
[ROW][C]67[/C][C]52.4[/C][C]52.3799850904706[/C][C]0.0200149095293582[/C][/ROW]
[ROW][C]68[/C][C]52.41[/C][C]52.4095254326108[/C][C]0.000474567389154856[/C][/ROW]
[ROW][C]69[/C][C]52.71[/C][C]52.4779713904202[/C][C]0.232028609579785[/C][/ROW]
[ROW][C]70[/C][C]53.17[/C][C]52.8347592809833[/C][C]0.33524071901671[/C][/ROW]
[ROW][C]71[/C][C]53.33[/C][C]53.1179653548523[/C][C]0.21203464514771[/C][/ROW]
[ROW][C]72[/C][C]53.32[/C][C]53.3214810443782[/C][C]-0.00148104437816698[/C][/ROW]
[ROW][C]73[/C][C]53.32[/C][C]53.4313092527852[/C][C]-0.111309252785155[/C][/ROW]
[ROW][C]74[/C][C]53.3[/C][C]53.4738667828506[/C][C]-0.173866782850602[/C][/ROW]
[ROW][C]75[/C][C]53.31[/C][C]53.5419980313716[/C][C]-0.231998031371624[/C][/ROW]
[ROW][C]76[/C][C]53.72[/C][C]53.4921683873874[/C][C]0.227831612612562[/C][/ROW]
[ROW][C]77[/C][C]53.87[/C][C]53.7724861698642[/C][C]0.0975138301357887[/C][/ROW]
[ROW][C]78[/C][C]53.91[/C][C]53.8647003772588[/C][C]0.0452996227411688[/C][/ROW]
[ROW][C]79[/C][C]53.91[/C][C]53.8955326320105[/C][C]0.014467367989468[/C][/ROW]
[ROW][C]80[/C][C]53.96[/C][C]53.9291458030961[/C][C]0.0308541969039027[/C][/ROW]
[ROW][C]81[/C][C]54.02[/C][C]54.1103310232845[/C][C]-0.0903310232845342[/C][/ROW]
[ROW][C]82[/C][C]54.33[/C][C]54.2920518970097[/C][C]0.0379481029902848[/C][/ROW]
[ROW][C]83[/C][C]54.48[/C][C]54.3328424799963[/C][C]0.147157520003667[/C][/ROW]
[ROW][C]84[/C][C]54.54[/C][C]54.4218558771116[/C][C]0.11814412288836[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232944&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232944&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
13	48.4	48.0974755456824	0.302524454317584
14	48.5	48.4180446782984	0.0819553217016207
15	48.41	48.4011353845685	0.00886461543149863
16	48.35	48.3599012742241	-0.00990127422406317
17	48.53	48.543152152768	-0.0131521527679865
18	48.52	48.5442555140545	-0.0242555140545235
19	48.52	48.6573947588819	-0.137394758881932
20	48.49	48.488833468595	0.00116653140504752
21	48.45	48.4482302517551	0.00176974824494636
22	48.65	48.5073140957999	0.142685904200142
23	48.74	48.7925170891644	-0.0525170891644109
24	48.74	48.9398375998561	-0.199837599856124
25	48.74	48.9100054396662	-0.170005439666213
26	48.79	48.8287909475206	-0.0387909475205603
27	48.82	48.6931755897683	0.126824410231741
28	48.82	48.7164227720463	0.103577227953664
29	49.2	48.9708624947968	0.229137505203241
30	49.3	49.1327526449417	0.167247355058294
31	49.3	49.3454665919582	-0.0454665919581743
32	49.34	49.2889343812242	0.051065618775759
33	49.47	49.2888314721187	0.181168527881297
34	49.65	49.5283322243154	0.121667775684571
35	49.7	49.7503745307362	-0.0503745307361783
36	49.75	49.865707304539	-0.115707304538986
37	49.75	49.9180481877385	-0.168048187738535
38	49.7	49.8945805029529	-0.194580502952867
39	50.09	49.7133706044651	0.376629395534884
40	50.19	49.9131638397459	0.276836160254085
41	50.53	50.3507240938616	0.179275906138422
42	50.55	50.4764202523037	0.0735797476963356
43	50.55	50.5740006291667	-0.0240006291666717
44	50.55	50.579004159375	-0.0290041593749777
45	50.58	50.5797288727598	0.000271127240225155
46	50.61	50.6886693232595	-0.0786693232595397
47	50.94	50.7258177103463	0.214182289653714
48	51.01	51.0150286579794	-0.00502865797945162
49	51.01	51.1429017810961	-0.132901781096102
50	51.04	51.1517793206041	-0.111779320604136
51	51.15	51.2285118747268	-0.0785118747268214
52	51.31	51.0895029767421	0.220497023257877
53	51.31	51.4662502404456	-0.15625024044558
54	51.34	51.3250489005762	0.0149510994238184
55	51.34	51.3465935400721	-0.00659354007210311
56	51.34	51.3573432260545	-0.0173432260544928
57	51.47	51.3712466662668	0.098753333733228
58	51.95	51.5214969306433	0.428503069356672
59	51.97	52.0118469840796	-0.0418469840795694
60	51.92	52.0619487888161	-0.141948788816116
61	51.92	52.0578582263177	-0.137858226317697
62	51.91	52.0723966639187	-0.162396663918699
63	51.97	52.1273273709331	-0.157327370933139
64	52.14	52.0268880238223	0.113111976177706
65	52.33	52.2069557358363	0.123044264163681
66	52.4	52.3125480299527	0.0874519700472618
67	52.4	52.3799850904706	0.0200149095293582
68	52.41	52.4095254326108	0.000474567389154856
69	52.71	52.4779713904202	0.232028609579785
70	53.17	52.8347592809833	0.33524071901671
71	53.33	53.1179653548523	0.21203464514771
72	53.32	53.3214810443782	-0.00148104437816698
73	53.32	53.4313092527852	-0.111309252785155
74	53.3	53.4738667828506	-0.173866782850602
75	53.31	53.5419980313716	-0.231998031371624
76	53.72	53.4921683873874	0.227831612612562
77	53.87	53.7724861698642	0.0975138301357887
78	53.91	53.8647003772588	0.0452996227411688
79	53.91	53.8955326320105	0.014467367989468
80	53.96	53.9291458030961	0.0308541969039027
81	54.02	54.1103310232845	-0.0903310232845342
82	54.33	54.2920518970097	0.0379481029902848
83	54.48	54.3328424799963	0.147157520003667
84	54.54	54.4218558771116	0.11814412288836

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
85	54.5806526128668	54.2912368349791	54.8700683907545
86	54.6847813273319	54.3303103470441	55.0392523076198
87	54.8647286857096	54.451475602028	55.2779817693911
88	55.1396736966	54.6710298597491	55.6083175334509
89	55.2329092376349	54.7124995299629	55.753318945307
90	55.2474353396609	54.6776751565196	55.8171955228021
91	55.2414155238666	54.6238486341467	55.8589824135864
92	55.2748098915687	54.6100781996741	55.9395415834632
93	55.4023545123537	54.6901822143941	56.1145268103133
94	55.6989112141353	54.9377806612397	56.4600417670309
95	55.753919538432	54.9470308062502	56.5608082706139
96	55.7334497726404	52.714314854495	58.7525846907859

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
85 & 54.5806526128668 & 54.2912368349791 & 54.8700683907545 \tabularnewline
86 & 54.6847813273319 & 54.3303103470441 & 55.0392523076198 \tabularnewline
87 & 54.8647286857096 & 54.451475602028 & 55.2779817693911 \tabularnewline
88 & 55.1396736966 & 54.6710298597491 & 55.6083175334509 \tabularnewline
89 & 55.2329092376349 & 54.7124995299629 & 55.753318945307 \tabularnewline
90 & 55.2474353396609 & 54.6776751565196 & 55.8171955228021 \tabularnewline
91 & 55.2414155238666 & 54.6238486341467 & 55.8589824135864 \tabularnewline
92 & 55.2748098915687 & 54.6100781996741 & 55.9395415834632 \tabularnewline
93 & 55.4023545123537 & 54.6901822143941 & 56.1145268103133 \tabularnewline
94 & 55.6989112141353 & 54.9377806612397 & 56.4600417670309 \tabularnewline
95 & 55.753919538432 & 54.9470308062502 & 56.5608082706139 \tabularnewline
96 & 55.7334497726404 & 52.714314854495 & 58.7525846907859 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232944&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]54.5806526128668[/C][C]54.2912368349791[/C][C]54.8700683907545[/C][/ROW]
[ROW][C]86[/C][C]54.6847813273319[/C][C]54.3303103470441[/C][C]55.0392523076198[/C][/ROW]
[ROW][C]87[/C][C]54.8647286857096[/C][C]54.451475602028[/C][C]55.2779817693911[/C][/ROW]
[ROW][C]88[/C][C]55.1396736966[/C][C]54.6710298597491[/C][C]55.6083175334509[/C][/ROW]
[ROW][C]89[/C][C]55.2329092376349[/C][C]54.7124995299629[/C][C]55.753318945307[/C][/ROW]
[ROW][C]90[/C][C]55.2474353396609[/C][C]54.6776751565196[/C][C]55.8171955228021[/C][/ROW]
[ROW][C]91[/C][C]55.2414155238666[/C][C]54.6238486341467[/C][C]55.8589824135864[/C][/ROW]
[ROW][C]92[/C][C]55.2748098915687[/C][C]54.6100781996741[/C][C]55.9395415834632[/C][/ROW]
[ROW][C]93[/C][C]55.4023545123537[/C][C]54.6901822143941[/C][C]56.1145268103133[/C][/ROW]
[ROW][C]94[/C][C]55.6989112141353[/C][C]54.9377806612397[/C][C]56.4600417670309[/C][/ROW]
[ROW][C]95[/C][C]55.753919538432[/C][C]54.9470308062502[/C][C]56.5608082706139[/C][/ROW]
[ROW][C]96[/C][C]55.7334497726404[/C][C]52.714314854495[/C][C]58.7525846907859[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232944&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232944&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	54.5806526128668	54.2912368349791	54.8700683907545
86	54.6847813273319	54.3303103470441	55.0392523076198
87	54.8647286857096	54.451475602028	55.2779817693911
88	55.1396736966	54.6710298597491	55.6083175334509
89	55.2329092376349	54.7124995299629	55.753318945307
90	55.2474353396609	54.6776751565196	55.8171955228021
91	55.2414155238666	54.6238486341467	55.8589824135864
92	55.2748098915687	54.6100781996741	55.9395415834632
93	55.4023545123537	54.6901822143941	56.1145268103133
94	55.6989112141353	54.9377806612397	56.4600417670309
95	55.753919538432	54.9470308062502	56.5608082706139
96	55.7334497726404	52.714314854495	58.7525846907859

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 = Triple ; par3 = multiplicative ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; par3 = multiplicative ;

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