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*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Wed, 25 Nov 2015 16:53:08 +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/25/t1448470443heq0m2nccu8zsrv.htm/, Retrieved Wed, 15 May 2024 13:08:12 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=284145, Retrieved Wed, 15 May 2024 13:08:12 +0000

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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] [] [2015-11-25 16:53:08] [bfd47bf74fe11b18de27163a0fd6b465] [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	1 seconds
R Server	'Herman Ole Andreas Wold' @ wold.wessa.net

\begin{tabular}{lllllllll}
\hline
Summary of computational transaction \tabularnewline
Raw Input & view raw input (R code)  \tabularnewline
Raw Output & view raw output of R engine  \tabularnewline
Computing time & 1 seconds \tabularnewline
R Server & 'Herman Ole Andreas Wold' @ wold.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284145&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]1 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Herman Ole Andreas Wold' @ wold.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284145&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284145&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	1 seconds
R Server	'Herman Ole Andreas Wold' @ wold.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.379538830805837
beta	0.092907864689236
gamma	1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.379538830805837 \tabularnewline
beta & 0.092907864689236 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284145&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]0.379538830805837[/C][/ROW]
[ROW][C]beta[/C][C]0.092907864689236[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284145&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284145&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	0.379538830805837
beta	0.092907864689236
gamma	1

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	84.45	83.9049465811966	0.545053418803363
14	84.28	83.9977747802205	0.282225219779505
15	84.28	84.1270513377549	0.152948662245066
16	84.7	84.5576557192905	0.142344280709494
17	85.04	84.9213380238772	0.118661976122851
18	85.31	85.2022995830956	0.10770041690445
19	85.18	85.4374818857083	-0.25748188570833
20	85.02	85.6837339612535	-0.663733961253527
21	86.31	85.301559584285	1.008440415715
22	86.56	85.6933500847317	0.866649915268312
23	86.4	86.5773021837546	-0.177302183754549
24	86.84	86.2354485355234	0.604551464476557
25	87.42	87.1556743014309	0.264325698569095
26	87.28	87.0652493618412	0.214750638158847
27	87.27	87.1726949369822	0.0973050630178136
28	87.73	87.6576280179042	0.0723719820958024
29	87.32	88.0596190074849	-0.739619007484947
30	87.15	88.0573233974159	-0.907323397415908
31	87.5	87.6941864092203	-0.194186409220308
32	87.43	87.7281329715949	-0.298132971594896
33	88.81	88.5508645434173	0.259135456582726
34	89.38	88.5724940255515	0.807505974448503
35	88.83	88.7863862299058	0.0436137700942254
36	88.91	89.0213978304559	-0.111397830455914
37	89.34	89.4414594928306	-0.1014594928306
38	89.56	89.1512104312291	0.408789568770857
39	89.32	89.2360380865502	0.0839619134498406
40	89.31	89.6765735976995	-0.366573597699457
41	89.45	89.3688173370082	0.08118266299185
42	88.92	89.5635955173957	-0.643595517395681
43	89.35	89.7419286659446	-0.391928665944647
44	88.89	89.6282580981932	-0.738258098193242
45	90.1	90.6061173001169	-0.506117300116898
46	90.49	90.6269705922952	-0.136970592295214
47	89.96	89.924551881328	0.0354481186719795
48	89.93	89.9761177496018	-0.0461177496017626
49	90.32	90.3452561340576	-0.025256134057642
50	90.24	90.3213400731094	-0.0813400731094305
51	90.61	89.922139666825	0.687860333175024
52	91.06	90.2371711565686	0.822828843431452
53	90.81	90.6254284273966	0.184571572603403
54	91.09	90.3801694575259	0.709830542474137
55	91.17	91.2464740227634	-0.0764740227633496
56	90.87	91.0669145469713	-0.196914546971257
57	91.92	92.4426257017117	-0.522625701711732
58	92.67	92.7340291941123	-0.0640291941123223
59	92.03	92.2166203428143	-0.18662034281428
60	92.09	92.1758101948516	-0.0858101948515753
61	92.61	92.5839439787536	0.0260560212464469
62	92.19	92.5876307446987	-0.397630744698716
63	92.68	92.5774174213312	0.102582578668844
64	92.66	92.7651905398037	-0.105190539803743
65	92.77	92.3836251585588	0.386374841441196
66	92.21	92.526387767671	-0.31638776767096
67	92.58	92.4646711370131	0.115328862986857
68	91.9	92.2392829679921	-0.339282967992105
69	93.81	93.3099517714951	0.500048228504866
70	94.05	94.261185847361	-0.211185847360994
71	94.51	93.5938180160363	0.91618198396371
72	94.49	94.0549558602258	0.435044139774234
73	94.36	94.7693920763949	-0.409392076394923
74	94.72	94.3687827053646	0.351217294635447
75	95.57	95.0034097319622	0.566590268037757
76	95.87	95.3049990367181	0.565000963281932
77	95.93	95.5730493767175	0.356950623282515
78	96.09	95.3578246719524	0.732175328047589
79	95.82	96.0881336688144	-0.2681336688144
80	96.06	95.5478076909859	0.512192309014139
81	97.09	97.6051117838665	-0.51511178386653
82	97.67	97.8366583151857	-0.166658315185728
83	98.53	97.9941467361867	0.535853263813308
84	98.12	98.1074648698839	0.0125351301161345
85	98.84	98.2177612123159	0.622238787684068
86	98.98	98.7971604910668	0.182839508933242
87	100.04	99.6121109133455	0.427889086654531
88	99.47	99.9657794679106	-0.495779467910609
89	99.84	99.7704377314633	0.0695622685367283
90	99.52	99.7371188618659	-0.217118861865913
91	99.81	99.5111753314907	0.298824668509269
92	99.55	99.7148805617133	-0.164880561713304
93	100.21	100.898618405373	-0.688618405372637
94	101.44	101.295207562923	0.144792437076561
95	101	102.032460494635	-1.03246049463463
96	101.32	101.196217674705	0.123782325294812
97	101.84	101.701330503879	0.138669496121011
98	101.81	101.781810991787	0.0281890082126353
99	101.83	102.641900706301	-0.811900706301486
100	102.18	101.859994191902	0.320005808098301
101	101.97	102.261887347421	-0.291887347420683
102	101.8	101.837604423358	-0.0376044233575072
103	101.69	101.930341202389	-0.240341202388947
104	101.91	101.55311349858	0.356886501420107
105	102.27	102.539734372477	-0.269734372477501
106	102.73	103.556987264204	-0.826987264203979
107	102.61	103.105287212059	-0.495287212059239
108	102.89	103.11958304263	-0.22958304262977

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 84.45 & 83.9049465811966 & 0.545053418803363 \tabularnewline
14 & 84.28 & 83.9977747802205 & 0.282225219779505 \tabularnewline
15 & 84.28 & 84.1270513377549 & 0.152948662245066 \tabularnewline
16 & 84.7 & 84.5576557192905 & 0.142344280709494 \tabularnewline
17 & 85.04 & 84.9213380238772 & 0.118661976122851 \tabularnewline
18 & 85.31 & 85.2022995830956 & 0.10770041690445 \tabularnewline
19 & 85.18 & 85.4374818857083 & -0.25748188570833 \tabularnewline
20 & 85.02 & 85.6837339612535 & -0.663733961253527 \tabularnewline
21 & 86.31 & 85.301559584285 & 1.008440415715 \tabularnewline
22 & 86.56 & 85.6933500847317 & 0.866649915268312 \tabularnewline
23 & 86.4 & 86.5773021837546 & -0.177302183754549 \tabularnewline
24 & 86.84 & 86.2354485355234 & 0.604551464476557 \tabularnewline
25 & 87.42 & 87.1556743014309 & 0.264325698569095 \tabularnewline
26 & 87.28 & 87.0652493618412 & 0.214750638158847 \tabularnewline
27 & 87.27 & 87.1726949369822 & 0.0973050630178136 \tabularnewline
28 & 87.73 & 87.6576280179042 & 0.0723719820958024 \tabularnewline
29 & 87.32 & 88.0596190074849 & -0.739619007484947 \tabularnewline
30 & 87.15 & 88.0573233974159 & -0.907323397415908 \tabularnewline
31 & 87.5 & 87.6941864092203 & -0.194186409220308 \tabularnewline
32 & 87.43 & 87.7281329715949 & -0.298132971594896 \tabularnewline
33 & 88.81 & 88.5508645434173 & 0.259135456582726 \tabularnewline
34 & 89.38 & 88.5724940255515 & 0.807505974448503 \tabularnewline
35 & 88.83 & 88.7863862299058 & 0.0436137700942254 \tabularnewline
36 & 88.91 & 89.0213978304559 & -0.111397830455914 \tabularnewline
37 & 89.34 & 89.4414594928306 & -0.1014594928306 \tabularnewline
38 & 89.56 & 89.1512104312291 & 0.408789568770857 \tabularnewline
39 & 89.32 & 89.2360380865502 & 0.0839619134498406 \tabularnewline
40 & 89.31 & 89.6765735976995 & -0.366573597699457 \tabularnewline
41 & 89.45 & 89.3688173370082 & 0.08118266299185 \tabularnewline
42 & 88.92 & 89.5635955173957 & -0.643595517395681 \tabularnewline
43 & 89.35 & 89.7419286659446 & -0.391928665944647 \tabularnewline
44 & 88.89 & 89.6282580981932 & -0.738258098193242 \tabularnewline
45 & 90.1 & 90.6061173001169 & -0.506117300116898 \tabularnewline
46 & 90.49 & 90.6269705922952 & -0.136970592295214 \tabularnewline
47 & 89.96 & 89.924551881328 & 0.0354481186719795 \tabularnewline
48 & 89.93 & 89.9761177496018 & -0.0461177496017626 \tabularnewline
49 & 90.32 & 90.3452561340576 & -0.025256134057642 \tabularnewline
50 & 90.24 & 90.3213400731094 & -0.0813400731094305 \tabularnewline
51 & 90.61 & 89.922139666825 & 0.687860333175024 \tabularnewline
52 & 91.06 & 90.2371711565686 & 0.822828843431452 \tabularnewline
53 & 90.81 & 90.6254284273966 & 0.184571572603403 \tabularnewline
54 & 91.09 & 90.3801694575259 & 0.709830542474137 \tabularnewline
55 & 91.17 & 91.2464740227634 & -0.0764740227633496 \tabularnewline
56 & 90.87 & 91.0669145469713 & -0.196914546971257 \tabularnewline
57 & 91.92 & 92.4426257017117 & -0.522625701711732 \tabularnewline
58 & 92.67 & 92.7340291941123 & -0.0640291941123223 \tabularnewline
59 & 92.03 & 92.2166203428143 & -0.18662034281428 \tabularnewline
60 & 92.09 & 92.1758101948516 & -0.0858101948515753 \tabularnewline
61 & 92.61 & 92.5839439787536 & 0.0260560212464469 \tabularnewline
62 & 92.19 & 92.5876307446987 & -0.397630744698716 \tabularnewline
63 & 92.68 & 92.5774174213312 & 0.102582578668844 \tabularnewline
64 & 92.66 & 92.7651905398037 & -0.105190539803743 \tabularnewline
65 & 92.77 & 92.3836251585588 & 0.386374841441196 \tabularnewline
66 & 92.21 & 92.526387767671 & -0.31638776767096 \tabularnewline
67 & 92.58 & 92.4646711370131 & 0.115328862986857 \tabularnewline
68 & 91.9 & 92.2392829679921 & -0.339282967992105 \tabularnewline
69 & 93.81 & 93.3099517714951 & 0.500048228504866 \tabularnewline
70 & 94.05 & 94.261185847361 & -0.211185847360994 \tabularnewline
71 & 94.51 & 93.5938180160363 & 0.91618198396371 \tabularnewline
72 & 94.49 & 94.0549558602258 & 0.435044139774234 \tabularnewline
73 & 94.36 & 94.7693920763949 & -0.409392076394923 \tabularnewline
74 & 94.72 & 94.3687827053646 & 0.351217294635447 \tabularnewline
75 & 95.57 & 95.0034097319622 & 0.566590268037757 \tabularnewline
76 & 95.87 & 95.3049990367181 & 0.565000963281932 \tabularnewline
77 & 95.93 & 95.5730493767175 & 0.356950623282515 \tabularnewline
78 & 96.09 & 95.3578246719524 & 0.732175328047589 \tabularnewline
79 & 95.82 & 96.0881336688144 & -0.2681336688144 \tabularnewline
80 & 96.06 & 95.5478076909859 & 0.512192309014139 \tabularnewline
81 & 97.09 & 97.6051117838665 & -0.51511178386653 \tabularnewline
82 & 97.67 & 97.8366583151857 & -0.166658315185728 \tabularnewline
83 & 98.53 & 97.9941467361867 & 0.535853263813308 \tabularnewline
84 & 98.12 & 98.1074648698839 & 0.0125351301161345 \tabularnewline
85 & 98.84 & 98.2177612123159 & 0.622238787684068 \tabularnewline
86 & 98.98 & 98.7971604910668 & 0.182839508933242 \tabularnewline
87 & 100.04 & 99.6121109133455 & 0.427889086654531 \tabularnewline
88 & 99.47 & 99.9657794679106 & -0.495779467910609 \tabularnewline
89 & 99.84 & 99.7704377314633 & 0.0695622685367283 \tabularnewline
90 & 99.52 & 99.7371188618659 & -0.217118861865913 \tabularnewline
91 & 99.81 & 99.5111753314907 & 0.298824668509269 \tabularnewline
92 & 99.55 & 99.7148805617133 & -0.164880561713304 \tabularnewline
93 & 100.21 & 100.898618405373 & -0.688618405372637 \tabularnewline
94 & 101.44 & 101.295207562923 & 0.144792437076561 \tabularnewline
95 & 101 & 102.032460494635 & -1.03246049463463 \tabularnewline
96 & 101.32 & 101.196217674705 & 0.123782325294812 \tabularnewline
97 & 101.84 & 101.701330503879 & 0.138669496121011 \tabularnewline
98 & 101.81 & 101.781810991787 & 0.0281890082126353 \tabularnewline
99 & 101.83 & 102.641900706301 & -0.811900706301486 \tabularnewline
100 & 102.18 & 101.859994191902 & 0.320005808098301 \tabularnewline
101 & 101.97 & 102.261887347421 & -0.291887347420683 \tabularnewline
102 & 101.8 & 101.837604423358 & -0.0376044233575072 \tabularnewline
103 & 101.69 & 101.930341202389 & -0.240341202388947 \tabularnewline
104 & 101.91 & 101.55311349858 & 0.356886501420107 \tabularnewline
105 & 102.27 & 102.539734372477 & -0.269734372477501 \tabularnewline
106 & 102.73 & 103.556987264204 & -0.826987264203979 \tabularnewline
107 & 102.61 & 103.105287212059 & -0.495287212059239 \tabularnewline
108 & 102.89 & 103.11958304263 & -0.22958304262977 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284145&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]84.45[/C][C]83.9049465811966[/C][C]0.545053418803363[/C][/ROW]
[ROW][C]14[/C][C]84.28[/C][C]83.9977747802205[/C][C]0.282225219779505[/C][/ROW]
[ROW][C]15[/C][C]84.28[/C][C]84.1270513377549[/C][C]0.152948662245066[/C][/ROW]
[ROW][C]16[/C][C]84.7[/C][C]84.5576557192905[/C][C]0.142344280709494[/C][/ROW]
[ROW][C]17[/C][C]85.04[/C][C]84.9213380238772[/C][C]0.118661976122851[/C][/ROW]
[ROW][C]18[/C][C]85.31[/C][C]85.2022995830956[/C][C]0.10770041690445[/C][/ROW]
[ROW][C]19[/C][C]85.18[/C][C]85.4374818857083[/C][C]-0.25748188570833[/C][/ROW]
[ROW][C]20[/C][C]85.02[/C][C]85.6837339612535[/C][C]-0.663733961253527[/C][/ROW]
[ROW][C]21[/C][C]86.31[/C][C]85.301559584285[/C][C]1.008440415715[/C][/ROW]
[ROW][C]22[/C][C]86.56[/C][C]85.6933500847317[/C][C]0.866649915268312[/C][/ROW]
[ROW][C]23[/C][C]86.4[/C][C]86.5773021837546[/C][C]-0.177302183754549[/C][/ROW]
[ROW][C]24[/C][C]86.84[/C][C]86.2354485355234[/C][C]0.604551464476557[/C][/ROW]
[ROW][C]25[/C][C]87.42[/C][C]87.1556743014309[/C][C]0.264325698569095[/C][/ROW]
[ROW][C]26[/C][C]87.28[/C][C]87.0652493618412[/C][C]0.214750638158847[/C][/ROW]
[ROW][C]27[/C][C]87.27[/C][C]87.1726949369822[/C][C]0.0973050630178136[/C][/ROW]
[ROW][C]28[/C][C]87.73[/C][C]87.6576280179042[/C][C]0.0723719820958024[/C][/ROW]
[ROW][C]29[/C][C]87.32[/C][C]88.0596190074849[/C][C]-0.739619007484947[/C][/ROW]
[ROW][C]30[/C][C]87.15[/C][C]88.0573233974159[/C][C]-0.907323397415908[/C][/ROW]
[ROW][C]31[/C][C]87.5[/C][C]87.6941864092203[/C][C]-0.194186409220308[/C][/ROW]
[ROW][C]32[/C][C]87.43[/C][C]87.7281329715949[/C][C]-0.298132971594896[/C][/ROW]
[ROW][C]33[/C][C]88.81[/C][C]88.5508645434173[/C][C]0.259135456582726[/C][/ROW]
[ROW][C]34[/C][C]89.38[/C][C]88.5724940255515[/C][C]0.807505974448503[/C][/ROW]
[ROW][C]35[/C][C]88.83[/C][C]88.7863862299058[/C][C]0.0436137700942254[/C][/ROW]
[ROW][C]36[/C][C]88.91[/C][C]89.0213978304559[/C][C]-0.111397830455914[/C][/ROW]
[ROW][C]37[/C][C]89.34[/C][C]89.4414594928306[/C][C]-0.1014594928306[/C][/ROW]
[ROW][C]38[/C][C]89.56[/C][C]89.1512104312291[/C][C]0.408789568770857[/C][/ROW]
[ROW][C]39[/C][C]89.32[/C][C]89.2360380865502[/C][C]0.0839619134498406[/C][/ROW]
[ROW][C]40[/C][C]89.31[/C][C]89.6765735976995[/C][C]-0.366573597699457[/C][/ROW]
[ROW][C]41[/C][C]89.45[/C][C]89.3688173370082[/C][C]0.08118266299185[/C][/ROW]
[ROW][C]42[/C][C]88.92[/C][C]89.5635955173957[/C][C]-0.643595517395681[/C][/ROW]
[ROW][C]43[/C][C]89.35[/C][C]89.7419286659446[/C][C]-0.391928665944647[/C][/ROW]
[ROW][C]44[/C][C]88.89[/C][C]89.6282580981932[/C][C]-0.738258098193242[/C][/ROW]
[ROW][C]45[/C][C]90.1[/C][C]90.6061173001169[/C][C]-0.506117300116898[/C][/ROW]
[ROW][C]46[/C][C]90.49[/C][C]90.6269705922952[/C][C]-0.136970592295214[/C][/ROW]
[ROW][C]47[/C][C]89.96[/C][C]89.924551881328[/C][C]0.0354481186719795[/C][/ROW]
[ROW][C]48[/C][C]89.93[/C][C]89.9761177496018[/C][C]-0.0461177496017626[/C][/ROW]
[ROW][C]49[/C][C]90.32[/C][C]90.3452561340576[/C][C]-0.025256134057642[/C][/ROW]
[ROW][C]50[/C][C]90.24[/C][C]90.3213400731094[/C][C]-0.0813400731094305[/C][/ROW]
[ROW][C]51[/C][C]90.61[/C][C]89.922139666825[/C][C]0.687860333175024[/C][/ROW]
[ROW][C]52[/C][C]91.06[/C][C]90.2371711565686[/C][C]0.822828843431452[/C][/ROW]
[ROW][C]53[/C][C]90.81[/C][C]90.6254284273966[/C][C]0.184571572603403[/C][/ROW]
[ROW][C]54[/C][C]91.09[/C][C]90.3801694575259[/C][C]0.709830542474137[/C][/ROW]
[ROW][C]55[/C][C]91.17[/C][C]91.2464740227634[/C][C]-0.0764740227633496[/C][/ROW]
[ROW][C]56[/C][C]90.87[/C][C]91.0669145469713[/C][C]-0.196914546971257[/C][/ROW]
[ROW][C]57[/C][C]91.92[/C][C]92.4426257017117[/C][C]-0.522625701711732[/C][/ROW]
[ROW][C]58[/C][C]92.67[/C][C]92.7340291941123[/C][C]-0.0640291941123223[/C][/ROW]
[ROW][C]59[/C][C]92.03[/C][C]92.2166203428143[/C][C]-0.18662034281428[/C][/ROW]
[ROW][C]60[/C][C]92.09[/C][C]92.1758101948516[/C][C]-0.0858101948515753[/C][/ROW]
[ROW][C]61[/C][C]92.61[/C][C]92.5839439787536[/C][C]0.0260560212464469[/C][/ROW]
[ROW][C]62[/C][C]92.19[/C][C]92.5876307446987[/C][C]-0.397630744698716[/C][/ROW]
[ROW][C]63[/C][C]92.68[/C][C]92.5774174213312[/C][C]0.102582578668844[/C][/ROW]
[ROW][C]64[/C][C]92.66[/C][C]92.7651905398037[/C][C]-0.105190539803743[/C][/ROW]
[ROW][C]65[/C][C]92.77[/C][C]92.3836251585588[/C][C]0.386374841441196[/C][/ROW]
[ROW][C]66[/C][C]92.21[/C][C]92.526387767671[/C][C]-0.31638776767096[/C][/ROW]
[ROW][C]67[/C][C]92.58[/C][C]92.4646711370131[/C][C]0.115328862986857[/C][/ROW]
[ROW][C]68[/C][C]91.9[/C][C]92.2392829679921[/C][C]-0.339282967992105[/C][/ROW]
[ROW][C]69[/C][C]93.81[/C][C]93.3099517714951[/C][C]0.500048228504866[/C][/ROW]
[ROW][C]70[/C][C]94.05[/C][C]94.261185847361[/C][C]-0.211185847360994[/C][/ROW]
[ROW][C]71[/C][C]94.51[/C][C]93.5938180160363[/C][C]0.91618198396371[/C][/ROW]
[ROW][C]72[/C][C]94.49[/C][C]94.0549558602258[/C][C]0.435044139774234[/C][/ROW]
[ROW][C]73[/C][C]94.36[/C][C]94.7693920763949[/C][C]-0.409392076394923[/C][/ROW]
[ROW][C]74[/C][C]94.72[/C][C]94.3687827053646[/C][C]0.351217294635447[/C][/ROW]
[ROW][C]75[/C][C]95.57[/C][C]95.0034097319622[/C][C]0.566590268037757[/C][/ROW]
[ROW][C]76[/C][C]95.87[/C][C]95.3049990367181[/C][C]0.565000963281932[/C][/ROW]
[ROW][C]77[/C][C]95.93[/C][C]95.5730493767175[/C][C]0.356950623282515[/C][/ROW]
[ROW][C]78[/C][C]96.09[/C][C]95.3578246719524[/C][C]0.732175328047589[/C][/ROW]
[ROW][C]79[/C][C]95.82[/C][C]96.0881336688144[/C][C]-0.2681336688144[/C][/ROW]
[ROW][C]80[/C][C]96.06[/C][C]95.5478076909859[/C][C]0.512192309014139[/C][/ROW]
[ROW][C]81[/C][C]97.09[/C][C]97.6051117838665[/C][C]-0.51511178386653[/C][/ROW]
[ROW][C]82[/C][C]97.67[/C][C]97.8366583151857[/C][C]-0.166658315185728[/C][/ROW]
[ROW][C]83[/C][C]98.53[/C][C]97.9941467361867[/C][C]0.535853263813308[/C][/ROW]
[ROW][C]84[/C][C]98.12[/C][C]98.1074648698839[/C][C]0.0125351301161345[/C][/ROW]
[ROW][C]85[/C][C]98.84[/C][C]98.2177612123159[/C][C]0.622238787684068[/C][/ROW]
[ROW][C]86[/C][C]98.98[/C][C]98.7971604910668[/C][C]0.182839508933242[/C][/ROW]
[ROW][C]87[/C][C]100.04[/C][C]99.6121109133455[/C][C]0.427889086654531[/C][/ROW]
[ROW][C]88[/C][C]99.47[/C][C]99.9657794679106[/C][C]-0.495779467910609[/C][/ROW]
[ROW][C]89[/C][C]99.84[/C][C]99.7704377314633[/C][C]0.0695622685367283[/C][/ROW]
[ROW][C]90[/C][C]99.52[/C][C]99.7371188618659[/C][C]-0.217118861865913[/C][/ROW]
[ROW][C]91[/C][C]99.81[/C][C]99.5111753314907[/C][C]0.298824668509269[/C][/ROW]
[ROW][C]92[/C][C]99.55[/C][C]99.7148805617133[/C][C]-0.164880561713304[/C][/ROW]
[ROW][C]93[/C][C]100.21[/C][C]100.898618405373[/C][C]-0.688618405372637[/C][/ROW]
[ROW][C]94[/C][C]101.44[/C][C]101.295207562923[/C][C]0.144792437076561[/C][/ROW]
[ROW][C]95[/C][C]101[/C][C]102.032460494635[/C][C]-1.03246049463463[/C][/ROW]
[ROW][C]96[/C][C]101.32[/C][C]101.196217674705[/C][C]0.123782325294812[/C][/ROW]
[ROW][C]97[/C][C]101.84[/C][C]101.701330503879[/C][C]0.138669496121011[/C][/ROW]
[ROW][C]98[/C][C]101.81[/C][C]101.781810991787[/C][C]0.0281890082126353[/C][/ROW]
[ROW][C]99[/C][C]101.83[/C][C]102.641900706301[/C][C]-0.811900706301486[/C][/ROW]
[ROW][C]100[/C][C]102.18[/C][C]101.859994191902[/C][C]0.320005808098301[/C][/ROW]
[ROW][C]101[/C][C]101.97[/C][C]102.261887347421[/C][C]-0.291887347420683[/C][/ROW]
[ROW][C]102[/C][C]101.8[/C][C]101.837604423358[/C][C]-0.0376044233575072[/C][/ROW]
[ROW][C]103[/C][C]101.69[/C][C]101.930341202389[/C][C]-0.240341202388947[/C][/ROW]
[ROW][C]104[/C][C]101.91[/C][C]101.55311349858[/C][C]0.356886501420107[/C][/ROW]
[ROW][C]105[/C][C]102.27[/C][C]102.539734372477[/C][C]-0.269734372477501[/C][/ROW]
[ROW][C]106[/C][C]102.73[/C][C]103.556987264204[/C][C]-0.826987264203979[/C][/ROW]
[ROW][C]107[/C][C]102.61[/C][C]103.105287212059[/C][C]-0.495287212059239[/C][/ROW]
[ROW][C]108[/C][C]102.89[/C][C]103.11958304263[/C][C]-0.22958304262977[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284145&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284145&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	84.45	83.9049465811966	0.545053418803363
14	84.28	83.9977747802205	0.282225219779505
15	84.28	84.1270513377549	0.152948662245066
16	84.7	84.5576557192905	0.142344280709494
17	85.04	84.9213380238772	0.118661976122851
18	85.31	85.2022995830956	0.10770041690445
19	85.18	85.4374818857083	-0.25748188570833
20	85.02	85.6837339612535	-0.663733961253527
21	86.31	85.301559584285	1.008440415715
22	86.56	85.6933500847317	0.866649915268312
23	86.4	86.5773021837546	-0.177302183754549
24	86.84	86.2354485355234	0.604551464476557
25	87.42	87.1556743014309	0.264325698569095
26	87.28	87.0652493618412	0.214750638158847
27	87.27	87.1726949369822	0.0973050630178136
28	87.73	87.6576280179042	0.0723719820958024
29	87.32	88.0596190074849	-0.739619007484947
30	87.15	88.0573233974159	-0.907323397415908
31	87.5	87.6941864092203	-0.194186409220308
32	87.43	87.7281329715949	-0.298132971594896
33	88.81	88.5508645434173	0.259135456582726
34	89.38	88.5724940255515	0.807505974448503
35	88.83	88.7863862299058	0.0436137700942254
36	88.91	89.0213978304559	-0.111397830455914
37	89.34	89.4414594928306	-0.1014594928306
38	89.56	89.1512104312291	0.408789568770857
39	89.32	89.2360380865502	0.0839619134498406
40	89.31	89.6765735976995	-0.366573597699457
41	89.45	89.3688173370082	0.08118266299185
42	88.92	89.5635955173957	-0.643595517395681
43	89.35	89.7419286659446	-0.391928665944647
44	88.89	89.6282580981932	-0.738258098193242
45	90.1	90.6061173001169	-0.506117300116898
46	90.49	90.6269705922952	-0.136970592295214
47	89.96	89.924551881328	0.0354481186719795
48	89.93	89.9761177496018	-0.0461177496017626
49	90.32	90.3452561340576	-0.025256134057642
50	90.24	90.3213400731094	-0.0813400731094305
51	90.61	89.922139666825	0.687860333175024
52	91.06	90.2371711565686	0.822828843431452
53	90.81	90.6254284273966	0.184571572603403
54	91.09	90.3801694575259	0.709830542474137
55	91.17	91.2464740227634	-0.0764740227633496
56	90.87	91.0669145469713	-0.196914546971257
57	91.92	92.4426257017117	-0.522625701711732
58	92.67	92.7340291941123	-0.0640291941123223
59	92.03	92.2166203428143	-0.18662034281428
60	92.09	92.1758101948516	-0.0858101948515753
61	92.61	92.5839439787536	0.0260560212464469
62	92.19	92.5876307446987	-0.397630744698716
63	92.68	92.5774174213312	0.102582578668844
64	92.66	92.7651905398037	-0.105190539803743
65	92.77	92.3836251585588	0.386374841441196
66	92.21	92.526387767671	-0.31638776767096
67	92.58	92.4646711370131	0.115328862986857
68	91.9	92.2392829679921	-0.339282967992105
69	93.81	93.3099517714951	0.500048228504866
70	94.05	94.261185847361	-0.211185847360994
71	94.51	93.5938180160363	0.91618198396371
72	94.49	94.0549558602258	0.435044139774234
73	94.36	94.7693920763949	-0.409392076394923
74	94.72	94.3687827053646	0.351217294635447
75	95.57	95.0034097319622	0.566590268037757
76	95.87	95.3049990367181	0.565000963281932
77	95.93	95.5730493767175	0.356950623282515
78	96.09	95.3578246719524	0.732175328047589
79	95.82	96.0881336688144	-0.2681336688144
80	96.06	95.5478076909859	0.512192309014139
81	97.09	97.6051117838665	-0.51511178386653
82	97.67	97.8366583151857	-0.166658315185728
83	98.53	97.9941467361867	0.535853263813308
84	98.12	98.1074648698839	0.0125351301161345
85	98.84	98.2177612123159	0.622238787684068
86	98.98	98.7971604910668	0.182839508933242
87	100.04	99.6121109133455	0.427889086654531
88	99.47	99.9657794679106	-0.495779467910609
89	99.84	99.7704377314633	0.0695622685367283
90	99.52	99.7371188618659	-0.217118861865913
91	99.81	99.5111753314907	0.298824668509269
92	99.55	99.7148805617133	-0.164880561713304
93	100.21	100.898618405373	-0.688618405372637
94	101.44	101.295207562923	0.144792437076561
95	101	102.032460494635	-1.03246049463463
96	101.32	101.196217674705	0.123782325294812
97	101.84	101.701330503879	0.138669496121011
98	101.81	101.781810991787	0.0281890082126353
99	101.83	102.641900706301	-0.811900706301486
100	102.18	101.859994191902	0.320005808098301
101	101.97	102.261887347421	-0.291887347420683
102	101.8	101.837604423358	-0.0376044233575072
103	101.69	101.930341202389	-0.240341202388947
104	101.91	101.55311349858	0.356886501420107
105	102.27	102.539734372477	-0.269734372477501
106	102.73	103.556987264204	-0.826987264203979
107	102.61	103.105287212059	-0.495287212059239
108	102.89	103.11958304263	-0.22958304262977

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
109	103.416613243696	102.555721420092	104.277505067299
110	103.287820976032	102.355804741855	104.219837210209
111	103.526881371567	102.51753695745	104.536225785683
112	103.694968650314	102.602566036757	104.787371263871
113	103.524009051531	102.343246876618	104.704771226444
114	103.306831782246	102.032788314129	104.580875250364
115	103.227927005566	101.856012691008	104.599841320123
116	103.260826070144	101.786740326791	104.734911813497
117	103.658967505949	102.078660726554	105.239274285345
118	104.378119464579	102.687760277079	106.06847865208
119	104.420539715861	102.616487095343	106.224592336378
120	104.779579805506	102.858359458166	106.700800152845

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
109 & 103.416613243696 & 102.555721420092 & 104.277505067299 \tabularnewline
110 & 103.287820976032 & 102.355804741855 & 104.219837210209 \tabularnewline
111 & 103.526881371567 & 102.51753695745 & 104.536225785683 \tabularnewline
112 & 103.694968650314 & 102.602566036757 & 104.787371263871 \tabularnewline
113 & 103.524009051531 & 102.343246876618 & 104.704771226444 \tabularnewline
114 & 103.306831782246 & 102.032788314129 & 104.580875250364 \tabularnewline
115 & 103.227927005566 & 101.856012691008 & 104.599841320123 \tabularnewline
116 & 103.260826070144 & 101.786740326791 & 104.734911813497 \tabularnewline
117 & 103.658967505949 & 102.078660726554 & 105.239274285345 \tabularnewline
118 & 104.378119464579 & 102.687760277079 & 106.06847865208 \tabularnewline
119 & 104.420539715861 & 102.616487095343 & 106.224592336378 \tabularnewline
120 & 104.779579805506 & 102.858359458166 & 106.700800152845 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284145&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]103.416613243696[/C][C]102.555721420092[/C][C]104.277505067299[/C][/ROW]
[ROW][C]110[/C][C]103.287820976032[/C][C]102.355804741855[/C][C]104.219837210209[/C][/ROW]
[ROW][C]111[/C][C]103.526881371567[/C][C]102.51753695745[/C][C]104.536225785683[/C][/ROW]
[ROW][C]112[/C][C]103.694968650314[/C][C]102.602566036757[/C][C]104.787371263871[/C][/ROW]
[ROW][C]113[/C][C]103.524009051531[/C][C]102.343246876618[/C][C]104.704771226444[/C][/ROW]
[ROW][C]114[/C][C]103.306831782246[/C][C]102.032788314129[/C][C]104.580875250364[/C][/ROW]
[ROW][C]115[/C][C]103.227927005566[/C][C]101.856012691008[/C][C]104.599841320123[/C][/ROW]
[ROW][C]116[/C][C]103.260826070144[/C][C]101.786740326791[/C][C]104.734911813497[/C][/ROW]
[ROW][C]117[/C][C]103.658967505949[/C][C]102.078660726554[/C][C]105.239274285345[/C][/ROW]
[ROW][C]118[/C][C]104.378119464579[/C][C]102.687760277079[/C][C]106.06847865208[/C][/ROW]
[ROW][C]119[/C][C]104.420539715861[/C][C]102.616487095343[/C][C]106.224592336378[/C][/ROW]
[ROW][C]120[/C][C]104.779579805506[/C][C]102.858359458166[/C][C]106.700800152845[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284145&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284145&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	103.416613243696	102.555721420092	104.277505067299
110	103.287820976032	102.355804741855	104.219837210209
111	103.526881371567	102.51753695745	104.536225785683
112	103.694968650314	102.602566036757	104.787371263871
113	103.524009051531	102.343246876618	104.704771226444
114	103.306831782246	102.032788314129	104.580875250364
115	103.227927005566	101.856012691008	104.599841320123
116	103.260826070144	101.786740326791	104.734911813497
117	103.658967505949	102.078660726554	105.239274285345
118	104.378119464579	102.687760277079	106.06847865208
119	104.420539715861	102.616487095343	106.224592336378
120	104.779579805506	102.858359458166	106.700800152845

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 = additive ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; 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