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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

Mon, 23 Nov 2015 14:16:33 +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/23/t1448288249ewtwrh6tmuipa29.htm/, Retrieved Tue, 14 May 2024 07:40:03 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=283929, Retrieved Tue, 14 May 2024 07:40:03 +0000

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IsPrivate?

No (this computation is public)

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Estimated Impact

155

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

-       [Exponential Smoothing] [exponential smoot...] [2015-11-23 14:16:33] [f80e7d2207e498094c9e16c0f447e302] [Current]

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Dataseries X:

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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=283929&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=283929&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=283929&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	0.931086591970149
beta	0.478906758365155
gamma	1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.931086591970149 \tabularnewline
beta & 0.478906758365155 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=283929&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]0.931086591970149[/C][/ROW]
[ROW][C]beta[/C][C]0.478906758365155[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=283929&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=283929&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	0.931086591970149
beta	0.478906758365155
gamma	1

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	79.09	76.4211591880342	2.66884081196581
14	80.84	81.789527323872	-0.949527323872005
15	81.82	82.5742336931936	-0.754233693193612
16	82.12	82.5423764447687	-0.422376444768659
17	81.01	81.2386678275391	-0.228667827539127
18	82.22	82.2979382183234	-0.0779382183234389
19	81.97	81.4519646570054	0.518035342994565
20	82.42	82.5498046107029	-0.129804610702863
21	82.77	83.1574024519091	-0.387402451909097
22	82.46	83.0894935585939	-0.629493558593907
23	83.35	83.2250667324638	0.124933267536235
24	84.27	83.3529514768724	0.917048523127562
25	87.34	86.7303649918175	0.609635008182465
26	91.03	89.3034487248892	1.72655127511082
27	93.52	93.1579161832249	0.362083816775112
28	94.24	95.2507285165794	-1.01072851657945
29	94.92	94.2126258537229	0.707374146277132
30	95.49	97.3712677691855	-1.88126776918554
31	96.55	95.3006456639846	1.24935433601536
32	98.07	97.7741967368173	0.295803263182691
33	102.87	99.6895351963806	3.1804648036194
34	104.12	105.447076195135	-1.32707619513512
35	103.49	107.194214852916	-3.70421485291632
36	103.31	104.313072518004	-1.00307251800409
37	103.92	105.52696719329	-1.60696719328965
38	103.69	104.7702468352	-1.08024683519973
39	103.41	103.322824541353	0.0871754586474083
40	102.83	102.34799800588	0.482001994119713
41	103	100.766700811106	2.23329918889378
42	103.42	103.796678295545	-0.376678295544735
43	102.57	102.642562427578	-0.072562427578049
44	102.72	102.529995860246	0.190004139754151
45	102.22	103.208855604062	-0.988855604061598
46	102.32	101.577890597468	0.742109402532165
47	102.48	102.814583265557	-0.334583265556503
48	101.56	102.486315478445	-0.926315478444934
49	101.02	102.993598241983	-1.97359824198334
50	101.41	101.031865676216	0.37813432378411
51	100.74	100.773126024045	-0.0331260240452593
52	99.76	99.4102068112508	0.349793188749246
53	99.76	97.4642568162884	2.29574318371164
54	99.17	100.038113817314	-0.868113817313585
55	99.11	97.8938548801476	1.21614511985241
56	99.69	99.0203886972529	0.6696113027471
57	99.4	100.299531296542	-0.899531296542065
58	99.79	99.1458179877204	0.6441820122796
59	99.72	100.448263367642	-0.728263367641929
60	98.74	99.7682537865778	-1.02825378657782
61	98.26	100.11858342301	-1.85858342300968
62	97.31	98.4874231370679	-1.17742313706792
63	96.73	96.1197722885269	0.610227711473087
64	96.18	95.0369220176645	1.14307798233553
65	95.92	93.9720821346623	1.9479178653377
66	96.13	95.8573461249402	0.272653875059746
67	95.64	95.2808412162411	0.3591587837589
68	94.52	95.5516168322016	-1.03161683220165
69	94.31	94.3598837003391	-0.0498837003390946
70	96.05	93.7037593349035	2.3462406650965
71	96.17	96.8554539156832	-0.685453915683183
72	95.14	96.572784269517	-1.43278426951704
73	95.37	96.6870125189829	-1.31701251898292
74	96.5	96.0463035329541	0.453696467045901
75	96.79	96.4871424097448	0.302857590255215
76	96.23	96.184350025588	0.0456499744120293
77	96	94.6933522882356	1.30664771176436
78	95.21	96.1203237664946	-0.910323766494614
79	94.77	94.1750652040744	0.59493479592561
80	96.84	94.401398622155	2.43860137784505
81	99.06	97.8876497248792	1.17235027512078
82	100.36	100.458910739328	-0.0989107393283035
83	100.09	101.958985879244	-1.86898587924389
84	100.03	100.829055825756	-0.799055825756469
85	100.49	102.130111608247	-1.6401116082466
86	101	101.755317166453	-0.755317166452997
87	102.11	100.965683415789	1.14431658421118
88	101.59	101.709465388908	-0.119465388908168
89	100.81	100.358833280852	0.451166719148219
90	100.86	100.662239388775	0.197760611224794
91	99.57	100.172275283666	-0.602275283665705
92	100.21	99.1969548911445	1.01304510885545
93	99.68	100.418966321608	-0.738966321607833
94	98.38	99.4210944168354	-1.04109441683543
95	97.93	97.7998851714716	0.130114828528434
96	97.37	97.3743819490477	-0.00438194904772615
97	99.08	98.4810943338719	0.598905666128061
98	99.15	100.374085487729	-1.22408548772924
99	99.44	99.1919649951919	0.248035004808145
100	99.48	98.5275513673589	0.952448632641136
101	99.62	98.2056703058842	1.41432969411576
102	98.95	99.8092614344207	-0.859261434420688
103	99.42	98.2295150419237	1.19048495807625
104	99.84	99.7836552292235	0.0563447707764624
105	99.27	100.316490862872	-1.04649086287218
106	99.16	99.1966721244512	-0.0366721244512291
107	99.04	99.2244604268286	-0.184460426828579
108	99.62	98.9896029104901	0.630397089509884

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 79.09 & 76.4211591880342 & 2.66884081196581 \tabularnewline
14 & 80.84 & 81.789527323872 & -0.949527323872005 \tabularnewline
15 & 81.82 & 82.5742336931936 & -0.754233693193612 \tabularnewline
16 & 82.12 & 82.5423764447687 & -0.422376444768659 \tabularnewline
17 & 81.01 & 81.2386678275391 & -0.228667827539127 \tabularnewline
18 & 82.22 & 82.2979382183234 & -0.0779382183234389 \tabularnewline
19 & 81.97 & 81.4519646570054 & 0.518035342994565 \tabularnewline
20 & 82.42 & 82.5498046107029 & -0.129804610702863 \tabularnewline
21 & 82.77 & 83.1574024519091 & -0.387402451909097 \tabularnewline
22 & 82.46 & 83.0894935585939 & -0.629493558593907 \tabularnewline
23 & 83.35 & 83.2250667324638 & 0.124933267536235 \tabularnewline
24 & 84.27 & 83.3529514768724 & 0.917048523127562 \tabularnewline
25 & 87.34 & 86.7303649918175 & 0.609635008182465 \tabularnewline
26 & 91.03 & 89.3034487248892 & 1.72655127511082 \tabularnewline
27 & 93.52 & 93.1579161832249 & 0.362083816775112 \tabularnewline
28 & 94.24 & 95.2507285165794 & -1.01072851657945 \tabularnewline
29 & 94.92 & 94.2126258537229 & 0.707374146277132 \tabularnewline
30 & 95.49 & 97.3712677691855 & -1.88126776918554 \tabularnewline
31 & 96.55 & 95.3006456639846 & 1.24935433601536 \tabularnewline
32 & 98.07 & 97.7741967368173 & 0.295803263182691 \tabularnewline
33 & 102.87 & 99.6895351963806 & 3.1804648036194 \tabularnewline
34 & 104.12 & 105.447076195135 & -1.32707619513512 \tabularnewline
35 & 103.49 & 107.194214852916 & -3.70421485291632 \tabularnewline
36 & 103.31 & 104.313072518004 & -1.00307251800409 \tabularnewline
37 & 103.92 & 105.52696719329 & -1.60696719328965 \tabularnewline
38 & 103.69 & 104.7702468352 & -1.08024683519973 \tabularnewline
39 & 103.41 & 103.322824541353 & 0.0871754586474083 \tabularnewline
40 & 102.83 & 102.34799800588 & 0.482001994119713 \tabularnewline
41 & 103 & 100.766700811106 & 2.23329918889378 \tabularnewline
42 & 103.42 & 103.796678295545 & -0.376678295544735 \tabularnewline
43 & 102.57 & 102.642562427578 & -0.072562427578049 \tabularnewline
44 & 102.72 & 102.529995860246 & 0.190004139754151 \tabularnewline
45 & 102.22 & 103.208855604062 & -0.988855604061598 \tabularnewline
46 & 102.32 & 101.577890597468 & 0.742109402532165 \tabularnewline
47 & 102.48 & 102.814583265557 & -0.334583265556503 \tabularnewline
48 & 101.56 & 102.486315478445 & -0.926315478444934 \tabularnewline
49 & 101.02 & 102.993598241983 & -1.97359824198334 \tabularnewline
50 & 101.41 & 101.031865676216 & 0.37813432378411 \tabularnewline
51 & 100.74 & 100.773126024045 & -0.0331260240452593 \tabularnewline
52 & 99.76 & 99.4102068112508 & 0.349793188749246 \tabularnewline
53 & 99.76 & 97.4642568162884 & 2.29574318371164 \tabularnewline
54 & 99.17 & 100.038113817314 & -0.868113817313585 \tabularnewline
55 & 99.11 & 97.8938548801476 & 1.21614511985241 \tabularnewline
56 & 99.69 & 99.0203886972529 & 0.6696113027471 \tabularnewline
57 & 99.4 & 100.299531296542 & -0.899531296542065 \tabularnewline
58 & 99.79 & 99.1458179877204 & 0.6441820122796 \tabularnewline
59 & 99.72 & 100.448263367642 & -0.728263367641929 \tabularnewline
60 & 98.74 & 99.7682537865778 & -1.02825378657782 \tabularnewline
61 & 98.26 & 100.11858342301 & -1.85858342300968 \tabularnewline
62 & 97.31 & 98.4874231370679 & -1.17742313706792 \tabularnewline
63 & 96.73 & 96.1197722885269 & 0.610227711473087 \tabularnewline
64 & 96.18 & 95.0369220176645 & 1.14307798233553 \tabularnewline
65 & 95.92 & 93.9720821346623 & 1.9479178653377 \tabularnewline
66 & 96.13 & 95.8573461249402 & 0.272653875059746 \tabularnewline
67 & 95.64 & 95.2808412162411 & 0.3591587837589 \tabularnewline
68 & 94.52 & 95.5516168322016 & -1.03161683220165 \tabularnewline
69 & 94.31 & 94.3598837003391 & -0.0498837003390946 \tabularnewline
70 & 96.05 & 93.7037593349035 & 2.3462406650965 \tabularnewline
71 & 96.17 & 96.8554539156832 & -0.685453915683183 \tabularnewline
72 & 95.14 & 96.572784269517 & -1.43278426951704 \tabularnewline
73 & 95.37 & 96.6870125189829 & -1.31701251898292 \tabularnewline
74 & 96.5 & 96.0463035329541 & 0.453696467045901 \tabularnewline
75 & 96.79 & 96.4871424097448 & 0.302857590255215 \tabularnewline
76 & 96.23 & 96.184350025588 & 0.0456499744120293 \tabularnewline
77 & 96 & 94.6933522882356 & 1.30664771176436 \tabularnewline
78 & 95.21 & 96.1203237664946 & -0.910323766494614 \tabularnewline
79 & 94.77 & 94.1750652040744 & 0.59493479592561 \tabularnewline
80 & 96.84 & 94.401398622155 & 2.43860137784505 \tabularnewline
81 & 99.06 & 97.8876497248792 & 1.17235027512078 \tabularnewline
82 & 100.36 & 100.458910739328 & -0.0989107393283035 \tabularnewline
83 & 100.09 & 101.958985879244 & -1.86898587924389 \tabularnewline
84 & 100.03 & 100.829055825756 & -0.799055825756469 \tabularnewline
85 & 100.49 & 102.130111608247 & -1.6401116082466 \tabularnewline
86 & 101 & 101.755317166453 & -0.755317166452997 \tabularnewline
87 & 102.11 & 100.965683415789 & 1.14431658421118 \tabularnewline
88 & 101.59 & 101.709465388908 & -0.119465388908168 \tabularnewline
89 & 100.81 & 100.358833280852 & 0.451166719148219 \tabularnewline
90 & 100.86 & 100.662239388775 & 0.197760611224794 \tabularnewline
91 & 99.57 & 100.172275283666 & -0.602275283665705 \tabularnewline
92 & 100.21 & 99.1969548911445 & 1.01304510885545 \tabularnewline
93 & 99.68 & 100.418966321608 & -0.738966321607833 \tabularnewline
94 & 98.38 & 99.4210944168354 & -1.04109441683543 \tabularnewline
95 & 97.93 & 97.7998851714716 & 0.130114828528434 \tabularnewline
96 & 97.37 & 97.3743819490477 & -0.00438194904772615 \tabularnewline
97 & 99.08 & 98.4810943338719 & 0.598905666128061 \tabularnewline
98 & 99.15 & 100.374085487729 & -1.22408548772924 \tabularnewline
99 & 99.44 & 99.1919649951919 & 0.248035004808145 \tabularnewline
100 & 99.48 & 98.5275513673589 & 0.952448632641136 \tabularnewline
101 & 99.62 & 98.2056703058842 & 1.41432969411576 \tabularnewline
102 & 98.95 & 99.8092614344207 & -0.859261434420688 \tabularnewline
103 & 99.42 & 98.2295150419237 & 1.19048495807625 \tabularnewline
104 & 99.84 & 99.7836552292235 & 0.0563447707764624 \tabularnewline
105 & 99.27 & 100.316490862872 & -1.04649086287218 \tabularnewline
106 & 99.16 & 99.1966721244512 & -0.0366721244512291 \tabularnewline
107 & 99.04 & 99.2244604268286 & -0.184460426828579 \tabularnewline
108 & 99.62 & 98.9896029104901 & 0.630397089509884 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=283929&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]79.09[/C][C]76.4211591880342[/C][C]2.66884081196581[/C][/ROW]
[ROW][C]14[/C][C]80.84[/C][C]81.789527323872[/C][C]-0.949527323872005[/C][/ROW]
[ROW][C]15[/C][C]81.82[/C][C]82.5742336931936[/C][C]-0.754233693193612[/C][/ROW]
[ROW][C]16[/C][C]82.12[/C][C]82.5423764447687[/C][C]-0.422376444768659[/C][/ROW]
[ROW][C]17[/C][C]81.01[/C][C]81.2386678275391[/C][C]-0.228667827539127[/C][/ROW]
[ROW][C]18[/C][C]82.22[/C][C]82.2979382183234[/C][C]-0.0779382183234389[/C][/ROW]
[ROW][C]19[/C][C]81.97[/C][C]81.4519646570054[/C][C]0.518035342994565[/C][/ROW]
[ROW][C]20[/C][C]82.42[/C][C]82.5498046107029[/C][C]-0.129804610702863[/C][/ROW]
[ROW][C]21[/C][C]82.77[/C][C]83.1574024519091[/C][C]-0.387402451909097[/C][/ROW]
[ROW][C]22[/C][C]82.46[/C][C]83.0894935585939[/C][C]-0.629493558593907[/C][/ROW]
[ROW][C]23[/C][C]83.35[/C][C]83.2250667324638[/C][C]0.124933267536235[/C][/ROW]
[ROW][C]24[/C][C]84.27[/C][C]83.3529514768724[/C][C]0.917048523127562[/C][/ROW]
[ROW][C]25[/C][C]87.34[/C][C]86.7303649918175[/C][C]0.609635008182465[/C][/ROW]
[ROW][C]26[/C][C]91.03[/C][C]89.3034487248892[/C][C]1.72655127511082[/C][/ROW]
[ROW][C]27[/C][C]93.52[/C][C]93.1579161832249[/C][C]0.362083816775112[/C][/ROW]
[ROW][C]28[/C][C]94.24[/C][C]95.2507285165794[/C][C]-1.01072851657945[/C][/ROW]
[ROW][C]29[/C][C]94.92[/C][C]94.2126258537229[/C][C]0.707374146277132[/C][/ROW]
[ROW][C]30[/C][C]95.49[/C][C]97.3712677691855[/C][C]-1.88126776918554[/C][/ROW]
[ROW][C]31[/C][C]96.55[/C][C]95.3006456639846[/C][C]1.24935433601536[/C][/ROW]
[ROW][C]32[/C][C]98.07[/C][C]97.7741967368173[/C][C]0.295803263182691[/C][/ROW]
[ROW][C]33[/C][C]102.87[/C][C]99.6895351963806[/C][C]3.1804648036194[/C][/ROW]
[ROW][C]34[/C][C]104.12[/C][C]105.447076195135[/C][C]-1.32707619513512[/C][/ROW]
[ROW][C]35[/C][C]103.49[/C][C]107.194214852916[/C][C]-3.70421485291632[/C][/ROW]
[ROW][C]36[/C][C]103.31[/C][C]104.313072518004[/C][C]-1.00307251800409[/C][/ROW]
[ROW][C]37[/C][C]103.92[/C][C]105.52696719329[/C][C]-1.60696719328965[/C][/ROW]
[ROW][C]38[/C][C]103.69[/C][C]104.7702468352[/C][C]-1.08024683519973[/C][/ROW]
[ROW][C]39[/C][C]103.41[/C][C]103.322824541353[/C][C]0.0871754586474083[/C][/ROW]
[ROW][C]40[/C][C]102.83[/C][C]102.34799800588[/C][C]0.482001994119713[/C][/ROW]
[ROW][C]41[/C][C]103[/C][C]100.766700811106[/C][C]2.23329918889378[/C][/ROW]
[ROW][C]42[/C][C]103.42[/C][C]103.796678295545[/C][C]-0.376678295544735[/C][/ROW]
[ROW][C]43[/C][C]102.57[/C][C]102.642562427578[/C][C]-0.072562427578049[/C][/ROW]
[ROW][C]44[/C][C]102.72[/C][C]102.529995860246[/C][C]0.190004139754151[/C][/ROW]
[ROW][C]45[/C][C]102.22[/C][C]103.208855604062[/C][C]-0.988855604061598[/C][/ROW]
[ROW][C]46[/C][C]102.32[/C][C]101.577890597468[/C][C]0.742109402532165[/C][/ROW]
[ROW][C]47[/C][C]102.48[/C][C]102.814583265557[/C][C]-0.334583265556503[/C][/ROW]
[ROW][C]48[/C][C]101.56[/C][C]102.486315478445[/C][C]-0.926315478444934[/C][/ROW]
[ROW][C]49[/C][C]101.02[/C][C]102.993598241983[/C][C]-1.97359824198334[/C][/ROW]
[ROW][C]50[/C][C]101.41[/C][C]101.031865676216[/C][C]0.37813432378411[/C][/ROW]
[ROW][C]51[/C][C]100.74[/C][C]100.773126024045[/C][C]-0.0331260240452593[/C][/ROW]
[ROW][C]52[/C][C]99.76[/C][C]99.4102068112508[/C][C]0.349793188749246[/C][/ROW]
[ROW][C]53[/C][C]99.76[/C][C]97.4642568162884[/C][C]2.29574318371164[/C][/ROW]
[ROW][C]54[/C][C]99.17[/C][C]100.038113817314[/C][C]-0.868113817313585[/C][/ROW]
[ROW][C]55[/C][C]99.11[/C][C]97.8938548801476[/C][C]1.21614511985241[/C][/ROW]
[ROW][C]56[/C][C]99.69[/C][C]99.0203886972529[/C][C]0.6696113027471[/C][/ROW]
[ROW][C]57[/C][C]99.4[/C][C]100.299531296542[/C][C]-0.899531296542065[/C][/ROW]
[ROW][C]58[/C][C]99.79[/C][C]99.1458179877204[/C][C]0.6441820122796[/C][/ROW]
[ROW][C]59[/C][C]99.72[/C][C]100.448263367642[/C][C]-0.728263367641929[/C][/ROW]
[ROW][C]60[/C][C]98.74[/C][C]99.7682537865778[/C][C]-1.02825378657782[/C][/ROW]
[ROW][C]61[/C][C]98.26[/C][C]100.11858342301[/C][C]-1.85858342300968[/C][/ROW]
[ROW][C]62[/C][C]97.31[/C][C]98.4874231370679[/C][C]-1.17742313706792[/C][/ROW]
[ROW][C]63[/C][C]96.73[/C][C]96.1197722885269[/C][C]0.610227711473087[/C][/ROW]
[ROW][C]64[/C][C]96.18[/C][C]95.0369220176645[/C][C]1.14307798233553[/C][/ROW]
[ROW][C]65[/C][C]95.92[/C][C]93.9720821346623[/C][C]1.9479178653377[/C][/ROW]
[ROW][C]66[/C][C]96.13[/C][C]95.8573461249402[/C][C]0.272653875059746[/C][/ROW]
[ROW][C]67[/C][C]95.64[/C][C]95.2808412162411[/C][C]0.3591587837589[/C][/ROW]
[ROW][C]68[/C][C]94.52[/C][C]95.5516168322016[/C][C]-1.03161683220165[/C][/ROW]
[ROW][C]69[/C][C]94.31[/C][C]94.3598837003391[/C][C]-0.0498837003390946[/C][/ROW]
[ROW][C]70[/C][C]96.05[/C][C]93.7037593349035[/C][C]2.3462406650965[/C][/ROW]
[ROW][C]71[/C][C]96.17[/C][C]96.8554539156832[/C][C]-0.685453915683183[/C][/ROW]
[ROW][C]72[/C][C]95.14[/C][C]96.572784269517[/C][C]-1.43278426951704[/C][/ROW]
[ROW][C]73[/C][C]95.37[/C][C]96.6870125189829[/C][C]-1.31701251898292[/C][/ROW]
[ROW][C]74[/C][C]96.5[/C][C]96.0463035329541[/C][C]0.453696467045901[/C][/ROW]
[ROW][C]75[/C][C]96.79[/C][C]96.4871424097448[/C][C]0.302857590255215[/C][/ROW]
[ROW][C]76[/C][C]96.23[/C][C]96.184350025588[/C][C]0.0456499744120293[/C][/ROW]
[ROW][C]77[/C][C]96[/C][C]94.6933522882356[/C][C]1.30664771176436[/C][/ROW]
[ROW][C]78[/C][C]95.21[/C][C]96.1203237664946[/C][C]-0.910323766494614[/C][/ROW]
[ROW][C]79[/C][C]94.77[/C][C]94.1750652040744[/C][C]0.59493479592561[/C][/ROW]
[ROW][C]80[/C][C]96.84[/C][C]94.401398622155[/C][C]2.43860137784505[/C][/ROW]
[ROW][C]81[/C][C]99.06[/C][C]97.8876497248792[/C][C]1.17235027512078[/C][/ROW]
[ROW][C]82[/C][C]100.36[/C][C]100.458910739328[/C][C]-0.0989107393283035[/C][/ROW]
[ROW][C]83[/C][C]100.09[/C][C]101.958985879244[/C][C]-1.86898587924389[/C][/ROW]
[ROW][C]84[/C][C]100.03[/C][C]100.829055825756[/C][C]-0.799055825756469[/C][/ROW]
[ROW][C]85[/C][C]100.49[/C][C]102.130111608247[/C][C]-1.6401116082466[/C][/ROW]
[ROW][C]86[/C][C]101[/C][C]101.755317166453[/C][C]-0.755317166452997[/C][/ROW]
[ROW][C]87[/C][C]102.11[/C][C]100.965683415789[/C][C]1.14431658421118[/C][/ROW]
[ROW][C]88[/C][C]101.59[/C][C]101.709465388908[/C][C]-0.119465388908168[/C][/ROW]
[ROW][C]89[/C][C]100.81[/C][C]100.358833280852[/C][C]0.451166719148219[/C][/ROW]
[ROW][C]90[/C][C]100.86[/C][C]100.662239388775[/C][C]0.197760611224794[/C][/ROW]
[ROW][C]91[/C][C]99.57[/C][C]100.172275283666[/C][C]-0.602275283665705[/C][/ROW]
[ROW][C]92[/C][C]100.21[/C][C]99.1969548911445[/C][C]1.01304510885545[/C][/ROW]
[ROW][C]93[/C][C]99.68[/C][C]100.418966321608[/C][C]-0.738966321607833[/C][/ROW]
[ROW][C]94[/C][C]98.38[/C][C]99.4210944168354[/C][C]-1.04109441683543[/C][/ROW]
[ROW][C]95[/C][C]97.93[/C][C]97.7998851714716[/C][C]0.130114828528434[/C][/ROW]
[ROW][C]96[/C][C]97.37[/C][C]97.3743819490477[/C][C]-0.00438194904772615[/C][/ROW]
[ROW][C]97[/C][C]99.08[/C][C]98.4810943338719[/C][C]0.598905666128061[/C][/ROW]
[ROW][C]98[/C][C]99.15[/C][C]100.374085487729[/C][C]-1.22408548772924[/C][/ROW]
[ROW][C]99[/C][C]99.44[/C][C]99.1919649951919[/C][C]0.248035004808145[/C][/ROW]
[ROW][C]100[/C][C]99.48[/C][C]98.5275513673589[/C][C]0.952448632641136[/C][/ROW]
[ROW][C]101[/C][C]99.62[/C][C]98.2056703058842[/C][C]1.41432969411576[/C][/ROW]
[ROW][C]102[/C][C]98.95[/C][C]99.8092614344207[/C][C]-0.859261434420688[/C][/ROW]
[ROW][C]103[/C][C]99.42[/C][C]98.2295150419237[/C][C]1.19048495807625[/C][/ROW]
[ROW][C]104[/C][C]99.84[/C][C]99.7836552292235[/C][C]0.0563447707764624[/C][/ROW]
[ROW][C]105[/C][C]99.27[/C][C]100.316490862872[/C][C]-1.04649086287218[/C][/ROW]
[ROW][C]106[/C][C]99.16[/C][C]99.1966721244512[/C][C]-0.0366721244512291[/C][/ROW]
[ROW][C]107[/C][C]99.04[/C][C]99.2244604268286[/C][C]-0.184460426828579[/C][/ROW]
[ROW][C]108[/C][C]99.62[/C][C]98.9896029104901[/C][C]0.630397089509884[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=283929&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=283929&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	79.09	76.4211591880342	2.66884081196581
14	80.84	81.789527323872	-0.949527323872005
15	81.82	82.5742336931936	-0.754233693193612
16	82.12	82.5423764447687	-0.422376444768659
17	81.01	81.2386678275391	-0.228667827539127
18	82.22	82.2979382183234	-0.0779382183234389
19	81.97	81.4519646570054	0.518035342994565
20	82.42	82.5498046107029	-0.129804610702863
21	82.77	83.1574024519091	-0.387402451909097
22	82.46	83.0894935585939	-0.629493558593907
23	83.35	83.2250667324638	0.124933267536235
24	84.27	83.3529514768724	0.917048523127562
25	87.34	86.7303649918175	0.609635008182465
26	91.03	89.3034487248892	1.72655127511082
27	93.52	93.1579161832249	0.362083816775112
28	94.24	95.2507285165794	-1.01072851657945
29	94.92	94.2126258537229	0.707374146277132
30	95.49	97.3712677691855	-1.88126776918554
31	96.55	95.3006456639846	1.24935433601536
32	98.07	97.7741967368173	0.295803263182691
33	102.87	99.6895351963806	3.1804648036194
34	104.12	105.447076195135	-1.32707619513512
35	103.49	107.194214852916	-3.70421485291632
36	103.31	104.313072518004	-1.00307251800409
37	103.92	105.52696719329	-1.60696719328965
38	103.69	104.7702468352	-1.08024683519973
39	103.41	103.322824541353	0.0871754586474083
40	102.83	102.34799800588	0.482001994119713
41	103	100.766700811106	2.23329918889378
42	103.42	103.796678295545	-0.376678295544735
43	102.57	102.642562427578	-0.072562427578049
44	102.72	102.529995860246	0.190004139754151
45	102.22	103.208855604062	-0.988855604061598
46	102.32	101.577890597468	0.742109402532165
47	102.48	102.814583265557	-0.334583265556503
48	101.56	102.486315478445	-0.926315478444934
49	101.02	102.993598241983	-1.97359824198334
50	101.41	101.031865676216	0.37813432378411
51	100.74	100.773126024045	-0.0331260240452593
52	99.76	99.4102068112508	0.349793188749246
53	99.76	97.4642568162884	2.29574318371164
54	99.17	100.038113817314	-0.868113817313585
55	99.11	97.8938548801476	1.21614511985241
56	99.69	99.0203886972529	0.6696113027471
57	99.4	100.299531296542	-0.899531296542065
58	99.79	99.1458179877204	0.6441820122796
59	99.72	100.448263367642	-0.728263367641929
60	98.74	99.7682537865778	-1.02825378657782
61	98.26	100.11858342301	-1.85858342300968
62	97.31	98.4874231370679	-1.17742313706792
63	96.73	96.1197722885269	0.610227711473087
64	96.18	95.0369220176645	1.14307798233553
65	95.92	93.9720821346623	1.9479178653377
66	96.13	95.8573461249402	0.272653875059746
67	95.64	95.2808412162411	0.3591587837589
68	94.52	95.5516168322016	-1.03161683220165
69	94.31	94.3598837003391	-0.0498837003390946
70	96.05	93.7037593349035	2.3462406650965
71	96.17	96.8554539156832	-0.685453915683183
72	95.14	96.572784269517	-1.43278426951704
73	95.37	96.6870125189829	-1.31701251898292
74	96.5	96.0463035329541	0.453696467045901
75	96.79	96.4871424097448	0.302857590255215
76	96.23	96.184350025588	0.0456499744120293
77	96	94.6933522882356	1.30664771176436
78	95.21	96.1203237664946	-0.910323766494614
79	94.77	94.1750652040744	0.59493479592561
80	96.84	94.401398622155	2.43860137784505
81	99.06	97.8876497248792	1.17235027512078
82	100.36	100.458910739328	-0.0989107393283035
83	100.09	101.958985879244	-1.86898587924389
84	100.03	100.829055825756	-0.799055825756469
85	100.49	102.130111608247	-1.6401116082466
86	101	101.755317166453	-0.755317166452997
87	102.11	100.965683415789	1.14431658421118
88	101.59	101.709465388908	-0.119465388908168
89	100.81	100.358833280852	0.451166719148219
90	100.86	100.662239388775	0.197760611224794
91	99.57	100.172275283666	-0.602275283665705
92	100.21	99.1969548911445	1.01304510885545
93	99.68	100.418966321608	-0.738966321607833
94	98.38	99.4210944168354	-1.04109441683543
95	97.93	97.7998851714716	0.130114828528434
96	97.37	97.3743819490477	-0.00438194904772615
97	99.08	98.4810943338719	0.598905666128061
98	99.15	100.374085487729	-1.22408548772924
99	99.44	99.1919649951919	0.248035004808145
100	99.48	98.5275513673589	0.952448632641136
101	99.62	98.2056703058842	1.41432969411576
102	98.95	99.8092614344207	-0.859261434420688
103	99.42	98.2295150419237	1.19048495807625
104	99.84	99.7836552292235	0.0563447707764624
105	99.27	100.316490862872	-1.04649086287218
106	99.16	99.1966721244512	-0.0366721244512291
107	99.04	99.2244604268286	-0.184460426828579
108	99.62	98.9896029104901	0.630397089509884

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
109	101.504785589936	99.2451435453345	103.764427634538
110	103.223322382929	99.3778775476087	107.06876721825
111	104.337011724545	98.7019137009745	109.972109748116
112	104.434231265256	96.8160241881193	112.052438342393
113	103.776699209841	93.9957923180802	113.557606101603
114	103.79542258063	91.6846027471142	115.906242414145
115	103.428802388234	88.8313718967747	118.026232879693
116	103.537323315859	86.3055026052157	120.769144026503
117	103.657555375526	83.651196215714	123.663914535337
118	103.764192855027	80.8497496351946	126.67863607486
119	104.014786275891	78.0644877347646	129.965084817018
120	104.288928368654	75.1801030511241	133.397753686184

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
109 & 101.504785589936 & 99.2451435453345 & 103.764427634538 \tabularnewline
110 & 103.223322382929 & 99.3778775476087 & 107.06876721825 \tabularnewline
111 & 104.337011724545 & 98.7019137009745 & 109.972109748116 \tabularnewline
112 & 104.434231265256 & 96.8160241881193 & 112.052438342393 \tabularnewline
113 & 103.776699209841 & 93.9957923180802 & 113.557606101603 \tabularnewline
114 & 103.79542258063 & 91.6846027471142 & 115.906242414145 \tabularnewline
115 & 103.428802388234 & 88.8313718967747 & 118.026232879693 \tabularnewline
116 & 103.537323315859 & 86.3055026052157 & 120.769144026503 \tabularnewline
117 & 103.657555375526 & 83.651196215714 & 123.663914535337 \tabularnewline
118 & 103.764192855027 & 80.8497496351946 & 126.67863607486 \tabularnewline
119 & 104.014786275891 & 78.0644877347646 & 129.965084817018 \tabularnewline
120 & 104.288928368654 & 75.1801030511241 & 133.397753686184 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=283929&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]101.504785589936[/C][C]99.2451435453345[/C][C]103.764427634538[/C][/ROW]
[ROW][C]110[/C][C]103.223322382929[/C][C]99.3778775476087[/C][C]107.06876721825[/C][/ROW]
[ROW][C]111[/C][C]104.337011724545[/C][C]98.7019137009745[/C][C]109.972109748116[/C][/ROW]
[ROW][C]112[/C][C]104.434231265256[/C][C]96.8160241881193[/C][C]112.052438342393[/C][/ROW]
[ROW][C]113[/C][C]103.776699209841[/C][C]93.9957923180802[/C][C]113.557606101603[/C][/ROW]
[ROW][C]114[/C][C]103.79542258063[/C][C]91.6846027471142[/C][C]115.906242414145[/C][/ROW]
[ROW][C]115[/C][C]103.428802388234[/C][C]88.8313718967747[/C][C]118.026232879693[/C][/ROW]
[ROW][C]116[/C][C]103.537323315859[/C][C]86.3055026052157[/C][C]120.769144026503[/C][/ROW]
[ROW][C]117[/C][C]103.657555375526[/C][C]83.651196215714[/C][C]123.663914535337[/C][/ROW]
[ROW][C]118[/C][C]103.764192855027[/C][C]80.8497496351946[/C][C]126.67863607486[/C][/ROW]
[ROW][C]119[/C][C]104.014786275891[/C][C]78.0644877347646[/C][C]129.965084817018[/C][/ROW]
[ROW][C]120[/C][C]104.288928368654[/C][C]75.1801030511241[/C][C]133.397753686184[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=283929&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=283929&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	101.504785589936	99.2451435453345	103.764427634538
110	103.223322382929	99.3778775476087	107.06876721825
111	104.337011724545	98.7019137009745	109.972109748116
112	104.434231265256	96.8160241881193	112.052438342393
113	103.776699209841	93.9957923180802	113.557606101603
114	103.79542258063	91.6846027471142	115.906242414145
115	103.428802388234	88.8313718967747	118.026232879693
116	103.537323315859	86.3055026052157	120.769144026503
117	103.657555375526	83.651196215714	123.663914535337
118	103.764192855027	80.8497496351946	126.67863607486
119	104.014786275891	78.0644877347646	129.965084817018
120	104.288928368654	75.1801030511241	133.397753686184

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