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*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Thu, 19 May 2011 21:52:51 +0000

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Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2011/May/19/t1305841740i1fq779vxsxyjz1.htm/, Retrieved Sat, 11 May 2024 10:45:26 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=122245, Retrieved Sat, 11 May 2024 10:45:26 +0000

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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] [oef 10.2 Thomas S...] [2011-05-19 21:52:51] [93a9440e82e53db41c1ce1bc7dd7ea5d] [Current]

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Dataseries X:

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17,1
13,4
15,3
14
9,7
13,7
13,7
12,5
9,8
7
-1,9
-2,9
-6,8
-10,4
-17,2
-19,8
-16,8
-23,2
-21,7
-17,6
-13
-12,6
-4
-0,2
3,1
6,5
19,2
26,6
26,6
31,4
31,2
26,4
20,7
20,7
15
13,3
8,7
10,2
4,3
-0,1
-4,6
-3,9
-3,5
-3,4
-2,5
-1,1
0,3
-0,9
3,6
2,7
-0,2
-1
5,8
6,4
9,6
13,2
10,6
10,9
12,9
15,9
12,2
9,1
9
17,4
14,7
17
13,7
9,5
14,8
13,6
12,6
8,9
10,2
12,7
16
10,4
9,9
9,5
8,6
10
3,5
-4,2
-4,4
-1,5
-0,1
0,8
-2,4
-1,2
0,2
-1,9
-1,6
-4,2
-2,2
6,2
5,7
3,1
1,1
-0,9
0,1
-4
-4
-5,3
-8
-6,3
-3,6
-3,5
-5,1
-3,3

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	'Sir Ronald Aylmer Fisher' @ 193.190.124.24

\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 & 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=122245&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]'Sir Ronald Aylmer Fisher' @ 193.190.124.24[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=122245&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=122245&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	'Sir Ronald Aylmer Fisher' @ 193.190.124.24

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	1
beta	0.0129118310986201
gamma	1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 1 \tabularnewline
beta & 0.0129118310986201 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=122245&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]0.0129118310986201[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=122245&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=122245&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	0.0129118310986201
gamma	1

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	-6.8	5.29011904502332	-12.0901190450233
14	-10.4	-8.84171324014721	-1.55828675985279
15	-17.2	-15.9777490065531	-1.22225099344691
16	-19.8	-19.2767864574917	-0.523213542508337
17	-16.8	-17.4447550613783	0.644755061378326
18	-23.2	-25.6515712032783	2.45157120327834
19	-21.7	-22.7216106860833	1.02161068608327
20	-17.6	-27.7644764890422	10.1644764890422
21	-13	-23.2666467148864	10.2666467148864
22	-12.6	-26.3010044782941	13.7010044782941
23	-4	-18.9086575698116	14.9086575698116
24	-0.2	-2.02357903063458	1.82357903063458
25	3.1	-1.49201857893718	4.59201857893718
26	6.5	2.01844508797077	4.48155491202923
27	19.2	7.01265005348185	12.1873499465182
28	26.6	17.6315183424416	8.96848165755836
29	26.6	20.0432882620052	6.55671173799476
30	31.4	35.4760470482716	-4.07604704827165
31	31.2	26.6707282575723	4.52927174242769
32	26.4	35.0400343978482	-8.6400343978482
33	20.7	29.121780244723	-8.42178024472298
34	20.7	31.8724152661828	-11.1724152661828
35	15	19.698985465602	-4.69898546560196
36	13.3	2.68691920634813	10.6130807936519
37	8.7	15.0018137763925	-6.30181377639252
38	10.2	8.18633685500812	2.01366314499188
39	4.3	12.1873122352001	-7.88731223520014
40	-0.1	2.85287939834127	-2.95287939834127
41	-4.6	-1.23202195284512	-3.36797804715488
42	-3.9	-8.0189175674334	4.1189175674334
43	-3.5	-4.88744331137933	1.38744331137933
44	-3.4	-5.72230740148329	2.32230740148329
45	-2.5	-5.85143471834727	3.35143471834727
46	-1.1	-7.32300181020793	6.22300181020793
47	0.3	-4.64616789106606	4.94616789106606
48	-0.9	-0.723371722037258	-0.176628277962742
49	3.6	-2.02736684196994	5.62736684196994
50	2.7	2.88068944693305	-0.180689446933046
51	-0.2	2.33993111446629	-2.53993111446629
52	-1	-1.45894998659693	0.458949986596932
53	5.8	-1.78705521406384	7.58705521406384
54	6.4	6.78395031443612	-0.383950314436121
55	9.6	4.57927487564638	5.02072512435362
56	13.2	9.98483733517275	3.21516266482725
57	10.6	14.0490822853714	-3.44908228537135
58	10.9	15.5107002009973	-4.61070020099728
59	12.9	9.52806161767183	3.37193838232817
60	15.9	2.42119499932248	13.4788050006775
61	12.2	18.4313153693455	-6.23131536934553
62	9.1	12.235734114758	-3.135734114758
63	9	11.0968677546165	-2.09686775461647
64	17.4	7.95731408297065	9.44268591702935
65	14.7	13.1140651690469	1.58593483095308
66	17	19.4266947755849	-2.42669477558491
67	13.7	14.265274203766	-0.565274203766021
68	9.5	14.9983918972925	-5.49839189729249
69	14.8	9.8646200732908	4.9353799267092
70	13.6	23.018654100887	-9.41865410088695
71	12.6	12.9705529867879	-0.37055298678793
72	8.9	2.43308277596981	6.46691722403019
73	10.2	10.0597855698704	0.140214430129612
74	12.7	10.1958385383712	2.50416146162881
75	16	16.0653780776154	-0.0653780776153816
76	10.4	15.1310834436457	-4.73108344364572
77	9.9	7.5980102770202	2.3019897229798
78	9.5	12.8440215409794	-3.34402154097945
79	8.6	7.6437147267147	0.956285273285293
80	10	9.11933041423105	0.880669585768947
81	3.5	10.6275335375779	-7.1275335375779
82	-4.2	3.97518243477542	-8.17518243477542
83	-4.4	-7.03640718058249	2.63640718058249
84	-1.5	-1.58532966425595	0.0853296642559509
85	-0.1	-2.43471972696672	2.33471972696672
86	0.8	-0.75450079413503	1.55450079413503
87	-2.4	0.227130211188902	-2.6271302111889
88	-1.2	-3.24300742534464	2.04300742534464
89	0.2	-1.61223726725314	1.81223726725314
90	-1.9	-0.605446798773695	-1.29455320122631
91	-1.6	-2.48633153195007	0.88633153195007
92	-4.2	-2.78478623755605	-1.41521376244395
93	-2.2	-6.00017033176583	3.80017033176583
94	6.2	-5.47963543102029	11.6796354310203
95	5.7	5.17096964284982	0.529030357150184
96	3.1	0.924394713266959	2.17560528673304
97	1.1	3.22217617686705	-2.12217617686705
98	-0.9	0.626064053175678	-1.52606405317568
99	0.1	-1.93194385717674	2.03194385717674
100	-4	-0.609061691802807	-3.39093830819719
101	-4	-3.7668836166656	-0.233116383334405
102	-5.3	-6.36284721069496	1.06284721069496
103	-8	-5.43241262096231	-2.56758737903769
104	-6.3	-10.2165708515616	3.91657085156157
105	-3.6	-8.33243326213262	4.73243326213262
106	-3.5	-7.61940988647601	4.11940988647601
107	-5.1	-5.62254278041096	0.522542780410958
108	-3.3	-1.60974105661271	-1.69025894338729

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & -6.8 & 5.29011904502332 & -12.0901190450233 \tabularnewline
14 & -10.4 & -8.84171324014721 & -1.55828675985279 \tabularnewline
15 & -17.2 & -15.9777490065531 & -1.22225099344691 \tabularnewline
16 & -19.8 & -19.2767864574917 & -0.523213542508337 \tabularnewline
17 & -16.8 & -17.4447550613783 & 0.644755061378326 \tabularnewline
18 & -23.2 & -25.6515712032783 & 2.45157120327834 \tabularnewline
19 & -21.7 & -22.7216106860833 & 1.02161068608327 \tabularnewline
20 & -17.6 & -27.7644764890422 & 10.1644764890422 \tabularnewline
21 & -13 & -23.2666467148864 & 10.2666467148864 \tabularnewline
22 & -12.6 & -26.3010044782941 & 13.7010044782941 \tabularnewline
23 & -4 & -18.9086575698116 & 14.9086575698116 \tabularnewline
24 & -0.2 & -2.02357903063458 & 1.82357903063458 \tabularnewline
25 & 3.1 & -1.49201857893718 & 4.59201857893718 \tabularnewline
26 & 6.5 & 2.01844508797077 & 4.48155491202923 \tabularnewline
27 & 19.2 & 7.01265005348185 & 12.1873499465182 \tabularnewline
28 & 26.6 & 17.6315183424416 & 8.96848165755836 \tabularnewline
29 & 26.6 & 20.0432882620052 & 6.55671173799476 \tabularnewline
30 & 31.4 & 35.4760470482716 & -4.07604704827165 \tabularnewline
31 & 31.2 & 26.6707282575723 & 4.52927174242769 \tabularnewline
32 & 26.4 & 35.0400343978482 & -8.6400343978482 \tabularnewline
33 & 20.7 & 29.121780244723 & -8.42178024472298 \tabularnewline
34 & 20.7 & 31.8724152661828 & -11.1724152661828 \tabularnewline
35 & 15 & 19.698985465602 & -4.69898546560196 \tabularnewline
36 & 13.3 & 2.68691920634813 & 10.6130807936519 \tabularnewline
37 & 8.7 & 15.0018137763925 & -6.30181377639252 \tabularnewline
38 & 10.2 & 8.18633685500812 & 2.01366314499188 \tabularnewline
39 & 4.3 & 12.1873122352001 & -7.88731223520014 \tabularnewline
40 & -0.1 & 2.85287939834127 & -2.95287939834127 \tabularnewline
41 & -4.6 & -1.23202195284512 & -3.36797804715488 \tabularnewline
42 & -3.9 & -8.0189175674334 & 4.1189175674334 \tabularnewline
43 & -3.5 & -4.88744331137933 & 1.38744331137933 \tabularnewline
44 & -3.4 & -5.72230740148329 & 2.32230740148329 \tabularnewline
45 & -2.5 & -5.85143471834727 & 3.35143471834727 \tabularnewline
46 & -1.1 & -7.32300181020793 & 6.22300181020793 \tabularnewline
47 & 0.3 & -4.64616789106606 & 4.94616789106606 \tabularnewline
48 & -0.9 & -0.723371722037258 & -0.176628277962742 \tabularnewline
49 & 3.6 & -2.02736684196994 & 5.62736684196994 \tabularnewline
50 & 2.7 & 2.88068944693305 & -0.180689446933046 \tabularnewline
51 & -0.2 & 2.33993111446629 & -2.53993111446629 \tabularnewline
52 & -1 & -1.45894998659693 & 0.458949986596932 \tabularnewline
53 & 5.8 & -1.78705521406384 & 7.58705521406384 \tabularnewline
54 & 6.4 & 6.78395031443612 & -0.383950314436121 \tabularnewline
55 & 9.6 & 4.57927487564638 & 5.02072512435362 \tabularnewline
56 & 13.2 & 9.98483733517275 & 3.21516266482725 \tabularnewline
57 & 10.6 & 14.0490822853714 & -3.44908228537135 \tabularnewline
58 & 10.9 & 15.5107002009973 & -4.61070020099728 \tabularnewline
59 & 12.9 & 9.52806161767183 & 3.37193838232817 \tabularnewline
60 & 15.9 & 2.42119499932248 & 13.4788050006775 \tabularnewline
61 & 12.2 & 18.4313153693455 & -6.23131536934553 \tabularnewline
62 & 9.1 & 12.235734114758 & -3.135734114758 \tabularnewline
63 & 9 & 11.0968677546165 & -2.09686775461647 \tabularnewline
64 & 17.4 & 7.95731408297065 & 9.44268591702935 \tabularnewline
65 & 14.7 & 13.1140651690469 & 1.58593483095308 \tabularnewline
66 & 17 & 19.4266947755849 & -2.42669477558491 \tabularnewline
67 & 13.7 & 14.265274203766 & -0.565274203766021 \tabularnewline
68 & 9.5 & 14.9983918972925 & -5.49839189729249 \tabularnewline
69 & 14.8 & 9.8646200732908 & 4.9353799267092 \tabularnewline
70 & 13.6 & 23.018654100887 & -9.41865410088695 \tabularnewline
71 & 12.6 & 12.9705529867879 & -0.37055298678793 \tabularnewline
72 & 8.9 & 2.43308277596981 & 6.46691722403019 \tabularnewline
73 & 10.2 & 10.0597855698704 & 0.140214430129612 \tabularnewline
74 & 12.7 & 10.1958385383712 & 2.50416146162881 \tabularnewline
75 & 16 & 16.0653780776154 & -0.0653780776153816 \tabularnewline
76 & 10.4 & 15.1310834436457 & -4.73108344364572 \tabularnewline
77 & 9.9 & 7.5980102770202 & 2.3019897229798 \tabularnewline
78 & 9.5 & 12.8440215409794 & -3.34402154097945 \tabularnewline
79 & 8.6 & 7.6437147267147 & 0.956285273285293 \tabularnewline
80 & 10 & 9.11933041423105 & 0.880669585768947 \tabularnewline
81 & 3.5 & 10.6275335375779 & -7.1275335375779 \tabularnewline
82 & -4.2 & 3.97518243477542 & -8.17518243477542 \tabularnewline
83 & -4.4 & -7.03640718058249 & 2.63640718058249 \tabularnewline
84 & -1.5 & -1.58532966425595 & 0.0853296642559509 \tabularnewline
85 & -0.1 & -2.43471972696672 & 2.33471972696672 \tabularnewline
86 & 0.8 & -0.75450079413503 & 1.55450079413503 \tabularnewline
87 & -2.4 & 0.227130211188902 & -2.6271302111889 \tabularnewline
88 & -1.2 & -3.24300742534464 & 2.04300742534464 \tabularnewline
89 & 0.2 & -1.61223726725314 & 1.81223726725314 \tabularnewline
90 & -1.9 & -0.605446798773695 & -1.29455320122631 \tabularnewline
91 & -1.6 & -2.48633153195007 & 0.88633153195007 \tabularnewline
92 & -4.2 & -2.78478623755605 & -1.41521376244395 \tabularnewline
93 & -2.2 & -6.00017033176583 & 3.80017033176583 \tabularnewline
94 & 6.2 & -5.47963543102029 & 11.6796354310203 \tabularnewline
95 & 5.7 & 5.17096964284982 & 0.529030357150184 \tabularnewline
96 & 3.1 & 0.924394713266959 & 2.17560528673304 \tabularnewline
97 & 1.1 & 3.22217617686705 & -2.12217617686705 \tabularnewline
98 & -0.9 & 0.626064053175678 & -1.52606405317568 \tabularnewline
99 & 0.1 & -1.93194385717674 & 2.03194385717674 \tabularnewline
100 & -4 & -0.609061691802807 & -3.39093830819719 \tabularnewline
101 & -4 & -3.7668836166656 & -0.233116383334405 \tabularnewline
102 & -5.3 & -6.36284721069496 & 1.06284721069496 \tabularnewline
103 & -8 & -5.43241262096231 & -2.56758737903769 \tabularnewline
104 & -6.3 & -10.2165708515616 & 3.91657085156157 \tabularnewline
105 & -3.6 & -8.33243326213262 & 4.73243326213262 \tabularnewline
106 & -3.5 & -7.61940988647601 & 4.11940988647601 \tabularnewline
107 & -5.1 & -5.62254278041096 & 0.522542780410958 \tabularnewline
108 & -3.3 & -1.60974105661271 & -1.69025894338729 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=122245&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]-6.8[/C][C]5.29011904502332[/C][C]-12.0901190450233[/C][/ROW]
[ROW][C]14[/C][C]-10.4[/C][C]-8.84171324014721[/C][C]-1.55828675985279[/C][/ROW]
[ROW][C]15[/C][C]-17.2[/C][C]-15.9777490065531[/C][C]-1.22225099344691[/C][/ROW]
[ROW][C]16[/C][C]-19.8[/C][C]-19.2767864574917[/C][C]-0.523213542508337[/C][/ROW]
[ROW][C]17[/C][C]-16.8[/C][C]-17.4447550613783[/C][C]0.644755061378326[/C][/ROW]
[ROW][C]18[/C][C]-23.2[/C][C]-25.6515712032783[/C][C]2.45157120327834[/C][/ROW]
[ROW][C]19[/C][C]-21.7[/C][C]-22.7216106860833[/C][C]1.02161068608327[/C][/ROW]
[ROW][C]20[/C][C]-17.6[/C][C]-27.7644764890422[/C][C]10.1644764890422[/C][/ROW]
[ROW][C]21[/C][C]-13[/C][C]-23.2666467148864[/C][C]10.2666467148864[/C][/ROW]
[ROW][C]22[/C][C]-12.6[/C][C]-26.3010044782941[/C][C]13.7010044782941[/C][/ROW]
[ROW][C]23[/C][C]-4[/C][C]-18.9086575698116[/C][C]14.9086575698116[/C][/ROW]
[ROW][C]24[/C][C]-0.2[/C][C]-2.02357903063458[/C][C]1.82357903063458[/C][/ROW]
[ROW][C]25[/C][C]3.1[/C][C]-1.49201857893718[/C][C]4.59201857893718[/C][/ROW]
[ROW][C]26[/C][C]6.5[/C][C]2.01844508797077[/C][C]4.48155491202923[/C][/ROW]
[ROW][C]27[/C][C]19.2[/C][C]7.01265005348185[/C][C]12.1873499465182[/C][/ROW]
[ROW][C]28[/C][C]26.6[/C][C]17.6315183424416[/C][C]8.96848165755836[/C][/ROW]
[ROW][C]29[/C][C]26.6[/C][C]20.0432882620052[/C][C]6.55671173799476[/C][/ROW]
[ROW][C]30[/C][C]31.4[/C][C]35.4760470482716[/C][C]-4.07604704827165[/C][/ROW]
[ROW][C]31[/C][C]31.2[/C][C]26.6707282575723[/C][C]4.52927174242769[/C][/ROW]
[ROW][C]32[/C][C]26.4[/C][C]35.0400343978482[/C][C]-8.6400343978482[/C][/ROW]
[ROW][C]33[/C][C]20.7[/C][C]29.121780244723[/C][C]-8.42178024472298[/C][/ROW]
[ROW][C]34[/C][C]20.7[/C][C]31.8724152661828[/C][C]-11.1724152661828[/C][/ROW]
[ROW][C]35[/C][C]15[/C][C]19.698985465602[/C][C]-4.69898546560196[/C][/ROW]
[ROW][C]36[/C][C]13.3[/C][C]2.68691920634813[/C][C]10.6130807936519[/C][/ROW]
[ROW][C]37[/C][C]8.7[/C][C]15.0018137763925[/C][C]-6.30181377639252[/C][/ROW]
[ROW][C]38[/C][C]10.2[/C][C]8.18633685500812[/C][C]2.01366314499188[/C][/ROW]
[ROW][C]39[/C][C]4.3[/C][C]12.1873122352001[/C][C]-7.88731223520014[/C][/ROW]
[ROW][C]40[/C][C]-0.1[/C][C]2.85287939834127[/C][C]-2.95287939834127[/C][/ROW]
[ROW][C]41[/C][C]-4.6[/C][C]-1.23202195284512[/C][C]-3.36797804715488[/C][/ROW]
[ROW][C]42[/C][C]-3.9[/C][C]-8.0189175674334[/C][C]4.1189175674334[/C][/ROW]
[ROW][C]43[/C][C]-3.5[/C][C]-4.88744331137933[/C][C]1.38744331137933[/C][/ROW]
[ROW][C]44[/C][C]-3.4[/C][C]-5.72230740148329[/C][C]2.32230740148329[/C][/ROW]
[ROW][C]45[/C][C]-2.5[/C][C]-5.85143471834727[/C][C]3.35143471834727[/C][/ROW]
[ROW][C]46[/C][C]-1.1[/C][C]-7.32300181020793[/C][C]6.22300181020793[/C][/ROW]
[ROW][C]47[/C][C]0.3[/C][C]-4.64616789106606[/C][C]4.94616789106606[/C][/ROW]
[ROW][C]48[/C][C]-0.9[/C][C]-0.723371722037258[/C][C]-0.176628277962742[/C][/ROW]
[ROW][C]49[/C][C]3.6[/C][C]-2.02736684196994[/C][C]5.62736684196994[/C][/ROW]
[ROW][C]50[/C][C]2.7[/C][C]2.88068944693305[/C][C]-0.180689446933046[/C][/ROW]
[ROW][C]51[/C][C]-0.2[/C][C]2.33993111446629[/C][C]-2.53993111446629[/C][/ROW]
[ROW][C]52[/C][C]-1[/C][C]-1.45894998659693[/C][C]0.458949986596932[/C][/ROW]
[ROW][C]53[/C][C]5.8[/C][C]-1.78705521406384[/C][C]7.58705521406384[/C][/ROW]
[ROW][C]54[/C][C]6.4[/C][C]6.78395031443612[/C][C]-0.383950314436121[/C][/ROW]
[ROW][C]55[/C][C]9.6[/C][C]4.57927487564638[/C][C]5.02072512435362[/C][/ROW]
[ROW][C]56[/C][C]13.2[/C][C]9.98483733517275[/C][C]3.21516266482725[/C][/ROW]
[ROW][C]57[/C][C]10.6[/C][C]14.0490822853714[/C][C]-3.44908228537135[/C][/ROW]
[ROW][C]58[/C][C]10.9[/C][C]15.5107002009973[/C][C]-4.61070020099728[/C][/ROW]
[ROW][C]59[/C][C]12.9[/C][C]9.52806161767183[/C][C]3.37193838232817[/C][/ROW]
[ROW][C]60[/C][C]15.9[/C][C]2.42119499932248[/C][C]13.4788050006775[/C][/ROW]
[ROW][C]61[/C][C]12.2[/C][C]18.4313153693455[/C][C]-6.23131536934553[/C][/ROW]
[ROW][C]62[/C][C]9.1[/C][C]12.235734114758[/C][C]-3.135734114758[/C][/ROW]
[ROW][C]63[/C][C]9[/C][C]11.0968677546165[/C][C]-2.09686775461647[/C][/ROW]
[ROW][C]64[/C][C]17.4[/C][C]7.95731408297065[/C][C]9.44268591702935[/C][/ROW]
[ROW][C]65[/C][C]14.7[/C][C]13.1140651690469[/C][C]1.58593483095308[/C][/ROW]
[ROW][C]66[/C][C]17[/C][C]19.4266947755849[/C][C]-2.42669477558491[/C][/ROW]
[ROW][C]67[/C][C]13.7[/C][C]14.265274203766[/C][C]-0.565274203766021[/C][/ROW]
[ROW][C]68[/C][C]9.5[/C][C]14.9983918972925[/C][C]-5.49839189729249[/C][/ROW]
[ROW][C]69[/C][C]14.8[/C][C]9.8646200732908[/C][C]4.9353799267092[/C][/ROW]
[ROW][C]70[/C][C]13.6[/C][C]23.018654100887[/C][C]-9.41865410088695[/C][/ROW]
[ROW][C]71[/C][C]12.6[/C][C]12.9705529867879[/C][C]-0.37055298678793[/C][/ROW]
[ROW][C]72[/C][C]8.9[/C][C]2.43308277596981[/C][C]6.46691722403019[/C][/ROW]
[ROW][C]73[/C][C]10.2[/C][C]10.0597855698704[/C][C]0.140214430129612[/C][/ROW]
[ROW][C]74[/C][C]12.7[/C][C]10.1958385383712[/C][C]2.50416146162881[/C][/ROW]
[ROW][C]75[/C][C]16[/C][C]16.0653780776154[/C][C]-0.0653780776153816[/C][/ROW]
[ROW][C]76[/C][C]10.4[/C][C]15.1310834436457[/C][C]-4.73108344364572[/C][/ROW]
[ROW][C]77[/C][C]9.9[/C][C]7.5980102770202[/C][C]2.3019897229798[/C][/ROW]
[ROW][C]78[/C][C]9.5[/C][C]12.8440215409794[/C][C]-3.34402154097945[/C][/ROW]
[ROW][C]79[/C][C]8.6[/C][C]7.6437147267147[/C][C]0.956285273285293[/C][/ROW]
[ROW][C]80[/C][C]10[/C][C]9.11933041423105[/C][C]0.880669585768947[/C][/ROW]
[ROW][C]81[/C][C]3.5[/C][C]10.6275335375779[/C][C]-7.1275335375779[/C][/ROW]
[ROW][C]82[/C][C]-4.2[/C][C]3.97518243477542[/C][C]-8.17518243477542[/C][/ROW]
[ROW][C]83[/C][C]-4.4[/C][C]-7.03640718058249[/C][C]2.63640718058249[/C][/ROW]
[ROW][C]84[/C][C]-1.5[/C][C]-1.58532966425595[/C][C]0.0853296642559509[/C][/ROW]
[ROW][C]85[/C][C]-0.1[/C][C]-2.43471972696672[/C][C]2.33471972696672[/C][/ROW]
[ROW][C]86[/C][C]0.8[/C][C]-0.75450079413503[/C][C]1.55450079413503[/C][/ROW]
[ROW][C]87[/C][C]-2.4[/C][C]0.227130211188902[/C][C]-2.6271302111889[/C][/ROW]
[ROW][C]88[/C][C]-1.2[/C][C]-3.24300742534464[/C][C]2.04300742534464[/C][/ROW]
[ROW][C]89[/C][C]0.2[/C][C]-1.61223726725314[/C][C]1.81223726725314[/C][/ROW]
[ROW][C]90[/C][C]-1.9[/C][C]-0.605446798773695[/C][C]-1.29455320122631[/C][/ROW]
[ROW][C]91[/C][C]-1.6[/C][C]-2.48633153195007[/C][C]0.88633153195007[/C][/ROW]
[ROW][C]92[/C][C]-4.2[/C][C]-2.78478623755605[/C][C]-1.41521376244395[/C][/ROW]
[ROW][C]93[/C][C]-2.2[/C][C]-6.00017033176583[/C][C]3.80017033176583[/C][/ROW]
[ROW][C]94[/C][C]6.2[/C][C]-5.47963543102029[/C][C]11.6796354310203[/C][/ROW]
[ROW][C]95[/C][C]5.7[/C][C]5.17096964284982[/C][C]0.529030357150184[/C][/ROW]
[ROW][C]96[/C][C]3.1[/C][C]0.924394713266959[/C][C]2.17560528673304[/C][/ROW]
[ROW][C]97[/C][C]1.1[/C][C]3.22217617686705[/C][C]-2.12217617686705[/C][/ROW]
[ROW][C]98[/C][C]-0.9[/C][C]0.626064053175678[/C][C]-1.52606405317568[/C][/ROW]
[ROW][C]99[/C][C]0.1[/C][C]-1.93194385717674[/C][C]2.03194385717674[/C][/ROW]
[ROW][C]100[/C][C]-4[/C][C]-0.609061691802807[/C][C]-3.39093830819719[/C][/ROW]
[ROW][C]101[/C][C]-4[/C][C]-3.7668836166656[/C][C]-0.233116383334405[/C][/ROW]
[ROW][C]102[/C][C]-5.3[/C][C]-6.36284721069496[/C][C]1.06284721069496[/C][/ROW]
[ROW][C]103[/C][C]-8[/C][C]-5.43241262096231[/C][C]-2.56758737903769[/C][/ROW]
[ROW][C]104[/C][C]-6.3[/C][C]-10.2165708515616[/C][C]3.91657085156157[/C][/ROW]
[ROW][C]105[/C][C]-3.6[/C][C]-8.33243326213262[/C][C]4.73243326213262[/C][/ROW]
[ROW][C]106[/C][C]-3.5[/C][C]-7.61940988647601[/C][C]4.11940988647601[/C][/ROW]
[ROW][C]107[/C][C]-5.1[/C][C]-5.62254278041096[/C][C]0.522542780410958[/C][/ROW]
[ROW][C]108[/C][C]-3.3[/C][C]-1.60974105661271[/C][C]-1.69025894338729[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=122245&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=122245&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	-6.8	5.29011904502332	-12.0901190450233
14	-10.4	-8.84171324014721	-1.55828675985279
15	-17.2	-15.9777490065531	-1.22225099344691
16	-19.8	-19.2767864574917	-0.523213542508337
17	-16.8	-17.4447550613783	0.644755061378326
18	-23.2	-25.6515712032783	2.45157120327834
19	-21.7	-22.7216106860833	1.02161068608327
20	-17.6	-27.7644764890422	10.1644764890422
21	-13	-23.2666467148864	10.2666467148864
22	-12.6	-26.3010044782941	13.7010044782941
23	-4	-18.9086575698116	14.9086575698116
24	-0.2	-2.02357903063458	1.82357903063458
25	3.1	-1.49201857893718	4.59201857893718
26	6.5	2.01844508797077	4.48155491202923
27	19.2	7.01265005348185	12.1873499465182
28	26.6	17.6315183424416	8.96848165755836
29	26.6	20.0432882620052	6.55671173799476
30	31.4	35.4760470482716	-4.07604704827165
31	31.2	26.6707282575723	4.52927174242769
32	26.4	35.0400343978482	-8.6400343978482
33	20.7	29.121780244723	-8.42178024472298
34	20.7	31.8724152661828	-11.1724152661828
35	15	19.698985465602	-4.69898546560196
36	13.3	2.68691920634813	10.6130807936519
37	8.7	15.0018137763925	-6.30181377639252
38	10.2	8.18633685500812	2.01366314499188
39	4.3	12.1873122352001	-7.88731223520014
40	-0.1	2.85287939834127	-2.95287939834127
41	-4.6	-1.23202195284512	-3.36797804715488
42	-3.9	-8.0189175674334	4.1189175674334
43	-3.5	-4.88744331137933	1.38744331137933
44	-3.4	-5.72230740148329	2.32230740148329
45	-2.5	-5.85143471834727	3.35143471834727
46	-1.1	-7.32300181020793	6.22300181020793
47	0.3	-4.64616789106606	4.94616789106606
48	-0.9	-0.723371722037258	-0.176628277962742
49	3.6	-2.02736684196994	5.62736684196994
50	2.7	2.88068944693305	-0.180689446933046
51	-0.2	2.33993111446629	-2.53993111446629
52	-1	-1.45894998659693	0.458949986596932
53	5.8	-1.78705521406384	7.58705521406384
54	6.4	6.78395031443612	-0.383950314436121
55	9.6	4.57927487564638	5.02072512435362
56	13.2	9.98483733517275	3.21516266482725
57	10.6	14.0490822853714	-3.44908228537135
58	10.9	15.5107002009973	-4.61070020099728
59	12.9	9.52806161767183	3.37193838232817
60	15.9	2.42119499932248	13.4788050006775
61	12.2	18.4313153693455	-6.23131536934553
62	9.1	12.235734114758	-3.135734114758
63	9	11.0968677546165	-2.09686775461647
64	17.4	7.95731408297065	9.44268591702935
65	14.7	13.1140651690469	1.58593483095308
66	17	19.4266947755849	-2.42669477558491
67	13.7	14.265274203766	-0.565274203766021
68	9.5	14.9983918972925	-5.49839189729249
69	14.8	9.8646200732908	4.9353799267092
70	13.6	23.018654100887	-9.41865410088695
71	12.6	12.9705529867879	-0.37055298678793
72	8.9	2.43308277596981	6.46691722403019
73	10.2	10.0597855698704	0.140214430129612
74	12.7	10.1958385383712	2.50416146162881
75	16	16.0653780776154	-0.0653780776153816
76	10.4	15.1310834436457	-4.73108344364572
77	9.9	7.5980102770202	2.3019897229798
78	9.5	12.8440215409794	-3.34402154097945
79	8.6	7.6437147267147	0.956285273285293
80	10	9.11933041423105	0.880669585768947
81	3.5	10.6275335375779	-7.1275335375779
82	-4.2	3.97518243477542	-8.17518243477542
83	-4.4	-7.03640718058249	2.63640718058249
84	-1.5	-1.58532966425595	0.0853296642559509
85	-0.1	-2.43471972696672	2.33471972696672
86	0.8	-0.75450079413503	1.55450079413503
87	-2.4	0.227130211188902	-2.6271302111889
88	-1.2	-3.24300742534464	2.04300742534464
89	0.2	-1.61223726725314	1.81223726725314
90	-1.9	-0.605446798773695	-1.29455320122631
91	-1.6	-2.48633153195007	0.88633153195007
92	-4.2	-2.78478623755605	-1.41521376244395
93	-2.2	-6.00017033176583	3.80017033176583
94	6.2	-5.47963543102029	11.6796354310203
95	5.7	5.17096964284982	0.529030357150184
96	3.1	0.924394713266959	2.17560528673304
97	1.1	3.22217617686705	-2.12217617686705
98	-0.9	0.626064053175678	-1.52606405317568
99	0.1	-1.93194385717674	2.03194385717674
100	-4	-0.609061691802807	-3.39093830819719
101	-4	-3.7668836166656	-0.233116383334405
102	-5.3	-6.36284721069496	1.06284721069496
103	-8	-5.43241262096231	-2.56758737903769
104	-6.3	-10.2165708515616	3.91657085156157
105	-3.6	-8.33243326213262	4.73243326213262
106	-3.5	-7.61940988647601	4.11940988647601
107	-5.1	-5.62254278041096	0.522542780410958
108	-3.3	-1.60974105661271	-1.69025894338729

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
109	-4.44284851504703	-15.1026372020344	6.21694017194033
110	-5.23428871886869	-20.8659350488255	10.3973576110881
111	-7.63572478088411	-31.1619943847527	15.8905448229845
112	-8.27753317344292	-34.2222250606486	17.6671587137628
113	-7.11374296291278	-30.5085123965361	16.2810264707105
114	-10.6058132489667	-45.068046409191	23.8564199112576
115	-10.1003986641228	-42.8773584730367	22.6765611447911
116	-12.5753399888747	-52.7271861864918	27.5765062087423
117	-15.6150514136752	-64.2753883226724	33.0452854953221
118	-27.9316081905319	-111.783414649533	55.9201982684687
119	-33.2535434044915	-129.522151514782	63.015064705799
120	-8.326146148085	-32.4065436990005	15.7542514028305

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
109 & -4.44284851504703 & -15.1026372020344 & 6.21694017194033 \tabularnewline
110 & -5.23428871886869 & -20.8659350488255 & 10.3973576110881 \tabularnewline
111 & -7.63572478088411 & -31.1619943847527 & 15.8905448229845 \tabularnewline
112 & -8.27753317344292 & -34.2222250606486 & 17.6671587137628 \tabularnewline
113 & -7.11374296291278 & -30.5085123965361 & 16.2810264707105 \tabularnewline
114 & -10.6058132489667 & -45.068046409191 & 23.8564199112576 \tabularnewline
115 & -10.1003986641228 & -42.8773584730367 & 22.6765611447911 \tabularnewline
116 & -12.5753399888747 & -52.7271861864918 & 27.5765062087423 \tabularnewline
117 & -15.6150514136752 & -64.2753883226724 & 33.0452854953221 \tabularnewline
118 & -27.9316081905319 & -111.783414649533 & 55.9201982684687 \tabularnewline
119 & -33.2535434044915 & -129.522151514782 & 63.015064705799 \tabularnewline
120 & -8.326146148085 & -32.4065436990005 & 15.7542514028305 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=122245&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]-4.44284851504703[/C][C]-15.1026372020344[/C][C]6.21694017194033[/C][/ROW]
[ROW][C]110[/C][C]-5.23428871886869[/C][C]-20.8659350488255[/C][C]10.3973576110881[/C][/ROW]
[ROW][C]111[/C][C]-7.63572478088411[/C][C]-31.1619943847527[/C][C]15.8905448229845[/C][/ROW]
[ROW][C]112[/C][C]-8.27753317344292[/C][C]-34.2222250606486[/C][C]17.6671587137628[/C][/ROW]
[ROW][C]113[/C][C]-7.11374296291278[/C][C]-30.5085123965361[/C][C]16.2810264707105[/C][/ROW]
[ROW][C]114[/C][C]-10.6058132489667[/C][C]-45.068046409191[/C][C]23.8564199112576[/C][/ROW]
[ROW][C]115[/C][C]-10.1003986641228[/C][C]-42.8773584730367[/C][C]22.6765611447911[/C][/ROW]
[ROW][C]116[/C][C]-12.5753399888747[/C][C]-52.7271861864918[/C][C]27.5765062087423[/C][/ROW]
[ROW][C]117[/C][C]-15.6150514136752[/C][C]-64.2753883226724[/C][C]33.0452854953221[/C][/ROW]
[ROW][C]118[/C][C]-27.9316081905319[/C][C]-111.783414649533[/C][C]55.9201982684687[/C][/ROW]
[ROW][C]119[/C][C]-33.2535434044915[/C][C]-129.522151514782[/C][C]63.015064705799[/C][/ROW]
[ROW][C]120[/C][C]-8.326146148085[/C][C]-32.4065436990005[/C][C]15.7542514028305[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=122245&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=122245&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	-4.44284851504703	-15.1026372020344	6.21694017194033
110	-5.23428871886869	-20.8659350488255	10.3973576110881
111	-7.63572478088411	-31.1619943847527	15.8905448229845
112	-8.27753317344292	-34.2222250606486	17.6671587137628
113	-7.11374296291278	-30.5085123965361	16.2810264707105
114	-10.6058132489667	-45.068046409191	23.8564199112576
115	-10.1003986641228	-42.8773584730367	22.6765611447911
116	-12.5753399888747	-52.7271861864918	27.5765062087423
117	-15.6150514136752	-64.2753883226724	33.0452854953221
118	-27.9316081905319	-111.783414649533	55.9201982684687
119	-33.2535434044915	-129.522151514782	63.015064705799
120	-8.326146148085	-32.4065436990005	15.7542514028305

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 = multiplicative ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; par3 = multiplicative ;

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