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*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Thu, 12 Dec 2013 09:47:43 -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/2013/Dec/12/t1386860062knqeufxu1fq62az.htm/, Retrieved Wed, 13 May 2026 05:03:55 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=232251, Retrieved Wed, 13 May 2026 05:03:55 +0000

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IsPrivate?

No (this computation is public)

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Estimated Impact

452

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

-       [Exponential Smoothing] [] [2013-12-12 14:47:43] [6bddb01df32adf96be73fb2a4c89eedb] [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	'Gertrude Mary Cox' @ cox.wessa.net

\begin{tabular}{lllllllll}
\hline
Summary of computational transaction \tabularnewline
Raw Input & view raw input (R code)  \tabularnewline
Raw Output & view raw output of R engine  \tabularnewline
Computing time & 4 seconds \tabularnewline
R Server & 'Gertrude Mary Cox' @ cox.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232251&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]'Gertrude Mary Cox' @ cox.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232251&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232251&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	'Gertrude Mary Cox' @ cox.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	1
beta	0.0718071265959284
gamma	FALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 1 \tabularnewline
beta & 0.0718071265959284 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232251&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]0.0718071265959284[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232251&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232251&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	0.0718071265959284
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
3	16.38	16.44	-0.0600000000000023
4	16.37	16.4456915724042	-0.0756915724042422
5	16.42	16.4302563780824	-0.0102563780823672
6	16.43	16.479519897043	-0.0495198970429946
7	16.44	16.485964015527	-0.0459640155270051
8	16.53	16.4926634716452	0.0373365283547962
9	16.55	16.5853445004634	-0.0353445004634274
10	16.56	16.6028065134442	-0.0428065134441837
11	16.6	16.6097327007142	-0.0097327007141601
12	16.61	16.6490338234419	-0.0390338234418657
13	16.62	16.6562309167404	-0.0362309167404469
14	16.64	16.6636292787154	-0.0236292787153829
15	16.61	16.6819325281073	-0.0719325281072969
16	16.74	16.6467672599551	0.0932327400448685
17	16.87	16.7834620351224	0.0865379648775857
18	16.89	16.9196760777217	-0.0296760777217351
19	16.89	16.9375451238519	-0.0475451238519007
20	16.99	16.9341310451244	0.0558689548755495
21	17.06	17.03814283424	0.0218571657600215
22	17.1	17.1097123345087	-0.00971233450873399
23	17.11	17.1490149196751	-0.0390149196751288
24	17.17	17.1562133703989	0.0137866296011175
25	17.17	17.217203348656	-0.0472033486559837
26	17.21	17.2138138118233	-0.00381381182329221
27	17.37	17.2535399529549	0.116460047045116
28	17.43	17.4219026142964	0.00809738570357865
29	17.44	17.4824840642967	-0.0424840642967297
30	17.46	17.4894334057135	-0.0294334057134691
31	17.42	17.5073198774233	-0.0873198774232513
32	17.47	17.4610496879308	0.00895031206921715
33	17.45	17.5116923841226	-0.0616923841226047
34	17.44	17.4872624312859	-0.0472624312859082
35	17.46	17.4738686518993	-0.0138686518993332
36	17.47	17.4928727838567	-0.022872783856684
37	17.47	17.5012303549707	-0.0312303549706812
38	17.56	17.4989877929177	0.0610122070823316
39	17.61	17.5933689041955	0.016631095804474
40	17.61	17.6445631353974	-0.0345631353973879
41	17.6	17.6420812559584	-0.042081255958351
42	17.57	17.6290595218844	-0.0590595218844392
43	17.59	17.5948186273198	-0.00481862731978566
44	17.59	17.6144726155378	-0.0244726155378139
45	17.68	17.6127153073358	0.0672846926642414
46	17.73	17.7075468277799	0.022453172220132
47	17.75	17.75915912556	-0.00915912555995746
48	17.75	17.7785014350714	-0.0285014350713659
49	17.75	17.776454828915	-0.0264548289150319
50	17.85	17.7745551836661	0.0754448163339454
51	18.06	17.8799726591436	0.180027340856444
52	18.05	18.1028999051992	-0.0528999051991583
53	18.16	18.0891013150096	0.0708986849903859
54	18.2	18.2041923458582	-0.00419234585820405
55	18.21	18.2438913055484	-0.0338913055484262
56	18.33	18.2514576682804	0.0785423317195857
57	18.36	18.3770975674373	-0.0170975674373395
58	18.37	18.4058698402479	-0.0358698402478836
59	18.4	18.4132941300882	-0.0132941300882301
60	18.47	18.442339516806	0.0276604831940013
61	18.49	18.5143257366244	-0.0243257366244158
62	18.5	18.5325789753751	-0.0325789753750847
63	18.53	18.540239572766	-0.0102395727659612
64	18.56	18.5695042984681	-0.0095042984680731
65	18.6	18.5988218221048	0.001178177895234
66	18.61	18.638906423674	-0.0289064236740444
67	18.62	18.6468307364498	-0.0268307364498419
68	18.61	18.6549040983609	-0.0449040983609308
69	18.65	18.6416796640852	0.00832033591474968
70	18.77	18.6822771234996	0.0877228765004006
71	18.78	18.8085762511978	-0.0285762511978227
72	18.78	18.8165242727104	-0.0365242727104267
73	18.8	18.8139015696361	-0.0139015696360829
74	18.85	18.8329033378653	0.0170966621346587
75	18.85	18.8841310000476	-0.0341310000476156
76	18.98	18.8816801510063	0.0983198489936505
77	19.06	19.0187402168499	0.0412597831500676
78	19.08	19.1017029633219	-0.021702963321907
79	19.19	19.1201445358871	0.0698554641128588
80	19.21	19.2351606560421	-0.0251606560421145
81	19.29	19.2533539416285	0.036646058371538
82	19.3	19.3359853897812	-0.0359853897811853
83	19.36	19.3434013823416	0.0165986176584347
84	19.36	19.4045932813811	-0.0445932813810828

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 16.38 & 16.44 & -0.0600000000000023 \tabularnewline
4 & 16.37 & 16.4456915724042 & -0.0756915724042422 \tabularnewline
5 & 16.42 & 16.4302563780824 & -0.0102563780823672 \tabularnewline
6 & 16.43 & 16.479519897043 & -0.0495198970429946 \tabularnewline
7 & 16.44 & 16.485964015527 & -0.0459640155270051 \tabularnewline
8 & 16.53 & 16.4926634716452 & 0.0373365283547962 \tabularnewline
9 & 16.55 & 16.5853445004634 & -0.0353445004634274 \tabularnewline
10 & 16.56 & 16.6028065134442 & -0.0428065134441837 \tabularnewline
11 & 16.6 & 16.6097327007142 & -0.0097327007141601 \tabularnewline
12 & 16.61 & 16.6490338234419 & -0.0390338234418657 \tabularnewline
13 & 16.62 & 16.6562309167404 & -0.0362309167404469 \tabularnewline
14 & 16.64 & 16.6636292787154 & -0.0236292787153829 \tabularnewline
15 & 16.61 & 16.6819325281073 & -0.0719325281072969 \tabularnewline
16 & 16.74 & 16.6467672599551 & 0.0932327400448685 \tabularnewline
17 & 16.87 & 16.7834620351224 & 0.0865379648775857 \tabularnewline
18 & 16.89 & 16.9196760777217 & -0.0296760777217351 \tabularnewline
19 & 16.89 & 16.9375451238519 & -0.0475451238519007 \tabularnewline
20 & 16.99 & 16.9341310451244 & 0.0558689548755495 \tabularnewline
21 & 17.06 & 17.03814283424 & 0.0218571657600215 \tabularnewline
22 & 17.1 & 17.1097123345087 & -0.00971233450873399 \tabularnewline
23 & 17.11 & 17.1490149196751 & -0.0390149196751288 \tabularnewline
24 & 17.17 & 17.1562133703989 & 0.0137866296011175 \tabularnewline
25 & 17.17 & 17.217203348656 & -0.0472033486559837 \tabularnewline
26 & 17.21 & 17.2138138118233 & -0.00381381182329221 \tabularnewline
27 & 17.37 & 17.2535399529549 & 0.116460047045116 \tabularnewline
28 & 17.43 & 17.4219026142964 & 0.00809738570357865 \tabularnewline
29 & 17.44 & 17.4824840642967 & -0.0424840642967297 \tabularnewline
30 & 17.46 & 17.4894334057135 & -0.0294334057134691 \tabularnewline
31 & 17.42 & 17.5073198774233 & -0.0873198774232513 \tabularnewline
32 & 17.47 & 17.4610496879308 & 0.00895031206921715 \tabularnewline
33 & 17.45 & 17.5116923841226 & -0.0616923841226047 \tabularnewline
34 & 17.44 & 17.4872624312859 & -0.0472624312859082 \tabularnewline
35 & 17.46 & 17.4738686518993 & -0.0138686518993332 \tabularnewline
36 & 17.47 & 17.4928727838567 & -0.022872783856684 \tabularnewline
37 & 17.47 & 17.5012303549707 & -0.0312303549706812 \tabularnewline
38 & 17.56 & 17.4989877929177 & 0.0610122070823316 \tabularnewline
39 & 17.61 & 17.5933689041955 & 0.016631095804474 \tabularnewline
40 & 17.61 & 17.6445631353974 & -0.0345631353973879 \tabularnewline
41 & 17.6 & 17.6420812559584 & -0.042081255958351 \tabularnewline
42 & 17.57 & 17.6290595218844 & -0.0590595218844392 \tabularnewline
43 & 17.59 & 17.5948186273198 & -0.00481862731978566 \tabularnewline
44 & 17.59 & 17.6144726155378 & -0.0244726155378139 \tabularnewline
45 & 17.68 & 17.6127153073358 & 0.0672846926642414 \tabularnewline
46 & 17.73 & 17.7075468277799 & 0.022453172220132 \tabularnewline
47 & 17.75 & 17.75915912556 & -0.00915912555995746 \tabularnewline
48 & 17.75 & 17.7785014350714 & -0.0285014350713659 \tabularnewline
49 & 17.75 & 17.776454828915 & -0.0264548289150319 \tabularnewline
50 & 17.85 & 17.7745551836661 & 0.0754448163339454 \tabularnewline
51 & 18.06 & 17.8799726591436 & 0.180027340856444 \tabularnewline
52 & 18.05 & 18.1028999051992 & -0.0528999051991583 \tabularnewline
53 & 18.16 & 18.0891013150096 & 0.0708986849903859 \tabularnewline
54 & 18.2 & 18.2041923458582 & -0.00419234585820405 \tabularnewline
55 & 18.21 & 18.2438913055484 & -0.0338913055484262 \tabularnewline
56 & 18.33 & 18.2514576682804 & 0.0785423317195857 \tabularnewline
57 & 18.36 & 18.3770975674373 & -0.0170975674373395 \tabularnewline
58 & 18.37 & 18.4058698402479 & -0.0358698402478836 \tabularnewline
59 & 18.4 & 18.4132941300882 & -0.0132941300882301 \tabularnewline
60 & 18.47 & 18.442339516806 & 0.0276604831940013 \tabularnewline
61 & 18.49 & 18.5143257366244 & -0.0243257366244158 \tabularnewline
62 & 18.5 & 18.5325789753751 & -0.0325789753750847 \tabularnewline
63 & 18.53 & 18.540239572766 & -0.0102395727659612 \tabularnewline
64 & 18.56 & 18.5695042984681 & -0.0095042984680731 \tabularnewline
65 & 18.6 & 18.5988218221048 & 0.001178177895234 \tabularnewline
66 & 18.61 & 18.638906423674 & -0.0289064236740444 \tabularnewline
67 & 18.62 & 18.6468307364498 & -0.0268307364498419 \tabularnewline
68 & 18.61 & 18.6549040983609 & -0.0449040983609308 \tabularnewline
69 & 18.65 & 18.6416796640852 & 0.00832033591474968 \tabularnewline
70 & 18.77 & 18.6822771234996 & 0.0877228765004006 \tabularnewline
71 & 18.78 & 18.8085762511978 & -0.0285762511978227 \tabularnewline
72 & 18.78 & 18.8165242727104 & -0.0365242727104267 \tabularnewline
73 & 18.8 & 18.8139015696361 & -0.0139015696360829 \tabularnewline
74 & 18.85 & 18.8329033378653 & 0.0170966621346587 \tabularnewline
75 & 18.85 & 18.8841310000476 & -0.0341310000476156 \tabularnewline
76 & 18.98 & 18.8816801510063 & 0.0983198489936505 \tabularnewline
77 & 19.06 & 19.0187402168499 & 0.0412597831500676 \tabularnewline
78 & 19.08 & 19.1017029633219 & -0.021702963321907 \tabularnewline
79 & 19.19 & 19.1201445358871 & 0.0698554641128588 \tabularnewline
80 & 19.21 & 19.2351606560421 & -0.0251606560421145 \tabularnewline
81 & 19.29 & 19.2533539416285 & 0.036646058371538 \tabularnewline
82 & 19.3 & 19.3359853897812 & -0.0359853897811853 \tabularnewline
83 & 19.36 & 19.3434013823416 & 0.0165986176584347 \tabularnewline
84 & 19.36 & 19.4045932813811 & -0.0445932813810828 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232251&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]16.38[/C][C]16.44[/C][C]-0.0600000000000023[/C][/ROW]
[ROW][C]4[/C][C]16.37[/C][C]16.4456915724042[/C][C]-0.0756915724042422[/C][/ROW]
[ROW][C]5[/C][C]16.42[/C][C]16.4302563780824[/C][C]-0.0102563780823672[/C][/ROW]
[ROW][C]6[/C][C]16.43[/C][C]16.479519897043[/C][C]-0.0495198970429946[/C][/ROW]
[ROW][C]7[/C][C]16.44[/C][C]16.485964015527[/C][C]-0.0459640155270051[/C][/ROW]
[ROW][C]8[/C][C]16.53[/C][C]16.4926634716452[/C][C]0.0373365283547962[/C][/ROW]
[ROW][C]9[/C][C]16.55[/C][C]16.5853445004634[/C][C]-0.0353445004634274[/C][/ROW]
[ROW][C]10[/C][C]16.56[/C][C]16.6028065134442[/C][C]-0.0428065134441837[/C][/ROW]
[ROW][C]11[/C][C]16.6[/C][C]16.6097327007142[/C][C]-0.0097327007141601[/C][/ROW]
[ROW][C]12[/C][C]16.61[/C][C]16.6490338234419[/C][C]-0.0390338234418657[/C][/ROW]
[ROW][C]13[/C][C]16.62[/C][C]16.6562309167404[/C][C]-0.0362309167404469[/C][/ROW]
[ROW][C]14[/C][C]16.64[/C][C]16.6636292787154[/C][C]-0.0236292787153829[/C][/ROW]
[ROW][C]15[/C][C]16.61[/C][C]16.6819325281073[/C][C]-0.0719325281072969[/C][/ROW]
[ROW][C]16[/C][C]16.74[/C][C]16.6467672599551[/C][C]0.0932327400448685[/C][/ROW]
[ROW][C]17[/C][C]16.87[/C][C]16.7834620351224[/C][C]0.0865379648775857[/C][/ROW]
[ROW][C]18[/C][C]16.89[/C][C]16.9196760777217[/C][C]-0.0296760777217351[/C][/ROW]
[ROW][C]19[/C][C]16.89[/C][C]16.9375451238519[/C][C]-0.0475451238519007[/C][/ROW]
[ROW][C]20[/C][C]16.99[/C][C]16.9341310451244[/C][C]0.0558689548755495[/C][/ROW]
[ROW][C]21[/C][C]17.06[/C][C]17.03814283424[/C][C]0.0218571657600215[/C][/ROW]
[ROW][C]22[/C][C]17.1[/C][C]17.1097123345087[/C][C]-0.00971233450873399[/C][/ROW]
[ROW][C]23[/C][C]17.11[/C][C]17.1490149196751[/C][C]-0.0390149196751288[/C][/ROW]
[ROW][C]24[/C][C]17.17[/C][C]17.1562133703989[/C][C]0.0137866296011175[/C][/ROW]
[ROW][C]25[/C][C]17.17[/C][C]17.217203348656[/C][C]-0.0472033486559837[/C][/ROW]
[ROW][C]26[/C][C]17.21[/C][C]17.2138138118233[/C][C]-0.00381381182329221[/C][/ROW]
[ROW][C]27[/C][C]17.37[/C][C]17.2535399529549[/C][C]0.116460047045116[/C][/ROW]
[ROW][C]28[/C][C]17.43[/C][C]17.4219026142964[/C][C]0.00809738570357865[/C][/ROW]
[ROW][C]29[/C][C]17.44[/C][C]17.4824840642967[/C][C]-0.0424840642967297[/C][/ROW]
[ROW][C]30[/C][C]17.46[/C][C]17.4894334057135[/C][C]-0.0294334057134691[/C][/ROW]
[ROW][C]31[/C][C]17.42[/C][C]17.5073198774233[/C][C]-0.0873198774232513[/C][/ROW]
[ROW][C]32[/C][C]17.47[/C][C]17.4610496879308[/C][C]0.00895031206921715[/C][/ROW]
[ROW][C]33[/C][C]17.45[/C][C]17.5116923841226[/C][C]-0.0616923841226047[/C][/ROW]
[ROW][C]34[/C][C]17.44[/C][C]17.4872624312859[/C][C]-0.0472624312859082[/C][/ROW]
[ROW][C]35[/C][C]17.46[/C][C]17.4738686518993[/C][C]-0.0138686518993332[/C][/ROW]
[ROW][C]36[/C][C]17.47[/C][C]17.4928727838567[/C][C]-0.022872783856684[/C][/ROW]
[ROW][C]37[/C][C]17.47[/C][C]17.5012303549707[/C][C]-0.0312303549706812[/C][/ROW]
[ROW][C]38[/C][C]17.56[/C][C]17.4989877929177[/C][C]0.0610122070823316[/C][/ROW]
[ROW][C]39[/C][C]17.61[/C][C]17.5933689041955[/C][C]0.016631095804474[/C][/ROW]
[ROW][C]40[/C][C]17.61[/C][C]17.6445631353974[/C][C]-0.0345631353973879[/C][/ROW]
[ROW][C]41[/C][C]17.6[/C][C]17.6420812559584[/C][C]-0.042081255958351[/C][/ROW]
[ROW][C]42[/C][C]17.57[/C][C]17.6290595218844[/C][C]-0.0590595218844392[/C][/ROW]
[ROW][C]43[/C][C]17.59[/C][C]17.5948186273198[/C][C]-0.00481862731978566[/C][/ROW]
[ROW][C]44[/C][C]17.59[/C][C]17.6144726155378[/C][C]-0.0244726155378139[/C][/ROW]
[ROW][C]45[/C][C]17.68[/C][C]17.6127153073358[/C][C]0.0672846926642414[/C][/ROW]
[ROW][C]46[/C][C]17.73[/C][C]17.7075468277799[/C][C]0.022453172220132[/C][/ROW]
[ROW][C]47[/C][C]17.75[/C][C]17.75915912556[/C][C]-0.00915912555995746[/C][/ROW]
[ROW][C]48[/C][C]17.75[/C][C]17.7785014350714[/C][C]-0.0285014350713659[/C][/ROW]
[ROW][C]49[/C][C]17.75[/C][C]17.776454828915[/C][C]-0.0264548289150319[/C][/ROW]
[ROW][C]50[/C][C]17.85[/C][C]17.7745551836661[/C][C]0.0754448163339454[/C][/ROW]
[ROW][C]51[/C][C]18.06[/C][C]17.8799726591436[/C][C]0.180027340856444[/C][/ROW]
[ROW][C]52[/C][C]18.05[/C][C]18.1028999051992[/C][C]-0.0528999051991583[/C][/ROW]
[ROW][C]53[/C][C]18.16[/C][C]18.0891013150096[/C][C]0.0708986849903859[/C][/ROW]
[ROW][C]54[/C][C]18.2[/C][C]18.2041923458582[/C][C]-0.00419234585820405[/C][/ROW]
[ROW][C]55[/C][C]18.21[/C][C]18.2438913055484[/C][C]-0.0338913055484262[/C][/ROW]
[ROW][C]56[/C][C]18.33[/C][C]18.2514576682804[/C][C]0.0785423317195857[/C][/ROW]
[ROW][C]57[/C][C]18.36[/C][C]18.3770975674373[/C][C]-0.0170975674373395[/C][/ROW]
[ROW][C]58[/C][C]18.37[/C][C]18.4058698402479[/C][C]-0.0358698402478836[/C][/ROW]
[ROW][C]59[/C][C]18.4[/C][C]18.4132941300882[/C][C]-0.0132941300882301[/C][/ROW]
[ROW][C]60[/C][C]18.47[/C][C]18.442339516806[/C][C]0.0276604831940013[/C][/ROW]
[ROW][C]61[/C][C]18.49[/C][C]18.5143257366244[/C][C]-0.0243257366244158[/C][/ROW]
[ROW][C]62[/C][C]18.5[/C][C]18.5325789753751[/C][C]-0.0325789753750847[/C][/ROW]
[ROW][C]63[/C][C]18.53[/C][C]18.540239572766[/C][C]-0.0102395727659612[/C][/ROW]
[ROW][C]64[/C][C]18.56[/C][C]18.5695042984681[/C][C]-0.0095042984680731[/C][/ROW]
[ROW][C]65[/C][C]18.6[/C][C]18.5988218221048[/C][C]0.001178177895234[/C][/ROW]
[ROW][C]66[/C][C]18.61[/C][C]18.638906423674[/C][C]-0.0289064236740444[/C][/ROW]
[ROW][C]67[/C][C]18.62[/C][C]18.6468307364498[/C][C]-0.0268307364498419[/C][/ROW]
[ROW][C]68[/C][C]18.61[/C][C]18.6549040983609[/C][C]-0.0449040983609308[/C][/ROW]
[ROW][C]69[/C][C]18.65[/C][C]18.6416796640852[/C][C]0.00832033591474968[/C][/ROW]
[ROW][C]70[/C][C]18.77[/C][C]18.6822771234996[/C][C]0.0877228765004006[/C][/ROW]
[ROW][C]71[/C][C]18.78[/C][C]18.8085762511978[/C][C]-0.0285762511978227[/C][/ROW]
[ROW][C]72[/C][C]18.78[/C][C]18.8165242727104[/C][C]-0.0365242727104267[/C][/ROW]
[ROW][C]73[/C][C]18.8[/C][C]18.8139015696361[/C][C]-0.0139015696360829[/C][/ROW]
[ROW][C]74[/C][C]18.85[/C][C]18.8329033378653[/C][C]0.0170966621346587[/C][/ROW]
[ROW][C]75[/C][C]18.85[/C][C]18.8841310000476[/C][C]-0.0341310000476156[/C][/ROW]
[ROW][C]76[/C][C]18.98[/C][C]18.8816801510063[/C][C]0.0983198489936505[/C][/ROW]
[ROW][C]77[/C][C]19.06[/C][C]19.0187402168499[/C][C]0.0412597831500676[/C][/ROW]
[ROW][C]78[/C][C]19.08[/C][C]19.1017029633219[/C][C]-0.021702963321907[/C][/ROW]
[ROW][C]79[/C][C]19.19[/C][C]19.1201445358871[/C][C]0.0698554641128588[/C][/ROW]
[ROW][C]80[/C][C]19.21[/C][C]19.2351606560421[/C][C]-0.0251606560421145[/C][/ROW]
[ROW][C]81[/C][C]19.29[/C][C]19.2533539416285[/C][C]0.036646058371538[/C][/ROW]
[ROW][C]82[/C][C]19.3[/C][C]19.3359853897812[/C][C]-0.0359853897811853[/C][/ROW]
[ROW][C]83[/C][C]19.36[/C][C]19.3434013823416[/C][C]0.0165986176584347[/C][/ROW]
[ROW][C]84[/C][C]19.36[/C][C]19.4045932813811[/C][C]-0.0445932813810828[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232251&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232251&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	16.38	16.44	-0.0600000000000023
4	16.37	16.4456915724042	-0.0756915724042422
5	16.42	16.4302563780824	-0.0102563780823672
6	16.43	16.479519897043	-0.0495198970429946
7	16.44	16.485964015527	-0.0459640155270051
8	16.53	16.4926634716452	0.0373365283547962
9	16.55	16.5853445004634	-0.0353445004634274
10	16.56	16.6028065134442	-0.0428065134441837
11	16.6	16.6097327007142	-0.0097327007141601
12	16.61	16.6490338234419	-0.0390338234418657
13	16.62	16.6562309167404	-0.0362309167404469
14	16.64	16.6636292787154	-0.0236292787153829
15	16.61	16.6819325281073	-0.0719325281072969
16	16.74	16.6467672599551	0.0932327400448685
17	16.87	16.7834620351224	0.0865379648775857
18	16.89	16.9196760777217	-0.0296760777217351
19	16.89	16.9375451238519	-0.0475451238519007
20	16.99	16.9341310451244	0.0558689548755495
21	17.06	17.03814283424	0.0218571657600215
22	17.1	17.1097123345087	-0.00971233450873399
23	17.11	17.1490149196751	-0.0390149196751288
24	17.17	17.1562133703989	0.0137866296011175
25	17.17	17.217203348656	-0.0472033486559837
26	17.21	17.2138138118233	-0.00381381182329221
27	17.37	17.2535399529549	0.116460047045116
28	17.43	17.4219026142964	0.00809738570357865
29	17.44	17.4824840642967	-0.0424840642967297
30	17.46	17.4894334057135	-0.0294334057134691
31	17.42	17.5073198774233	-0.0873198774232513
32	17.47	17.4610496879308	0.00895031206921715
33	17.45	17.5116923841226	-0.0616923841226047
34	17.44	17.4872624312859	-0.0472624312859082
35	17.46	17.4738686518993	-0.0138686518993332
36	17.47	17.4928727838567	-0.022872783856684
37	17.47	17.5012303549707	-0.0312303549706812
38	17.56	17.4989877929177	0.0610122070823316
39	17.61	17.5933689041955	0.016631095804474
40	17.61	17.6445631353974	-0.0345631353973879
41	17.6	17.6420812559584	-0.042081255958351
42	17.57	17.6290595218844	-0.0590595218844392
43	17.59	17.5948186273198	-0.00481862731978566
44	17.59	17.6144726155378	-0.0244726155378139
45	17.68	17.6127153073358	0.0672846926642414
46	17.73	17.7075468277799	0.022453172220132
47	17.75	17.75915912556	-0.00915912555995746
48	17.75	17.7785014350714	-0.0285014350713659
49	17.75	17.776454828915	-0.0264548289150319
50	17.85	17.7745551836661	0.0754448163339454
51	18.06	17.8799726591436	0.180027340856444
52	18.05	18.1028999051992	-0.0528999051991583
53	18.16	18.0891013150096	0.0708986849903859
54	18.2	18.2041923458582	-0.00419234585820405
55	18.21	18.2438913055484	-0.0338913055484262
56	18.33	18.2514576682804	0.0785423317195857
57	18.36	18.3770975674373	-0.0170975674373395
58	18.37	18.4058698402479	-0.0358698402478836
59	18.4	18.4132941300882	-0.0132941300882301
60	18.47	18.442339516806	0.0276604831940013
61	18.49	18.5143257366244	-0.0243257366244158
62	18.5	18.5325789753751	-0.0325789753750847
63	18.53	18.540239572766	-0.0102395727659612
64	18.56	18.5695042984681	-0.0095042984680731
65	18.6	18.5988218221048	0.001178177895234
66	18.61	18.638906423674	-0.0289064236740444
67	18.62	18.6468307364498	-0.0268307364498419
68	18.61	18.6549040983609	-0.0449040983609308
69	18.65	18.6416796640852	0.00832033591474968
70	18.77	18.6822771234996	0.0877228765004006
71	18.78	18.8085762511978	-0.0285762511978227
72	18.78	18.8165242727104	-0.0365242727104267
73	18.8	18.8139015696361	-0.0139015696360829
74	18.85	18.8329033378653	0.0170966621346587
75	18.85	18.8841310000476	-0.0341310000476156
76	18.98	18.8816801510063	0.0983198489936505
77	19.06	19.0187402168499	0.0412597831500676
78	19.08	19.1017029633219	-0.021702963321907
79	19.19	19.1201445358871	0.0698554641128588
80	19.21	19.2351606560421	-0.0251606560421145
81	19.29	19.2533539416285	0.036646058371538
82	19.3	19.3359853897812	-0.0359853897811853
83	19.36	19.3434013823416	0.0165986176584347
84	19.36	19.4045932813811	-0.0445932813810828

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
85	19.4013911659796	19.3048223795813	19.4979599523779
86	19.4427823319592	19.3012251880064	19.5843394759121
87	19.4841734979389	19.3046328248957	19.663714170982
88	19.5255646639185	19.311062950386	19.7400663774509
89	19.5669558298981	19.319038524567	19.8148731352292
90	19.6083469958777	19.327834895277	19.8888590964785
91	19.6497381618574	19.3370413441655	19.9624349795493
92	19.691129327837	19.3464034313627	20.0358552243113
93	19.7325204938166	19.3557538019721	20.1092871856611
94	19.7739116597962	19.3649776079875	20.182845711605
95	19.8153028257759	19.3739935584591	20.2566120930926
96	19.8566939917555	19.3827427896891	20.3306451938218

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
85 & 19.4013911659796 & 19.3048223795813 & 19.4979599523779 \tabularnewline
86 & 19.4427823319592 & 19.3012251880064 & 19.5843394759121 \tabularnewline
87 & 19.4841734979389 & 19.3046328248957 & 19.663714170982 \tabularnewline
88 & 19.5255646639185 & 19.311062950386 & 19.7400663774509 \tabularnewline
89 & 19.5669558298981 & 19.319038524567 & 19.8148731352292 \tabularnewline
90 & 19.6083469958777 & 19.327834895277 & 19.8888590964785 \tabularnewline
91 & 19.6497381618574 & 19.3370413441655 & 19.9624349795493 \tabularnewline
92 & 19.691129327837 & 19.3464034313627 & 20.0358552243113 \tabularnewline
93 & 19.7325204938166 & 19.3557538019721 & 20.1092871856611 \tabularnewline
94 & 19.7739116597962 & 19.3649776079875 & 20.182845711605 \tabularnewline
95 & 19.8153028257759 & 19.3739935584591 & 20.2566120930926 \tabularnewline
96 & 19.8566939917555 & 19.3827427896891 & 20.3306451938218 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232251&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]19.4013911659796[/C][C]19.3048223795813[/C][C]19.4979599523779[/C][/ROW]
[ROW][C]86[/C][C]19.4427823319592[/C][C]19.3012251880064[/C][C]19.5843394759121[/C][/ROW]
[ROW][C]87[/C][C]19.4841734979389[/C][C]19.3046328248957[/C][C]19.663714170982[/C][/ROW]
[ROW][C]88[/C][C]19.5255646639185[/C][C]19.311062950386[/C][C]19.7400663774509[/C][/ROW]
[ROW][C]89[/C][C]19.5669558298981[/C][C]19.319038524567[/C][C]19.8148731352292[/C][/ROW]
[ROW][C]90[/C][C]19.6083469958777[/C][C]19.327834895277[/C][C]19.8888590964785[/C][/ROW]
[ROW][C]91[/C][C]19.6497381618574[/C][C]19.3370413441655[/C][C]19.9624349795493[/C][/ROW]
[ROW][C]92[/C][C]19.691129327837[/C][C]19.3464034313627[/C][C]20.0358552243113[/C][/ROW]
[ROW][C]93[/C][C]19.7325204938166[/C][C]19.3557538019721[/C][C]20.1092871856611[/C][/ROW]
[ROW][C]94[/C][C]19.7739116597962[/C][C]19.3649776079875[/C][C]20.182845711605[/C][/ROW]
[ROW][C]95[/C][C]19.8153028257759[/C][C]19.3739935584591[/C][C]20.2566120930926[/C][/ROW]
[ROW][C]96[/C][C]19.8566939917555[/C][C]19.3827427896891[/C][C]20.3306451938218[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232251&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232251&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	19.4013911659796	19.3048223795813	19.4979599523779
86	19.4427823319592	19.3012251880064	19.5843394759121
87	19.4841734979389	19.3046328248957	19.663714170982
88	19.5255646639185	19.311062950386	19.7400663774509
89	19.5669558298981	19.319038524567	19.8148731352292
90	19.6083469958777	19.327834895277	19.8888590964785
91	19.6497381618574	19.3370413441655	19.9624349795493
92	19.691129327837	19.3464034313627	20.0358552243113
93	19.7325204938166	19.3557538019721	20.1092871856611
94	19.7739116597962	19.3649776079875	20.182845711605
95	19.8153028257759	19.3739935584591	20.2566120930926
96	19.8566939917555	19.3827427896891	20.3306451938218

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