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Author's title

Author

*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Wed, 01 Apr 2015 13:57:17 +0100

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/Apr/01/t1427893112fx52aeunlup86w7.htm/, Retrieved Mon, 27 Jul 2026 19:35:07 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=278508, Retrieved Mon, 27 Jul 2026 19:35:07 +0000

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Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords

Estimated Impact

496

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

-       [Exponential Smoothing] [] [2015-04-01 12:57:17] [7657461249ddfb44f7ff766614f926c4] [Current]

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Dataseries X:

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8,09
8,18
8,26
8,28
8,28
8,28
8,29
8,3
8,3
8,31
8,33
8,33
8,34
8,48
8,59
8,67
8,67
8,67
8,71
8,72
8,72
8,72
8,74
8,74
8,74
8,74
8,79
8,85
8,86
8,87
8,92
8,96
8,97
8,99
8,98
8,98
9,01
9,01
9,03
9,05
9,05
9,05
9,13
9,13
9,13
9,14
9,16
9,16
9,16
9,16
9,22
9,22
9,25
9,25
9,26
9,26
9,29
9,28
9,37
9,41
9,41
9,45
9,49
9,49
9,56
9,54
9,55
9,55
9,57
9,57
9,58
9,59
9,59
9,59
9,61
9,61
9,62
9,62
9,63
9,65
9,65
9,68
9,73
9,73
9,73
9,74
9,68
9,68
9,74
9,74
9,76
9,78
9,79
9,81
9,84
9,84

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	'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 & 1 seconds \tabularnewline
R Server & 'George Udny Yule' @ yule.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=278508&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]'George Udny Yule' @ yule.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=278508&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=278508&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	'George Udny Yule' @ yule.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	1
beta	0.302009667782302
gamma	FALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 1 \tabularnewline
beta & 0.302009667782302 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=278508&T=1

[TABLE]
[ROW][C]Estimated Parameters of Exponential Smoothing[/C][/ROW]
[ROW][C]Parameter[/C][C]Value[/C][/ROW]
[ROW][C]alpha[/C][C]1[/C][/ROW]
[ROW][C]beta[/C][C]0.302009667782302[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=278508&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=278508&T=1

As an alternative you can also use a QR Code:

The GUIDs for individual cells are displayed in the table below:

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	1
beta	0.302009667782302
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
3	8.26	8.27	-0.00999999999999979
4	8.28	8.34697990332218	-0.0669799033221778
5	8.28	8.34675132497176	-0.0667513249717562
6	8.28	8.32659177949301	-0.0465917794930064
7	8.29	8.31252061164694	-0.0225206116469376
8	8.3	8.31571916920519	-0.015719169205191
9	8.3	8.32097182813572	-0.0209718281357176
10	8.31	8.31463813328766	-0.00463813328766349
11	8.33	8.32323737219433	0.00676262780567427
12	8.33	8.34527975117125	-0.0152797511712528
13	8.34	8.34066511859623	-0.000665118596225867
14	8.48	8.35046424634994	0.129535753650057
15	8.59	8.52958529627573	0.0604147037242715
16	8.67	8.65783112087666	0.0121688791233385
17	8.67	8.74150624001798	-0.0715062400179836
18	8.67	8.71991066422579	-0.0499106642257914
19	8.71	8.70483716110417	0.00516283889583491
20	8.72	8.74639638836391	-0.02639638836391
21	8.72	8.74842442388347	-0.0284244238834734
22	8.72	8.73983997306952	-0.0198399730695229
23	8.74	8.73384810939399	0.00615189060601296
24	8.74	8.75570603983214	-0.0157060398321409
25	8.74	8.75096266396026	-0.0109626639602602
26	8.74	8.74765183345961	-0.0076518334596134
27	8.79	8.74534090577855	0.0446590942214495
28	8.85	8.80882838398783	0.0411716160121713
29	8.86	8.88126261006173	-0.0212626100617257
30	8.87	8.8848410962608	-0.014841096260799
31	8.92	8.89035894170955	0.0296410582904514
32	8.96	8.94931082787656	0.0106891721234366
33	8.97	8.99253906119843	-0.0225390611984313
34	8.99	8.99573204681377	-0.00573204681376893
35	8.98	9.01400091325983	-0.0340009132598293
36	8.98	8.99373230874193	-0.0137323087419343
37	9.01	8.9895850187409	0.0204149812591012
38	9.01	9.02575054044874	-0.0157505404487406
39	9.03	9.02099372496042	0.00900627503957452
40	9.05	9.04371370709308	0.00628629290691762
41	9.05	9.06561222832548	-0.0156122283254838
42	9.05	9.06089718443556	-0.0108971844355636
43	9.13	9.05760612938442	0.0723938706155831
44	9.13	9.1594697781985	-0.0294697781985036
45	9.13	9.15056962027516	-0.0205696202751557
46	9.14	9.14435739608945	-0.00435739608944807
47	9.16	9.15304142034408	0.00695857965592239
48	9.16	9.1751429786742	-0.0151429786741986
49	9.16	9.17056965271557	-0.010569652715569
50	9.16	9.16737751541037	-0.0073775154103668
51	9.22	9.16514943443222	0.0548505655677776
52	9.22	9.24171483551702	-0.0217148355170185
53	9.25	9.23515674525658	0.0148432547434236
54	9.25	9.26963955169045	-0.019639551690446
55	9.26	9.26370821720902	-0.00370821720902192
56	9.26	9.27258829976166	-0.01258829976166
57	9.29	9.2687865115327	0.0212134884673016
58	9.28	9.30519319013721	-0.0251931901372107
59	9.37	9.28758460315349	0.0824153968465051
60	9.41	9.40247484977525	0.00752515022474576
61	9.41	9.44474751789464	-0.0347475178946421
62	9.45	9.43425343155902	0.0157465684409779
63	9.49	9.47900904746259	0.0109909525374086
64	9.49	9.52232842138703	-0.0323284213870263
65	9.56	9.51256492558401	0.0474350744159953
66	9.54	9.59689077664961	-0.0568907766496096
67	9.55	9.55970921209378	-0.00970921209378162
68	9.55	9.56677693617491	-0.0167769361749119
69	9.57	9.56171013925432	0.00828986074567872
70	9.57	9.58421375734409	-0.0142137573440859
71	9.58	9.57992106521066	7.89347893395131e-05
72	9.59	9.58994490428016	5.50957198353075e-05
73	9.59	9.59996154372021	-0.00996154372020897
74	9.59	9.59695306121067	-0.00695306121066963
75	9.61	9.59485316950436	0.0151468304956346
76	9.61	9.61942765875031	-0.00942765875030638
77	9.62	9.61658041466316	0.00341958533683773
78	9.62	9.62761316249469	-0.00761316249469246
79	9.63	9.6253139138189	0.00468608618110267
80	9.65	9.63672915714965	0.0132708428503463
81	9.65	9.66073707999008	-0.0107370799900774
82	9.68	9.65749437802932	0.0225056219706783
83	9.73	9.69429129344392	0.0357087065560808
84	9.73	9.75507566804786	-0.0250756680478581
85	9.73	9.74750257387131	-0.0175025738713046
86	9.74	9.7422166273511	-0.00221662735109796
87	9.68	9.7515471844612	-0.0715471844611955
88	9.68	9.66993924305131	0.0100607569486897
89	9.74	9.67297768891502	0.0670223110849779
90	9.74	9.7532190748198	-0.0132190748197996
91	9.76	9.74922678642508	0.0107732135749181
92	9.78	9.77248040107779	0.0075195989222081
93	9.79	9.79475139265014	-0.00475139265014413
94	9.81	9.80331642613437	0.00668357386563123
95	9.84	9.82533493005713	0.0146650699428719
96	9.84	9.85976392295858	-0.0197639229585782

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 8.26 & 8.27 & -0.00999999999999979 \tabularnewline
4 & 8.28 & 8.34697990332218 & -0.0669799033221778 \tabularnewline
5 & 8.28 & 8.34675132497176 & -0.0667513249717562 \tabularnewline
6 & 8.28 & 8.32659177949301 & -0.0465917794930064 \tabularnewline
7 & 8.29 & 8.31252061164694 & -0.0225206116469376 \tabularnewline
8 & 8.3 & 8.31571916920519 & -0.015719169205191 \tabularnewline
9 & 8.3 & 8.32097182813572 & -0.0209718281357176 \tabularnewline
10 & 8.31 & 8.31463813328766 & -0.00463813328766349 \tabularnewline
11 & 8.33 & 8.32323737219433 & 0.00676262780567427 \tabularnewline
12 & 8.33 & 8.34527975117125 & -0.0152797511712528 \tabularnewline
13 & 8.34 & 8.34066511859623 & -0.000665118596225867 \tabularnewline
14 & 8.48 & 8.35046424634994 & 0.129535753650057 \tabularnewline
15 & 8.59 & 8.52958529627573 & 0.0604147037242715 \tabularnewline
16 & 8.67 & 8.65783112087666 & 0.0121688791233385 \tabularnewline
17 & 8.67 & 8.74150624001798 & -0.0715062400179836 \tabularnewline
18 & 8.67 & 8.71991066422579 & -0.0499106642257914 \tabularnewline
19 & 8.71 & 8.70483716110417 & 0.00516283889583491 \tabularnewline
20 & 8.72 & 8.74639638836391 & -0.02639638836391 \tabularnewline
21 & 8.72 & 8.74842442388347 & -0.0284244238834734 \tabularnewline
22 & 8.72 & 8.73983997306952 & -0.0198399730695229 \tabularnewline
23 & 8.74 & 8.73384810939399 & 0.00615189060601296 \tabularnewline
24 & 8.74 & 8.75570603983214 & -0.0157060398321409 \tabularnewline
25 & 8.74 & 8.75096266396026 & -0.0109626639602602 \tabularnewline
26 & 8.74 & 8.74765183345961 & -0.0076518334596134 \tabularnewline
27 & 8.79 & 8.74534090577855 & 0.0446590942214495 \tabularnewline
28 & 8.85 & 8.80882838398783 & 0.0411716160121713 \tabularnewline
29 & 8.86 & 8.88126261006173 & -0.0212626100617257 \tabularnewline
30 & 8.87 & 8.8848410962608 & -0.014841096260799 \tabularnewline
31 & 8.92 & 8.89035894170955 & 0.0296410582904514 \tabularnewline
32 & 8.96 & 8.94931082787656 & 0.0106891721234366 \tabularnewline
33 & 8.97 & 8.99253906119843 & -0.0225390611984313 \tabularnewline
34 & 8.99 & 8.99573204681377 & -0.00573204681376893 \tabularnewline
35 & 8.98 & 9.01400091325983 & -0.0340009132598293 \tabularnewline
36 & 8.98 & 8.99373230874193 & -0.0137323087419343 \tabularnewline
37 & 9.01 & 8.9895850187409 & 0.0204149812591012 \tabularnewline
38 & 9.01 & 9.02575054044874 & -0.0157505404487406 \tabularnewline
39 & 9.03 & 9.02099372496042 & 0.00900627503957452 \tabularnewline
40 & 9.05 & 9.04371370709308 & 0.00628629290691762 \tabularnewline
41 & 9.05 & 9.06561222832548 & -0.0156122283254838 \tabularnewline
42 & 9.05 & 9.06089718443556 & -0.0108971844355636 \tabularnewline
43 & 9.13 & 9.05760612938442 & 0.0723938706155831 \tabularnewline
44 & 9.13 & 9.1594697781985 & -0.0294697781985036 \tabularnewline
45 & 9.13 & 9.15056962027516 & -0.0205696202751557 \tabularnewline
46 & 9.14 & 9.14435739608945 & -0.00435739608944807 \tabularnewline
47 & 9.16 & 9.15304142034408 & 0.00695857965592239 \tabularnewline
48 & 9.16 & 9.1751429786742 & -0.0151429786741986 \tabularnewline
49 & 9.16 & 9.17056965271557 & -0.010569652715569 \tabularnewline
50 & 9.16 & 9.16737751541037 & -0.0073775154103668 \tabularnewline
51 & 9.22 & 9.16514943443222 & 0.0548505655677776 \tabularnewline
52 & 9.22 & 9.24171483551702 & -0.0217148355170185 \tabularnewline
53 & 9.25 & 9.23515674525658 & 0.0148432547434236 \tabularnewline
54 & 9.25 & 9.26963955169045 & -0.019639551690446 \tabularnewline
55 & 9.26 & 9.26370821720902 & -0.00370821720902192 \tabularnewline
56 & 9.26 & 9.27258829976166 & -0.01258829976166 \tabularnewline
57 & 9.29 & 9.2687865115327 & 0.0212134884673016 \tabularnewline
58 & 9.28 & 9.30519319013721 & -0.0251931901372107 \tabularnewline
59 & 9.37 & 9.28758460315349 & 0.0824153968465051 \tabularnewline
60 & 9.41 & 9.40247484977525 & 0.00752515022474576 \tabularnewline
61 & 9.41 & 9.44474751789464 & -0.0347475178946421 \tabularnewline
62 & 9.45 & 9.43425343155902 & 0.0157465684409779 \tabularnewline
63 & 9.49 & 9.47900904746259 & 0.0109909525374086 \tabularnewline
64 & 9.49 & 9.52232842138703 & -0.0323284213870263 \tabularnewline
65 & 9.56 & 9.51256492558401 & 0.0474350744159953 \tabularnewline
66 & 9.54 & 9.59689077664961 & -0.0568907766496096 \tabularnewline
67 & 9.55 & 9.55970921209378 & -0.00970921209378162 \tabularnewline
68 & 9.55 & 9.56677693617491 & -0.0167769361749119 \tabularnewline
69 & 9.57 & 9.56171013925432 & 0.00828986074567872 \tabularnewline
70 & 9.57 & 9.58421375734409 & -0.0142137573440859 \tabularnewline
71 & 9.58 & 9.57992106521066 & 7.89347893395131e-05 \tabularnewline
72 & 9.59 & 9.58994490428016 & 5.50957198353075e-05 \tabularnewline
73 & 9.59 & 9.59996154372021 & -0.00996154372020897 \tabularnewline
74 & 9.59 & 9.59695306121067 & -0.00695306121066963 \tabularnewline
75 & 9.61 & 9.59485316950436 & 0.0151468304956346 \tabularnewline
76 & 9.61 & 9.61942765875031 & -0.00942765875030638 \tabularnewline
77 & 9.62 & 9.61658041466316 & 0.00341958533683773 \tabularnewline
78 & 9.62 & 9.62761316249469 & -0.00761316249469246 \tabularnewline
79 & 9.63 & 9.6253139138189 & 0.00468608618110267 \tabularnewline
80 & 9.65 & 9.63672915714965 & 0.0132708428503463 \tabularnewline
81 & 9.65 & 9.66073707999008 & -0.0107370799900774 \tabularnewline
82 & 9.68 & 9.65749437802932 & 0.0225056219706783 \tabularnewline
83 & 9.73 & 9.69429129344392 & 0.0357087065560808 \tabularnewline
84 & 9.73 & 9.75507566804786 & -0.0250756680478581 \tabularnewline
85 & 9.73 & 9.74750257387131 & -0.0175025738713046 \tabularnewline
86 & 9.74 & 9.7422166273511 & -0.00221662735109796 \tabularnewline
87 & 9.68 & 9.7515471844612 & -0.0715471844611955 \tabularnewline
88 & 9.68 & 9.66993924305131 & 0.0100607569486897 \tabularnewline
89 & 9.74 & 9.67297768891502 & 0.0670223110849779 \tabularnewline
90 & 9.74 & 9.7532190748198 & -0.0132190748197996 \tabularnewline
91 & 9.76 & 9.74922678642508 & 0.0107732135749181 \tabularnewline
92 & 9.78 & 9.77248040107779 & 0.0075195989222081 \tabularnewline
93 & 9.79 & 9.79475139265014 & -0.00475139265014413 \tabularnewline
94 & 9.81 & 9.80331642613437 & 0.00668357386563123 \tabularnewline
95 & 9.84 & 9.82533493005713 & 0.0146650699428719 \tabularnewline
96 & 9.84 & 9.85976392295858 & -0.0197639229585782 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=278508&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]8.26[/C][C]8.27[/C][C]-0.00999999999999979[/C][/ROW]
[ROW][C]4[/C][C]8.28[/C][C]8.34697990332218[/C][C]-0.0669799033221778[/C][/ROW]
[ROW][C]5[/C][C]8.28[/C][C]8.34675132497176[/C][C]-0.0667513249717562[/C][/ROW]
[ROW][C]6[/C][C]8.28[/C][C]8.32659177949301[/C][C]-0.0465917794930064[/C][/ROW]
[ROW][C]7[/C][C]8.29[/C][C]8.31252061164694[/C][C]-0.0225206116469376[/C][/ROW]
[ROW][C]8[/C][C]8.3[/C][C]8.31571916920519[/C][C]-0.015719169205191[/C][/ROW]
[ROW][C]9[/C][C]8.3[/C][C]8.32097182813572[/C][C]-0.0209718281357176[/C][/ROW]
[ROW][C]10[/C][C]8.31[/C][C]8.31463813328766[/C][C]-0.00463813328766349[/C][/ROW]
[ROW][C]11[/C][C]8.33[/C][C]8.32323737219433[/C][C]0.00676262780567427[/C][/ROW]
[ROW][C]12[/C][C]8.33[/C][C]8.34527975117125[/C][C]-0.0152797511712528[/C][/ROW]
[ROW][C]13[/C][C]8.34[/C][C]8.34066511859623[/C][C]-0.000665118596225867[/C][/ROW]
[ROW][C]14[/C][C]8.48[/C][C]8.35046424634994[/C][C]0.129535753650057[/C][/ROW]
[ROW][C]15[/C][C]8.59[/C][C]8.52958529627573[/C][C]0.0604147037242715[/C][/ROW]
[ROW][C]16[/C][C]8.67[/C][C]8.65783112087666[/C][C]0.0121688791233385[/C][/ROW]
[ROW][C]17[/C][C]8.67[/C][C]8.74150624001798[/C][C]-0.0715062400179836[/C][/ROW]
[ROW][C]18[/C][C]8.67[/C][C]8.71991066422579[/C][C]-0.0499106642257914[/C][/ROW]
[ROW][C]19[/C][C]8.71[/C][C]8.70483716110417[/C][C]0.00516283889583491[/C][/ROW]
[ROW][C]20[/C][C]8.72[/C][C]8.74639638836391[/C][C]-0.02639638836391[/C][/ROW]
[ROW][C]21[/C][C]8.72[/C][C]8.74842442388347[/C][C]-0.0284244238834734[/C][/ROW]
[ROW][C]22[/C][C]8.72[/C][C]8.73983997306952[/C][C]-0.0198399730695229[/C][/ROW]
[ROW][C]23[/C][C]8.74[/C][C]8.73384810939399[/C][C]0.00615189060601296[/C][/ROW]
[ROW][C]24[/C][C]8.74[/C][C]8.75570603983214[/C][C]-0.0157060398321409[/C][/ROW]
[ROW][C]25[/C][C]8.74[/C][C]8.75096266396026[/C][C]-0.0109626639602602[/C][/ROW]
[ROW][C]26[/C][C]8.74[/C][C]8.74765183345961[/C][C]-0.0076518334596134[/C][/ROW]
[ROW][C]27[/C][C]8.79[/C][C]8.74534090577855[/C][C]0.0446590942214495[/C][/ROW]
[ROW][C]28[/C][C]8.85[/C][C]8.80882838398783[/C][C]0.0411716160121713[/C][/ROW]
[ROW][C]29[/C][C]8.86[/C][C]8.88126261006173[/C][C]-0.0212626100617257[/C][/ROW]
[ROW][C]30[/C][C]8.87[/C][C]8.8848410962608[/C][C]-0.014841096260799[/C][/ROW]
[ROW][C]31[/C][C]8.92[/C][C]8.89035894170955[/C][C]0.0296410582904514[/C][/ROW]
[ROW][C]32[/C][C]8.96[/C][C]8.94931082787656[/C][C]0.0106891721234366[/C][/ROW]
[ROW][C]33[/C][C]8.97[/C][C]8.99253906119843[/C][C]-0.0225390611984313[/C][/ROW]
[ROW][C]34[/C][C]8.99[/C][C]8.99573204681377[/C][C]-0.00573204681376893[/C][/ROW]
[ROW][C]35[/C][C]8.98[/C][C]9.01400091325983[/C][C]-0.0340009132598293[/C][/ROW]
[ROW][C]36[/C][C]8.98[/C][C]8.99373230874193[/C][C]-0.0137323087419343[/C][/ROW]
[ROW][C]37[/C][C]9.01[/C][C]8.9895850187409[/C][C]0.0204149812591012[/C][/ROW]
[ROW][C]38[/C][C]9.01[/C][C]9.02575054044874[/C][C]-0.0157505404487406[/C][/ROW]
[ROW][C]39[/C][C]9.03[/C][C]9.02099372496042[/C][C]0.00900627503957452[/C][/ROW]
[ROW][C]40[/C][C]9.05[/C][C]9.04371370709308[/C][C]0.00628629290691762[/C][/ROW]
[ROW][C]41[/C][C]9.05[/C][C]9.06561222832548[/C][C]-0.0156122283254838[/C][/ROW]
[ROW][C]42[/C][C]9.05[/C][C]9.06089718443556[/C][C]-0.0108971844355636[/C][/ROW]
[ROW][C]43[/C][C]9.13[/C][C]9.05760612938442[/C][C]0.0723938706155831[/C][/ROW]
[ROW][C]44[/C][C]9.13[/C][C]9.1594697781985[/C][C]-0.0294697781985036[/C][/ROW]
[ROW][C]45[/C][C]9.13[/C][C]9.15056962027516[/C][C]-0.0205696202751557[/C][/ROW]
[ROW][C]46[/C][C]9.14[/C][C]9.14435739608945[/C][C]-0.00435739608944807[/C][/ROW]
[ROW][C]47[/C][C]9.16[/C][C]9.15304142034408[/C][C]0.00695857965592239[/C][/ROW]
[ROW][C]48[/C][C]9.16[/C][C]9.1751429786742[/C][C]-0.0151429786741986[/C][/ROW]
[ROW][C]49[/C][C]9.16[/C][C]9.17056965271557[/C][C]-0.010569652715569[/C][/ROW]
[ROW][C]50[/C][C]9.16[/C][C]9.16737751541037[/C][C]-0.0073775154103668[/C][/ROW]
[ROW][C]51[/C][C]9.22[/C][C]9.16514943443222[/C][C]0.0548505655677776[/C][/ROW]
[ROW][C]52[/C][C]9.22[/C][C]9.24171483551702[/C][C]-0.0217148355170185[/C][/ROW]
[ROW][C]53[/C][C]9.25[/C][C]9.23515674525658[/C][C]0.0148432547434236[/C][/ROW]
[ROW][C]54[/C][C]9.25[/C][C]9.26963955169045[/C][C]-0.019639551690446[/C][/ROW]
[ROW][C]55[/C][C]9.26[/C][C]9.26370821720902[/C][C]-0.00370821720902192[/C][/ROW]
[ROW][C]56[/C][C]9.26[/C][C]9.27258829976166[/C][C]-0.01258829976166[/C][/ROW]
[ROW][C]57[/C][C]9.29[/C][C]9.2687865115327[/C][C]0.0212134884673016[/C][/ROW]
[ROW][C]58[/C][C]9.28[/C][C]9.30519319013721[/C][C]-0.0251931901372107[/C][/ROW]
[ROW][C]59[/C][C]9.37[/C][C]9.28758460315349[/C][C]0.0824153968465051[/C][/ROW]
[ROW][C]60[/C][C]9.41[/C][C]9.40247484977525[/C][C]0.00752515022474576[/C][/ROW]
[ROW][C]61[/C][C]9.41[/C][C]9.44474751789464[/C][C]-0.0347475178946421[/C][/ROW]
[ROW][C]62[/C][C]9.45[/C][C]9.43425343155902[/C][C]0.0157465684409779[/C][/ROW]
[ROW][C]63[/C][C]9.49[/C][C]9.47900904746259[/C][C]0.0109909525374086[/C][/ROW]
[ROW][C]64[/C][C]9.49[/C][C]9.52232842138703[/C][C]-0.0323284213870263[/C][/ROW]
[ROW][C]65[/C][C]9.56[/C][C]9.51256492558401[/C][C]0.0474350744159953[/C][/ROW]
[ROW][C]66[/C][C]9.54[/C][C]9.59689077664961[/C][C]-0.0568907766496096[/C][/ROW]
[ROW][C]67[/C][C]9.55[/C][C]9.55970921209378[/C][C]-0.00970921209378162[/C][/ROW]
[ROW][C]68[/C][C]9.55[/C][C]9.56677693617491[/C][C]-0.0167769361749119[/C][/ROW]
[ROW][C]69[/C][C]9.57[/C][C]9.56171013925432[/C][C]0.00828986074567872[/C][/ROW]
[ROW][C]70[/C][C]9.57[/C][C]9.58421375734409[/C][C]-0.0142137573440859[/C][/ROW]
[ROW][C]71[/C][C]9.58[/C][C]9.57992106521066[/C][C]7.89347893395131e-05[/C][/ROW]
[ROW][C]72[/C][C]9.59[/C][C]9.58994490428016[/C][C]5.50957198353075e-05[/C][/ROW]
[ROW][C]73[/C][C]9.59[/C][C]9.59996154372021[/C][C]-0.00996154372020897[/C][/ROW]
[ROW][C]74[/C][C]9.59[/C][C]9.59695306121067[/C][C]-0.00695306121066963[/C][/ROW]
[ROW][C]75[/C][C]9.61[/C][C]9.59485316950436[/C][C]0.0151468304956346[/C][/ROW]
[ROW][C]76[/C][C]9.61[/C][C]9.61942765875031[/C][C]-0.00942765875030638[/C][/ROW]
[ROW][C]77[/C][C]9.62[/C][C]9.61658041466316[/C][C]0.00341958533683773[/C][/ROW]
[ROW][C]78[/C][C]9.62[/C][C]9.62761316249469[/C][C]-0.00761316249469246[/C][/ROW]
[ROW][C]79[/C][C]9.63[/C][C]9.6253139138189[/C][C]0.00468608618110267[/C][/ROW]
[ROW][C]80[/C][C]9.65[/C][C]9.63672915714965[/C][C]0.0132708428503463[/C][/ROW]
[ROW][C]81[/C][C]9.65[/C][C]9.66073707999008[/C][C]-0.0107370799900774[/C][/ROW]
[ROW][C]82[/C][C]9.68[/C][C]9.65749437802932[/C][C]0.0225056219706783[/C][/ROW]
[ROW][C]83[/C][C]9.73[/C][C]9.69429129344392[/C][C]0.0357087065560808[/C][/ROW]
[ROW][C]84[/C][C]9.73[/C][C]9.75507566804786[/C][C]-0.0250756680478581[/C][/ROW]
[ROW][C]85[/C][C]9.73[/C][C]9.74750257387131[/C][C]-0.0175025738713046[/C][/ROW]
[ROW][C]86[/C][C]9.74[/C][C]9.7422166273511[/C][C]-0.00221662735109796[/C][/ROW]
[ROW][C]87[/C][C]9.68[/C][C]9.7515471844612[/C][C]-0.0715471844611955[/C][/ROW]
[ROW][C]88[/C][C]9.68[/C][C]9.66993924305131[/C][C]0.0100607569486897[/C][/ROW]
[ROW][C]89[/C][C]9.74[/C][C]9.67297768891502[/C][C]0.0670223110849779[/C][/ROW]
[ROW][C]90[/C][C]9.74[/C][C]9.7532190748198[/C][C]-0.0132190748197996[/C][/ROW]
[ROW][C]91[/C][C]9.76[/C][C]9.74922678642508[/C][C]0.0107732135749181[/C][/ROW]
[ROW][C]92[/C][C]9.78[/C][C]9.77248040107779[/C][C]0.0075195989222081[/C][/ROW]
[ROW][C]93[/C][C]9.79[/C][C]9.79475139265014[/C][C]-0.00475139265014413[/C][/ROW]
[ROW][C]94[/C][C]9.81[/C][C]9.80331642613437[/C][C]0.00668357386563123[/C][/ROW]
[ROW][C]95[/C][C]9.84[/C][C]9.82533493005713[/C][C]0.0146650699428719[/C][/ROW]
[ROW][C]96[/C][C]9.84[/C][C]9.85976392295858[/C][C]-0.0197639229585782[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=278508&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=278508&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	8.26	8.27	-0.00999999999999979
4	8.28	8.34697990332218	-0.0669799033221778
5	8.28	8.34675132497176	-0.0667513249717562
6	8.28	8.32659177949301	-0.0465917794930064
7	8.29	8.31252061164694	-0.0225206116469376
8	8.3	8.31571916920519	-0.015719169205191
9	8.3	8.32097182813572	-0.0209718281357176
10	8.31	8.31463813328766	-0.00463813328766349
11	8.33	8.32323737219433	0.00676262780567427
12	8.33	8.34527975117125	-0.0152797511712528
13	8.34	8.34066511859623	-0.000665118596225867
14	8.48	8.35046424634994	0.129535753650057
15	8.59	8.52958529627573	0.0604147037242715
16	8.67	8.65783112087666	0.0121688791233385
17	8.67	8.74150624001798	-0.0715062400179836
18	8.67	8.71991066422579	-0.0499106642257914
19	8.71	8.70483716110417	0.00516283889583491
20	8.72	8.74639638836391	-0.02639638836391
21	8.72	8.74842442388347	-0.0284244238834734
22	8.72	8.73983997306952	-0.0198399730695229
23	8.74	8.73384810939399	0.00615189060601296
24	8.74	8.75570603983214	-0.0157060398321409
25	8.74	8.75096266396026	-0.0109626639602602
26	8.74	8.74765183345961	-0.0076518334596134
27	8.79	8.74534090577855	0.0446590942214495
28	8.85	8.80882838398783	0.0411716160121713
29	8.86	8.88126261006173	-0.0212626100617257
30	8.87	8.8848410962608	-0.014841096260799
31	8.92	8.89035894170955	0.0296410582904514
32	8.96	8.94931082787656	0.0106891721234366
33	8.97	8.99253906119843	-0.0225390611984313
34	8.99	8.99573204681377	-0.00573204681376893
35	8.98	9.01400091325983	-0.0340009132598293
36	8.98	8.99373230874193	-0.0137323087419343
37	9.01	8.9895850187409	0.0204149812591012
38	9.01	9.02575054044874	-0.0157505404487406
39	9.03	9.02099372496042	0.00900627503957452
40	9.05	9.04371370709308	0.00628629290691762
41	9.05	9.06561222832548	-0.0156122283254838
42	9.05	9.06089718443556	-0.0108971844355636
43	9.13	9.05760612938442	0.0723938706155831
44	9.13	9.1594697781985	-0.0294697781985036
45	9.13	9.15056962027516	-0.0205696202751557
46	9.14	9.14435739608945	-0.00435739608944807
47	9.16	9.15304142034408	0.00695857965592239
48	9.16	9.1751429786742	-0.0151429786741986
49	9.16	9.17056965271557	-0.010569652715569
50	9.16	9.16737751541037	-0.0073775154103668
51	9.22	9.16514943443222	0.0548505655677776
52	9.22	9.24171483551702	-0.0217148355170185
53	9.25	9.23515674525658	0.0148432547434236
54	9.25	9.26963955169045	-0.019639551690446
55	9.26	9.26370821720902	-0.00370821720902192
56	9.26	9.27258829976166	-0.01258829976166
57	9.29	9.2687865115327	0.0212134884673016
58	9.28	9.30519319013721	-0.0251931901372107
59	9.37	9.28758460315349	0.0824153968465051
60	9.41	9.40247484977525	0.00752515022474576
61	9.41	9.44474751789464	-0.0347475178946421
62	9.45	9.43425343155902	0.0157465684409779
63	9.49	9.47900904746259	0.0109909525374086
64	9.49	9.52232842138703	-0.0323284213870263
65	9.56	9.51256492558401	0.0474350744159953
66	9.54	9.59689077664961	-0.0568907766496096
67	9.55	9.55970921209378	-0.00970921209378162
68	9.55	9.56677693617491	-0.0167769361749119
69	9.57	9.56171013925432	0.00828986074567872
70	9.57	9.58421375734409	-0.0142137573440859
71	9.58	9.57992106521066	7.89347893395131e-05
72	9.59	9.58994490428016	5.50957198353075e-05
73	9.59	9.59996154372021	-0.00996154372020897
74	9.59	9.59695306121067	-0.00695306121066963
75	9.61	9.59485316950436	0.0151468304956346
76	9.61	9.61942765875031	-0.00942765875030638
77	9.62	9.61658041466316	0.00341958533683773
78	9.62	9.62761316249469	-0.00761316249469246
79	9.63	9.6253139138189	0.00468608618110267
80	9.65	9.63672915714965	0.0132708428503463
81	9.65	9.66073707999008	-0.0107370799900774
82	9.68	9.65749437802932	0.0225056219706783
83	9.73	9.69429129344392	0.0357087065560808
84	9.73	9.75507566804786	-0.0250756680478581
85	9.73	9.74750257387131	-0.0175025738713046
86	9.74	9.7422166273511	-0.00221662735109796
87	9.68	9.7515471844612	-0.0715471844611955
88	9.68	9.66993924305131	0.0100607569486897
89	9.74	9.67297768891502	0.0670223110849779
90	9.74	9.7532190748198	-0.0132190748197996
91	9.76	9.74922678642508	0.0107732135749181
92	9.78	9.77248040107779	0.0075195989222081
93	9.79	9.79475139265014	-0.00475139265014413
94	9.81	9.80331642613437	0.00668357386563123
95	9.84	9.82533493005713	0.0146650699428719
96	9.84	9.85976392295858	-0.0197639229585782

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
97	9.85379502715178	9.79173768170916	9.91585237259441
98	9.86759005430357	9.76570955989175	9.96947054871539
99	9.88138508145535	9.73894884289119	10.0238213200195
100	9.89518010860713	9.71003429075216	10.0803259264621
101	9.90897513575892	9.67863879563451	10.1393114758833
102	9.9227701629107	9.64470871611136	10.20083160971
103	9.93656519006248	9.60827947804692	10.264850902078
104	9.95036021721427	9.56941747855681	10.3313029558717
105	9.96415524436605	9.52819851318204	10.4001119755501
106	9.97795027151783	9.48469908631738	10.4712014567183
107	9.99174529866962	9.43899284969401	10.5444977476452
108	10.0055403258214	9.39114924116363	10.6199314104792

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
97 & 9.85379502715178 & 9.79173768170916 & 9.91585237259441 \tabularnewline
98 & 9.86759005430357 & 9.76570955989175 & 9.96947054871539 \tabularnewline
99 & 9.88138508145535 & 9.73894884289119 & 10.0238213200195 \tabularnewline
100 & 9.89518010860713 & 9.71003429075216 & 10.0803259264621 \tabularnewline
101 & 9.90897513575892 & 9.67863879563451 & 10.1393114758833 \tabularnewline
102 & 9.9227701629107 & 9.64470871611136 & 10.20083160971 \tabularnewline
103 & 9.93656519006248 & 9.60827947804692 & 10.264850902078 \tabularnewline
104 & 9.95036021721427 & 9.56941747855681 & 10.3313029558717 \tabularnewline
105 & 9.96415524436605 & 9.52819851318204 & 10.4001119755501 \tabularnewline
106 & 9.97795027151783 & 9.48469908631738 & 10.4712014567183 \tabularnewline
107 & 9.99174529866962 & 9.43899284969401 & 10.5444977476452 \tabularnewline
108 & 10.0055403258214 & 9.39114924116363 & 10.6199314104792 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=278508&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]97[/C][C]9.85379502715178[/C][C]9.79173768170916[/C][C]9.91585237259441[/C][/ROW]
[ROW][C]98[/C][C]9.86759005430357[/C][C]9.76570955989175[/C][C]9.96947054871539[/C][/ROW]
[ROW][C]99[/C][C]9.88138508145535[/C][C]9.73894884289119[/C][C]10.0238213200195[/C][/ROW]
[ROW][C]100[/C][C]9.89518010860713[/C][C]9.71003429075216[/C][C]10.0803259264621[/C][/ROW]
[ROW][C]101[/C][C]9.90897513575892[/C][C]9.67863879563451[/C][C]10.1393114758833[/C][/ROW]
[ROW][C]102[/C][C]9.9227701629107[/C][C]9.64470871611136[/C][C]10.20083160971[/C][/ROW]
[ROW][C]103[/C][C]9.93656519006248[/C][C]9.60827947804692[/C][C]10.264850902078[/C][/ROW]
[ROW][C]104[/C][C]9.95036021721427[/C][C]9.56941747855681[/C][C]10.3313029558717[/C][/ROW]
[ROW][C]105[/C][C]9.96415524436605[/C][C]9.52819851318204[/C][C]10.4001119755501[/C][/ROW]
[ROW][C]106[/C][C]9.97795027151783[/C][C]9.48469908631738[/C][C]10.4712014567183[/C][/ROW]
[ROW][C]107[/C][C]9.99174529866962[/C][C]9.43899284969401[/C][C]10.5444977476452[/C][/ROW]
[ROW][C]108[/C][C]10.0055403258214[/C][C]9.39114924116363[/C][C]10.6199314104792[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=278508&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=278508&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
97	9.85379502715178	9.79173768170916	9.91585237259441
98	9.86759005430357	9.76570955989175	9.96947054871539
99	9.88138508145535	9.73894884289119	10.0238213200195
100	9.89518010860713	9.71003429075216	10.0803259264621
101	9.90897513575892	9.67863879563451	10.1393114758833
102	9.9227701629107	9.64470871611136	10.20083160971
103	9.93656519006248	9.60827947804692	10.264850902078
104	9.95036021721427	9.56941747855681	10.3313029558717
105	9.96415524436605	9.52819851318204	10.4001119755501
106	9.97795027151783	9.48469908631738	10.4712014567183
107	9.99174529866962	9.43899284969401	10.5444977476452
108	10.0055403258214	9.39114924116363	10.6199314104792

Figure 1

PNG link

Postscript link

PDF link

Figure 2

PNG link

Postscript link

PDF link

Figure 3

PNG link

Postscript link

PDF link

Parameters (Session):

par1 = 12 ; par2 = Double ; par3 = additive ;

Parameters (R input):

par1 = 12 ; par2 = Double ; par3 = additive ;

R code (references can be found in the software module):

par1 <- as.numeric(par1)
if (par2 == 'Single') K <- 1
if (par2 == 'Double') K <- 2
if (par2 == 'Triple') K <- par1
nx <- length(x)
nxmK <- nx - K
x <- ts(x, frequency = par1)
if (par2 == 'Single') fit <- HoltWinters(x, gamma=F, beta=F)
if (par2 == 'Double') fit <- HoltWinters(x, gamma=F)
if (par2 == 'Triple') fit <- HoltWinters(x, seasonal=par3)
fit
myresid <- x - fit$fitted[,'xhat']
bitmap(file='test1.png')
op <- par(mfrow=c(2,1))
plot(fit,ylab='Observed (black) / Fitted (red)',main='Interpolation Fit of Exponential Smoothing')
plot(myresid,ylab='Residuals',main='Interpolation Prediction Errors')
par(op)
dev.off()
bitmap(file='test2.png')
p <- predict(fit, par1, prediction.interval=TRUE)
np <- length(p[,1])
plot(fit,p,ylab='Observed (black) / Fitted (red)',main='Extrapolation Fit of Exponential Smoothing')
dev.off()
bitmap(file='test3.png')
op <- par(mfrow = c(2,2))
acf(as.numeric(myresid),lag.max = nx/2,main='Residual ACF')
spectrum(myresid,main='Residals Periodogram')
cpgram(myresid,main='Residal Cumulative Periodogram')
qqnorm(myresid,main='Residual Normal QQ Plot')
qqline(myresid)
par(op)
dev.off()
load(file='createtable')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Estimated Parameters of Exponential Smoothing',2,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Parameter',header=TRUE)
a<-table.element(a,'Value',header=TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'alpha',header=TRUE)
a<-table.element(a,fit$alpha)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'beta',header=TRUE)
a<-table.element(a,fit$beta)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'gamma',header=TRUE)
a<-table.element(a,fit$gamma)
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file='mytable.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Interpolation Forecasts of Exponential Smoothing',4,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'t',header=TRUE)
a<-table.element(a,'Observed',header=TRUE)
a<-table.element(a,'Fitted',header=TRUE)
a<-table.element(a,'Residuals',header=TRUE)
a<-table.row.end(a)
for (i in 1:nxmK) {
a<-table.row.start(a)
a<-table.element(a,i+K,header=TRUE)
a<-table.element(a,x[i+K])
a<-table.element(a,fit$fitted[i,'xhat'])
a<-table.element(a,myresid[i])
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable1.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Extrapolation Forecasts of Exponential Smoothing',4,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'t',header=TRUE)
a<-table.element(a,'Forecast',header=TRUE)
a<-table.element(a,'95% Lower Bound',header=TRUE)
a<-table.element(a,'95% Upper Bound',header=TRUE)
a<-table.row.end(a)
for (i in 1:np) {
a<-table.row.start(a)
a<-table.element(a,nx+i,header=TRUE)
a<-table.element(a,p[i,'fit'])
a<-table.element(a,p[i,'lwr'])
a<-table.element(a,p[i,'upr'])
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable2.tab')

Free Statistics

Description of Statistical Computation

Tree of Dependent Computations

Dataset

Tables (Output of Computation)

Figures (Output of Computation)

Input Parameters & R Code