Repository of Reproducible Computations

Free Statistics

of Irreproducible Research!

Author's title

Author

*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Sun, 01 Jun 2008 07:40:54 -0600

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/2008/Jun/01/t12123277331za3i6d3wcomwlu.htm/, Retrieved Wed, 11 Feb 2026 00:30:36 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=13679, Retrieved Wed, 11 Feb 2026 00:30:36 +0000

QR Codes:

Paste this QR Code to cite your computation.

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords

Estimated Impact

531

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

-       [Exponential Smoothing] [Exponentail smoot...] [2008-06-01 13:40:54] [d41d8cd98f00b204e9800998ecf8427e] [Current]

Feedback Forum

Post a new message

Dataseries X:

Download CSV

Histogram

Boxplots

4,43
4,43
4,44
4,44
4,44
4,45
4,47
4,48
4,48
4,5
4,52
4,52
4,53
4,53
4,63
4,66
4,67
4,68
4,69
4,69
4,7
4,71
4,72
4,72
4,72
4,73
4,74
4,76
4,81
4,82
4,83
4,83
4,84
4,89
4,92
4,95
4,95
5,01
5,05
5,08
5,11
5,14
5,17
5,18
5,2
5,22
5,24
5,28
5,29
5,33
5,4
5,43
5,46
5,46
5,46
5,47
5,49
5,5
5,54
5,55
5,55
5,56
5,6
5,61
5,63
5,64
5,66
5,67
5,69
5,77
5,77
5,78
5,8
5,82
5,85
5,87
5,88
5,9
5,91
5,94
5,97
5,98
6
6,01
6,02

Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	5 seconds
R Server	'Herman Ole Andreas Wold' @ 193.190.124.10:1001

\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 & 5 seconds \tabularnewline
R Server & 'Herman Ole Andreas Wold' @ 193.190.124.10:1001 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=13679&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]5 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Herman Ole Andreas Wold' @ 193.190.124.10:1001[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=13679&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=13679&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	5 seconds
R Server	'Herman Ole Andreas Wold' @ 193.190.124.10:1001

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.686614575696178
beta	0.0358454934700404
gamma	1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.686614575696178 \tabularnewline
beta & 0.0358454934700404 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=13679&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]0.686614575696178[/C][/ROW]
[ROW][C]beta[/C][C]0.0358454934700404[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=13679&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=13679&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	0.686614575696178
beta	0.0358454934700404
gamma	1

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	4.53	4.43497253424839	0.0950274657516115
14	4.53	4.49969420656947	0.0303057934305313
15	4.63	4.62043652430831	0.00956347569169314
16	4.66	4.65733725550416	0.00266274449583559
17	4.67	4.67041323329498	-0.000413233294983328
18	4.68	4.68178288945708	-0.00178288945708438
19	4.69	4.69258405979525	-0.00258405979524579
20	4.69	4.69276267518553	-0.00276267518552942
21	4.7	4.70608656021938	-0.00608656021937559
22	4.71	4.72115041863392	-0.0111504186339193
23	4.72	4.73121151664716	-0.0112115166471591
24	4.72	4.72926397189902	-0.00926397189901795
25	4.72	4.74322464567819	-0.0232246456781917
26	4.73	4.70414731942536	0.0258526805746389
27	4.74	4.81776722251646	-0.0777672225164565
28	4.76	4.78992051151631	-0.0299205115163144
29	4.81	4.77576818901657	0.0342318109834272
30	4.82	4.80745383907355	0.0125461609264512
31	4.83	4.82516389657769	0.00483610342231255
32	4.83	4.82760246919499	0.00239753080501082
33	4.84	4.84114082662566	-0.00114082662566428
34	4.89	4.85595198394663	0.0340480160533714
35	4.92	4.89608776863383	0.0239122313661735
36	4.95	4.91834370470068	0.0316562952993156
37	4.95	4.95688087403988	-0.00688087403987847
38	5.01	4.94451247598575	0.0654875240142481
39	5.05	5.05744208713877	-0.00744208713877459
40	5.08	5.09855530081831	-0.0185553008183108
41	5.11	5.11741625263689	-0.00741625263689105
42	5.14	5.11613306159569	0.0238669384043142
43	5.17	5.14219806131586	0.0278019386841359
44	5.18	5.16258046388381	0.0174195361161900
45	5.2	5.18946415994598	0.0105358400540219
46	5.22	5.22885782567948	-0.00885782567948201
47	5.24	5.23985876271087	0.000141237289129847
48	5.28	5.25073755444733	0.0292624455526678
49	5.29	5.27777215560207	0.0122278443979278
50	5.33	5.304365128821	0.0256348711790002
51	5.4	5.37124064663802	0.028759353361977
52	5.43	5.43876597642108	-0.00876597642108301
53	5.46	5.47268372609452	-0.0126837260945214
54	5.46	5.48079886887492	-0.0207988688749188
55	5.46	5.47935076906593	-0.0193507690659267
56	5.47	5.4641615070581	0.00583849294190397
57	5.49	5.48154856751204	0.00845143248795921
58	5.5	5.51471987397711	-0.0147198739771062
59	5.54	5.52533266134113	0.0146673386588736
60	5.55	5.55644755958811	-0.00644755958811327
61	5.55	5.55293705142872	-0.00293705142872369
62	5.56	5.57329014389258	-0.0132901438925819
63	5.6	5.61460454049775	-0.0146045404977500
64	5.61	5.63900210036783	-0.0290021003678271
65	5.63	5.65575402171078	-0.0257540217107826
66	5.64	5.6491591406135	-0.00915914061350254
67	5.66	5.65323440948609	0.00676559051390768
68	5.67	5.66134137391522	0.00865862608477563
69	5.69	5.6793189664412	0.0106810335587983
70	5.77	5.7048506699643	0.0651493300357009
71	5.77	5.7800140278497	-0.0100140278497012
72	5.78	5.78677245691401	-0.00677245691401218
73	5.8	5.78283388605958	0.0171661139404202
74	5.82	5.81364219221143	0.0063578077885742
75	5.85	5.86985498492767	-0.0198549849276661
76	5.87	5.88687706850462	-0.0168770685046233
77	5.88	5.91443214609609	-0.0344321460960897
78	5.9	5.90736974328775	-0.00736974328775375
79	5.91	5.91796678378416	-0.00796678378416171
80	5.94	5.91598839847011	0.0240116015298897
81	5.97	5.94531779416014	0.0246822058398593
82	5.98	5.99894912596645	-0.0189491259664516
83	6	5.99104085165187	0.0089591483481275
84	6.01	6.0108270811518	-0.000827081151795639
85	6.02	6.0173444149626	0.00265558503740237

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 4.53 & 4.43497253424839 & 0.0950274657516115 \tabularnewline
14 & 4.53 & 4.49969420656947 & 0.0303057934305313 \tabularnewline
15 & 4.63 & 4.62043652430831 & 0.00956347569169314 \tabularnewline
16 & 4.66 & 4.65733725550416 & 0.00266274449583559 \tabularnewline
17 & 4.67 & 4.67041323329498 & -0.000413233294983328 \tabularnewline
18 & 4.68 & 4.68178288945708 & -0.00178288945708438 \tabularnewline
19 & 4.69 & 4.69258405979525 & -0.00258405979524579 \tabularnewline
20 & 4.69 & 4.69276267518553 & -0.00276267518552942 \tabularnewline
21 & 4.7 & 4.70608656021938 & -0.00608656021937559 \tabularnewline
22 & 4.71 & 4.72115041863392 & -0.0111504186339193 \tabularnewline
23 & 4.72 & 4.73121151664716 & -0.0112115166471591 \tabularnewline
24 & 4.72 & 4.72926397189902 & -0.00926397189901795 \tabularnewline
25 & 4.72 & 4.74322464567819 & -0.0232246456781917 \tabularnewline
26 & 4.73 & 4.70414731942536 & 0.0258526805746389 \tabularnewline
27 & 4.74 & 4.81776722251646 & -0.0777672225164565 \tabularnewline
28 & 4.76 & 4.78992051151631 & -0.0299205115163144 \tabularnewline
29 & 4.81 & 4.77576818901657 & 0.0342318109834272 \tabularnewline
30 & 4.82 & 4.80745383907355 & 0.0125461609264512 \tabularnewline
31 & 4.83 & 4.82516389657769 & 0.00483610342231255 \tabularnewline
32 & 4.83 & 4.82760246919499 & 0.00239753080501082 \tabularnewline
33 & 4.84 & 4.84114082662566 & -0.00114082662566428 \tabularnewline
34 & 4.89 & 4.85595198394663 & 0.0340480160533714 \tabularnewline
35 & 4.92 & 4.89608776863383 & 0.0239122313661735 \tabularnewline
36 & 4.95 & 4.91834370470068 & 0.0316562952993156 \tabularnewline
37 & 4.95 & 4.95688087403988 & -0.00688087403987847 \tabularnewline
38 & 5.01 & 4.94451247598575 & 0.0654875240142481 \tabularnewline
39 & 5.05 & 5.05744208713877 & -0.00744208713877459 \tabularnewline
40 & 5.08 & 5.09855530081831 & -0.0185553008183108 \tabularnewline
41 & 5.11 & 5.11741625263689 & -0.00741625263689105 \tabularnewline
42 & 5.14 & 5.11613306159569 & 0.0238669384043142 \tabularnewline
43 & 5.17 & 5.14219806131586 & 0.0278019386841359 \tabularnewline
44 & 5.18 & 5.16258046388381 & 0.0174195361161900 \tabularnewline
45 & 5.2 & 5.18946415994598 & 0.0105358400540219 \tabularnewline
46 & 5.22 & 5.22885782567948 & -0.00885782567948201 \tabularnewline
47 & 5.24 & 5.23985876271087 & 0.000141237289129847 \tabularnewline
48 & 5.28 & 5.25073755444733 & 0.0292624455526678 \tabularnewline
49 & 5.29 & 5.27777215560207 & 0.0122278443979278 \tabularnewline
50 & 5.33 & 5.304365128821 & 0.0256348711790002 \tabularnewline
51 & 5.4 & 5.37124064663802 & 0.028759353361977 \tabularnewline
52 & 5.43 & 5.43876597642108 & -0.00876597642108301 \tabularnewline
53 & 5.46 & 5.47268372609452 & -0.0126837260945214 \tabularnewline
54 & 5.46 & 5.48079886887492 & -0.0207988688749188 \tabularnewline
55 & 5.46 & 5.47935076906593 & -0.0193507690659267 \tabularnewline
56 & 5.47 & 5.4641615070581 & 0.00583849294190397 \tabularnewline
57 & 5.49 & 5.48154856751204 & 0.00845143248795921 \tabularnewline
58 & 5.5 & 5.51471987397711 & -0.0147198739771062 \tabularnewline
59 & 5.54 & 5.52533266134113 & 0.0146673386588736 \tabularnewline
60 & 5.55 & 5.55644755958811 & -0.00644755958811327 \tabularnewline
61 & 5.55 & 5.55293705142872 & -0.00293705142872369 \tabularnewline
62 & 5.56 & 5.57329014389258 & -0.0132901438925819 \tabularnewline
63 & 5.6 & 5.61460454049775 & -0.0146045404977500 \tabularnewline
64 & 5.61 & 5.63900210036783 & -0.0290021003678271 \tabularnewline
65 & 5.63 & 5.65575402171078 & -0.0257540217107826 \tabularnewline
66 & 5.64 & 5.6491591406135 & -0.00915914061350254 \tabularnewline
67 & 5.66 & 5.65323440948609 & 0.00676559051390768 \tabularnewline
68 & 5.67 & 5.66134137391522 & 0.00865862608477563 \tabularnewline
69 & 5.69 & 5.6793189664412 & 0.0106810335587983 \tabularnewline
70 & 5.77 & 5.7048506699643 & 0.0651493300357009 \tabularnewline
71 & 5.77 & 5.7800140278497 & -0.0100140278497012 \tabularnewline
72 & 5.78 & 5.78677245691401 & -0.00677245691401218 \tabularnewline
73 & 5.8 & 5.78283388605958 & 0.0171661139404202 \tabularnewline
74 & 5.82 & 5.81364219221143 & 0.0063578077885742 \tabularnewline
75 & 5.85 & 5.86985498492767 & -0.0198549849276661 \tabularnewline
76 & 5.87 & 5.88687706850462 & -0.0168770685046233 \tabularnewline
77 & 5.88 & 5.91443214609609 & -0.0344321460960897 \tabularnewline
78 & 5.9 & 5.90736974328775 & -0.00736974328775375 \tabularnewline
79 & 5.91 & 5.91796678378416 & -0.00796678378416171 \tabularnewline
80 & 5.94 & 5.91598839847011 & 0.0240116015298897 \tabularnewline
81 & 5.97 & 5.94531779416014 & 0.0246822058398593 \tabularnewline
82 & 5.98 & 5.99894912596645 & -0.0189491259664516 \tabularnewline
83 & 6 & 5.99104085165187 & 0.0089591483481275 \tabularnewline
84 & 6.01 & 6.0108270811518 & -0.000827081151795639 \tabularnewline
85 & 6.02 & 6.0173444149626 & 0.00265558503740237 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=13679&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]4.53[/C][C]4.43497253424839[/C][C]0.0950274657516115[/C][/ROW]
[ROW][C]14[/C][C]4.53[/C][C]4.49969420656947[/C][C]0.0303057934305313[/C][/ROW]
[ROW][C]15[/C][C]4.63[/C][C]4.62043652430831[/C][C]0.00956347569169314[/C][/ROW]
[ROW][C]16[/C][C]4.66[/C][C]4.65733725550416[/C][C]0.00266274449583559[/C][/ROW]
[ROW][C]17[/C][C]4.67[/C][C]4.67041323329498[/C][C]-0.000413233294983328[/C][/ROW]
[ROW][C]18[/C][C]4.68[/C][C]4.68178288945708[/C][C]-0.00178288945708438[/C][/ROW]
[ROW][C]19[/C][C]4.69[/C][C]4.69258405979525[/C][C]-0.00258405979524579[/C][/ROW]
[ROW][C]20[/C][C]4.69[/C][C]4.69276267518553[/C][C]-0.00276267518552942[/C][/ROW]
[ROW][C]21[/C][C]4.7[/C][C]4.70608656021938[/C][C]-0.00608656021937559[/C][/ROW]
[ROW][C]22[/C][C]4.71[/C][C]4.72115041863392[/C][C]-0.0111504186339193[/C][/ROW]
[ROW][C]23[/C][C]4.72[/C][C]4.73121151664716[/C][C]-0.0112115166471591[/C][/ROW]
[ROW][C]24[/C][C]4.72[/C][C]4.72926397189902[/C][C]-0.00926397189901795[/C][/ROW]
[ROW][C]25[/C][C]4.72[/C][C]4.74322464567819[/C][C]-0.0232246456781917[/C][/ROW]
[ROW][C]26[/C][C]4.73[/C][C]4.70414731942536[/C][C]0.0258526805746389[/C][/ROW]
[ROW][C]27[/C][C]4.74[/C][C]4.81776722251646[/C][C]-0.0777672225164565[/C][/ROW]
[ROW][C]28[/C][C]4.76[/C][C]4.78992051151631[/C][C]-0.0299205115163144[/C][/ROW]
[ROW][C]29[/C][C]4.81[/C][C]4.77576818901657[/C][C]0.0342318109834272[/C][/ROW]
[ROW][C]30[/C][C]4.82[/C][C]4.80745383907355[/C][C]0.0125461609264512[/C][/ROW]
[ROW][C]31[/C][C]4.83[/C][C]4.82516389657769[/C][C]0.00483610342231255[/C][/ROW]
[ROW][C]32[/C][C]4.83[/C][C]4.82760246919499[/C][C]0.00239753080501082[/C][/ROW]
[ROW][C]33[/C][C]4.84[/C][C]4.84114082662566[/C][C]-0.00114082662566428[/C][/ROW]
[ROW][C]34[/C][C]4.89[/C][C]4.85595198394663[/C][C]0.0340480160533714[/C][/ROW]
[ROW][C]35[/C][C]4.92[/C][C]4.89608776863383[/C][C]0.0239122313661735[/C][/ROW]
[ROW][C]36[/C][C]4.95[/C][C]4.91834370470068[/C][C]0.0316562952993156[/C][/ROW]
[ROW][C]37[/C][C]4.95[/C][C]4.95688087403988[/C][C]-0.00688087403987847[/C][/ROW]
[ROW][C]38[/C][C]5.01[/C][C]4.94451247598575[/C][C]0.0654875240142481[/C][/ROW]
[ROW][C]39[/C][C]5.05[/C][C]5.05744208713877[/C][C]-0.00744208713877459[/C][/ROW]
[ROW][C]40[/C][C]5.08[/C][C]5.09855530081831[/C][C]-0.0185553008183108[/C][/ROW]
[ROW][C]41[/C][C]5.11[/C][C]5.11741625263689[/C][C]-0.00741625263689105[/C][/ROW]
[ROW][C]42[/C][C]5.14[/C][C]5.11613306159569[/C][C]0.0238669384043142[/C][/ROW]
[ROW][C]43[/C][C]5.17[/C][C]5.14219806131586[/C][C]0.0278019386841359[/C][/ROW]
[ROW][C]44[/C][C]5.18[/C][C]5.16258046388381[/C][C]0.0174195361161900[/C][/ROW]
[ROW][C]45[/C][C]5.2[/C][C]5.18946415994598[/C][C]0.0105358400540219[/C][/ROW]
[ROW][C]46[/C][C]5.22[/C][C]5.22885782567948[/C][C]-0.00885782567948201[/C][/ROW]
[ROW][C]47[/C][C]5.24[/C][C]5.23985876271087[/C][C]0.000141237289129847[/C][/ROW]
[ROW][C]48[/C][C]5.28[/C][C]5.25073755444733[/C][C]0.0292624455526678[/C][/ROW]
[ROW][C]49[/C][C]5.29[/C][C]5.27777215560207[/C][C]0.0122278443979278[/C][/ROW]
[ROW][C]50[/C][C]5.33[/C][C]5.304365128821[/C][C]0.0256348711790002[/C][/ROW]
[ROW][C]51[/C][C]5.4[/C][C]5.37124064663802[/C][C]0.028759353361977[/C][/ROW]
[ROW][C]52[/C][C]5.43[/C][C]5.43876597642108[/C][C]-0.00876597642108301[/C][/ROW]
[ROW][C]53[/C][C]5.46[/C][C]5.47268372609452[/C][C]-0.0126837260945214[/C][/ROW]
[ROW][C]54[/C][C]5.46[/C][C]5.48079886887492[/C][C]-0.0207988688749188[/C][/ROW]
[ROW][C]55[/C][C]5.46[/C][C]5.47935076906593[/C][C]-0.0193507690659267[/C][/ROW]
[ROW][C]56[/C][C]5.47[/C][C]5.4641615070581[/C][C]0.00583849294190397[/C][/ROW]
[ROW][C]57[/C][C]5.49[/C][C]5.48154856751204[/C][C]0.00845143248795921[/C][/ROW]
[ROW][C]58[/C][C]5.5[/C][C]5.51471987397711[/C][C]-0.0147198739771062[/C][/ROW]
[ROW][C]59[/C][C]5.54[/C][C]5.52533266134113[/C][C]0.0146673386588736[/C][/ROW]
[ROW][C]60[/C][C]5.55[/C][C]5.55644755958811[/C][C]-0.00644755958811327[/C][/ROW]
[ROW][C]61[/C][C]5.55[/C][C]5.55293705142872[/C][C]-0.00293705142872369[/C][/ROW]
[ROW][C]62[/C][C]5.56[/C][C]5.57329014389258[/C][C]-0.0132901438925819[/C][/ROW]
[ROW][C]63[/C][C]5.6[/C][C]5.61460454049775[/C][C]-0.0146045404977500[/C][/ROW]
[ROW][C]64[/C][C]5.61[/C][C]5.63900210036783[/C][C]-0.0290021003678271[/C][/ROW]
[ROW][C]65[/C][C]5.63[/C][C]5.65575402171078[/C][C]-0.0257540217107826[/C][/ROW]
[ROW][C]66[/C][C]5.64[/C][C]5.6491591406135[/C][C]-0.00915914061350254[/C][/ROW]
[ROW][C]67[/C][C]5.66[/C][C]5.65323440948609[/C][C]0.00676559051390768[/C][/ROW]
[ROW][C]68[/C][C]5.67[/C][C]5.66134137391522[/C][C]0.00865862608477563[/C][/ROW]
[ROW][C]69[/C][C]5.69[/C][C]5.6793189664412[/C][C]0.0106810335587983[/C][/ROW]
[ROW][C]70[/C][C]5.77[/C][C]5.7048506699643[/C][C]0.0651493300357009[/C][/ROW]
[ROW][C]71[/C][C]5.77[/C][C]5.7800140278497[/C][C]-0.0100140278497012[/C][/ROW]
[ROW][C]72[/C][C]5.78[/C][C]5.78677245691401[/C][C]-0.00677245691401218[/C][/ROW]
[ROW][C]73[/C][C]5.8[/C][C]5.78283388605958[/C][C]0.0171661139404202[/C][/ROW]
[ROW][C]74[/C][C]5.82[/C][C]5.81364219221143[/C][C]0.0063578077885742[/C][/ROW]
[ROW][C]75[/C][C]5.85[/C][C]5.86985498492767[/C][C]-0.0198549849276661[/C][/ROW]
[ROW][C]76[/C][C]5.87[/C][C]5.88687706850462[/C][C]-0.0168770685046233[/C][/ROW]
[ROW][C]77[/C][C]5.88[/C][C]5.91443214609609[/C][C]-0.0344321460960897[/C][/ROW]
[ROW][C]78[/C][C]5.9[/C][C]5.90736974328775[/C][C]-0.00736974328775375[/C][/ROW]
[ROW][C]79[/C][C]5.91[/C][C]5.91796678378416[/C][C]-0.00796678378416171[/C][/ROW]
[ROW][C]80[/C][C]5.94[/C][C]5.91598839847011[/C][C]0.0240116015298897[/C][/ROW]
[ROW][C]81[/C][C]5.97[/C][C]5.94531779416014[/C][C]0.0246822058398593[/C][/ROW]
[ROW][C]82[/C][C]5.98[/C][C]5.99894912596645[/C][C]-0.0189491259664516[/C][/ROW]
[ROW][C]83[/C][C]6[/C][C]5.99104085165187[/C][C]0.0089591483481275[/C][/ROW]
[ROW][C]84[/C][C]6.01[/C][C]6.0108270811518[/C][C]-0.000827081151795639[/C][/ROW]
[ROW][C]85[/C][C]6.02[/C][C]6.0173444149626[/C][C]0.00265558503740237[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=13679&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=13679&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	4.53	4.43497253424839	0.0950274657516115
14	4.53	4.49969420656947	0.0303057934305313
15	4.63	4.62043652430831	0.00956347569169314
16	4.66	4.65733725550416	0.00266274449583559
17	4.67	4.67041323329498	-0.000413233294983328
18	4.68	4.68178288945708	-0.00178288945708438
19	4.69	4.69258405979525	-0.00258405979524579
20	4.69	4.69276267518553	-0.00276267518552942
21	4.7	4.70608656021938	-0.00608656021937559
22	4.71	4.72115041863392	-0.0111504186339193
23	4.72	4.73121151664716	-0.0112115166471591
24	4.72	4.72926397189902	-0.00926397189901795
25	4.72	4.74322464567819	-0.0232246456781917
26	4.73	4.70414731942536	0.0258526805746389
27	4.74	4.81776722251646	-0.0777672225164565
28	4.76	4.78992051151631	-0.0299205115163144
29	4.81	4.77576818901657	0.0342318109834272
30	4.82	4.80745383907355	0.0125461609264512
31	4.83	4.82516389657769	0.00483610342231255
32	4.83	4.82760246919499	0.00239753080501082
33	4.84	4.84114082662566	-0.00114082662566428
34	4.89	4.85595198394663	0.0340480160533714
35	4.92	4.89608776863383	0.0239122313661735
36	4.95	4.91834370470068	0.0316562952993156
37	4.95	4.95688087403988	-0.00688087403987847
38	5.01	4.94451247598575	0.0654875240142481
39	5.05	5.05744208713877	-0.00744208713877459
40	5.08	5.09855530081831	-0.0185553008183108
41	5.11	5.11741625263689	-0.00741625263689105
42	5.14	5.11613306159569	0.0238669384043142
43	5.17	5.14219806131586	0.0278019386841359
44	5.18	5.16258046388381	0.0174195361161900
45	5.2	5.18946415994598	0.0105358400540219
46	5.22	5.22885782567948	-0.00885782567948201
47	5.24	5.23985876271087	0.000141237289129847
48	5.28	5.25073755444733	0.0292624455526678
49	5.29	5.27777215560207	0.0122278443979278
50	5.33	5.304365128821	0.0256348711790002
51	5.4	5.37124064663802	0.028759353361977
52	5.43	5.43876597642108	-0.00876597642108301
53	5.46	5.47268372609452	-0.0126837260945214
54	5.46	5.48079886887492	-0.0207988688749188
55	5.46	5.47935076906593	-0.0193507690659267
56	5.47	5.4641615070581	0.00583849294190397
57	5.49	5.48154856751204	0.00845143248795921
58	5.5	5.51471987397711	-0.0147198739771062
59	5.54	5.52533266134113	0.0146673386588736
60	5.55	5.55644755958811	-0.00644755958811327
61	5.55	5.55293705142872	-0.00293705142872369
62	5.56	5.57329014389258	-0.0132901438925819
63	5.6	5.61460454049775	-0.0146045404977500
64	5.61	5.63900210036783	-0.0290021003678271
65	5.63	5.65575402171078	-0.0257540217107826
66	5.64	5.6491591406135	-0.00915914061350254
67	5.66	5.65323440948609	0.00676559051390768
68	5.67	5.66134137391522	0.00865862608477563
69	5.69	5.6793189664412	0.0106810335587983
70	5.77	5.7048506699643	0.0651493300357009
71	5.77	5.7800140278497	-0.0100140278497012
72	5.78	5.78677245691401	-0.00677245691401218
73	5.8	5.78283388605958	0.0171661139404202
74	5.82	5.81364219221143	0.0063578077885742
75	5.85	5.86985498492767	-0.0198549849276661
76	5.87	5.88687706850462	-0.0168770685046233
77	5.88	5.91443214609609	-0.0344321460960897
78	5.9	5.90736974328775	-0.00736974328775375
79	5.91	5.91796678378416	-0.00796678378416171
80	5.94	5.91598839847011	0.0240116015298897
81	5.97	5.94531779416014	0.0246822058398593
82	5.98	5.99894912596645	-0.0189491259664516
83	6	5.99104085165187	0.0089591483481275
84	6.01	6.0108270811518	-0.000827081151795639
85	6.02	6.0173444149626	0.00265558503740237

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
86	6.0336189737387	5.98556034061836	6.08167760685904
87	6.0769192576933	6.01787041815344	6.13596809723316
88	6.10826210684619	6.03936690523253	6.17715730845985
89	6.14215582659063	6.06405461868499	6.22025703449627
90	6.16806290881736	6.08126095064475	6.25486486698997
91	6.18413775367375	6.08904946722908	6.27922604011843
92	6.19834637286976	6.09522235252433	6.30147039321519
93	6.21147424919997	6.1004981903968	6.32245030800313
94	6.23436081234307	6.11550292746226	6.35321869722388
95	6.24819683261748	6.1216949653809	6.37469869985406
96	6.25839865906047	6.12438819753698	6.39240912058397
97	6.26613330853258	3.93193264423224	8.60033397283292

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
86 & 6.0336189737387 & 5.98556034061836 & 6.08167760685904 \tabularnewline
87 & 6.0769192576933 & 6.01787041815344 & 6.13596809723316 \tabularnewline
88 & 6.10826210684619 & 6.03936690523253 & 6.17715730845985 \tabularnewline
89 & 6.14215582659063 & 6.06405461868499 & 6.22025703449627 \tabularnewline
90 & 6.16806290881736 & 6.08126095064475 & 6.25486486698997 \tabularnewline
91 & 6.18413775367375 & 6.08904946722908 & 6.27922604011843 \tabularnewline
92 & 6.19834637286976 & 6.09522235252433 & 6.30147039321519 \tabularnewline
93 & 6.21147424919997 & 6.1004981903968 & 6.32245030800313 \tabularnewline
94 & 6.23436081234307 & 6.11550292746226 & 6.35321869722388 \tabularnewline
95 & 6.24819683261748 & 6.1216949653809 & 6.37469869985406 \tabularnewline
96 & 6.25839865906047 & 6.12438819753698 & 6.39240912058397 \tabularnewline
97 & 6.26613330853258 & 3.93193264423224 & 8.60033397283292 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=13679&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]86[/C][C]6.0336189737387[/C][C]5.98556034061836[/C][C]6.08167760685904[/C][/ROW]
[ROW][C]87[/C][C]6.0769192576933[/C][C]6.01787041815344[/C][C]6.13596809723316[/C][/ROW]
[ROW][C]88[/C][C]6.10826210684619[/C][C]6.03936690523253[/C][C]6.17715730845985[/C][/ROW]
[ROW][C]89[/C][C]6.14215582659063[/C][C]6.06405461868499[/C][C]6.22025703449627[/C][/ROW]
[ROW][C]90[/C][C]6.16806290881736[/C][C]6.08126095064475[/C][C]6.25486486698997[/C][/ROW]
[ROW][C]91[/C][C]6.18413775367375[/C][C]6.08904946722908[/C][C]6.27922604011843[/C][/ROW]
[ROW][C]92[/C][C]6.19834637286976[/C][C]6.09522235252433[/C][C]6.30147039321519[/C][/ROW]
[ROW][C]93[/C][C]6.21147424919997[/C][C]6.1004981903968[/C][C]6.32245030800313[/C][/ROW]
[ROW][C]94[/C][C]6.23436081234307[/C][C]6.11550292746226[/C][C]6.35321869722388[/C][/ROW]
[ROW][C]95[/C][C]6.24819683261748[/C][C]6.1216949653809[/C][C]6.37469869985406[/C][/ROW]
[ROW][C]96[/C][C]6.25839865906047[/C][C]6.12438819753698[/C][C]6.39240912058397[/C][/ROW]
[ROW][C]97[/C][C]6.26613330853258[/C][C]3.93193264423224[/C][C]8.60033397283292[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=13679&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=13679&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
86	6.0336189737387	5.98556034061836	6.08167760685904
87	6.0769192576933	6.01787041815344	6.13596809723316
88	6.10826210684619	6.03936690523253	6.17715730845985
89	6.14215582659063	6.06405461868499	6.22025703449627
90	6.16806290881736	6.08126095064475	6.25486486698997
91	6.18413775367375	6.08904946722908	6.27922604011843
92	6.19834637286976	6.09522235252433	6.30147039321519
93	6.21147424919997	6.1004981903968	6.32245030800313
94	6.23436081234307	6.11550292746226	6.35321869722388
95	6.24819683261748	6.1216949653809	6.37469869985406
96	6.25839865906047	6.12438819753698	6.39240912058397
97	6.26613330853258	3.93193264423224	8.60033397283292

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=0, beta=0)
if (par2 == 'Double') fit <- HoltWinters(x, gamma=0)
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