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Author

*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Thu, 26 Nov 2015 12:36:13 +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/26/t1448541404j9fsccvxz0mdb48.htm/, Retrieved Wed, 11 Feb 2026 15:07:40 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=284211, Retrieved Wed, 11 Feb 2026 15:07:40 +0000

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IsPrivate?

No (this computation is public)

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Estimated Impact

340

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

-       [Exponential Smoothing] [] [2015-11-26 12:36:13] [237b8e3b7b7bc12136ba0893525d9132] [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	'Gwilym Jenkins' @ jenkins.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 & 'Gwilym Jenkins' @ jenkins.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284211&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]'Gwilym Jenkins' @ jenkins.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284211&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284211&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	'Gwilym Jenkins' @ jenkins.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.884123289345126
beta	0.213481062357375
gamma	1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.884123289345126 \tabularnewline
beta & 0.213481062357375 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284211&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]0.884123289345126[/C][/ROW]
[ROW][C]beta[/C][C]0.213481062357375[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284211&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284211&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	0.884123289345126
beta	0.213481062357375
gamma	1

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	77.15	80.3096260683761	-3.15962606837611
14	76.33	76.3894445164792	-0.0594445164792035
15	70.19	69.8239859050165	0.366014094983527
16	68.42	68.0191846375422	0.400815362457791
17	66.49	65.8162199738616	0.673780026138417
18	63.41	62.4621780468113	0.947821953188694
19	62.92	68.8201516108171	-5.90015161081708
20	65.53	61.4242231966847	4.10577680331528
21	65.26	64.7580414670371	0.501958532962902
22	68.25	65.0941315240568	3.15586847594324
23	74.39	68.3572550806957	6.03274491930435
24	78.71	76.4754506357966	2.2345493642034
25	82.15	79.1266194787051	3.02338052129491
26	86.05	83.9545552291246	2.09544477087545
27	89.46	82.6726450211182	6.78735497888178
28	89.32	91.0901801492725	-1.77018014927252
29	88.94	91.1307032918918	-2.19070329189177
30	93.35	88.8664924468356	4.4835075531644
31	94.72	101.814897715236	-7.09489771523614
32	96.11	98.5545902340917	-2.44459023409175
33	104.06	98.475607844204	5.584392155796
34	104.11	107.368128898875	-3.2581288988746
35	103.9	107.83865686781	-3.93865686781027
36	110.75	107.363549296523	3.38645070347705
37	110.82	112.004729910127	-1.18472991012688
38	107.59	113.090579067478	-5.50057906747804
39	96.03	104.288757659191	-8.25875765919146
40	95.69	94.2244252776737	1.46557472232629
41	90.63	93.5001238875273	-2.87012388752731
42	75.87	87.7034688598038	-11.8334688598038
43	75.57	78.0991250152341	-2.52912501523409
44	78.78	73.4912833896202	5.28871661037979
45	74.93	76.7163793088096	-1.78637930880964
46	75.85	72.2129111657573	3.63708883424269
47	75.49	74.1475562167618	1.3424437832382
48	76.87	75.6339277749369	1.2360722250631
49	78.18	73.881870871137	4.29812912886302
50	79.37	76.3876469811253	2.9823530188747
51	80.59	73.4397836807535	7.15021631924648
52	81.18	79.7076615576134	1.47233844238659
53	81.02	80.0701640947689	0.94983590523114
54	82.75	78.9164048936744	3.83359510632562
55	83.63	89.5031150044683	-5.87311500446826
56	85.35	87.4748041508472	-2.12480415084725
57	90.52	86.5564648531823	3.96353514681772
58	90.66	92.0812130613927	-1.42121306139275
59	90.69	92.6392070980686	-1.9492070980686
60	92.56	93.9431569138225	-1.38315691382255
61	92.87	92.4759663763942	0.394033623605807
62	93.82	92.8864667264436	0.933533273556435
63	96.32	89.7323446993533	6.58765530064666
64	96.03	95.8609279221096	0.169072077890434
65	96.53	95.780665736759	0.749334263240968
66	102.96	95.5159847953348	7.44401520466521
67	102.38	109.58359927212	-7.20359927211965
68	102.66	107.975827372673	-5.31582737267348
69	106.83	105.501950864233	1.32804913576658
70	106.5	108.135430544277	-1.63543054427686
71	106.78	108.465208471736	-1.68520847173562
72	108.49	110.140346469005	-1.65034646900489
73	108.77	108.664620918124	0.105379081876308
74	110.43	108.849707320041	1.58029267995924
75	110.84	107.011929904003	3.82807009599699
76	110.52	109.525429774328	0.994570225671524
77	110.11	109.966550581871	0.143449418128753
78	109.42	109.551895452204	-0.131895452203793
79	109.06	113.404194110299	-4.34419411029916
80	108.98	113.26297268606	-4.28297268606048
81	108.36	111.386817286358	-3.02681728635794
82	108.11	107.919386458338	0.190613541662231
83	108.44	108.295225078498	0.144774921501835
84	107.76	110.375112013764	-2.61511201376371
85	106.27	106.850547482494	-0.580547482493586
86	101.07	105.071319140424	-4.00131914042356
87	100.79	95.9769011533363	4.81309884666372
88	100.97	96.6365964682487	4.33340353175134
89	99.33	98.1648610595406	1.16513894045936
90	99.35	97.0482652884637	2.30173471153626
91	99.23	101.45008372169	-2.22008372168955
92	98.14	102.480842078008	-4.34084207800787
93	98.17	99.9750698603638	-1.8050698603638
94	98.48	97.4672243717983	1.01277562820169
95	99	98.2264065118693	0.773593488130658
96	99.19	100.322887933783	-1.13288793378341
97	99.1	98.4047591219387	0.695240878061298
98	100.13	97.6581023201761	2.47189767982394
99	100.07	96.8309750603315	3.23902493966852
100	95.26	97.2690963335927	-2.00909633359274
101	94.72	92.8512617510251	1.86873824897494
102	94.25	92.6498200950695	1.60017990493049
103	89.46	95.9363707744928	-6.47637077449282
104	88.38	92.183919902685	-3.80391990268497
105	88.57	89.7736504252291	-1.2036504252291
106	93.82	87.5645309858545	6.25546901414546
107	93.94	93.3611839748699	0.578816025130095
108	93.92	95.4577774810964	-1.53777748109643

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 77.15 & 80.3096260683761 & -3.15962606837611 \tabularnewline
14 & 76.33 & 76.3894445164792 & -0.0594445164792035 \tabularnewline
15 & 70.19 & 69.8239859050165 & 0.366014094983527 \tabularnewline
16 & 68.42 & 68.0191846375422 & 0.400815362457791 \tabularnewline
17 & 66.49 & 65.8162199738616 & 0.673780026138417 \tabularnewline
18 & 63.41 & 62.4621780468113 & 0.947821953188694 \tabularnewline
19 & 62.92 & 68.8201516108171 & -5.90015161081708 \tabularnewline
20 & 65.53 & 61.4242231966847 & 4.10577680331528 \tabularnewline
21 & 65.26 & 64.7580414670371 & 0.501958532962902 \tabularnewline
22 & 68.25 & 65.0941315240568 & 3.15586847594324 \tabularnewline
23 & 74.39 & 68.3572550806957 & 6.03274491930435 \tabularnewline
24 & 78.71 & 76.4754506357966 & 2.2345493642034 \tabularnewline
25 & 82.15 & 79.1266194787051 & 3.02338052129491 \tabularnewline
26 & 86.05 & 83.9545552291246 & 2.09544477087545 \tabularnewline
27 & 89.46 & 82.6726450211182 & 6.78735497888178 \tabularnewline
28 & 89.32 & 91.0901801492725 & -1.77018014927252 \tabularnewline
29 & 88.94 & 91.1307032918918 & -2.19070329189177 \tabularnewline
30 & 93.35 & 88.8664924468356 & 4.4835075531644 \tabularnewline
31 & 94.72 & 101.814897715236 & -7.09489771523614 \tabularnewline
32 & 96.11 & 98.5545902340917 & -2.44459023409175 \tabularnewline
33 & 104.06 & 98.475607844204 & 5.584392155796 \tabularnewline
34 & 104.11 & 107.368128898875 & -3.2581288988746 \tabularnewline
35 & 103.9 & 107.83865686781 & -3.93865686781027 \tabularnewline
36 & 110.75 & 107.363549296523 & 3.38645070347705 \tabularnewline
37 & 110.82 & 112.004729910127 & -1.18472991012688 \tabularnewline
38 & 107.59 & 113.090579067478 & -5.50057906747804 \tabularnewline
39 & 96.03 & 104.288757659191 & -8.25875765919146 \tabularnewline
40 & 95.69 & 94.2244252776737 & 1.46557472232629 \tabularnewline
41 & 90.63 & 93.5001238875273 & -2.87012388752731 \tabularnewline
42 & 75.87 & 87.7034688598038 & -11.8334688598038 \tabularnewline
43 & 75.57 & 78.0991250152341 & -2.52912501523409 \tabularnewline
44 & 78.78 & 73.4912833896202 & 5.28871661037979 \tabularnewline
45 & 74.93 & 76.7163793088096 & -1.78637930880964 \tabularnewline
46 & 75.85 & 72.2129111657573 & 3.63708883424269 \tabularnewline
47 & 75.49 & 74.1475562167618 & 1.3424437832382 \tabularnewline
48 & 76.87 & 75.6339277749369 & 1.2360722250631 \tabularnewline
49 & 78.18 & 73.881870871137 & 4.29812912886302 \tabularnewline
50 & 79.37 & 76.3876469811253 & 2.9823530188747 \tabularnewline
51 & 80.59 & 73.4397836807535 & 7.15021631924648 \tabularnewline
52 & 81.18 & 79.7076615576134 & 1.47233844238659 \tabularnewline
53 & 81.02 & 80.0701640947689 & 0.94983590523114 \tabularnewline
54 & 82.75 & 78.9164048936744 & 3.83359510632562 \tabularnewline
55 & 83.63 & 89.5031150044683 & -5.87311500446826 \tabularnewline
56 & 85.35 & 87.4748041508472 & -2.12480415084725 \tabularnewline
57 & 90.52 & 86.5564648531823 & 3.96353514681772 \tabularnewline
58 & 90.66 & 92.0812130613927 & -1.42121306139275 \tabularnewline
59 & 90.69 & 92.6392070980686 & -1.9492070980686 \tabularnewline
60 & 92.56 & 93.9431569138225 & -1.38315691382255 \tabularnewline
61 & 92.87 & 92.4759663763942 & 0.394033623605807 \tabularnewline
62 & 93.82 & 92.8864667264436 & 0.933533273556435 \tabularnewline
63 & 96.32 & 89.7323446993533 & 6.58765530064666 \tabularnewline
64 & 96.03 & 95.8609279221096 & 0.169072077890434 \tabularnewline
65 & 96.53 & 95.780665736759 & 0.749334263240968 \tabularnewline
66 & 102.96 & 95.5159847953348 & 7.44401520466521 \tabularnewline
67 & 102.38 & 109.58359927212 & -7.20359927211965 \tabularnewline
68 & 102.66 & 107.975827372673 & -5.31582737267348 \tabularnewline
69 & 106.83 & 105.501950864233 & 1.32804913576658 \tabularnewline
70 & 106.5 & 108.135430544277 & -1.63543054427686 \tabularnewline
71 & 106.78 & 108.465208471736 & -1.68520847173562 \tabularnewline
72 & 108.49 & 110.140346469005 & -1.65034646900489 \tabularnewline
73 & 108.77 & 108.664620918124 & 0.105379081876308 \tabularnewline
74 & 110.43 & 108.849707320041 & 1.58029267995924 \tabularnewline
75 & 110.84 & 107.011929904003 & 3.82807009599699 \tabularnewline
76 & 110.52 & 109.525429774328 & 0.994570225671524 \tabularnewline
77 & 110.11 & 109.966550581871 & 0.143449418128753 \tabularnewline
78 & 109.42 & 109.551895452204 & -0.131895452203793 \tabularnewline
79 & 109.06 & 113.404194110299 & -4.34419411029916 \tabularnewline
80 & 108.98 & 113.26297268606 & -4.28297268606048 \tabularnewline
81 & 108.36 & 111.386817286358 & -3.02681728635794 \tabularnewline
82 & 108.11 & 107.919386458338 & 0.190613541662231 \tabularnewline
83 & 108.44 & 108.295225078498 & 0.144774921501835 \tabularnewline
84 & 107.76 & 110.375112013764 & -2.61511201376371 \tabularnewline
85 & 106.27 & 106.850547482494 & -0.580547482493586 \tabularnewline
86 & 101.07 & 105.071319140424 & -4.00131914042356 \tabularnewline
87 & 100.79 & 95.9769011533363 & 4.81309884666372 \tabularnewline
88 & 100.97 & 96.6365964682487 & 4.33340353175134 \tabularnewline
89 & 99.33 & 98.1648610595406 & 1.16513894045936 \tabularnewline
90 & 99.35 & 97.0482652884637 & 2.30173471153626 \tabularnewline
91 & 99.23 & 101.45008372169 & -2.22008372168955 \tabularnewline
92 & 98.14 & 102.480842078008 & -4.34084207800787 \tabularnewline
93 & 98.17 & 99.9750698603638 & -1.8050698603638 \tabularnewline
94 & 98.48 & 97.4672243717983 & 1.01277562820169 \tabularnewline
95 & 99 & 98.2264065118693 & 0.773593488130658 \tabularnewline
96 & 99.19 & 100.322887933783 & -1.13288793378341 \tabularnewline
97 & 99.1 & 98.4047591219387 & 0.695240878061298 \tabularnewline
98 & 100.13 & 97.6581023201761 & 2.47189767982394 \tabularnewline
99 & 100.07 & 96.8309750603315 & 3.23902493966852 \tabularnewline
100 & 95.26 & 97.2690963335927 & -2.00909633359274 \tabularnewline
101 & 94.72 & 92.8512617510251 & 1.86873824897494 \tabularnewline
102 & 94.25 & 92.6498200950695 & 1.60017990493049 \tabularnewline
103 & 89.46 & 95.9363707744928 & -6.47637077449282 \tabularnewline
104 & 88.38 & 92.183919902685 & -3.80391990268497 \tabularnewline
105 & 88.57 & 89.7736504252291 & -1.2036504252291 \tabularnewline
106 & 93.82 & 87.5645309858545 & 6.25546901414546 \tabularnewline
107 & 93.94 & 93.3611839748699 & 0.578816025130095 \tabularnewline
108 & 93.92 & 95.4577774810964 & -1.53777748109643 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284211&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]77.15[/C][C]80.3096260683761[/C][C]-3.15962606837611[/C][/ROW]
[ROW][C]14[/C][C]76.33[/C][C]76.3894445164792[/C][C]-0.0594445164792035[/C][/ROW]
[ROW][C]15[/C][C]70.19[/C][C]69.8239859050165[/C][C]0.366014094983527[/C][/ROW]
[ROW][C]16[/C][C]68.42[/C][C]68.0191846375422[/C][C]0.400815362457791[/C][/ROW]
[ROW][C]17[/C][C]66.49[/C][C]65.8162199738616[/C][C]0.673780026138417[/C][/ROW]
[ROW][C]18[/C][C]63.41[/C][C]62.4621780468113[/C][C]0.947821953188694[/C][/ROW]
[ROW][C]19[/C][C]62.92[/C][C]68.8201516108171[/C][C]-5.90015161081708[/C][/ROW]
[ROW][C]20[/C][C]65.53[/C][C]61.4242231966847[/C][C]4.10577680331528[/C][/ROW]
[ROW][C]21[/C][C]65.26[/C][C]64.7580414670371[/C][C]0.501958532962902[/C][/ROW]
[ROW][C]22[/C][C]68.25[/C][C]65.0941315240568[/C][C]3.15586847594324[/C][/ROW]
[ROW][C]23[/C][C]74.39[/C][C]68.3572550806957[/C][C]6.03274491930435[/C][/ROW]
[ROW][C]24[/C][C]78.71[/C][C]76.4754506357966[/C][C]2.2345493642034[/C][/ROW]
[ROW][C]25[/C][C]82.15[/C][C]79.1266194787051[/C][C]3.02338052129491[/C][/ROW]
[ROW][C]26[/C][C]86.05[/C][C]83.9545552291246[/C][C]2.09544477087545[/C][/ROW]
[ROW][C]27[/C][C]89.46[/C][C]82.6726450211182[/C][C]6.78735497888178[/C][/ROW]
[ROW][C]28[/C][C]89.32[/C][C]91.0901801492725[/C][C]-1.77018014927252[/C][/ROW]
[ROW][C]29[/C][C]88.94[/C][C]91.1307032918918[/C][C]-2.19070329189177[/C][/ROW]
[ROW][C]30[/C][C]93.35[/C][C]88.8664924468356[/C][C]4.4835075531644[/C][/ROW]
[ROW][C]31[/C][C]94.72[/C][C]101.814897715236[/C][C]-7.09489771523614[/C][/ROW]
[ROW][C]32[/C][C]96.11[/C][C]98.5545902340917[/C][C]-2.44459023409175[/C][/ROW]
[ROW][C]33[/C][C]104.06[/C][C]98.475607844204[/C][C]5.584392155796[/C][/ROW]
[ROW][C]34[/C][C]104.11[/C][C]107.368128898875[/C][C]-3.2581288988746[/C][/ROW]
[ROW][C]35[/C][C]103.9[/C][C]107.83865686781[/C][C]-3.93865686781027[/C][/ROW]
[ROW][C]36[/C][C]110.75[/C][C]107.363549296523[/C][C]3.38645070347705[/C][/ROW]
[ROW][C]37[/C][C]110.82[/C][C]112.004729910127[/C][C]-1.18472991012688[/C][/ROW]
[ROW][C]38[/C][C]107.59[/C][C]113.090579067478[/C][C]-5.50057906747804[/C][/ROW]
[ROW][C]39[/C][C]96.03[/C][C]104.288757659191[/C][C]-8.25875765919146[/C][/ROW]
[ROW][C]40[/C][C]95.69[/C][C]94.2244252776737[/C][C]1.46557472232629[/C][/ROW]
[ROW][C]41[/C][C]90.63[/C][C]93.5001238875273[/C][C]-2.87012388752731[/C][/ROW]
[ROW][C]42[/C][C]75.87[/C][C]87.7034688598038[/C][C]-11.8334688598038[/C][/ROW]
[ROW][C]43[/C][C]75.57[/C][C]78.0991250152341[/C][C]-2.52912501523409[/C][/ROW]
[ROW][C]44[/C][C]78.78[/C][C]73.4912833896202[/C][C]5.28871661037979[/C][/ROW]
[ROW][C]45[/C][C]74.93[/C][C]76.7163793088096[/C][C]-1.78637930880964[/C][/ROW]
[ROW][C]46[/C][C]75.85[/C][C]72.2129111657573[/C][C]3.63708883424269[/C][/ROW]
[ROW][C]47[/C][C]75.49[/C][C]74.1475562167618[/C][C]1.3424437832382[/C][/ROW]
[ROW][C]48[/C][C]76.87[/C][C]75.6339277749369[/C][C]1.2360722250631[/C][/ROW]
[ROW][C]49[/C][C]78.18[/C][C]73.881870871137[/C][C]4.29812912886302[/C][/ROW]
[ROW][C]50[/C][C]79.37[/C][C]76.3876469811253[/C][C]2.9823530188747[/C][/ROW]
[ROW][C]51[/C][C]80.59[/C][C]73.4397836807535[/C][C]7.15021631924648[/C][/ROW]
[ROW][C]52[/C][C]81.18[/C][C]79.7076615576134[/C][C]1.47233844238659[/C][/ROW]
[ROW][C]53[/C][C]81.02[/C][C]80.0701640947689[/C][C]0.94983590523114[/C][/ROW]
[ROW][C]54[/C][C]82.75[/C][C]78.9164048936744[/C][C]3.83359510632562[/C][/ROW]
[ROW][C]55[/C][C]83.63[/C][C]89.5031150044683[/C][C]-5.87311500446826[/C][/ROW]
[ROW][C]56[/C][C]85.35[/C][C]87.4748041508472[/C][C]-2.12480415084725[/C][/ROW]
[ROW][C]57[/C][C]90.52[/C][C]86.5564648531823[/C][C]3.96353514681772[/C][/ROW]
[ROW][C]58[/C][C]90.66[/C][C]92.0812130613927[/C][C]-1.42121306139275[/C][/ROW]
[ROW][C]59[/C][C]90.69[/C][C]92.6392070980686[/C][C]-1.9492070980686[/C][/ROW]
[ROW][C]60[/C][C]92.56[/C][C]93.9431569138225[/C][C]-1.38315691382255[/C][/ROW]
[ROW][C]61[/C][C]92.87[/C][C]92.4759663763942[/C][C]0.394033623605807[/C][/ROW]
[ROW][C]62[/C][C]93.82[/C][C]92.8864667264436[/C][C]0.933533273556435[/C][/ROW]
[ROW][C]63[/C][C]96.32[/C][C]89.7323446993533[/C][C]6.58765530064666[/C][/ROW]
[ROW][C]64[/C][C]96.03[/C][C]95.8609279221096[/C][C]0.169072077890434[/C][/ROW]
[ROW][C]65[/C][C]96.53[/C][C]95.780665736759[/C][C]0.749334263240968[/C][/ROW]
[ROW][C]66[/C][C]102.96[/C][C]95.5159847953348[/C][C]7.44401520466521[/C][/ROW]
[ROW][C]67[/C][C]102.38[/C][C]109.58359927212[/C][C]-7.20359927211965[/C][/ROW]
[ROW][C]68[/C][C]102.66[/C][C]107.975827372673[/C][C]-5.31582737267348[/C][/ROW]
[ROW][C]69[/C][C]106.83[/C][C]105.501950864233[/C][C]1.32804913576658[/C][/ROW]
[ROW][C]70[/C][C]106.5[/C][C]108.135430544277[/C][C]-1.63543054427686[/C][/ROW]
[ROW][C]71[/C][C]106.78[/C][C]108.465208471736[/C][C]-1.68520847173562[/C][/ROW]
[ROW][C]72[/C][C]108.49[/C][C]110.140346469005[/C][C]-1.65034646900489[/C][/ROW]
[ROW][C]73[/C][C]108.77[/C][C]108.664620918124[/C][C]0.105379081876308[/C][/ROW]
[ROW][C]74[/C][C]110.43[/C][C]108.849707320041[/C][C]1.58029267995924[/C][/ROW]
[ROW][C]75[/C][C]110.84[/C][C]107.011929904003[/C][C]3.82807009599699[/C][/ROW]
[ROW][C]76[/C][C]110.52[/C][C]109.525429774328[/C][C]0.994570225671524[/C][/ROW]
[ROW][C]77[/C][C]110.11[/C][C]109.966550581871[/C][C]0.143449418128753[/C][/ROW]
[ROW][C]78[/C][C]109.42[/C][C]109.551895452204[/C][C]-0.131895452203793[/C][/ROW]
[ROW][C]79[/C][C]109.06[/C][C]113.404194110299[/C][C]-4.34419411029916[/C][/ROW]
[ROW][C]80[/C][C]108.98[/C][C]113.26297268606[/C][C]-4.28297268606048[/C][/ROW]
[ROW][C]81[/C][C]108.36[/C][C]111.386817286358[/C][C]-3.02681728635794[/C][/ROW]
[ROW][C]82[/C][C]108.11[/C][C]107.919386458338[/C][C]0.190613541662231[/C][/ROW]
[ROW][C]83[/C][C]108.44[/C][C]108.295225078498[/C][C]0.144774921501835[/C][/ROW]
[ROW][C]84[/C][C]107.76[/C][C]110.375112013764[/C][C]-2.61511201376371[/C][/ROW]
[ROW][C]85[/C][C]106.27[/C][C]106.850547482494[/C][C]-0.580547482493586[/C][/ROW]
[ROW][C]86[/C][C]101.07[/C][C]105.071319140424[/C][C]-4.00131914042356[/C][/ROW]
[ROW][C]87[/C][C]100.79[/C][C]95.9769011533363[/C][C]4.81309884666372[/C][/ROW]
[ROW][C]88[/C][C]100.97[/C][C]96.6365964682487[/C][C]4.33340353175134[/C][/ROW]
[ROW][C]89[/C][C]99.33[/C][C]98.1648610595406[/C][C]1.16513894045936[/C][/ROW]
[ROW][C]90[/C][C]99.35[/C][C]97.0482652884637[/C][C]2.30173471153626[/C][/ROW]
[ROW][C]91[/C][C]99.23[/C][C]101.45008372169[/C][C]-2.22008372168955[/C][/ROW]
[ROW][C]92[/C][C]98.14[/C][C]102.480842078008[/C][C]-4.34084207800787[/C][/ROW]
[ROW][C]93[/C][C]98.17[/C][C]99.9750698603638[/C][C]-1.8050698603638[/C][/ROW]
[ROW][C]94[/C][C]98.48[/C][C]97.4672243717983[/C][C]1.01277562820169[/C][/ROW]
[ROW][C]95[/C][C]99[/C][C]98.2264065118693[/C][C]0.773593488130658[/C][/ROW]
[ROW][C]96[/C][C]99.19[/C][C]100.322887933783[/C][C]-1.13288793378341[/C][/ROW]
[ROW][C]97[/C][C]99.1[/C][C]98.4047591219387[/C][C]0.695240878061298[/C][/ROW]
[ROW][C]98[/C][C]100.13[/C][C]97.6581023201761[/C][C]2.47189767982394[/C][/ROW]
[ROW][C]99[/C][C]100.07[/C][C]96.8309750603315[/C][C]3.23902493966852[/C][/ROW]
[ROW][C]100[/C][C]95.26[/C][C]97.2690963335927[/C][C]-2.00909633359274[/C][/ROW]
[ROW][C]101[/C][C]94.72[/C][C]92.8512617510251[/C][C]1.86873824897494[/C][/ROW]
[ROW][C]102[/C][C]94.25[/C][C]92.6498200950695[/C][C]1.60017990493049[/C][/ROW]
[ROW][C]103[/C][C]89.46[/C][C]95.9363707744928[/C][C]-6.47637077449282[/C][/ROW]
[ROW][C]104[/C][C]88.38[/C][C]92.183919902685[/C][C]-3.80391990268497[/C][/ROW]
[ROW][C]105[/C][C]88.57[/C][C]89.7736504252291[/C][C]-1.2036504252291[/C][/ROW]
[ROW][C]106[/C][C]93.82[/C][C]87.5645309858545[/C][C]6.25546901414546[/C][/ROW]
[ROW][C]107[/C][C]93.94[/C][C]93.3611839748699[/C][C]0.578816025130095[/C][/ROW]
[ROW][C]108[/C][C]93.92[/C][C]95.4577774810964[/C][C]-1.53777748109643[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284211&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284211&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	77.15	80.3096260683761	-3.15962606837611
14	76.33	76.3894445164792	-0.0594445164792035
15	70.19	69.8239859050165	0.366014094983527
16	68.42	68.0191846375422	0.400815362457791
17	66.49	65.8162199738616	0.673780026138417
18	63.41	62.4621780468113	0.947821953188694
19	62.92	68.8201516108171	-5.90015161081708
20	65.53	61.4242231966847	4.10577680331528
21	65.26	64.7580414670371	0.501958532962902
22	68.25	65.0941315240568	3.15586847594324
23	74.39	68.3572550806957	6.03274491930435
24	78.71	76.4754506357966	2.2345493642034
25	82.15	79.1266194787051	3.02338052129491
26	86.05	83.9545552291246	2.09544477087545
27	89.46	82.6726450211182	6.78735497888178
28	89.32	91.0901801492725	-1.77018014927252
29	88.94	91.1307032918918	-2.19070329189177
30	93.35	88.8664924468356	4.4835075531644
31	94.72	101.814897715236	-7.09489771523614
32	96.11	98.5545902340917	-2.44459023409175
33	104.06	98.475607844204	5.584392155796
34	104.11	107.368128898875	-3.2581288988746
35	103.9	107.83865686781	-3.93865686781027
36	110.75	107.363549296523	3.38645070347705
37	110.82	112.004729910127	-1.18472991012688
38	107.59	113.090579067478	-5.50057906747804
39	96.03	104.288757659191	-8.25875765919146
40	95.69	94.2244252776737	1.46557472232629
41	90.63	93.5001238875273	-2.87012388752731
42	75.87	87.7034688598038	-11.8334688598038
43	75.57	78.0991250152341	-2.52912501523409
44	78.78	73.4912833896202	5.28871661037979
45	74.93	76.7163793088096	-1.78637930880964
46	75.85	72.2129111657573	3.63708883424269
47	75.49	74.1475562167618	1.3424437832382
48	76.87	75.6339277749369	1.2360722250631
49	78.18	73.881870871137	4.29812912886302
50	79.37	76.3876469811253	2.9823530188747
51	80.59	73.4397836807535	7.15021631924648
52	81.18	79.7076615576134	1.47233844238659
53	81.02	80.0701640947689	0.94983590523114
54	82.75	78.9164048936744	3.83359510632562
55	83.63	89.5031150044683	-5.87311500446826
56	85.35	87.4748041508472	-2.12480415084725
57	90.52	86.5564648531823	3.96353514681772
58	90.66	92.0812130613927	-1.42121306139275
59	90.69	92.6392070980686	-1.9492070980686
60	92.56	93.9431569138225	-1.38315691382255
61	92.87	92.4759663763942	0.394033623605807
62	93.82	92.8864667264436	0.933533273556435
63	96.32	89.7323446993533	6.58765530064666
64	96.03	95.8609279221096	0.169072077890434
65	96.53	95.780665736759	0.749334263240968
66	102.96	95.5159847953348	7.44401520466521
67	102.38	109.58359927212	-7.20359927211965
68	102.66	107.975827372673	-5.31582737267348
69	106.83	105.501950864233	1.32804913576658
70	106.5	108.135430544277	-1.63543054427686
71	106.78	108.465208471736	-1.68520847173562
72	108.49	110.140346469005	-1.65034646900489
73	108.77	108.664620918124	0.105379081876308
74	110.43	108.849707320041	1.58029267995924
75	110.84	107.011929904003	3.82807009599699
76	110.52	109.525429774328	0.994570225671524
77	110.11	109.966550581871	0.143449418128753
78	109.42	109.551895452204	-0.131895452203793
79	109.06	113.404194110299	-4.34419411029916
80	108.98	113.26297268606	-4.28297268606048
81	108.36	111.386817286358	-3.02681728635794
82	108.11	107.919386458338	0.190613541662231
83	108.44	108.295225078498	0.144774921501835
84	107.76	110.375112013764	-2.61511201376371
85	106.27	106.850547482494	-0.580547482493586
86	101.07	105.071319140424	-4.00131914042356
87	100.79	95.9769011533363	4.81309884666372
88	100.97	96.6365964682487	4.33340353175134
89	99.33	98.1648610595406	1.16513894045936
90	99.35	97.0482652884637	2.30173471153626
91	99.23	101.45008372169	-2.22008372168955
92	98.14	102.480842078008	-4.34084207800787
93	98.17	99.9750698603638	-1.8050698603638
94	98.48	97.4672243717983	1.01277562820169
95	99	98.2264065118693	0.773593488130658
96	99.19	100.322887933783	-1.13288793378341
97	99.1	98.4047591219387	0.695240878061298
98	100.13	97.6581023201761	2.47189767982394
99	100.07	96.8309750603315	3.23902493966852
100	95.26	97.2690963335927	-2.00909633359274
101	94.72	92.8512617510251	1.86873824897494
102	94.25	92.6498200950695	1.60017990493049
103	89.46	95.9363707744928	-6.47637077449282
104	88.38	92.183919902685	-3.80391990268497
105	88.57	89.7736504252291	-1.2036504252291
106	93.82	87.5645309858545	6.25546901414546
107	93.94	93.3611839748699	0.578816025130095
108	93.92	95.4577774810964	-1.53777748109643

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
109	93.7103298136311	86.5313602903912	100.889299336871
110	92.7404611237848	82.2114717412097	103.26945050636
111	89.5358025424476	75.6473178610288	103.424287223866
112	85.6097850443168	68.2517685430922	102.967801545541
113	82.9044877121972	61.9339423399611	103.875033084433
114	80.1539167211494	55.4162716986137	104.891561743685
115	79.9219886009526	51.2595632592095	108.584413942696
116	82.259657824685	49.5154007053465	115.003914944024
117	84.286333701732	47.3055509869445	121.26711641652
118	85.0054096536211	43.6365525309952	126.374266776247
119	84.4326671081736	38.5276511724683	130.337683043879
120	85.4820063674548	34.8962899948264	136.067722740083

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
109 & 93.7103298136311 & 86.5313602903912 & 100.889299336871 \tabularnewline
110 & 92.7404611237848 & 82.2114717412097 & 103.26945050636 \tabularnewline
111 & 89.5358025424476 & 75.6473178610288 & 103.424287223866 \tabularnewline
112 & 85.6097850443168 & 68.2517685430922 & 102.967801545541 \tabularnewline
113 & 82.9044877121972 & 61.9339423399611 & 103.875033084433 \tabularnewline
114 & 80.1539167211494 & 55.4162716986137 & 104.891561743685 \tabularnewline
115 & 79.9219886009526 & 51.2595632592095 & 108.584413942696 \tabularnewline
116 & 82.259657824685 & 49.5154007053465 & 115.003914944024 \tabularnewline
117 & 84.286333701732 & 47.3055509869445 & 121.26711641652 \tabularnewline
118 & 85.0054096536211 & 43.6365525309952 & 126.374266776247 \tabularnewline
119 & 84.4326671081736 & 38.5276511724683 & 130.337683043879 \tabularnewline
120 & 85.4820063674548 & 34.8962899948264 & 136.067722740083 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284211&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]93.7103298136311[/C][C]86.5313602903912[/C][C]100.889299336871[/C][/ROW]
[ROW][C]110[/C][C]92.7404611237848[/C][C]82.2114717412097[/C][C]103.26945050636[/C][/ROW]
[ROW][C]111[/C][C]89.5358025424476[/C][C]75.6473178610288[/C][C]103.424287223866[/C][/ROW]
[ROW][C]112[/C][C]85.6097850443168[/C][C]68.2517685430922[/C][C]102.967801545541[/C][/ROW]
[ROW][C]113[/C][C]82.9044877121972[/C][C]61.9339423399611[/C][C]103.875033084433[/C][/ROW]
[ROW][C]114[/C][C]80.1539167211494[/C][C]55.4162716986137[/C][C]104.891561743685[/C][/ROW]
[ROW][C]115[/C][C]79.9219886009526[/C][C]51.2595632592095[/C][C]108.584413942696[/C][/ROW]
[ROW][C]116[/C][C]82.259657824685[/C][C]49.5154007053465[/C][C]115.003914944024[/C][/ROW]
[ROW][C]117[/C][C]84.286333701732[/C][C]47.3055509869445[/C][C]121.26711641652[/C][/ROW]
[ROW][C]118[/C][C]85.0054096536211[/C][C]43.6365525309952[/C][C]126.374266776247[/C][/ROW]
[ROW][C]119[/C][C]84.4326671081736[/C][C]38.5276511724683[/C][C]130.337683043879[/C][/ROW]
[ROW][C]120[/C][C]85.4820063674548[/C][C]34.8962899948264[/C][C]136.067722740083[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284211&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284211&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	93.7103298136311	86.5313602903912	100.889299336871
110	92.7404611237848	82.2114717412097	103.26945050636
111	89.5358025424476	75.6473178610288	103.424287223866
112	85.6097850443168	68.2517685430922	102.967801545541
113	82.9044877121972	61.9339423399611	103.875033084433
114	80.1539167211494	55.4162716986137	104.891561743685
115	79.9219886009526	51.2595632592095	108.584413942696
116	82.259657824685	49.5154007053465	115.003914944024
117	84.286333701732	47.3055509869445	121.26711641652
118	85.0054096536211	43.6365525309952	126.374266776247
119	84.4326671081736	38.5276511724683	130.337683043879
120	85.4820063674548	34.8962899948264	136.067722740083

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