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*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Sun, 29 Nov 2015 22:17:32 +0000

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/2015/Nov/29/t1448835519bea8illys26t9t0.htm/, Retrieved Tue, 21 May 2024 13:36:54 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=284544, Retrieved Tue, 21 May 2024 13:36:54 +0000

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No (this computation is public)

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Estimated Impact

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

-       [Exponential Smoothing] [Exponential smoot...] [2015-11-29 22:17:32] [442c3b7d1457f8bc4e82a9331e05e70d] [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	2 seconds
R Server	'George Udny Yule' @ yule.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 & 'George Udny Yule' @ yule.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284544&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]'George Udny Yule' @ yule.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284544&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284544&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	'George Udny Yule' @ yule.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.995146583346137
beta	0.268454698791468
gamma	FALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.995146583346137 \tabularnewline
beta & 0.268454698791468 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284544&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]0.995146583346137[/C][/ROW]
[ROW][C]beta[/C][C]0.268454698791468[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284544&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284544&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	0.995146583346137
beta	0.268454698791468
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
3	85.47	85.95	-0.480000000000018
4	85.78	85.7540967873768	0.0259032126231915
5	86.07	86.0685615175632	0.00143848243678235
6	86.05	86.3590645482303	-0.309064548230253
7	86.32	86.2580044057638	0.0619955942362225
8	86.43	86.5427657294108	-0.11276572941081
9	86.41	86.6234883540132	-0.213488354013251
10	86.38	86.546943409886	-0.166943409885988
11	86.59	86.4721182793888	0.117881720611209
12	86.68	86.7122282154099	-0.032228215409873
13	86.87	86.79434693648	0.0756530635199795
14	87.32	87.004034193985	0.315965806014972
15	87.13	87.5372786824409	-0.407278682440861
16	87.42	87.2419836658286	0.178016334171389
17	87.22	87.5767003651293	-0.356700365129313
18	87.17	87.2840024319168	-0.114002431916788
19	87.52	87.2023685655384	0.317631434461561
20	87.49	87.6351294684635	-0.145129468463509
21	87.53	87.5686038446451	-0.0386038446451096
22	87.93	87.59777374574	0.332226254259979
23	88.54	88.0847287867165	0.45527121328351
24	88.96	88.8157581115836	0.144241888416431
25	89.3	89.2758021431939	0.024197856806083
26	90.01	89.6228492673242	0.387150732675821
27	90.52	90.4345157117165	0.0854842882835101
28	90.64	90.968817104121	-0.328817104121043
29	91.25	91.0029838079597	0.247016192040334
30	91.59	91.6761798635251	-0.086179863525075
31	92.09	91.9947738991894	0.0952261008105921
32	91.81	92.5193332824413	-0.709333282441293
33	92.03	92.0537384979683	-0.0237384979683242
34	92.15	92.264069238925	-0.114069238924969
35	91.98	92.3540338518492	-0.374033851849148
36	92.11	92.0853717605185	0.024628239481487
37	92.28	92.2200163652094	0.0599836347905693
38	92.53	92.4058695053274	0.124130494672585
39	91.97	92.6887198560324	-0.718719856032365
40	92.05	91.9408032737704	0.109196726229555
41	91.87	92.0459571490189	-0.175957149018856
42	91.49	91.8203338546754	-0.330333854675388
43	91.48	91.3528338331058	0.127166166894213
44	91.63	91.3745860622512	0.255413937748813
45	91.46	91.592197909542	-0.132197909541986
46	91.61	91.3887622449815	0.221237755018535
47	91.7	91.5961509336757	0.103849066324344
48	91.87	91.7144641324265	0.155535867573505
49	92.21	91.9257649581429	0.284235041857087
50	92.65	92.3410742237418	0.308925776258206
51	92.83	92.863484459188	-0.0334844591880454
52	93.02	93.0362008859772	-0.0162008859771703
53	93.33	93.2217889061289	0.108211093871105
54	93.35	93.5600938688954	-0.210093868895427
55	93.45	93.5255117852406	-0.0755117852406073
56	93.51	93.604685494757	-0.0946854947569733
57	93.8	93.6394831546453	0.160516845354735
58	93.94	93.9711269117179	-0.0311269117179336
59	94.02	94.1037414289644	-0.0837414289643732
60	94.26	94.1616251176421	0.0983748823579162
61	94.71	94.4270222558676	0.28297774413241
62	95.26	94.9517243082581	0.308275691741869
63	95.54	95.5839579254124	-0.0439579254123998
64	95.69	95.8539240240602	-0.163924024060208
65	96.03	95.9607136753177	0.069286324682281
66	96.4	96.3180917730384	0.0819082269615734
67	96.55	96.7099124420129	-0.15991244201291
68	96.45	96.818365205541	-0.368365205541039
69	96.65	96.620967494673	0.0290325053269953
70	96.84	96.826794843373	0.0132051566269951
71	97.21	97.0203994411398	0.189600558860178
72	97.31	97.4401954468412	-0.130195446841171
73	97.91	97.5069656052132	0.403034394786829
74	98.51	98.2120489730909	0.297951026909075
75	98.54	98.8921571325435	-0.352157132543496
76	98.52	98.8312329748208	-0.311232974820783
77	98.66	98.7278879107701	-0.0678879107701391
78	98.53	98.8485704798331	-0.318570479833085
79	98.71	98.6346804772291	0.0753195227709256
80	98.92	98.8328905092279	0.0871094907721357
81	98.96	99.0661047427919	-0.106104742791942
82	99.25	99.0786964214617	0.171303578538314
83	99.32	99.4131140985856	-0.0931140985856018
84	99.41	99.459521830909	-0.049521830908958
85	99.36	99.5360804143788	-0.176080414378788
86	99.58	99.4396544604451	0.140345539554872
87	99.77	99.6956122736364	0.0743877263635824
88	99.77	99.9058052076173	-0.135805207617281
89	100.03	99.8705447590598	0.159455240940204
90	100.2	100.171710487937	0.0282895120632389
91	100.24	100.349904683268	-0.109904683267658
92	100.1	100.36121416592	-0.261214165919696
93	100.03	100.152164705466	-0.122164705466275
94	100.18	100.048853322434	0.131146677565511
95	100.29	100.232659964615	0.0573400353849394
96	100.41	100.358336671307	0.051663328693266
97	100.6	100.492166172759	0.107833827241237
98	100.75	100.710701552417	0.0392984475831923
99	100.79	100.871532833247	-0.081532833246925
100	100.44	100.89033763657	-0.450337636570126
101	100.29	100.421819100407	-0.131819100406645
102	100.34	100.235057490417	0.104942509583339
103	100.46	100.311943965519	0.148056034481186
104	100.12	100.47128815022	-0.351288150220483
105	100.06	100.039864422283	0.0201355777169283
106	100.28	99.9834410035299	0.296558996470125
107	100.28	100.281325668186	-0.00132566818624014
108	100.4	100.282417271968	0.117582728032104

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 85.47 & 85.95 & -0.480000000000018 \tabularnewline
4 & 85.78 & 85.7540967873768 & 0.0259032126231915 \tabularnewline
5 & 86.07 & 86.0685615175632 & 0.00143848243678235 \tabularnewline
6 & 86.05 & 86.3590645482303 & -0.309064548230253 \tabularnewline
7 & 86.32 & 86.2580044057638 & 0.0619955942362225 \tabularnewline
8 & 86.43 & 86.5427657294108 & -0.11276572941081 \tabularnewline
9 & 86.41 & 86.6234883540132 & -0.213488354013251 \tabularnewline
10 & 86.38 & 86.546943409886 & -0.166943409885988 \tabularnewline
11 & 86.59 & 86.4721182793888 & 0.117881720611209 \tabularnewline
12 & 86.68 & 86.7122282154099 & -0.032228215409873 \tabularnewline
13 & 86.87 & 86.79434693648 & 0.0756530635199795 \tabularnewline
14 & 87.32 & 87.004034193985 & 0.315965806014972 \tabularnewline
15 & 87.13 & 87.5372786824409 & -0.407278682440861 \tabularnewline
16 & 87.42 & 87.2419836658286 & 0.178016334171389 \tabularnewline
17 & 87.22 & 87.5767003651293 & -0.356700365129313 \tabularnewline
18 & 87.17 & 87.2840024319168 & -0.114002431916788 \tabularnewline
19 & 87.52 & 87.2023685655384 & 0.317631434461561 \tabularnewline
20 & 87.49 & 87.6351294684635 & -0.145129468463509 \tabularnewline
21 & 87.53 & 87.5686038446451 & -0.0386038446451096 \tabularnewline
22 & 87.93 & 87.59777374574 & 0.332226254259979 \tabularnewline
23 & 88.54 & 88.0847287867165 & 0.45527121328351 \tabularnewline
24 & 88.96 & 88.8157581115836 & 0.144241888416431 \tabularnewline
25 & 89.3 & 89.2758021431939 & 0.024197856806083 \tabularnewline
26 & 90.01 & 89.6228492673242 & 0.387150732675821 \tabularnewline
27 & 90.52 & 90.4345157117165 & 0.0854842882835101 \tabularnewline
28 & 90.64 & 90.968817104121 & -0.328817104121043 \tabularnewline
29 & 91.25 & 91.0029838079597 & 0.247016192040334 \tabularnewline
30 & 91.59 & 91.6761798635251 & -0.086179863525075 \tabularnewline
31 & 92.09 & 91.9947738991894 & 0.0952261008105921 \tabularnewline
32 & 91.81 & 92.5193332824413 & -0.709333282441293 \tabularnewline
33 & 92.03 & 92.0537384979683 & -0.0237384979683242 \tabularnewline
34 & 92.15 & 92.264069238925 & -0.114069238924969 \tabularnewline
35 & 91.98 & 92.3540338518492 & -0.374033851849148 \tabularnewline
36 & 92.11 & 92.0853717605185 & 0.024628239481487 \tabularnewline
37 & 92.28 & 92.2200163652094 & 0.0599836347905693 \tabularnewline
38 & 92.53 & 92.4058695053274 & 0.124130494672585 \tabularnewline
39 & 91.97 & 92.6887198560324 & -0.718719856032365 \tabularnewline
40 & 92.05 & 91.9408032737704 & 0.109196726229555 \tabularnewline
41 & 91.87 & 92.0459571490189 & -0.175957149018856 \tabularnewline
42 & 91.49 & 91.8203338546754 & -0.330333854675388 \tabularnewline
43 & 91.48 & 91.3528338331058 & 0.127166166894213 \tabularnewline
44 & 91.63 & 91.3745860622512 & 0.255413937748813 \tabularnewline
45 & 91.46 & 91.592197909542 & -0.132197909541986 \tabularnewline
46 & 91.61 & 91.3887622449815 & 0.221237755018535 \tabularnewline
47 & 91.7 & 91.5961509336757 & 0.103849066324344 \tabularnewline
48 & 91.87 & 91.7144641324265 & 0.155535867573505 \tabularnewline
49 & 92.21 & 91.9257649581429 & 0.284235041857087 \tabularnewline
50 & 92.65 & 92.3410742237418 & 0.308925776258206 \tabularnewline
51 & 92.83 & 92.863484459188 & -0.0334844591880454 \tabularnewline
52 & 93.02 & 93.0362008859772 & -0.0162008859771703 \tabularnewline
53 & 93.33 & 93.2217889061289 & 0.108211093871105 \tabularnewline
54 & 93.35 & 93.5600938688954 & -0.210093868895427 \tabularnewline
55 & 93.45 & 93.5255117852406 & -0.0755117852406073 \tabularnewline
56 & 93.51 & 93.604685494757 & -0.0946854947569733 \tabularnewline
57 & 93.8 & 93.6394831546453 & 0.160516845354735 \tabularnewline
58 & 93.94 & 93.9711269117179 & -0.0311269117179336 \tabularnewline
59 & 94.02 & 94.1037414289644 & -0.0837414289643732 \tabularnewline
60 & 94.26 & 94.1616251176421 & 0.0983748823579162 \tabularnewline
61 & 94.71 & 94.4270222558676 & 0.28297774413241 \tabularnewline
62 & 95.26 & 94.9517243082581 & 0.308275691741869 \tabularnewline
63 & 95.54 & 95.5839579254124 & -0.0439579254123998 \tabularnewline
64 & 95.69 & 95.8539240240602 & -0.163924024060208 \tabularnewline
65 & 96.03 & 95.9607136753177 & 0.069286324682281 \tabularnewline
66 & 96.4 & 96.3180917730384 & 0.0819082269615734 \tabularnewline
67 & 96.55 & 96.7099124420129 & -0.15991244201291 \tabularnewline
68 & 96.45 & 96.818365205541 & -0.368365205541039 \tabularnewline
69 & 96.65 & 96.620967494673 & 0.0290325053269953 \tabularnewline
70 & 96.84 & 96.826794843373 & 0.0132051566269951 \tabularnewline
71 & 97.21 & 97.0203994411398 & 0.189600558860178 \tabularnewline
72 & 97.31 & 97.4401954468412 & -0.130195446841171 \tabularnewline
73 & 97.91 & 97.5069656052132 & 0.403034394786829 \tabularnewline
74 & 98.51 & 98.2120489730909 & 0.297951026909075 \tabularnewline
75 & 98.54 & 98.8921571325435 & -0.352157132543496 \tabularnewline
76 & 98.52 & 98.8312329748208 & -0.311232974820783 \tabularnewline
77 & 98.66 & 98.7278879107701 & -0.0678879107701391 \tabularnewline
78 & 98.53 & 98.8485704798331 & -0.318570479833085 \tabularnewline
79 & 98.71 & 98.6346804772291 & 0.0753195227709256 \tabularnewline
80 & 98.92 & 98.8328905092279 & 0.0871094907721357 \tabularnewline
81 & 98.96 & 99.0661047427919 & -0.106104742791942 \tabularnewline
82 & 99.25 & 99.0786964214617 & 0.171303578538314 \tabularnewline
83 & 99.32 & 99.4131140985856 & -0.0931140985856018 \tabularnewline
84 & 99.41 & 99.459521830909 & -0.049521830908958 \tabularnewline
85 & 99.36 & 99.5360804143788 & -0.176080414378788 \tabularnewline
86 & 99.58 & 99.4396544604451 & 0.140345539554872 \tabularnewline
87 & 99.77 & 99.6956122736364 & 0.0743877263635824 \tabularnewline
88 & 99.77 & 99.9058052076173 & -0.135805207617281 \tabularnewline
89 & 100.03 & 99.8705447590598 & 0.159455240940204 \tabularnewline
90 & 100.2 & 100.171710487937 & 0.0282895120632389 \tabularnewline
91 & 100.24 & 100.349904683268 & -0.109904683267658 \tabularnewline
92 & 100.1 & 100.36121416592 & -0.261214165919696 \tabularnewline
93 & 100.03 & 100.152164705466 & -0.122164705466275 \tabularnewline
94 & 100.18 & 100.048853322434 & 0.131146677565511 \tabularnewline
95 & 100.29 & 100.232659964615 & 0.0573400353849394 \tabularnewline
96 & 100.41 & 100.358336671307 & 0.051663328693266 \tabularnewline
97 & 100.6 & 100.492166172759 & 0.107833827241237 \tabularnewline
98 & 100.75 & 100.710701552417 & 0.0392984475831923 \tabularnewline
99 & 100.79 & 100.871532833247 & -0.081532833246925 \tabularnewline
100 & 100.44 & 100.89033763657 & -0.450337636570126 \tabularnewline
101 & 100.29 & 100.421819100407 & -0.131819100406645 \tabularnewline
102 & 100.34 & 100.235057490417 & 0.104942509583339 \tabularnewline
103 & 100.46 & 100.311943965519 & 0.148056034481186 \tabularnewline
104 & 100.12 & 100.47128815022 & -0.351288150220483 \tabularnewline
105 & 100.06 & 100.039864422283 & 0.0201355777169283 \tabularnewline
106 & 100.28 & 99.9834410035299 & 0.296558996470125 \tabularnewline
107 & 100.28 & 100.281325668186 & -0.00132566818624014 \tabularnewline
108 & 100.4 & 100.282417271968 & 0.117582728032104 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284544&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]85.47[/C][C]85.95[/C][C]-0.480000000000018[/C][/ROW]
[ROW][C]4[/C][C]85.78[/C][C]85.7540967873768[/C][C]0.0259032126231915[/C][/ROW]
[ROW][C]5[/C][C]86.07[/C][C]86.0685615175632[/C][C]0.00143848243678235[/C][/ROW]
[ROW][C]6[/C][C]86.05[/C][C]86.3590645482303[/C][C]-0.309064548230253[/C][/ROW]
[ROW][C]7[/C][C]86.32[/C][C]86.2580044057638[/C][C]0.0619955942362225[/C][/ROW]
[ROW][C]8[/C][C]86.43[/C][C]86.5427657294108[/C][C]-0.11276572941081[/C][/ROW]
[ROW][C]9[/C][C]86.41[/C][C]86.6234883540132[/C][C]-0.213488354013251[/C][/ROW]
[ROW][C]10[/C][C]86.38[/C][C]86.546943409886[/C][C]-0.166943409885988[/C][/ROW]
[ROW][C]11[/C][C]86.59[/C][C]86.4721182793888[/C][C]0.117881720611209[/C][/ROW]
[ROW][C]12[/C][C]86.68[/C][C]86.7122282154099[/C][C]-0.032228215409873[/C][/ROW]
[ROW][C]13[/C][C]86.87[/C][C]86.79434693648[/C][C]0.0756530635199795[/C][/ROW]
[ROW][C]14[/C][C]87.32[/C][C]87.004034193985[/C][C]0.315965806014972[/C][/ROW]
[ROW][C]15[/C][C]87.13[/C][C]87.5372786824409[/C][C]-0.407278682440861[/C][/ROW]
[ROW][C]16[/C][C]87.42[/C][C]87.2419836658286[/C][C]0.178016334171389[/C][/ROW]
[ROW][C]17[/C][C]87.22[/C][C]87.5767003651293[/C][C]-0.356700365129313[/C][/ROW]
[ROW][C]18[/C][C]87.17[/C][C]87.2840024319168[/C][C]-0.114002431916788[/C][/ROW]
[ROW][C]19[/C][C]87.52[/C][C]87.2023685655384[/C][C]0.317631434461561[/C][/ROW]
[ROW][C]20[/C][C]87.49[/C][C]87.6351294684635[/C][C]-0.145129468463509[/C][/ROW]
[ROW][C]21[/C][C]87.53[/C][C]87.5686038446451[/C][C]-0.0386038446451096[/C][/ROW]
[ROW][C]22[/C][C]87.93[/C][C]87.59777374574[/C][C]0.332226254259979[/C][/ROW]
[ROW][C]23[/C][C]88.54[/C][C]88.0847287867165[/C][C]0.45527121328351[/C][/ROW]
[ROW][C]24[/C][C]88.96[/C][C]88.8157581115836[/C][C]0.144241888416431[/C][/ROW]
[ROW][C]25[/C][C]89.3[/C][C]89.2758021431939[/C][C]0.024197856806083[/C][/ROW]
[ROW][C]26[/C][C]90.01[/C][C]89.6228492673242[/C][C]0.387150732675821[/C][/ROW]
[ROW][C]27[/C][C]90.52[/C][C]90.4345157117165[/C][C]0.0854842882835101[/C][/ROW]
[ROW][C]28[/C][C]90.64[/C][C]90.968817104121[/C][C]-0.328817104121043[/C][/ROW]
[ROW][C]29[/C][C]91.25[/C][C]91.0029838079597[/C][C]0.247016192040334[/C][/ROW]
[ROW][C]30[/C][C]91.59[/C][C]91.6761798635251[/C][C]-0.086179863525075[/C][/ROW]
[ROW][C]31[/C][C]92.09[/C][C]91.9947738991894[/C][C]0.0952261008105921[/C][/ROW]
[ROW][C]32[/C][C]91.81[/C][C]92.5193332824413[/C][C]-0.709333282441293[/C][/ROW]
[ROW][C]33[/C][C]92.03[/C][C]92.0537384979683[/C][C]-0.0237384979683242[/C][/ROW]
[ROW][C]34[/C][C]92.15[/C][C]92.264069238925[/C][C]-0.114069238924969[/C][/ROW]
[ROW][C]35[/C][C]91.98[/C][C]92.3540338518492[/C][C]-0.374033851849148[/C][/ROW]
[ROW][C]36[/C][C]92.11[/C][C]92.0853717605185[/C][C]0.024628239481487[/C][/ROW]
[ROW][C]37[/C][C]92.28[/C][C]92.2200163652094[/C][C]0.0599836347905693[/C][/ROW]
[ROW][C]38[/C][C]92.53[/C][C]92.4058695053274[/C][C]0.124130494672585[/C][/ROW]
[ROW][C]39[/C][C]91.97[/C][C]92.6887198560324[/C][C]-0.718719856032365[/C][/ROW]
[ROW][C]40[/C][C]92.05[/C][C]91.9408032737704[/C][C]0.109196726229555[/C][/ROW]
[ROW][C]41[/C][C]91.87[/C][C]92.0459571490189[/C][C]-0.175957149018856[/C][/ROW]
[ROW][C]42[/C][C]91.49[/C][C]91.8203338546754[/C][C]-0.330333854675388[/C][/ROW]
[ROW][C]43[/C][C]91.48[/C][C]91.3528338331058[/C][C]0.127166166894213[/C][/ROW]
[ROW][C]44[/C][C]91.63[/C][C]91.3745860622512[/C][C]0.255413937748813[/C][/ROW]
[ROW][C]45[/C][C]91.46[/C][C]91.592197909542[/C][C]-0.132197909541986[/C][/ROW]
[ROW][C]46[/C][C]91.61[/C][C]91.3887622449815[/C][C]0.221237755018535[/C][/ROW]
[ROW][C]47[/C][C]91.7[/C][C]91.5961509336757[/C][C]0.103849066324344[/C][/ROW]
[ROW][C]48[/C][C]91.87[/C][C]91.7144641324265[/C][C]0.155535867573505[/C][/ROW]
[ROW][C]49[/C][C]92.21[/C][C]91.9257649581429[/C][C]0.284235041857087[/C][/ROW]
[ROW][C]50[/C][C]92.65[/C][C]92.3410742237418[/C][C]0.308925776258206[/C][/ROW]
[ROW][C]51[/C][C]92.83[/C][C]92.863484459188[/C][C]-0.0334844591880454[/C][/ROW]
[ROW][C]52[/C][C]93.02[/C][C]93.0362008859772[/C][C]-0.0162008859771703[/C][/ROW]
[ROW][C]53[/C][C]93.33[/C][C]93.2217889061289[/C][C]0.108211093871105[/C][/ROW]
[ROW][C]54[/C][C]93.35[/C][C]93.5600938688954[/C][C]-0.210093868895427[/C][/ROW]
[ROW][C]55[/C][C]93.45[/C][C]93.5255117852406[/C][C]-0.0755117852406073[/C][/ROW]
[ROW][C]56[/C][C]93.51[/C][C]93.604685494757[/C][C]-0.0946854947569733[/C][/ROW]
[ROW][C]57[/C][C]93.8[/C][C]93.6394831546453[/C][C]0.160516845354735[/C][/ROW]
[ROW][C]58[/C][C]93.94[/C][C]93.9711269117179[/C][C]-0.0311269117179336[/C][/ROW]
[ROW][C]59[/C][C]94.02[/C][C]94.1037414289644[/C][C]-0.0837414289643732[/C][/ROW]
[ROW][C]60[/C][C]94.26[/C][C]94.1616251176421[/C][C]0.0983748823579162[/C][/ROW]
[ROW][C]61[/C][C]94.71[/C][C]94.4270222558676[/C][C]0.28297774413241[/C][/ROW]
[ROW][C]62[/C][C]95.26[/C][C]94.9517243082581[/C][C]0.308275691741869[/C][/ROW]
[ROW][C]63[/C][C]95.54[/C][C]95.5839579254124[/C][C]-0.0439579254123998[/C][/ROW]
[ROW][C]64[/C][C]95.69[/C][C]95.8539240240602[/C][C]-0.163924024060208[/C][/ROW]
[ROW][C]65[/C][C]96.03[/C][C]95.9607136753177[/C][C]0.069286324682281[/C][/ROW]
[ROW][C]66[/C][C]96.4[/C][C]96.3180917730384[/C][C]0.0819082269615734[/C][/ROW]
[ROW][C]67[/C][C]96.55[/C][C]96.7099124420129[/C][C]-0.15991244201291[/C][/ROW]
[ROW][C]68[/C][C]96.45[/C][C]96.818365205541[/C][C]-0.368365205541039[/C][/ROW]
[ROW][C]69[/C][C]96.65[/C][C]96.620967494673[/C][C]0.0290325053269953[/C][/ROW]
[ROW][C]70[/C][C]96.84[/C][C]96.826794843373[/C][C]0.0132051566269951[/C][/ROW]
[ROW][C]71[/C][C]97.21[/C][C]97.0203994411398[/C][C]0.189600558860178[/C][/ROW]
[ROW][C]72[/C][C]97.31[/C][C]97.4401954468412[/C][C]-0.130195446841171[/C][/ROW]
[ROW][C]73[/C][C]97.91[/C][C]97.5069656052132[/C][C]0.403034394786829[/C][/ROW]
[ROW][C]74[/C][C]98.51[/C][C]98.2120489730909[/C][C]0.297951026909075[/C][/ROW]
[ROW][C]75[/C][C]98.54[/C][C]98.8921571325435[/C][C]-0.352157132543496[/C][/ROW]
[ROW][C]76[/C][C]98.52[/C][C]98.8312329748208[/C][C]-0.311232974820783[/C][/ROW]
[ROW][C]77[/C][C]98.66[/C][C]98.7278879107701[/C][C]-0.0678879107701391[/C][/ROW]
[ROW][C]78[/C][C]98.53[/C][C]98.8485704798331[/C][C]-0.318570479833085[/C][/ROW]
[ROW][C]79[/C][C]98.71[/C][C]98.6346804772291[/C][C]0.0753195227709256[/C][/ROW]
[ROW][C]80[/C][C]98.92[/C][C]98.8328905092279[/C][C]0.0871094907721357[/C][/ROW]
[ROW][C]81[/C][C]98.96[/C][C]99.0661047427919[/C][C]-0.106104742791942[/C][/ROW]
[ROW][C]82[/C][C]99.25[/C][C]99.0786964214617[/C][C]0.171303578538314[/C][/ROW]
[ROW][C]83[/C][C]99.32[/C][C]99.4131140985856[/C][C]-0.0931140985856018[/C][/ROW]
[ROW][C]84[/C][C]99.41[/C][C]99.459521830909[/C][C]-0.049521830908958[/C][/ROW]
[ROW][C]85[/C][C]99.36[/C][C]99.5360804143788[/C][C]-0.176080414378788[/C][/ROW]
[ROW][C]86[/C][C]99.58[/C][C]99.4396544604451[/C][C]0.140345539554872[/C][/ROW]
[ROW][C]87[/C][C]99.77[/C][C]99.6956122736364[/C][C]0.0743877263635824[/C][/ROW]
[ROW][C]88[/C][C]99.77[/C][C]99.9058052076173[/C][C]-0.135805207617281[/C][/ROW]
[ROW][C]89[/C][C]100.03[/C][C]99.8705447590598[/C][C]0.159455240940204[/C][/ROW]
[ROW][C]90[/C][C]100.2[/C][C]100.171710487937[/C][C]0.0282895120632389[/C][/ROW]
[ROW][C]91[/C][C]100.24[/C][C]100.349904683268[/C][C]-0.109904683267658[/C][/ROW]
[ROW][C]92[/C][C]100.1[/C][C]100.36121416592[/C][C]-0.261214165919696[/C][/ROW]
[ROW][C]93[/C][C]100.03[/C][C]100.152164705466[/C][C]-0.122164705466275[/C][/ROW]
[ROW][C]94[/C][C]100.18[/C][C]100.048853322434[/C][C]0.131146677565511[/C][/ROW]
[ROW][C]95[/C][C]100.29[/C][C]100.232659964615[/C][C]0.0573400353849394[/C][/ROW]
[ROW][C]96[/C][C]100.41[/C][C]100.358336671307[/C][C]0.051663328693266[/C][/ROW]
[ROW][C]97[/C][C]100.6[/C][C]100.492166172759[/C][C]0.107833827241237[/C][/ROW]
[ROW][C]98[/C][C]100.75[/C][C]100.710701552417[/C][C]0.0392984475831923[/C][/ROW]
[ROW][C]99[/C][C]100.79[/C][C]100.871532833247[/C][C]-0.081532833246925[/C][/ROW]
[ROW][C]100[/C][C]100.44[/C][C]100.89033763657[/C][C]-0.450337636570126[/C][/ROW]
[ROW][C]101[/C][C]100.29[/C][C]100.421819100407[/C][C]-0.131819100406645[/C][/ROW]
[ROW][C]102[/C][C]100.34[/C][C]100.235057490417[/C][C]0.104942509583339[/C][/ROW]
[ROW][C]103[/C][C]100.46[/C][C]100.311943965519[/C][C]0.148056034481186[/C][/ROW]
[ROW][C]104[/C][C]100.12[/C][C]100.47128815022[/C][C]-0.351288150220483[/C][/ROW]
[ROW][C]105[/C][C]100.06[/C][C]100.039864422283[/C][C]0.0201355777169283[/C][/ROW]
[ROW][C]106[/C][C]100.28[/C][C]99.9834410035299[/C][C]0.296558996470125[/C][/ROW]
[ROW][C]107[/C][C]100.28[/C][C]100.281325668186[/C][C]-0.00132566818624014[/C][/ROW]
[ROW][C]108[/C][C]100.4[/C][C]100.282417271968[/C][C]0.117582728032104[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284544&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284544&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	85.47	85.95	-0.480000000000018
4	85.78	85.7540967873768	0.0259032126231915
5	86.07	86.0685615175632	0.00143848243678235
6	86.05	86.3590645482303	-0.309064548230253
7	86.32	86.2580044057638	0.0619955942362225
8	86.43	86.5427657294108	-0.11276572941081
9	86.41	86.6234883540132	-0.213488354013251
10	86.38	86.546943409886	-0.166943409885988
11	86.59	86.4721182793888	0.117881720611209
12	86.68	86.7122282154099	-0.032228215409873
13	86.87	86.79434693648	0.0756530635199795
14	87.32	87.004034193985	0.315965806014972
15	87.13	87.5372786824409	-0.407278682440861
16	87.42	87.2419836658286	0.178016334171389
17	87.22	87.5767003651293	-0.356700365129313
18	87.17	87.2840024319168	-0.114002431916788
19	87.52	87.2023685655384	0.317631434461561
20	87.49	87.6351294684635	-0.145129468463509
21	87.53	87.5686038446451	-0.0386038446451096
22	87.93	87.59777374574	0.332226254259979
23	88.54	88.0847287867165	0.45527121328351
24	88.96	88.8157581115836	0.144241888416431
25	89.3	89.2758021431939	0.024197856806083
26	90.01	89.6228492673242	0.387150732675821
27	90.52	90.4345157117165	0.0854842882835101
28	90.64	90.968817104121	-0.328817104121043
29	91.25	91.0029838079597	0.247016192040334
30	91.59	91.6761798635251	-0.086179863525075
31	92.09	91.9947738991894	0.0952261008105921
32	91.81	92.5193332824413	-0.709333282441293
33	92.03	92.0537384979683	-0.0237384979683242
34	92.15	92.264069238925	-0.114069238924969
35	91.98	92.3540338518492	-0.374033851849148
36	92.11	92.0853717605185	0.024628239481487
37	92.28	92.2200163652094	0.0599836347905693
38	92.53	92.4058695053274	0.124130494672585
39	91.97	92.6887198560324	-0.718719856032365
40	92.05	91.9408032737704	0.109196726229555
41	91.87	92.0459571490189	-0.175957149018856
42	91.49	91.8203338546754	-0.330333854675388
43	91.48	91.3528338331058	0.127166166894213
44	91.63	91.3745860622512	0.255413937748813
45	91.46	91.592197909542	-0.132197909541986
46	91.61	91.3887622449815	0.221237755018535
47	91.7	91.5961509336757	0.103849066324344
48	91.87	91.7144641324265	0.155535867573505
49	92.21	91.9257649581429	0.284235041857087
50	92.65	92.3410742237418	0.308925776258206
51	92.83	92.863484459188	-0.0334844591880454
52	93.02	93.0362008859772	-0.0162008859771703
53	93.33	93.2217889061289	0.108211093871105
54	93.35	93.5600938688954	-0.210093868895427
55	93.45	93.5255117852406	-0.0755117852406073
56	93.51	93.604685494757	-0.0946854947569733
57	93.8	93.6394831546453	0.160516845354735
58	93.94	93.9711269117179	-0.0311269117179336
59	94.02	94.1037414289644	-0.0837414289643732
60	94.26	94.1616251176421	0.0983748823579162
61	94.71	94.4270222558676	0.28297774413241
62	95.26	94.9517243082581	0.308275691741869
63	95.54	95.5839579254124	-0.0439579254123998
64	95.69	95.8539240240602	-0.163924024060208
65	96.03	95.9607136753177	0.069286324682281
66	96.4	96.3180917730384	0.0819082269615734
67	96.55	96.7099124420129	-0.15991244201291
68	96.45	96.818365205541	-0.368365205541039
69	96.65	96.620967494673	0.0290325053269953
70	96.84	96.826794843373	0.0132051566269951
71	97.21	97.0203994411398	0.189600558860178
72	97.31	97.4401954468412	-0.130195446841171
73	97.91	97.5069656052132	0.403034394786829
74	98.51	98.2120489730909	0.297951026909075
75	98.54	98.8921571325435	-0.352157132543496
76	98.52	98.8312329748208	-0.311232974820783
77	98.66	98.7278879107701	-0.0678879107701391
78	98.53	98.8485704798331	-0.318570479833085
79	98.71	98.6346804772291	0.0753195227709256
80	98.92	98.8328905092279	0.0871094907721357
81	98.96	99.0661047427919	-0.106104742791942
82	99.25	99.0786964214617	0.171303578538314
83	99.32	99.4131140985856	-0.0931140985856018
84	99.41	99.459521830909	-0.049521830908958
85	99.36	99.5360804143788	-0.176080414378788
86	99.58	99.4396544604451	0.140345539554872
87	99.77	99.6956122736364	0.0743877263635824
88	99.77	99.9058052076173	-0.135805207617281
89	100.03	99.8705447590598	0.159455240940204
90	100.2	100.171710487937	0.0282895120632389
91	100.24	100.349904683268	-0.109904683267658
92	100.1	100.36121416592	-0.261214165919696
93	100.03	100.152164705466	-0.122164705466275
94	100.18	100.048853322434	0.131146677565511
95	100.29	100.232659964615	0.0573400353849394
96	100.41	100.358336671307	0.051663328693266
97	100.6	100.492166172759	0.107833827241237
98	100.75	100.710701552417	0.0392984475831923
99	100.79	100.871532833247	-0.081532833246925
100	100.44	100.89033763657	-0.450337636570126
101	100.29	100.421819100407	-0.131819100406645
102	100.34	100.235057490417	0.104942509583339
103	100.46	100.311943965519	0.148056034481186
104	100.12	100.47128815022	-0.351288150220483
105	100.06	100.039864422283	0.0201355777169283
106	100.28	99.9834410035299	0.296558996470125
107	100.28	100.281325668186	-0.00132566818624014
108	100.4	100.282417271968	0.117582728032104

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
109	100.433252594632	99.9940637125963	100.872441476667
110	100.467075867234	99.7598048659003	101.174346868567
111	100.500899139836	99.5254829912873	101.476315288384
112	100.534722412438	99.2801177915752	101.789327033301
113	100.56854568504	99.0207881960031	102.116303174077
114	100.602368957642	98.7466936537677	102.458044261517
115	100.636192230244	98.4577757823793	102.814608678109
116	100.670015502846	98.1542674817505	103.185763523942
117	100.703838775449	97.8365171699449	103.571160380952
118	100.737662048051	97.5049130365176	103.970411059584
119	100.771485320653	97.1598486068885	104.383122034417
120	100.805308593255	96.8017068858753	104.808910300635

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
109 & 100.433252594632 & 99.9940637125963 & 100.872441476667 \tabularnewline
110 & 100.467075867234 & 99.7598048659003 & 101.174346868567 \tabularnewline
111 & 100.500899139836 & 99.5254829912873 & 101.476315288384 \tabularnewline
112 & 100.534722412438 & 99.2801177915752 & 101.789327033301 \tabularnewline
113 & 100.56854568504 & 99.0207881960031 & 102.116303174077 \tabularnewline
114 & 100.602368957642 & 98.7466936537677 & 102.458044261517 \tabularnewline
115 & 100.636192230244 & 98.4577757823793 & 102.814608678109 \tabularnewline
116 & 100.670015502846 & 98.1542674817505 & 103.185763523942 \tabularnewline
117 & 100.703838775449 & 97.8365171699449 & 103.571160380952 \tabularnewline
118 & 100.737662048051 & 97.5049130365176 & 103.970411059584 \tabularnewline
119 & 100.771485320653 & 97.1598486068885 & 104.383122034417 \tabularnewline
120 & 100.805308593255 & 96.8017068858753 & 104.808910300635 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284544&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]109[/C][C]100.433252594632[/C][C]99.9940637125963[/C][C]100.872441476667[/C][/ROW]
[ROW][C]110[/C][C]100.467075867234[/C][C]99.7598048659003[/C][C]101.174346868567[/C][/ROW]
[ROW][C]111[/C][C]100.500899139836[/C][C]99.5254829912873[/C][C]101.476315288384[/C][/ROW]
[ROW][C]112[/C][C]100.534722412438[/C][C]99.2801177915752[/C][C]101.789327033301[/C][/ROW]
[ROW][C]113[/C][C]100.56854568504[/C][C]99.0207881960031[/C][C]102.116303174077[/C][/ROW]
[ROW][C]114[/C][C]100.602368957642[/C][C]98.7466936537677[/C][C]102.458044261517[/C][/ROW]
[ROW][C]115[/C][C]100.636192230244[/C][C]98.4577757823793[/C][C]102.814608678109[/C][/ROW]
[ROW][C]116[/C][C]100.670015502846[/C][C]98.1542674817505[/C][C]103.185763523942[/C][/ROW]
[ROW][C]117[/C][C]100.703838775449[/C][C]97.8365171699449[/C][C]103.571160380952[/C][/ROW]
[ROW][C]118[/C][C]100.737662048051[/C][C]97.5049130365176[/C][C]103.970411059584[/C][/ROW]
[ROW][C]119[/C][C]100.771485320653[/C][C]97.1598486068885[/C][C]104.383122034417[/C][/ROW]
[ROW][C]120[/C][C]100.805308593255[/C][C]96.8017068858753[/C][C]104.808910300635[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284544&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284544&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
109	100.433252594632	99.9940637125963	100.872441476667
110	100.467075867234	99.7598048659003	101.174346868567
111	100.500899139836	99.5254829912873	101.476315288384
112	100.534722412438	99.2801177915752	101.789327033301
113	100.56854568504	99.0207881960031	102.116303174077
114	100.602368957642	98.7466936537677	102.458044261517
115	100.636192230244	98.4577757823793	102.814608678109
116	100.670015502846	98.1542674817505	103.185763523942
117	100.703838775449	97.8365171699449	103.571160380952
118	100.737662048051	97.5049130365176	103.970411059584
119	100.771485320653	97.1598486068885	104.383122034417
120	100.805308593255	96.8017068858753	104.808910300635

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 = multiplicative ;

Parameters (R input):

par1 = 12 ; par2 = Double ; 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