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Author

*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Mon, 19 Dec 2011 13:11:54 -0500

Cite this page as follows

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2011/Dec/19/t132431985891bx1s2pg2zf9sq.htm/, Retrieved Mon, 20 May 2024 11:41:17 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=157599, Retrieved Mon, 20 May 2024 11:41:17 +0000

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Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords

KDGP2W102

Estimated Impact

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

-       [Exponential Smoothing] [Exponential Smoot...] [2011-12-19 18:11:54] [76c30f62b7052b57088120e90a652e05] [Current]

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Dataseries X:

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7,6
7,4
7,3
7,1
6,9
6,8
7,5
7,6
7,8
8
8,1
8,2
8,3
8,2
8
7,9
7,6
7,6
8,3
8,4
8,4
8,4
8,4
8,6
8,9
8,8
8,3
7,5
7,2
7,4
8,8
9,3
9,3
8,7
8,2
8,3
8,5
8,6
8,5
8,2
8,1
7,9
8,6
8,7
8,7
8,5
8,4
8,5
8,7
8,7
8,6
8,5
8,3
8
8,2
8,1
8,1
8
7,9
7,9
8
8
7,9
8
7,7
7,2
7,5
7,3
7
7
7
7,2
7,3
7,1
6,8
6,4
6,1
6,5
7,7
7,9
7,5
6,9
6,6
6,9
7,7
8
8
7,7
7,3
7,4
8,1
8,3
8,1
7,9
7,9
8,3
8,6
8,7
8,5
8,3
8
8
8,8
8,7
8,5
8,1
7,8
7,6

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
R Framework error message	Warning: there are blank lines in the 'Data' field. Please, use NA for missing data - blank lines are simply deleted and are NOT treated as missing values.

\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
R Framework error message & Warning: there are blank lines in the 'Data' field.
Please, use NA for missing data - blank lines are simply
 deleted and are NOT treated as missing values. \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=157599&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]
[ROW][C]R Framework error message[/C][C]Warning: there are blank lines in the 'Data' field.
Please, use NA for missing data - blank lines are simply
 deleted and are NOT treated as missing values.[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=157599&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=157599&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
R Framework error message	Warning: there are blank lines in the 'Data' field. Please, use NA for missing data - blank lines are simply deleted and are NOT treated as missing values.

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.953634145370514
beta	0.0168544582380706
gamma	1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.953634145370514 \tabularnewline
beta & 0.0168544582380706 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=157599&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]0.953634145370514[/C][/ROW]
[ROW][C]beta[/C][C]0.0168544582380706[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=157599&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=157599&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	0.953634145370514
beta	0.0168544582380706
gamma	1

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	8.3	7.92409188034188	0.37590811965812
14	8.2	8.17943434336599	0.0205656566340071
15	8	8.00457398521279	-0.00457398521279373
16	7.9	7.92233275525446	-0.0223327552544621
17	7.6	7.63529720172247	-0.0352972017224715
18	7.6	7.63533097790266	-0.0353309779026567
19	8.3	8.16809800295325	0.131901997046748
20	8.4	8.39579749554848	0.00420250445152348
21	8.4	8.60178593845731	-0.201785938457309
22	8.4	8.60809346591568	-0.208093465915676
23	8.4	8.50504123626992	-0.105041236269916
24	8.6	8.49857480515875	0.101425194841251
25	8.9	8.70389465301716	0.196105346982836
26	8.8	8.76875166730097	0.0312483326990343
27	8.3	8.60054112703961	-0.300541127039606
28	7.5	8.22810312240219	-0.728103122402194
29	7.2	7.24894690070226	-0.0489469007022629
30	7.4	7.21727006074236	0.182729939257644
31	8.8	7.95055398117613	0.849446018823867
32	9.3	8.85295279248321	0.447047207516793
33	9.3	9.47516580708207	-0.175165807082074
34	8.7	9.51045818384497	-0.810458183844975
35	8.2	8.83195813267965	-0.63195813267965
36	8.3	8.31831925831401	-0.0183192583140119
37	8.5	8.39765249047486	0.10234750952514
38	8.6	8.34776398209236	0.252236017907638
39	8.5	8.36077196334733	0.139228036652671
40	8.2	8.38081779701586	-0.180817797015859
41	8.1	7.95678694241383	0.14321305758617
42	7.9	8.12391661412454	-0.223916614124535
43	8.6	8.49859965165502	0.101400348344978
44	8.7	8.65523397093818	0.0447660290618241
45	8.7	8.84475758586611	-0.144757585866108
46	8.5	8.85987026395997	-0.359870263959975
47	8.4	8.60686269710329	-0.206862697103292
48	8.5	8.52141394033713	-0.0214139403371281
49	8.7	8.59769375943323	0.102306240566772
50	8.7	8.54901790460307	0.150982095396927
51	8.6	8.45290182365037	0.147098176349633
52	8.5	8.45841503659826	0.0415849634017427
53	8.3	8.25787503643893	0.0421249635610668
54	8	8.3063326019833	-0.306332601983302
55	8.2	8.61093110024137	-0.410931100241372
56	8.1	8.26155462302422	-0.161554623024218
57	8.1	8.22741207094934	-0.127412070949338
58	8	8.2312466115092	-0.231246611509205
59	7.9	8.09221511474571	-0.192215114745709
60	7.9	8.0137905497711	-0.113790549771098
61	8	7.9906857707795	0.00931422922049663
62	8	7.83706429589337	0.162935704106633
63	7.9	7.73383747293811	0.166162527061887
64	8	7.73461528224774	0.265384717752259
65	7.7	7.73309692850261	-0.0330969285026095
66	7.2	7.67802827735418	-0.478028277354182
67	7.5	7.79564693675134	-0.29564693675134
68	7.3	7.55122970714629	-0.251229707146294
69	7	7.41516941441635	-0.415169414416348
70	7	7.11716566239292	-0.117165662392916
71	7	7.06796031728709	-0.0679603172870937
72	7.2	7.09288767214233	0.10711232785767
73	7.3	7.27092392767661	0.029076072323388
74	7.1	7.12836109328781	-0.0283610932878142
75	6.8	6.82487229709344	-0.0248722970934407
76	6.4	6.62701836683365	-0.227018366833653
77	6.1	6.11311894343518	-0.0131189434351837
78	6.5	6.02782414636701	0.472175853632994
79	7.7	7.04667058300473	0.653329416995274
80	7.9	7.71116634090699	0.188833659093008
81	7.5	7.99611471903952	-0.496114719039524
82	6.9	7.64238534977164	-0.742385349771641
83	6.6	6.9968308529772	-0.396830852977204
84	6.9	6.70856773927733	0.191432260722672
85	7.7	6.95706572829244	0.742934271707565
86	8	7.49774274203451	0.502257257965486
87	8	7.71410352408582	0.285896475914176
88	7.7	7.82190365352116	-0.121903653521155
89	7.3	7.4385193686203	-0.138519368620303
90	7.4	7.27448052200785	0.125519477992152
91	8.1	7.98391210827477	0.116087891725227
92	8.3	8.11867335096217	0.181326649037832
93	8.1	8.36871800135363	-0.268718001353635
94	7.9	8.22809173311791	-0.328091733117907
95	7.9	8.00797101556812	-0.107971015568124
96	8.3	8.04141997906722	0.258580020932779
97	8.6	8.39957264314755	0.200427356852453
98	8.7	8.42306705290342	0.276932947096578
99	8.5	8.42222720061923	0.0777727993807691
100	8.3	8.31700838987043	-0.0170083898704263
101	8	8.03893429417733	-0.038934294177329
102	8	7.98975507722988	0.0102449227701209
103	8.8	8.59461631710185	0.20538368289815
104	8.7	8.82478988522208	-0.124789885222082
105	8.5	8.76435640340052	-0.264356403400518
106	8.1	8.62751844614867	-0.527518446148669
107	7.8	8.22660016386622	-0.42660016386622
108	7.6	7.96724409491284	-0.367244094912841

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 8.3 & 7.92409188034188 & 0.37590811965812 \tabularnewline
14 & 8.2 & 8.17943434336599 & 0.0205656566340071 \tabularnewline
15 & 8 & 8.00457398521279 & -0.00457398521279373 \tabularnewline
16 & 7.9 & 7.92233275525446 & -0.0223327552544621 \tabularnewline
17 & 7.6 & 7.63529720172247 & -0.0352972017224715 \tabularnewline
18 & 7.6 & 7.63533097790266 & -0.0353309779026567 \tabularnewline
19 & 8.3 & 8.16809800295325 & 0.131901997046748 \tabularnewline
20 & 8.4 & 8.39579749554848 & 0.00420250445152348 \tabularnewline
21 & 8.4 & 8.60178593845731 & -0.201785938457309 \tabularnewline
22 & 8.4 & 8.60809346591568 & -0.208093465915676 \tabularnewline
23 & 8.4 & 8.50504123626992 & -0.105041236269916 \tabularnewline
24 & 8.6 & 8.49857480515875 & 0.101425194841251 \tabularnewline
25 & 8.9 & 8.70389465301716 & 0.196105346982836 \tabularnewline
26 & 8.8 & 8.76875166730097 & 0.0312483326990343 \tabularnewline
27 & 8.3 & 8.60054112703961 & -0.300541127039606 \tabularnewline
28 & 7.5 & 8.22810312240219 & -0.728103122402194 \tabularnewline
29 & 7.2 & 7.24894690070226 & -0.0489469007022629 \tabularnewline
30 & 7.4 & 7.21727006074236 & 0.182729939257644 \tabularnewline
31 & 8.8 & 7.95055398117613 & 0.849446018823867 \tabularnewline
32 & 9.3 & 8.85295279248321 & 0.447047207516793 \tabularnewline
33 & 9.3 & 9.47516580708207 & -0.175165807082074 \tabularnewline
34 & 8.7 & 9.51045818384497 & -0.810458183844975 \tabularnewline
35 & 8.2 & 8.83195813267965 & -0.63195813267965 \tabularnewline
36 & 8.3 & 8.31831925831401 & -0.0183192583140119 \tabularnewline
37 & 8.5 & 8.39765249047486 & 0.10234750952514 \tabularnewline
38 & 8.6 & 8.34776398209236 & 0.252236017907638 \tabularnewline
39 & 8.5 & 8.36077196334733 & 0.139228036652671 \tabularnewline
40 & 8.2 & 8.38081779701586 & -0.180817797015859 \tabularnewline
41 & 8.1 & 7.95678694241383 & 0.14321305758617 \tabularnewline
42 & 7.9 & 8.12391661412454 & -0.223916614124535 \tabularnewline
43 & 8.6 & 8.49859965165502 & 0.101400348344978 \tabularnewline
44 & 8.7 & 8.65523397093818 & 0.0447660290618241 \tabularnewline
45 & 8.7 & 8.84475758586611 & -0.144757585866108 \tabularnewline
46 & 8.5 & 8.85987026395997 & -0.359870263959975 \tabularnewline
47 & 8.4 & 8.60686269710329 & -0.206862697103292 \tabularnewline
48 & 8.5 & 8.52141394033713 & -0.0214139403371281 \tabularnewline
49 & 8.7 & 8.59769375943323 & 0.102306240566772 \tabularnewline
50 & 8.7 & 8.54901790460307 & 0.150982095396927 \tabularnewline
51 & 8.6 & 8.45290182365037 & 0.147098176349633 \tabularnewline
52 & 8.5 & 8.45841503659826 & 0.0415849634017427 \tabularnewline
53 & 8.3 & 8.25787503643893 & 0.0421249635610668 \tabularnewline
54 & 8 & 8.3063326019833 & -0.306332601983302 \tabularnewline
55 & 8.2 & 8.61093110024137 & -0.410931100241372 \tabularnewline
56 & 8.1 & 8.26155462302422 & -0.161554623024218 \tabularnewline
57 & 8.1 & 8.22741207094934 & -0.127412070949338 \tabularnewline
58 & 8 & 8.2312466115092 & -0.231246611509205 \tabularnewline
59 & 7.9 & 8.09221511474571 & -0.192215114745709 \tabularnewline
60 & 7.9 & 8.0137905497711 & -0.113790549771098 \tabularnewline
61 & 8 & 7.9906857707795 & 0.00931422922049663 \tabularnewline
62 & 8 & 7.83706429589337 & 0.162935704106633 \tabularnewline
63 & 7.9 & 7.73383747293811 & 0.166162527061887 \tabularnewline
64 & 8 & 7.73461528224774 & 0.265384717752259 \tabularnewline
65 & 7.7 & 7.73309692850261 & -0.0330969285026095 \tabularnewline
66 & 7.2 & 7.67802827735418 & -0.478028277354182 \tabularnewline
67 & 7.5 & 7.79564693675134 & -0.29564693675134 \tabularnewline
68 & 7.3 & 7.55122970714629 & -0.251229707146294 \tabularnewline
69 & 7 & 7.41516941441635 & -0.415169414416348 \tabularnewline
70 & 7 & 7.11716566239292 & -0.117165662392916 \tabularnewline
71 & 7 & 7.06796031728709 & -0.0679603172870937 \tabularnewline
72 & 7.2 & 7.09288767214233 & 0.10711232785767 \tabularnewline
73 & 7.3 & 7.27092392767661 & 0.029076072323388 \tabularnewline
74 & 7.1 & 7.12836109328781 & -0.0283610932878142 \tabularnewline
75 & 6.8 & 6.82487229709344 & -0.0248722970934407 \tabularnewline
76 & 6.4 & 6.62701836683365 & -0.227018366833653 \tabularnewline
77 & 6.1 & 6.11311894343518 & -0.0131189434351837 \tabularnewline
78 & 6.5 & 6.02782414636701 & 0.472175853632994 \tabularnewline
79 & 7.7 & 7.04667058300473 & 0.653329416995274 \tabularnewline
80 & 7.9 & 7.71116634090699 & 0.188833659093008 \tabularnewline
81 & 7.5 & 7.99611471903952 & -0.496114719039524 \tabularnewline
82 & 6.9 & 7.64238534977164 & -0.742385349771641 \tabularnewline
83 & 6.6 & 6.9968308529772 & -0.396830852977204 \tabularnewline
84 & 6.9 & 6.70856773927733 & 0.191432260722672 \tabularnewline
85 & 7.7 & 6.95706572829244 & 0.742934271707565 \tabularnewline
86 & 8 & 7.49774274203451 & 0.502257257965486 \tabularnewline
87 & 8 & 7.71410352408582 & 0.285896475914176 \tabularnewline
88 & 7.7 & 7.82190365352116 & -0.121903653521155 \tabularnewline
89 & 7.3 & 7.4385193686203 & -0.138519368620303 \tabularnewline
90 & 7.4 & 7.27448052200785 & 0.125519477992152 \tabularnewline
91 & 8.1 & 7.98391210827477 & 0.116087891725227 \tabularnewline
92 & 8.3 & 8.11867335096217 & 0.181326649037832 \tabularnewline
93 & 8.1 & 8.36871800135363 & -0.268718001353635 \tabularnewline
94 & 7.9 & 8.22809173311791 & -0.328091733117907 \tabularnewline
95 & 7.9 & 8.00797101556812 & -0.107971015568124 \tabularnewline
96 & 8.3 & 8.04141997906722 & 0.258580020932779 \tabularnewline
97 & 8.6 & 8.39957264314755 & 0.200427356852453 \tabularnewline
98 & 8.7 & 8.42306705290342 & 0.276932947096578 \tabularnewline
99 & 8.5 & 8.42222720061923 & 0.0777727993807691 \tabularnewline
100 & 8.3 & 8.31700838987043 & -0.0170083898704263 \tabularnewline
101 & 8 & 8.03893429417733 & -0.038934294177329 \tabularnewline
102 & 8 & 7.98975507722988 & 0.0102449227701209 \tabularnewline
103 & 8.8 & 8.59461631710185 & 0.20538368289815 \tabularnewline
104 & 8.7 & 8.82478988522208 & -0.124789885222082 \tabularnewline
105 & 8.5 & 8.76435640340052 & -0.264356403400518 \tabularnewline
106 & 8.1 & 8.62751844614867 & -0.527518446148669 \tabularnewline
107 & 7.8 & 8.22660016386622 & -0.42660016386622 \tabularnewline
108 & 7.6 & 7.96724409491284 & -0.367244094912841 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=157599&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]8.3[/C][C]7.92409188034188[/C][C]0.37590811965812[/C][/ROW]
[ROW][C]14[/C][C]8.2[/C][C]8.17943434336599[/C][C]0.0205656566340071[/C][/ROW]
[ROW][C]15[/C][C]8[/C][C]8.00457398521279[/C][C]-0.00457398521279373[/C][/ROW]
[ROW][C]16[/C][C]7.9[/C][C]7.92233275525446[/C][C]-0.0223327552544621[/C][/ROW]
[ROW][C]17[/C][C]7.6[/C][C]7.63529720172247[/C][C]-0.0352972017224715[/C][/ROW]
[ROW][C]18[/C][C]7.6[/C][C]7.63533097790266[/C][C]-0.0353309779026567[/C][/ROW]
[ROW][C]19[/C][C]8.3[/C][C]8.16809800295325[/C][C]0.131901997046748[/C][/ROW]
[ROW][C]20[/C][C]8.4[/C][C]8.39579749554848[/C][C]0.00420250445152348[/C][/ROW]
[ROW][C]21[/C][C]8.4[/C][C]8.60178593845731[/C][C]-0.201785938457309[/C][/ROW]
[ROW][C]22[/C][C]8.4[/C][C]8.60809346591568[/C][C]-0.208093465915676[/C][/ROW]
[ROW][C]23[/C][C]8.4[/C][C]8.50504123626992[/C][C]-0.105041236269916[/C][/ROW]
[ROW][C]24[/C][C]8.6[/C][C]8.49857480515875[/C][C]0.101425194841251[/C][/ROW]
[ROW][C]25[/C][C]8.9[/C][C]8.70389465301716[/C][C]0.196105346982836[/C][/ROW]
[ROW][C]26[/C][C]8.8[/C][C]8.76875166730097[/C][C]0.0312483326990343[/C][/ROW]
[ROW][C]27[/C][C]8.3[/C][C]8.60054112703961[/C][C]-0.300541127039606[/C][/ROW]
[ROW][C]28[/C][C]7.5[/C][C]8.22810312240219[/C][C]-0.728103122402194[/C][/ROW]
[ROW][C]29[/C][C]7.2[/C][C]7.24894690070226[/C][C]-0.0489469007022629[/C][/ROW]
[ROW][C]30[/C][C]7.4[/C][C]7.21727006074236[/C][C]0.182729939257644[/C][/ROW]
[ROW][C]31[/C][C]8.8[/C][C]7.95055398117613[/C][C]0.849446018823867[/C][/ROW]
[ROW][C]32[/C][C]9.3[/C][C]8.85295279248321[/C][C]0.447047207516793[/C][/ROW]
[ROW][C]33[/C][C]9.3[/C][C]9.47516580708207[/C][C]-0.175165807082074[/C][/ROW]
[ROW][C]34[/C][C]8.7[/C][C]9.51045818384497[/C][C]-0.810458183844975[/C][/ROW]
[ROW][C]35[/C][C]8.2[/C][C]8.83195813267965[/C][C]-0.63195813267965[/C][/ROW]
[ROW][C]36[/C][C]8.3[/C][C]8.31831925831401[/C][C]-0.0183192583140119[/C][/ROW]
[ROW][C]37[/C][C]8.5[/C][C]8.39765249047486[/C][C]0.10234750952514[/C][/ROW]
[ROW][C]38[/C][C]8.6[/C][C]8.34776398209236[/C][C]0.252236017907638[/C][/ROW]
[ROW][C]39[/C][C]8.5[/C][C]8.36077196334733[/C][C]0.139228036652671[/C][/ROW]
[ROW][C]40[/C][C]8.2[/C][C]8.38081779701586[/C][C]-0.180817797015859[/C][/ROW]
[ROW][C]41[/C][C]8.1[/C][C]7.95678694241383[/C][C]0.14321305758617[/C][/ROW]
[ROW][C]42[/C][C]7.9[/C][C]8.12391661412454[/C][C]-0.223916614124535[/C][/ROW]
[ROW][C]43[/C][C]8.6[/C][C]8.49859965165502[/C][C]0.101400348344978[/C][/ROW]
[ROW][C]44[/C][C]8.7[/C][C]8.65523397093818[/C][C]0.0447660290618241[/C][/ROW]
[ROW][C]45[/C][C]8.7[/C][C]8.84475758586611[/C][C]-0.144757585866108[/C][/ROW]
[ROW][C]46[/C][C]8.5[/C][C]8.85987026395997[/C][C]-0.359870263959975[/C][/ROW]
[ROW][C]47[/C][C]8.4[/C][C]8.60686269710329[/C][C]-0.206862697103292[/C][/ROW]
[ROW][C]48[/C][C]8.5[/C][C]8.52141394033713[/C][C]-0.0214139403371281[/C][/ROW]
[ROW][C]49[/C][C]8.7[/C][C]8.59769375943323[/C][C]0.102306240566772[/C][/ROW]
[ROW][C]50[/C][C]8.7[/C][C]8.54901790460307[/C][C]0.150982095396927[/C][/ROW]
[ROW][C]51[/C][C]8.6[/C][C]8.45290182365037[/C][C]0.147098176349633[/C][/ROW]
[ROW][C]52[/C][C]8.5[/C][C]8.45841503659826[/C][C]0.0415849634017427[/C][/ROW]
[ROW][C]53[/C][C]8.3[/C][C]8.25787503643893[/C][C]0.0421249635610668[/C][/ROW]
[ROW][C]54[/C][C]8[/C][C]8.3063326019833[/C][C]-0.306332601983302[/C][/ROW]
[ROW][C]55[/C][C]8.2[/C][C]8.61093110024137[/C][C]-0.410931100241372[/C][/ROW]
[ROW][C]56[/C][C]8.1[/C][C]8.26155462302422[/C][C]-0.161554623024218[/C][/ROW]
[ROW][C]57[/C][C]8.1[/C][C]8.22741207094934[/C][C]-0.127412070949338[/C][/ROW]
[ROW][C]58[/C][C]8[/C][C]8.2312466115092[/C][C]-0.231246611509205[/C][/ROW]
[ROW][C]59[/C][C]7.9[/C][C]8.09221511474571[/C][C]-0.192215114745709[/C][/ROW]
[ROW][C]60[/C][C]7.9[/C][C]8.0137905497711[/C][C]-0.113790549771098[/C][/ROW]
[ROW][C]61[/C][C]8[/C][C]7.9906857707795[/C][C]0.00931422922049663[/C][/ROW]
[ROW][C]62[/C][C]8[/C][C]7.83706429589337[/C][C]0.162935704106633[/C][/ROW]
[ROW][C]63[/C][C]7.9[/C][C]7.73383747293811[/C][C]0.166162527061887[/C][/ROW]
[ROW][C]64[/C][C]8[/C][C]7.73461528224774[/C][C]0.265384717752259[/C][/ROW]
[ROW][C]65[/C][C]7.7[/C][C]7.73309692850261[/C][C]-0.0330969285026095[/C][/ROW]
[ROW][C]66[/C][C]7.2[/C][C]7.67802827735418[/C][C]-0.478028277354182[/C][/ROW]
[ROW][C]67[/C][C]7.5[/C][C]7.79564693675134[/C][C]-0.29564693675134[/C][/ROW]
[ROW][C]68[/C][C]7.3[/C][C]7.55122970714629[/C][C]-0.251229707146294[/C][/ROW]
[ROW][C]69[/C][C]7[/C][C]7.41516941441635[/C][C]-0.415169414416348[/C][/ROW]
[ROW][C]70[/C][C]7[/C][C]7.11716566239292[/C][C]-0.117165662392916[/C][/ROW]
[ROW][C]71[/C][C]7[/C][C]7.06796031728709[/C][C]-0.0679603172870937[/C][/ROW]
[ROW][C]72[/C][C]7.2[/C][C]7.09288767214233[/C][C]0.10711232785767[/C][/ROW]
[ROW][C]73[/C][C]7.3[/C][C]7.27092392767661[/C][C]0.029076072323388[/C][/ROW]
[ROW][C]74[/C][C]7.1[/C][C]7.12836109328781[/C][C]-0.0283610932878142[/C][/ROW]
[ROW][C]75[/C][C]6.8[/C][C]6.82487229709344[/C][C]-0.0248722970934407[/C][/ROW]
[ROW][C]76[/C][C]6.4[/C][C]6.62701836683365[/C][C]-0.227018366833653[/C][/ROW]
[ROW][C]77[/C][C]6.1[/C][C]6.11311894343518[/C][C]-0.0131189434351837[/C][/ROW]
[ROW][C]78[/C][C]6.5[/C][C]6.02782414636701[/C][C]0.472175853632994[/C][/ROW]
[ROW][C]79[/C][C]7.7[/C][C]7.04667058300473[/C][C]0.653329416995274[/C][/ROW]
[ROW][C]80[/C][C]7.9[/C][C]7.71116634090699[/C][C]0.188833659093008[/C][/ROW]
[ROW][C]81[/C][C]7.5[/C][C]7.99611471903952[/C][C]-0.496114719039524[/C][/ROW]
[ROW][C]82[/C][C]6.9[/C][C]7.64238534977164[/C][C]-0.742385349771641[/C][/ROW]
[ROW][C]83[/C][C]6.6[/C][C]6.9968308529772[/C][C]-0.396830852977204[/C][/ROW]
[ROW][C]84[/C][C]6.9[/C][C]6.70856773927733[/C][C]0.191432260722672[/C][/ROW]
[ROW][C]85[/C][C]7.7[/C][C]6.95706572829244[/C][C]0.742934271707565[/C][/ROW]
[ROW][C]86[/C][C]8[/C][C]7.49774274203451[/C][C]0.502257257965486[/C][/ROW]
[ROW][C]87[/C][C]8[/C][C]7.71410352408582[/C][C]0.285896475914176[/C][/ROW]
[ROW][C]88[/C][C]7.7[/C][C]7.82190365352116[/C][C]-0.121903653521155[/C][/ROW]
[ROW][C]89[/C][C]7.3[/C][C]7.4385193686203[/C][C]-0.138519368620303[/C][/ROW]
[ROW][C]90[/C][C]7.4[/C][C]7.27448052200785[/C][C]0.125519477992152[/C][/ROW]
[ROW][C]91[/C][C]8.1[/C][C]7.98391210827477[/C][C]0.116087891725227[/C][/ROW]
[ROW][C]92[/C][C]8.3[/C][C]8.11867335096217[/C][C]0.181326649037832[/C][/ROW]
[ROW][C]93[/C][C]8.1[/C][C]8.36871800135363[/C][C]-0.268718001353635[/C][/ROW]
[ROW][C]94[/C][C]7.9[/C][C]8.22809173311791[/C][C]-0.328091733117907[/C][/ROW]
[ROW][C]95[/C][C]7.9[/C][C]8.00797101556812[/C][C]-0.107971015568124[/C][/ROW]
[ROW][C]96[/C][C]8.3[/C][C]8.04141997906722[/C][C]0.258580020932779[/C][/ROW]
[ROW][C]97[/C][C]8.6[/C][C]8.39957264314755[/C][C]0.200427356852453[/C][/ROW]
[ROW][C]98[/C][C]8.7[/C][C]8.42306705290342[/C][C]0.276932947096578[/C][/ROW]
[ROW][C]99[/C][C]8.5[/C][C]8.42222720061923[/C][C]0.0777727993807691[/C][/ROW]
[ROW][C]100[/C][C]8.3[/C][C]8.31700838987043[/C][C]-0.0170083898704263[/C][/ROW]
[ROW][C]101[/C][C]8[/C][C]8.03893429417733[/C][C]-0.038934294177329[/C][/ROW]
[ROW][C]102[/C][C]8[/C][C]7.98975507722988[/C][C]0.0102449227701209[/C][/ROW]
[ROW][C]103[/C][C]8.8[/C][C]8.59461631710185[/C][C]0.20538368289815[/C][/ROW]
[ROW][C]104[/C][C]8.7[/C][C]8.82478988522208[/C][C]-0.124789885222082[/C][/ROW]
[ROW][C]105[/C][C]8.5[/C][C]8.76435640340052[/C][C]-0.264356403400518[/C][/ROW]
[ROW][C]106[/C][C]8.1[/C][C]8.62751844614867[/C][C]-0.527518446148669[/C][/ROW]
[ROW][C]107[/C][C]7.8[/C][C]8.22660016386622[/C][C]-0.42660016386622[/C][/ROW]
[ROW][C]108[/C][C]7.6[/C][C]7.96724409491284[/C][C]-0.367244094912841[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=157599&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=157599&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	8.3	7.92409188034188	0.37590811965812
14	8.2	8.17943434336599	0.0205656566340071
15	8	8.00457398521279	-0.00457398521279373
16	7.9	7.92233275525446	-0.0223327552544621
17	7.6	7.63529720172247	-0.0352972017224715
18	7.6	7.63533097790266	-0.0353309779026567
19	8.3	8.16809800295325	0.131901997046748
20	8.4	8.39579749554848	0.00420250445152348
21	8.4	8.60178593845731	-0.201785938457309
22	8.4	8.60809346591568	-0.208093465915676
23	8.4	8.50504123626992	-0.105041236269916
24	8.6	8.49857480515875	0.101425194841251
25	8.9	8.70389465301716	0.196105346982836
26	8.8	8.76875166730097	0.0312483326990343
27	8.3	8.60054112703961	-0.300541127039606
28	7.5	8.22810312240219	-0.728103122402194
29	7.2	7.24894690070226	-0.0489469007022629
30	7.4	7.21727006074236	0.182729939257644
31	8.8	7.95055398117613	0.849446018823867
32	9.3	8.85295279248321	0.447047207516793
33	9.3	9.47516580708207	-0.175165807082074
34	8.7	9.51045818384497	-0.810458183844975
35	8.2	8.83195813267965	-0.63195813267965
36	8.3	8.31831925831401	-0.0183192583140119
37	8.5	8.39765249047486	0.10234750952514
38	8.6	8.34776398209236	0.252236017907638
39	8.5	8.36077196334733	0.139228036652671
40	8.2	8.38081779701586	-0.180817797015859
41	8.1	7.95678694241383	0.14321305758617
42	7.9	8.12391661412454	-0.223916614124535
43	8.6	8.49859965165502	0.101400348344978
44	8.7	8.65523397093818	0.0447660290618241
45	8.7	8.84475758586611	-0.144757585866108
46	8.5	8.85987026395997	-0.359870263959975
47	8.4	8.60686269710329	-0.206862697103292
48	8.5	8.52141394033713	-0.0214139403371281
49	8.7	8.59769375943323	0.102306240566772
50	8.7	8.54901790460307	0.150982095396927
51	8.6	8.45290182365037	0.147098176349633
52	8.5	8.45841503659826	0.0415849634017427
53	8.3	8.25787503643893	0.0421249635610668
54	8	8.3063326019833	-0.306332601983302
55	8.2	8.61093110024137	-0.410931100241372
56	8.1	8.26155462302422	-0.161554623024218
57	8.1	8.22741207094934	-0.127412070949338
58	8	8.2312466115092	-0.231246611509205
59	7.9	8.09221511474571	-0.192215114745709
60	7.9	8.0137905497711	-0.113790549771098
61	8	7.9906857707795	0.00931422922049663
62	8	7.83706429589337	0.162935704106633
63	7.9	7.73383747293811	0.166162527061887
64	8	7.73461528224774	0.265384717752259
65	7.7	7.73309692850261	-0.0330969285026095
66	7.2	7.67802827735418	-0.478028277354182
67	7.5	7.79564693675134	-0.29564693675134
68	7.3	7.55122970714629	-0.251229707146294
69	7	7.41516941441635	-0.415169414416348
70	7	7.11716566239292	-0.117165662392916
71	7	7.06796031728709	-0.0679603172870937
72	7.2	7.09288767214233	0.10711232785767
73	7.3	7.27092392767661	0.029076072323388
74	7.1	7.12836109328781	-0.0283610932878142
75	6.8	6.82487229709344	-0.0248722970934407
76	6.4	6.62701836683365	-0.227018366833653
77	6.1	6.11311894343518	-0.0131189434351837
78	6.5	6.02782414636701	0.472175853632994
79	7.7	7.04667058300473	0.653329416995274
80	7.9	7.71116634090699	0.188833659093008
81	7.5	7.99611471903952	-0.496114719039524
82	6.9	7.64238534977164	-0.742385349771641
83	6.6	6.9968308529772	-0.396830852977204
84	6.9	6.70856773927733	0.191432260722672
85	7.7	6.95706572829244	0.742934271707565
86	8	7.49774274203451	0.502257257965486
87	8	7.71410352408582	0.285896475914176
88	7.7	7.82190365352116	-0.121903653521155
89	7.3	7.4385193686203	-0.138519368620303
90	7.4	7.27448052200785	0.125519477992152
91	8.1	7.98391210827477	0.116087891725227
92	8.3	8.11867335096217	0.181326649037832
93	8.1	8.36871800135363	-0.268718001353635
94	7.9	8.22809173311791	-0.328091733117907
95	7.9	8.00797101556812	-0.107971015568124
96	8.3	8.04141997906722	0.258580020932779
97	8.6	8.39957264314755	0.200427356852453
98	8.7	8.42306705290342	0.276932947096578
99	8.5	8.42222720061923	0.0777727993807691
100	8.3	8.31700838987043	-0.0170083898704263
101	8	8.03893429417733	-0.038934294177329
102	8	7.98975507722988	0.0102449227701209
103	8.8	8.59461631710185	0.20538368289815
104	8.7	8.82478988522208	-0.124789885222082
105	8.5	8.76435640340052	-0.264356403400518
106	8.1	8.62751844614867	-0.527518446148669
107	7.8	8.22660016386622	-0.42660016386622
108	7.6	7.96724409491284	-0.367244094912841

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
109	7.70988950335799	7.1326388621104	8.28714014460558
110	7.52657161095261	6.72248572619362	8.3306574957116
111	7.22872849618141	6.24365837575455	8.21379861660827
112	7.0200219186351	5.87773580433385	8.16230803293635
113	6.73249800773118	5.44793989218759	8.01705612327476
114	6.69870090670298	5.28224361266076	8.11515820074519
115	7.27864815442218	5.73794218777175	8.8193541210726
116	7.27015906135956	5.61108324949121	8.92923487322792
117	7.29677111177001	5.52397770721156	9.06956451632845
118	7.37859246891339	5.49584480640858	9.2613401314182
119	7.4726535032427	5.48304760416918	9.46225940231623
120	7.61696730231889	5.52308400997291	9.71085059466487

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
109 & 7.70988950335799 & 7.1326388621104 & 8.28714014460558 \tabularnewline
110 & 7.52657161095261 & 6.72248572619362 & 8.3306574957116 \tabularnewline
111 & 7.22872849618141 & 6.24365837575455 & 8.21379861660827 \tabularnewline
112 & 7.0200219186351 & 5.87773580433385 & 8.16230803293635 \tabularnewline
113 & 6.73249800773118 & 5.44793989218759 & 8.01705612327476 \tabularnewline
114 & 6.69870090670298 & 5.28224361266076 & 8.11515820074519 \tabularnewline
115 & 7.27864815442218 & 5.73794218777175 & 8.8193541210726 \tabularnewline
116 & 7.27015906135956 & 5.61108324949121 & 8.92923487322792 \tabularnewline
117 & 7.29677111177001 & 5.52397770721156 & 9.06956451632845 \tabularnewline
118 & 7.37859246891339 & 5.49584480640858 & 9.2613401314182 \tabularnewline
119 & 7.4726535032427 & 5.48304760416918 & 9.46225940231623 \tabularnewline
120 & 7.61696730231889 & 5.52308400997291 & 9.71085059466487 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=157599&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]7.70988950335799[/C][C]7.1326388621104[/C][C]8.28714014460558[/C][/ROW]
[ROW][C]110[/C][C]7.52657161095261[/C][C]6.72248572619362[/C][C]8.3306574957116[/C][/ROW]
[ROW][C]111[/C][C]7.22872849618141[/C][C]6.24365837575455[/C][C]8.21379861660827[/C][/ROW]
[ROW][C]112[/C][C]7.0200219186351[/C][C]5.87773580433385[/C][C]8.16230803293635[/C][/ROW]
[ROW][C]113[/C][C]6.73249800773118[/C][C]5.44793989218759[/C][C]8.01705612327476[/C][/ROW]
[ROW][C]114[/C][C]6.69870090670298[/C][C]5.28224361266076[/C][C]8.11515820074519[/C][/ROW]
[ROW][C]115[/C][C]7.27864815442218[/C][C]5.73794218777175[/C][C]8.8193541210726[/C][/ROW]
[ROW][C]116[/C][C]7.27015906135956[/C][C]5.61108324949121[/C][C]8.92923487322792[/C][/ROW]
[ROW][C]117[/C][C]7.29677111177001[/C][C]5.52397770721156[/C][C]9.06956451632845[/C][/ROW]
[ROW][C]118[/C][C]7.37859246891339[/C][C]5.49584480640858[/C][C]9.2613401314182[/C][/ROW]
[ROW][C]119[/C][C]7.4726535032427[/C][C]5.48304760416918[/C][C]9.46225940231623[/C][/ROW]
[ROW][C]120[/C][C]7.61696730231889[/C][C]5.52308400997291[/C][C]9.71085059466487[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=157599&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=157599&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	7.70988950335799	7.1326388621104	8.28714014460558
110	7.52657161095261	6.72248572619362	8.3306574957116
111	7.22872849618141	6.24365837575455	8.21379861660827
112	7.0200219186351	5.87773580433385	8.16230803293635
113	6.73249800773118	5.44793989218759	8.01705612327476
114	6.69870090670298	5.28224361266076	8.11515820074519
115	7.27864815442218	5.73794218777175	8.8193541210726
116	7.27015906135956	5.61108324949121	8.92923487322792
117	7.29677111177001	5.52397770721156	9.06956451632845
118	7.37859246891339	5.49584480640858	9.2613401314182
119	7.4726535032427	5.48304760416918	9.46225940231623
120	7.61696730231889	5.52308400997291	9.71085059466487

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