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Description of Statistical Computation

Author's title

Author*Unverified author*
R Software Modulerwasp_exponentialsmoothing.wasp
Title produced by softwareExponential Smoothing
Date of computationSun, 16 Jan 2011 18:04:42 +0000
Cite this page as followsStatistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2011/Jan/16/t1295200954kajej70ki2zsc1m.htm/, Retrieved Thu, 16 May 2024 16:08:26 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=117449, Retrieved Thu, 16 May 2024 16:08:26 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywordsKDGP2W102
Estimated Impact108

Tree of Dependent Computations

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-     [Classical Decomposition] [Decomposition van...] [2010-12-15 21:27:25] [d60cc6cf1b020023427d252a04cb3706]
- RMP     [Exponential Smoothing] [Exponential smoot...] [2011-01-16 18:04:42] [ba5a1e709d9e55625879b8034818fd90] [Current]
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Dataset

Dataseries X:
Download CSV
Histogram
Boxplots 
98,6
100,1
98,8
98,3
102,8
103,6
105,2
100,1
98,2
98,4
97,4
98,4
100,3
101,1
104,1
107,3
110,1
112,6
114,3
115,3
109,9
108,2
103,2
101,8
105,6
108,2
109,8
114,6
117,2
116,5
116,1
112,1
106,8
106,9
104,5
103
105,9
107,7
107,1
112,5
114,5
114,6
113,1
112,8
111,9
112
112,4
110
112,3
109,6
111,9
110,8
110,4
110,8
114
108,4
110,5
105,1
102,3
104,3
103,4
102,4
104,5
107,3
110,1
111,8
111,8
105,7
106
106,4
107,1
111,5
109,6
109,9
109,3
111,4
112,9
115,5
118,4
116,2
113,3
113,8
114,1
117,1
115,5
115,2
114,2
115,3
118,8
118
118,1
111,8
112
114,3
115
118,5
117,6
119,1
120,6
123,6
122,7
123,8
123,1
124,5
120,7
118,7
119
122,3
118,6
118,1
118,2
120,8
119,7
119,7
117,1
114,5
116,5
116,4
114,9
115,5

Tables (Output of Computation)





Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time2 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=117449&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=117449&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=117449&T=0

As an alternative you can also use a QR Code:  
QR Code


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

Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time2 seconds
R Server'Sir Ronald Aylmer Fisher' @ 193.190.124.24






Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.952517560780428
beta0
gamma0.374732084602801

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.952517560780428 \tabularnewline
beta & 0 \tabularnewline
gamma & 0.374732084602801 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=117449&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]0.952517560780428[/C][/ROW]
[ROW][C]beta[/C][C]0[/C][/ROW]
[ROW][C]gamma[/C][C]0.374732084602801[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=117449&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=117449&T=1

As an alternative you can also use a QR Code:  
QR Code


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

Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.952517560780428
beta0
gamma0.374732084602801






Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
13100.396.09567550505054.20432449494946
14101.199.88786841770921.21213158229078
15104.1102.9216117024841.17838829751557
16107.3106.3482139159530.951786084047157
17110.1109.4048068751140.695193124885819
18112.6112.1836572013680.416342798631561
19114.3112.3260643617031.97393563829721
20115.3108.9937727210316.30622727896878
21109.9112.838064946522-2.93806494652156
22108.2109.643673156913-1.44367315691304
23103.2106.589382456259-3.38938245625938
24101.8103.681769479805-1.88176947980459
25105.6103.1099926972682.49000730273224
26108.2105.2160277123152.98397228768540
27109.8109.936879983091-0.136879983090779
28114.6112.1066340851282.49336591487192
29117.2116.6270432963400.57295670366041
30116.5119.284499647016-2.78449964701640
31116.1116.405762719262-0.305762719262262
32112.1110.9791036628941.12089633710600
33106.8109.719791628103-2.91979162810317
34106.9106.5693954973560.330604502644263
35104.5105.170515043207-0.670515043207473
36103104.879496272427-1.87949627242708
37105.9104.3876726297641.51232737023554
38107.7105.5712395548112.12876044518912
39107.1109.421957599299-2.32195759929917
40112.5109.5571873742632.94281262573732
41114.5114.4715322576470.0284677423528308
42114.6116.550613440744-1.95061344074445
43113.1114.510272413776-1.41027241377645
44112.8108.0569332978814.74306670211908
45111.9110.1759054013451.72409459865489
46112111.5067273789320.49327262106786
47112.4110.2449780504352.15502194956522
48110112.623821247039-2.62382124703942
49112.3111.4833663692880.816633630711905
50109.6112.015241103453-2.41524110345301
51111.9111.4585253895750.441474610425246
52110.8114.319649947310-3.51964994730972
53110.4113.026530241747-2.62653024174656
54110.8112.541465046846-1.74146504684649
55114110.7099559302873.29004406971326
56108.4108.843238359275-0.443238359275142
57110.5105.9684466615984.53155333840212
58105.1109.951522136435-4.85152213643458
59102.3103.628329769884-1.32832976988441
60104.3102.6041884019661.69581159803396
61103.4105.639476344401-2.2394763444007
62102.4103.202847287306-0.802847287305596
63104.5104.2327950934750.267204906524682
64107.3106.8574436424740.4425563575262
65110.1109.3542864733800.745713526620278
66111.8112.097090823293-0.297090823293118
67111.8111.7309001339660.0690998660337954
68105.7106.729749630265-1.02974963026527
69106103.3848130783292.61518692167115
70106.4105.375561011441.02443898855991
71107.1104.7120136461822.38798635381784
72111.5107.2815377798234.21846222017699
73109.6112.649673399997-3.04967339999706
74109.9109.4668796384550.433120361545008
75109.3111.693147980377-2.39314798037726
76111.4111.786883749015-0.386883749014771
77112.9113.499064446974-0.599064446974438
78115.5114.9423893421800.557610657820305
79118.4115.3968325246063.00316747539389
80116.2113.1708809016993.02911909830122
81113.3113.756943151736-0.45694315173597
82113.8112.7931287552521.00687124474820
83114.1112.1371096675321.96289033246804
84117.1114.3342923869802.76570761302027
85115.5118.189330389361-2.68933038936144
86115.2115.411739696433-0.211739696432915
87114.2116.973479169590-2.77347916959022
88115.3116.740640649132-1.44064064913178
89118.8117.4453240227351.35467597726462
90118120.770201925111-2.77020192511111
91118.1118.0983594488100.00164055118968065
92111.8113.013862483689-1.21386248368944
93112109.4963820868862.50361791311403
94114.3111.3786000271382.92139997286236
95115112.5632138061162.43678619388353
96118.5115.2260753387063.27392466129390
97117.6119.468136457908-1.86813645790784
98119.1117.5168316559371.58316834406324
99120.6120.742671031687-0.142671031687485
100123.6123.0394389152320.56056108476804
101122.7125.700039692473-3.00003969247309
102123.8124.803579696823-1.00357969682265
103123.1123.863795845572-0.763795845572133
104124.5118.0285795902606.47142040974016
105120.7121.897611956789-1.19761195678892
106118.7120.261777126474-1.56177712647398
107119117.1674631635381.83253683646230
108122.3119.2696619387723.03033806122771
109118.6123.188208855713-4.58820885571336
110118.1118.707397062168-0.607397062168431
111118.2119.815976226602-1.6159762266022
112120.8120.7219077846760.0780922153237924
113119.7122.859594032883-3.15959403288261
114119.7121.846679124973-2.14667912497278
115117.1119.822339515138-2.72233951513807
116114.5112.2503136233532.24968637664685
117116.5111.9616136100924.53838638990806
118116.4115.7829382723080.617061727691592
119114.9114.8244022649830.0755977350171406
120115.5115.2743984017530.225601598246982

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 100.3 & 96.0956755050505 & 4.20432449494946 \tabularnewline
14 & 101.1 & 99.8878684177092 & 1.21213158229078 \tabularnewline
15 & 104.1 & 102.921611702484 & 1.17838829751557 \tabularnewline
16 & 107.3 & 106.348213915953 & 0.951786084047157 \tabularnewline
17 & 110.1 & 109.404806875114 & 0.695193124885819 \tabularnewline
18 & 112.6 & 112.183657201368 & 0.416342798631561 \tabularnewline
19 & 114.3 & 112.326064361703 & 1.97393563829721 \tabularnewline
20 & 115.3 & 108.993772721031 & 6.30622727896878 \tabularnewline
21 & 109.9 & 112.838064946522 & -2.93806494652156 \tabularnewline
22 & 108.2 & 109.643673156913 & -1.44367315691304 \tabularnewline
23 & 103.2 & 106.589382456259 & -3.38938245625938 \tabularnewline
24 & 101.8 & 103.681769479805 & -1.88176947980459 \tabularnewline
25 & 105.6 & 103.109992697268 & 2.49000730273224 \tabularnewline
26 & 108.2 & 105.216027712315 & 2.98397228768540 \tabularnewline
27 & 109.8 & 109.936879983091 & -0.136879983090779 \tabularnewline
28 & 114.6 & 112.106634085128 & 2.49336591487192 \tabularnewline
29 & 117.2 & 116.627043296340 & 0.57295670366041 \tabularnewline
30 & 116.5 & 119.284499647016 & -2.78449964701640 \tabularnewline
31 & 116.1 & 116.405762719262 & -0.305762719262262 \tabularnewline
32 & 112.1 & 110.979103662894 & 1.12089633710600 \tabularnewline
33 & 106.8 & 109.719791628103 & -2.91979162810317 \tabularnewline
34 & 106.9 & 106.569395497356 & 0.330604502644263 \tabularnewline
35 & 104.5 & 105.170515043207 & -0.670515043207473 \tabularnewline
36 & 103 & 104.879496272427 & -1.87949627242708 \tabularnewline
37 & 105.9 & 104.387672629764 & 1.51232737023554 \tabularnewline
38 & 107.7 & 105.571239554811 & 2.12876044518912 \tabularnewline
39 & 107.1 & 109.421957599299 & -2.32195759929917 \tabularnewline
40 & 112.5 & 109.557187374263 & 2.94281262573732 \tabularnewline
41 & 114.5 & 114.471532257647 & 0.0284677423528308 \tabularnewline
42 & 114.6 & 116.550613440744 & -1.95061344074445 \tabularnewline
43 & 113.1 & 114.510272413776 & -1.41027241377645 \tabularnewline
44 & 112.8 & 108.056933297881 & 4.74306670211908 \tabularnewline
45 & 111.9 & 110.175905401345 & 1.72409459865489 \tabularnewline
46 & 112 & 111.506727378932 & 0.49327262106786 \tabularnewline
47 & 112.4 & 110.244978050435 & 2.15502194956522 \tabularnewline
48 & 110 & 112.623821247039 & -2.62382124703942 \tabularnewline
49 & 112.3 & 111.483366369288 & 0.816633630711905 \tabularnewline
50 & 109.6 & 112.015241103453 & -2.41524110345301 \tabularnewline
51 & 111.9 & 111.458525389575 & 0.441474610425246 \tabularnewline
52 & 110.8 & 114.319649947310 & -3.51964994730972 \tabularnewline
53 & 110.4 & 113.026530241747 & -2.62653024174656 \tabularnewline
54 & 110.8 & 112.541465046846 & -1.74146504684649 \tabularnewline
55 & 114 & 110.709955930287 & 3.29004406971326 \tabularnewline
56 & 108.4 & 108.843238359275 & -0.443238359275142 \tabularnewline
57 & 110.5 & 105.968446661598 & 4.53155333840212 \tabularnewline
58 & 105.1 & 109.951522136435 & -4.85152213643458 \tabularnewline
59 & 102.3 & 103.628329769884 & -1.32832976988441 \tabularnewline
60 & 104.3 & 102.604188401966 & 1.69581159803396 \tabularnewline
61 & 103.4 & 105.639476344401 & -2.2394763444007 \tabularnewline
62 & 102.4 & 103.202847287306 & -0.802847287305596 \tabularnewline
63 & 104.5 & 104.232795093475 & 0.267204906524682 \tabularnewline
64 & 107.3 & 106.857443642474 & 0.4425563575262 \tabularnewline
65 & 110.1 & 109.354286473380 & 0.745713526620278 \tabularnewline
66 & 111.8 & 112.097090823293 & -0.297090823293118 \tabularnewline
67 & 111.8 & 111.730900133966 & 0.0690998660337954 \tabularnewline
68 & 105.7 & 106.729749630265 & -1.02974963026527 \tabularnewline
69 & 106 & 103.384813078329 & 2.61518692167115 \tabularnewline
70 & 106.4 & 105.37556101144 & 1.02443898855991 \tabularnewline
71 & 107.1 & 104.712013646182 & 2.38798635381784 \tabularnewline
72 & 111.5 & 107.281537779823 & 4.21846222017699 \tabularnewline
73 & 109.6 & 112.649673399997 & -3.04967339999706 \tabularnewline
74 & 109.9 & 109.466879638455 & 0.433120361545008 \tabularnewline
75 & 109.3 & 111.693147980377 & -2.39314798037726 \tabularnewline
76 & 111.4 & 111.786883749015 & -0.386883749014771 \tabularnewline
77 & 112.9 & 113.499064446974 & -0.599064446974438 \tabularnewline
78 & 115.5 & 114.942389342180 & 0.557610657820305 \tabularnewline
79 & 118.4 & 115.396832524606 & 3.00316747539389 \tabularnewline
80 & 116.2 & 113.170880901699 & 3.02911909830122 \tabularnewline
81 & 113.3 & 113.756943151736 & -0.45694315173597 \tabularnewline
82 & 113.8 & 112.793128755252 & 1.00687124474820 \tabularnewline
83 & 114.1 & 112.137109667532 & 1.96289033246804 \tabularnewline
84 & 117.1 & 114.334292386980 & 2.76570761302027 \tabularnewline
85 & 115.5 & 118.189330389361 & -2.68933038936144 \tabularnewline
86 & 115.2 & 115.411739696433 & -0.211739696432915 \tabularnewline
87 & 114.2 & 116.973479169590 & -2.77347916959022 \tabularnewline
88 & 115.3 & 116.740640649132 & -1.44064064913178 \tabularnewline
89 & 118.8 & 117.445324022735 & 1.35467597726462 \tabularnewline
90 & 118 & 120.770201925111 & -2.77020192511111 \tabularnewline
91 & 118.1 & 118.098359448810 & 0.00164055118968065 \tabularnewline
92 & 111.8 & 113.013862483689 & -1.21386248368944 \tabularnewline
93 & 112 & 109.496382086886 & 2.50361791311403 \tabularnewline
94 & 114.3 & 111.378600027138 & 2.92139997286236 \tabularnewline
95 & 115 & 112.563213806116 & 2.43678619388353 \tabularnewline
96 & 118.5 & 115.226075338706 & 3.27392466129390 \tabularnewline
97 & 117.6 & 119.468136457908 & -1.86813645790784 \tabularnewline
98 & 119.1 & 117.516831655937 & 1.58316834406324 \tabularnewline
99 & 120.6 & 120.742671031687 & -0.142671031687485 \tabularnewline
100 & 123.6 & 123.039438915232 & 0.56056108476804 \tabularnewline
101 & 122.7 & 125.700039692473 & -3.00003969247309 \tabularnewline
102 & 123.8 & 124.803579696823 & -1.00357969682265 \tabularnewline
103 & 123.1 & 123.863795845572 & -0.763795845572133 \tabularnewline
104 & 124.5 & 118.028579590260 & 6.47142040974016 \tabularnewline
105 & 120.7 & 121.897611956789 & -1.19761195678892 \tabularnewline
106 & 118.7 & 120.261777126474 & -1.56177712647398 \tabularnewline
107 & 119 & 117.167463163538 & 1.83253683646230 \tabularnewline
108 & 122.3 & 119.269661938772 & 3.03033806122771 \tabularnewline
109 & 118.6 & 123.188208855713 & -4.58820885571336 \tabularnewline
110 & 118.1 & 118.707397062168 & -0.607397062168431 \tabularnewline
111 & 118.2 & 119.815976226602 & -1.6159762266022 \tabularnewline
112 & 120.8 & 120.721907784676 & 0.0780922153237924 \tabularnewline
113 & 119.7 & 122.859594032883 & -3.15959403288261 \tabularnewline
114 & 119.7 & 121.846679124973 & -2.14667912497278 \tabularnewline
115 & 117.1 & 119.822339515138 & -2.72233951513807 \tabularnewline
116 & 114.5 & 112.250313623353 & 2.24968637664685 \tabularnewline
117 & 116.5 & 111.961613610092 & 4.53838638990806 \tabularnewline
118 & 116.4 & 115.782938272308 & 0.617061727691592 \tabularnewline
119 & 114.9 & 114.824402264983 & 0.0755977350171406 \tabularnewline
120 & 115.5 & 115.274398401753 & 0.225601598246982 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=117449&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]100.3[/C][C]96.0956755050505[/C][C]4.20432449494946[/C][/ROW]
[ROW][C]14[/C][C]101.1[/C][C]99.8878684177092[/C][C]1.21213158229078[/C][/ROW]
[ROW][C]15[/C][C]104.1[/C][C]102.921611702484[/C][C]1.17838829751557[/C][/ROW]
[ROW][C]16[/C][C]107.3[/C][C]106.348213915953[/C][C]0.951786084047157[/C][/ROW]
[ROW][C]17[/C][C]110.1[/C][C]109.404806875114[/C][C]0.695193124885819[/C][/ROW]
[ROW][C]18[/C][C]112.6[/C][C]112.183657201368[/C][C]0.416342798631561[/C][/ROW]
[ROW][C]19[/C][C]114.3[/C][C]112.326064361703[/C][C]1.97393563829721[/C][/ROW]
[ROW][C]20[/C][C]115.3[/C][C]108.993772721031[/C][C]6.30622727896878[/C][/ROW]
[ROW][C]21[/C][C]109.9[/C][C]112.838064946522[/C][C]-2.93806494652156[/C][/ROW]
[ROW][C]22[/C][C]108.2[/C][C]109.643673156913[/C][C]-1.44367315691304[/C][/ROW]
[ROW][C]23[/C][C]103.2[/C][C]106.589382456259[/C][C]-3.38938245625938[/C][/ROW]
[ROW][C]24[/C][C]101.8[/C][C]103.681769479805[/C][C]-1.88176947980459[/C][/ROW]
[ROW][C]25[/C][C]105.6[/C][C]103.109992697268[/C][C]2.49000730273224[/C][/ROW]
[ROW][C]26[/C][C]108.2[/C][C]105.216027712315[/C][C]2.98397228768540[/C][/ROW]
[ROW][C]27[/C][C]109.8[/C][C]109.936879983091[/C][C]-0.136879983090779[/C][/ROW]
[ROW][C]28[/C][C]114.6[/C][C]112.106634085128[/C][C]2.49336591487192[/C][/ROW]
[ROW][C]29[/C][C]117.2[/C][C]116.627043296340[/C][C]0.57295670366041[/C][/ROW]
[ROW][C]30[/C][C]116.5[/C][C]119.284499647016[/C][C]-2.78449964701640[/C][/ROW]
[ROW][C]31[/C][C]116.1[/C][C]116.405762719262[/C][C]-0.305762719262262[/C][/ROW]
[ROW][C]32[/C][C]112.1[/C][C]110.979103662894[/C][C]1.12089633710600[/C][/ROW]
[ROW][C]33[/C][C]106.8[/C][C]109.719791628103[/C][C]-2.91979162810317[/C][/ROW]
[ROW][C]34[/C][C]106.9[/C][C]106.569395497356[/C][C]0.330604502644263[/C][/ROW]
[ROW][C]35[/C][C]104.5[/C][C]105.170515043207[/C][C]-0.670515043207473[/C][/ROW]
[ROW][C]36[/C][C]103[/C][C]104.879496272427[/C][C]-1.87949627242708[/C][/ROW]
[ROW][C]37[/C][C]105.9[/C][C]104.387672629764[/C][C]1.51232737023554[/C][/ROW]
[ROW][C]38[/C][C]107.7[/C][C]105.571239554811[/C][C]2.12876044518912[/C][/ROW]
[ROW][C]39[/C][C]107.1[/C][C]109.421957599299[/C][C]-2.32195759929917[/C][/ROW]
[ROW][C]40[/C][C]112.5[/C][C]109.557187374263[/C][C]2.94281262573732[/C][/ROW]
[ROW][C]41[/C][C]114.5[/C][C]114.471532257647[/C][C]0.0284677423528308[/C][/ROW]
[ROW][C]42[/C][C]114.6[/C][C]116.550613440744[/C][C]-1.95061344074445[/C][/ROW]
[ROW][C]43[/C][C]113.1[/C][C]114.510272413776[/C][C]-1.41027241377645[/C][/ROW]
[ROW][C]44[/C][C]112.8[/C][C]108.056933297881[/C][C]4.74306670211908[/C][/ROW]
[ROW][C]45[/C][C]111.9[/C][C]110.175905401345[/C][C]1.72409459865489[/C][/ROW]
[ROW][C]46[/C][C]112[/C][C]111.506727378932[/C][C]0.49327262106786[/C][/ROW]
[ROW][C]47[/C][C]112.4[/C][C]110.244978050435[/C][C]2.15502194956522[/C][/ROW]
[ROW][C]48[/C][C]110[/C][C]112.623821247039[/C][C]-2.62382124703942[/C][/ROW]
[ROW][C]49[/C][C]112.3[/C][C]111.483366369288[/C][C]0.816633630711905[/C][/ROW]
[ROW][C]50[/C][C]109.6[/C][C]112.015241103453[/C][C]-2.41524110345301[/C][/ROW]
[ROW][C]51[/C][C]111.9[/C][C]111.458525389575[/C][C]0.441474610425246[/C][/ROW]
[ROW][C]52[/C][C]110.8[/C][C]114.319649947310[/C][C]-3.51964994730972[/C][/ROW]
[ROW][C]53[/C][C]110.4[/C][C]113.026530241747[/C][C]-2.62653024174656[/C][/ROW]
[ROW][C]54[/C][C]110.8[/C][C]112.541465046846[/C][C]-1.74146504684649[/C][/ROW]
[ROW][C]55[/C][C]114[/C][C]110.709955930287[/C][C]3.29004406971326[/C][/ROW]
[ROW][C]56[/C][C]108.4[/C][C]108.843238359275[/C][C]-0.443238359275142[/C][/ROW]
[ROW][C]57[/C][C]110.5[/C][C]105.968446661598[/C][C]4.53155333840212[/C][/ROW]
[ROW][C]58[/C][C]105.1[/C][C]109.951522136435[/C][C]-4.85152213643458[/C][/ROW]
[ROW][C]59[/C][C]102.3[/C][C]103.628329769884[/C][C]-1.32832976988441[/C][/ROW]
[ROW][C]60[/C][C]104.3[/C][C]102.604188401966[/C][C]1.69581159803396[/C][/ROW]
[ROW][C]61[/C][C]103.4[/C][C]105.639476344401[/C][C]-2.2394763444007[/C][/ROW]
[ROW][C]62[/C][C]102.4[/C][C]103.202847287306[/C][C]-0.802847287305596[/C][/ROW]
[ROW][C]63[/C][C]104.5[/C][C]104.232795093475[/C][C]0.267204906524682[/C][/ROW]
[ROW][C]64[/C][C]107.3[/C][C]106.857443642474[/C][C]0.4425563575262[/C][/ROW]
[ROW][C]65[/C][C]110.1[/C][C]109.354286473380[/C][C]0.745713526620278[/C][/ROW]
[ROW][C]66[/C][C]111.8[/C][C]112.097090823293[/C][C]-0.297090823293118[/C][/ROW]
[ROW][C]67[/C][C]111.8[/C][C]111.730900133966[/C][C]0.0690998660337954[/C][/ROW]
[ROW][C]68[/C][C]105.7[/C][C]106.729749630265[/C][C]-1.02974963026527[/C][/ROW]
[ROW][C]69[/C][C]106[/C][C]103.384813078329[/C][C]2.61518692167115[/C][/ROW]
[ROW][C]70[/C][C]106.4[/C][C]105.37556101144[/C][C]1.02443898855991[/C][/ROW]
[ROW][C]71[/C][C]107.1[/C][C]104.712013646182[/C][C]2.38798635381784[/C][/ROW]
[ROW][C]72[/C][C]111.5[/C][C]107.281537779823[/C][C]4.21846222017699[/C][/ROW]
[ROW][C]73[/C][C]109.6[/C][C]112.649673399997[/C][C]-3.04967339999706[/C][/ROW]
[ROW][C]74[/C][C]109.9[/C][C]109.466879638455[/C][C]0.433120361545008[/C][/ROW]
[ROW][C]75[/C][C]109.3[/C][C]111.693147980377[/C][C]-2.39314798037726[/C][/ROW]
[ROW][C]76[/C][C]111.4[/C][C]111.786883749015[/C][C]-0.386883749014771[/C][/ROW]
[ROW][C]77[/C][C]112.9[/C][C]113.499064446974[/C][C]-0.599064446974438[/C][/ROW]
[ROW][C]78[/C][C]115.5[/C][C]114.942389342180[/C][C]0.557610657820305[/C][/ROW]
[ROW][C]79[/C][C]118.4[/C][C]115.396832524606[/C][C]3.00316747539389[/C][/ROW]
[ROW][C]80[/C][C]116.2[/C][C]113.170880901699[/C][C]3.02911909830122[/C][/ROW]
[ROW][C]81[/C][C]113.3[/C][C]113.756943151736[/C][C]-0.45694315173597[/C][/ROW]
[ROW][C]82[/C][C]113.8[/C][C]112.793128755252[/C][C]1.00687124474820[/C][/ROW]
[ROW][C]83[/C][C]114.1[/C][C]112.137109667532[/C][C]1.96289033246804[/C][/ROW]
[ROW][C]84[/C][C]117.1[/C][C]114.334292386980[/C][C]2.76570761302027[/C][/ROW]
[ROW][C]85[/C][C]115.5[/C][C]118.189330389361[/C][C]-2.68933038936144[/C][/ROW]
[ROW][C]86[/C][C]115.2[/C][C]115.411739696433[/C][C]-0.211739696432915[/C][/ROW]
[ROW][C]87[/C][C]114.2[/C][C]116.973479169590[/C][C]-2.77347916959022[/C][/ROW]
[ROW][C]88[/C][C]115.3[/C][C]116.740640649132[/C][C]-1.44064064913178[/C][/ROW]
[ROW][C]89[/C][C]118.8[/C][C]117.445324022735[/C][C]1.35467597726462[/C][/ROW]
[ROW][C]90[/C][C]118[/C][C]120.770201925111[/C][C]-2.77020192511111[/C][/ROW]
[ROW][C]91[/C][C]118.1[/C][C]118.098359448810[/C][C]0.00164055118968065[/C][/ROW]
[ROW][C]92[/C][C]111.8[/C][C]113.013862483689[/C][C]-1.21386248368944[/C][/ROW]
[ROW][C]93[/C][C]112[/C][C]109.496382086886[/C][C]2.50361791311403[/C][/ROW]
[ROW][C]94[/C][C]114.3[/C][C]111.378600027138[/C][C]2.92139997286236[/C][/ROW]
[ROW][C]95[/C][C]115[/C][C]112.563213806116[/C][C]2.43678619388353[/C][/ROW]
[ROW][C]96[/C][C]118.5[/C][C]115.226075338706[/C][C]3.27392466129390[/C][/ROW]
[ROW][C]97[/C][C]117.6[/C][C]119.468136457908[/C][C]-1.86813645790784[/C][/ROW]
[ROW][C]98[/C][C]119.1[/C][C]117.516831655937[/C][C]1.58316834406324[/C][/ROW]
[ROW][C]99[/C][C]120.6[/C][C]120.742671031687[/C][C]-0.142671031687485[/C][/ROW]
[ROW][C]100[/C][C]123.6[/C][C]123.039438915232[/C][C]0.56056108476804[/C][/ROW]
[ROW][C]101[/C][C]122.7[/C][C]125.700039692473[/C][C]-3.00003969247309[/C][/ROW]
[ROW][C]102[/C][C]123.8[/C][C]124.803579696823[/C][C]-1.00357969682265[/C][/ROW]
[ROW][C]103[/C][C]123.1[/C][C]123.863795845572[/C][C]-0.763795845572133[/C][/ROW]
[ROW][C]104[/C][C]124.5[/C][C]118.028579590260[/C][C]6.47142040974016[/C][/ROW]
[ROW][C]105[/C][C]120.7[/C][C]121.897611956789[/C][C]-1.19761195678892[/C][/ROW]
[ROW][C]106[/C][C]118.7[/C][C]120.261777126474[/C][C]-1.56177712647398[/C][/ROW]
[ROW][C]107[/C][C]119[/C][C]117.167463163538[/C][C]1.83253683646230[/C][/ROW]
[ROW][C]108[/C][C]122.3[/C][C]119.269661938772[/C][C]3.03033806122771[/C][/ROW]
[ROW][C]109[/C][C]118.6[/C][C]123.188208855713[/C][C]-4.58820885571336[/C][/ROW]
[ROW][C]110[/C][C]118.1[/C][C]118.707397062168[/C][C]-0.607397062168431[/C][/ROW]
[ROW][C]111[/C][C]118.2[/C][C]119.815976226602[/C][C]-1.6159762266022[/C][/ROW]
[ROW][C]112[/C][C]120.8[/C][C]120.721907784676[/C][C]0.0780922153237924[/C][/ROW]
[ROW][C]113[/C][C]119.7[/C][C]122.859594032883[/C][C]-3.15959403288261[/C][/ROW]
[ROW][C]114[/C][C]119.7[/C][C]121.846679124973[/C][C]-2.14667912497278[/C][/ROW]
[ROW][C]115[/C][C]117.1[/C][C]119.822339515138[/C][C]-2.72233951513807[/C][/ROW]
[ROW][C]116[/C][C]114.5[/C][C]112.250313623353[/C][C]2.24968637664685[/C][/ROW]
[ROW][C]117[/C][C]116.5[/C][C]111.961613610092[/C][C]4.53838638990806[/C][/ROW]
[ROW][C]118[/C][C]116.4[/C][C]115.782938272308[/C][C]0.617061727691592[/C][/ROW]
[ROW][C]119[/C][C]114.9[/C][C]114.824402264983[/C][C]0.0755977350171406[/C][/ROW]
[ROW][C]120[/C][C]115.5[/C][C]115.274398401753[/C][C]0.225601598246982[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=117449&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=117449&T=2

As an alternative you can also use a QR Code:  
QR Code


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

Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
13100.396.09567550505054.20432449494946
14101.199.88786841770921.21213158229078
15104.1102.9216117024841.17838829751557
16107.3106.3482139159530.951786084047157
17110.1109.4048068751140.695193124885819
18112.6112.1836572013680.416342798631561
19114.3112.3260643617031.97393563829721
20115.3108.9937727210316.30622727896878
21109.9112.838064946522-2.93806494652156
22108.2109.643673156913-1.44367315691304
23103.2106.589382456259-3.38938245625938
24101.8103.681769479805-1.88176947980459
25105.6103.1099926972682.49000730273224
26108.2105.2160277123152.98397228768540
27109.8109.936879983091-0.136879983090779
28114.6112.1066340851282.49336591487192
29117.2116.6270432963400.57295670366041
30116.5119.284499647016-2.78449964701640
31116.1116.405762719262-0.305762719262262
32112.1110.9791036628941.12089633710600
33106.8109.719791628103-2.91979162810317
34106.9106.5693954973560.330604502644263
35104.5105.170515043207-0.670515043207473
36103104.879496272427-1.87949627242708
37105.9104.3876726297641.51232737023554
38107.7105.5712395548112.12876044518912
39107.1109.421957599299-2.32195759929917
40112.5109.5571873742632.94281262573732
41114.5114.4715322576470.0284677423528308
42114.6116.550613440744-1.95061344074445
43113.1114.510272413776-1.41027241377645
44112.8108.0569332978814.74306670211908
45111.9110.1759054013451.72409459865489
46112111.5067273789320.49327262106786
47112.4110.2449780504352.15502194956522
48110112.623821247039-2.62382124703942
49112.3111.4833663692880.816633630711905
50109.6112.015241103453-2.41524110345301
51111.9111.4585253895750.441474610425246
52110.8114.319649947310-3.51964994730972
53110.4113.026530241747-2.62653024174656
54110.8112.541465046846-1.74146504684649
55114110.7099559302873.29004406971326
56108.4108.843238359275-0.443238359275142
57110.5105.9684466615984.53155333840212
58105.1109.951522136435-4.85152213643458
59102.3103.628329769884-1.32832976988441
60104.3102.6041884019661.69581159803396
61103.4105.639476344401-2.2394763444007
62102.4103.202847287306-0.802847287305596
63104.5104.2327950934750.267204906524682
64107.3106.8574436424740.4425563575262
65110.1109.3542864733800.745713526620278
66111.8112.097090823293-0.297090823293118
67111.8111.7309001339660.0690998660337954
68105.7106.729749630265-1.02974963026527
69106103.3848130783292.61518692167115
70106.4105.375561011441.02443898855991
71107.1104.7120136461822.38798635381784
72111.5107.2815377798234.21846222017699
73109.6112.649673399997-3.04967339999706
74109.9109.4668796384550.433120361545008
75109.3111.693147980377-2.39314798037726
76111.4111.786883749015-0.386883749014771
77112.9113.499064446974-0.599064446974438
78115.5114.9423893421800.557610657820305
79118.4115.3968325246063.00316747539389
80116.2113.1708809016993.02911909830122
81113.3113.756943151736-0.45694315173597
82113.8112.7931287552521.00687124474820
83114.1112.1371096675321.96289033246804
84117.1114.3342923869802.76570761302027
85115.5118.189330389361-2.68933038936144
86115.2115.411739696433-0.211739696432915
87114.2116.973479169590-2.77347916959022
88115.3116.740640649132-1.44064064913178
89118.8117.4453240227351.35467597726462
90118120.770201925111-2.77020192511111
91118.1118.0983594488100.00164055118968065
92111.8113.013862483689-1.21386248368944
93112109.4963820868862.50361791311403
94114.3111.3786000271382.92139997286236
95115112.5632138061162.43678619388353
96118.5115.2260753387063.27392466129390
97117.6119.468136457908-1.86813645790784
98119.1117.5168316559371.58316834406324
99120.6120.742671031687-0.142671031687485
100123.6123.0394389152320.56056108476804
101122.7125.700039692473-3.00003969247309
102123.8124.803579696823-1.00357969682265
103123.1123.863795845572-0.763795845572133
104124.5118.0285795902606.47142040974016
105120.7121.897611956789-1.19761195678892
106118.7120.261777126474-1.56177712647398
107119117.1674631635381.83253683646230
108122.3119.2696619387723.03033806122771
109118.6123.188208855713-4.58820885571336
110118.1118.707397062168-0.607397062168431
111118.2119.815976226602-1.6159762266022
112120.8120.7219077846760.0780922153237924
113119.7122.859594032883-3.15959403288261
114119.7121.846679124973-2.14667912497278
115117.1119.822339515138-2.72233951513807
116114.5112.2503136233532.24968637664685
117116.5111.9616136100924.53838638990806
118116.4115.7829382723080.617061727691592
119114.9114.8244022649830.0755977350171406
120115.5115.2743984017530.225601598246982






Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
121116.385826305389111.837896432622120.933756178156
122116.346195373694110.065292298675122.627098448712
123118.015385062046110.385455786623125.645314337469
124120.490705241235111.716772368336129.264638114133
125122.496398505302112.711309064306132.281487946299
126124.511075389536113.809951491623135.212199287448
127124.521242506926112.976541125398136.065943888454
128119.630761028152107.300059211189131.961462845114
129117.239918516927104.170401134521130.309435899332
130116.668577556929102.899831720101130.437323393758
131115.112645044334100.678503600197129.546786488472
132115.493302058699100.423115711493130.563488405906

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
121 & 116.385826305389 & 111.837896432622 & 120.933756178156 \tabularnewline
122 & 116.346195373694 & 110.065292298675 & 122.627098448712 \tabularnewline
123 & 118.015385062046 & 110.385455786623 & 125.645314337469 \tabularnewline
124 & 120.490705241235 & 111.716772368336 & 129.264638114133 \tabularnewline
125 & 122.496398505302 & 112.711309064306 & 132.281487946299 \tabularnewline
126 & 124.511075389536 & 113.809951491623 & 135.212199287448 \tabularnewline
127 & 124.521242506926 & 112.976541125398 & 136.065943888454 \tabularnewline
128 & 119.630761028152 & 107.300059211189 & 131.961462845114 \tabularnewline
129 & 117.239918516927 & 104.170401134521 & 130.309435899332 \tabularnewline
130 & 116.668577556929 & 102.899831720101 & 130.437323393758 \tabularnewline
131 & 115.112645044334 & 100.678503600197 & 129.546786488472 \tabularnewline
132 & 115.493302058699 & 100.423115711493 & 130.563488405906 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=117449&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]121[/C][C]116.385826305389[/C][C]111.837896432622[/C][C]120.933756178156[/C][/ROW]
[ROW][C]122[/C][C]116.346195373694[/C][C]110.065292298675[/C][C]122.627098448712[/C][/ROW]
[ROW][C]123[/C][C]118.015385062046[/C][C]110.385455786623[/C][C]125.645314337469[/C][/ROW]
[ROW][C]124[/C][C]120.490705241235[/C][C]111.716772368336[/C][C]129.264638114133[/C][/ROW]
[ROW][C]125[/C][C]122.496398505302[/C][C]112.711309064306[/C][C]132.281487946299[/C][/ROW]
[ROW][C]126[/C][C]124.511075389536[/C][C]113.809951491623[/C][C]135.212199287448[/C][/ROW]
[ROW][C]127[/C][C]124.521242506926[/C][C]112.976541125398[/C][C]136.065943888454[/C][/ROW]
[ROW][C]128[/C][C]119.630761028152[/C][C]107.300059211189[/C][C]131.961462845114[/C][/ROW]
[ROW][C]129[/C][C]117.239918516927[/C][C]104.170401134521[/C][C]130.309435899332[/C][/ROW]
[ROW][C]130[/C][C]116.668577556929[/C][C]102.899831720101[/C][C]130.437323393758[/C][/ROW]
[ROW][C]131[/C][C]115.112645044334[/C][C]100.678503600197[/C][C]129.546786488472[/C][/ROW]
[ROW][C]132[/C][C]115.493302058699[/C][C]100.423115711493[/C][C]130.563488405906[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=117449&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=117449&T=3

As an alternative you can also use a QR Code:  
QR Code


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

Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
121116.385826305389111.837896432622120.933756178156
122116.346195373694110.065292298675122.627098448712
123118.015385062046110.385455786623125.645314337469
124120.490705241235111.716772368336129.264638114133
125122.496398505302112.711309064306132.281487946299
126124.511075389536113.809951491623135.212199287448
127124.521242506926112.976541125398136.065943888454
128119.630761028152107.300059211189131.961462845114
129117.239918516927104.170401134521130.309435899332
130116.668577556929102.899831720101130.437323393758
131115.112645044334100.678503600197129.546786488472
132115.493302058699100.423115711493130.563488405906


Figures (Output of Computation)

Input Parameters & R Code

Parameters (Session):
par1 = 4 ;
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')