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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 computationThu, 26 Nov 2015 09:19:24 +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/2015/Nov/26/t1448529616yuyy96rx103xfi1.htm/, Retrieved Tue, 14 May 2024 22:12:23 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=284172, Retrieved Tue, 14 May 2024 22:12:23 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact139

Tree of Dependent Computations

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [OPDRACHT 10 OEF 2] [2015-11-26 09:19:24] [48da048a5e5e3f4e8c34faa6148f9354] [Current]
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Dataset

Dataseries X:
Download CSV
Histogram
Boxplots 
78,46
78,59
81,37
83,61
84,65
84,56
83,85
84,08
85,41
85,75
86,38
88,87
90,37
92,21
95,75
97,29
98,29
99,51
99,04
98,9
100,74
100,3
101,68
101,3
103,13
104,17
105,98
106,25
104,01
101,68
101,93
104,41
105,51
104,71
103,14
102,66
102,68
101,89
101,37
101,16
99,34
99,35
99,88
99,31
99,91
98,39
98,02
98,7
98,01
98,42
98,2
93,5
93,17
93,42
93,13
92,31
92,09
92,62
91,43
89,38

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' @ fisher.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 & 'Sir Ronald Aylmer Fisher' @ fisher.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284172&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' @ fisher.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284172&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284172&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' @ fisher.wessa.net






Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.963396342115305
beta0.133116958456148
gamma1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.963396342115305 \tabularnewline
beta & 0.133116958456148 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284172&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]0.963396342115305[/C][/ROW]
[ROW][C]beta[/C][C]0.133116958456148[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284172&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284172&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
alpha0.963396342115305
beta0.133116958456148
gamma1






Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
1390.3783.43453508082826.93546491917181
1492.2192.8083404384583-0.59834043845828
1595.7596.5626461243587-0.81264612435865
1697.2998.0192383452437-0.729238345243687
1798.2998.9221041734142-0.63210417341422
1899.51100.148000054915-0.638000054914542
1999.0498.60776152025790.432238479742153
2098.999.7654426634382-0.865442663438188
21100.74100.741032654764-0.00103265476410286
22100.3101.376925319696-1.07692531969587
23101.68101.2195269574980.46047304250169
24101.3104.729632767722-3.42963276772221
25103.13103.0913525927050.0386474072948459
26104.17104.585258292681-0.415258292681074
27105.98107.795291702497-1.81529170249664
28106.25107.181075248621-0.93107524862134
29104.01106.708942717444-2.6989427174444
30101.68104.520631576916-2.84063157691622
31101.9399.17542444115742.75457555884256
32104.41101.1422507191183.26774928088165
33105.51105.3206730275820.189326972418016
34104.71105.26189000101-0.551890001009696
35103.14104.923605697648-1.78360569764793
36102.66105.123889958828-2.46388995882815
37102.68103.647671318669-0.967671318669176
38101.89103.116121409581-1.22612140958078
39101.37104.269963137682-2.89996313768162
40101.16101.316284467844-0.156284467844301
4199.34100.331353032498-0.991353032497557
4299.3598.78778729163040.562212708369572
4399.8896.43716228420923.44283771579082
4499.3198.65500893967870.654991060321279
4599.9199.41088900384950.499110996150478
4698.3998.9412394114001-0.55123941140009
4798.0297.85725378914020.162746210859765
4898.799.3447126496292-0.644712649629241
4998.0199.3854930362606-1.37549303626058
5098.4298.12222446188450.297775538115459
5198.2100.47768087204-2.2776808720398
5293.598.1692596270892-4.66925962708916
5393.1792.21463009401590.955369905984099
5493.4292.22845993809511.19154006190486
5593.1390.43892999595182.69107000404821
5692.3191.53150920496290.778490795037058
5792.0992.03205622421880.0579437757811689
5892.6290.76881800253231.85118199746773
5991.4391.9526685751149-0.522668575114878
6089.3892.4696973145884-3.08969731458845

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 90.37 & 83.4345350808282 & 6.93546491917181 \tabularnewline
14 & 92.21 & 92.8083404384583 & -0.59834043845828 \tabularnewline
15 & 95.75 & 96.5626461243587 & -0.81264612435865 \tabularnewline
16 & 97.29 & 98.0192383452437 & -0.729238345243687 \tabularnewline
17 & 98.29 & 98.9221041734142 & -0.63210417341422 \tabularnewline
18 & 99.51 & 100.148000054915 & -0.638000054914542 \tabularnewline
19 & 99.04 & 98.6077615202579 & 0.432238479742153 \tabularnewline
20 & 98.9 & 99.7654426634382 & -0.865442663438188 \tabularnewline
21 & 100.74 & 100.741032654764 & -0.00103265476410286 \tabularnewline
22 & 100.3 & 101.376925319696 & -1.07692531969587 \tabularnewline
23 & 101.68 & 101.219526957498 & 0.46047304250169 \tabularnewline
24 & 101.3 & 104.729632767722 & -3.42963276772221 \tabularnewline
25 & 103.13 & 103.091352592705 & 0.0386474072948459 \tabularnewline
26 & 104.17 & 104.585258292681 & -0.415258292681074 \tabularnewline
27 & 105.98 & 107.795291702497 & -1.81529170249664 \tabularnewline
28 & 106.25 & 107.181075248621 & -0.93107524862134 \tabularnewline
29 & 104.01 & 106.708942717444 & -2.6989427174444 \tabularnewline
30 & 101.68 & 104.520631576916 & -2.84063157691622 \tabularnewline
31 & 101.93 & 99.1754244411574 & 2.75457555884256 \tabularnewline
32 & 104.41 & 101.142250719118 & 3.26774928088165 \tabularnewline
33 & 105.51 & 105.320673027582 & 0.189326972418016 \tabularnewline
34 & 104.71 & 105.26189000101 & -0.551890001009696 \tabularnewline
35 & 103.14 & 104.923605697648 & -1.78360569764793 \tabularnewline
36 & 102.66 & 105.123889958828 & -2.46388995882815 \tabularnewline
37 & 102.68 & 103.647671318669 & -0.967671318669176 \tabularnewline
38 & 101.89 & 103.116121409581 & -1.22612140958078 \tabularnewline
39 & 101.37 & 104.269963137682 & -2.89996313768162 \tabularnewline
40 & 101.16 & 101.316284467844 & -0.156284467844301 \tabularnewline
41 & 99.34 & 100.331353032498 & -0.991353032497557 \tabularnewline
42 & 99.35 & 98.7877872916304 & 0.562212708369572 \tabularnewline
43 & 99.88 & 96.4371622842092 & 3.44283771579082 \tabularnewline
44 & 99.31 & 98.6550089396787 & 0.654991060321279 \tabularnewline
45 & 99.91 & 99.4108890038495 & 0.499110996150478 \tabularnewline
46 & 98.39 & 98.9412394114001 & -0.55123941140009 \tabularnewline
47 & 98.02 & 97.8572537891402 & 0.162746210859765 \tabularnewline
48 & 98.7 & 99.3447126496292 & -0.644712649629241 \tabularnewline
49 & 98.01 & 99.3854930362606 & -1.37549303626058 \tabularnewline
50 & 98.42 & 98.1222244618845 & 0.297775538115459 \tabularnewline
51 & 98.2 & 100.47768087204 & -2.2776808720398 \tabularnewline
52 & 93.5 & 98.1692596270892 & -4.66925962708916 \tabularnewline
53 & 93.17 & 92.2146300940159 & 0.955369905984099 \tabularnewline
54 & 93.42 & 92.2284599380951 & 1.19154006190486 \tabularnewline
55 & 93.13 & 90.4389299959518 & 2.69107000404821 \tabularnewline
56 & 92.31 & 91.5315092049629 & 0.778490795037058 \tabularnewline
57 & 92.09 & 92.0320562242188 & 0.0579437757811689 \tabularnewline
58 & 92.62 & 90.7688180025323 & 1.85118199746773 \tabularnewline
59 & 91.43 & 91.9526685751149 & -0.522668575114878 \tabularnewline
60 & 89.38 & 92.4696973145884 & -3.08969731458845 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284172&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]90.37[/C][C]83.4345350808282[/C][C]6.93546491917181[/C][/ROW]
[ROW][C]14[/C][C]92.21[/C][C]92.8083404384583[/C][C]-0.59834043845828[/C][/ROW]
[ROW][C]15[/C][C]95.75[/C][C]96.5626461243587[/C][C]-0.81264612435865[/C][/ROW]
[ROW][C]16[/C][C]97.29[/C][C]98.0192383452437[/C][C]-0.729238345243687[/C][/ROW]
[ROW][C]17[/C][C]98.29[/C][C]98.9221041734142[/C][C]-0.63210417341422[/C][/ROW]
[ROW][C]18[/C][C]99.51[/C][C]100.148000054915[/C][C]-0.638000054914542[/C][/ROW]
[ROW][C]19[/C][C]99.04[/C][C]98.6077615202579[/C][C]0.432238479742153[/C][/ROW]
[ROW][C]20[/C][C]98.9[/C][C]99.7654426634382[/C][C]-0.865442663438188[/C][/ROW]
[ROW][C]21[/C][C]100.74[/C][C]100.741032654764[/C][C]-0.00103265476410286[/C][/ROW]
[ROW][C]22[/C][C]100.3[/C][C]101.376925319696[/C][C]-1.07692531969587[/C][/ROW]
[ROW][C]23[/C][C]101.68[/C][C]101.219526957498[/C][C]0.46047304250169[/C][/ROW]
[ROW][C]24[/C][C]101.3[/C][C]104.729632767722[/C][C]-3.42963276772221[/C][/ROW]
[ROW][C]25[/C][C]103.13[/C][C]103.091352592705[/C][C]0.0386474072948459[/C][/ROW]
[ROW][C]26[/C][C]104.17[/C][C]104.585258292681[/C][C]-0.415258292681074[/C][/ROW]
[ROW][C]27[/C][C]105.98[/C][C]107.795291702497[/C][C]-1.81529170249664[/C][/ROW]
[ROW][C]28[/C][C]106.25[/C][C]107.181075248621[/C][C]-0.93107524862134[/C][/ROW]
[ROW][C]29[/C][C]104.01[/C][C]106.708942717444[/C][C]-2.6989427174444[/C][/ROW]
[ROW][C]30[/C][C]101.68[/C][C]104.520631576916[/C][C]-2.84063157691622[/C][/ROW]
[ROW][C]31[/C][C]101.93[/C][C]99.1754244411574[/C][C]2.75457555884256[/C][/ROW]
[ROW][C]32[/C][C]104.41[/C][C]101.142250719118[/C][C]3.26774928088165[/C][/ROW]
[ROW][C]33[/C][C]105.51[/C][C]105.320673027582[/C][C]0.189326972418016[/C][/ROW]
[ROW][C]34[/C][C]104.71[/C][C]105.26189000101[/C][C]-0.551890001009696[/C][/ROW]
[ROW][C]35[/C][C]103.14[/C][C]104.923605697648[/C][C]-1.78360569764793[/C][/ROW]
[ROW][C]36[/C][C]102.66[/C][C]105.123889958828[/C][C]-2.46388995882815[/C][/ROW]
[ROW][C]37[/C][C]102.68[/C][C]103.647671318669[/C][C]-0.967671318669176[/C][/ROW]
[ROW][C]38[/C][C]101.89[/C][C]103.116121409581[/C][C]-1.22612140958078[/C][/ROW]
[ROW][C]39[/C][C]101.37[/C][C]104.269963137682[/C][C]-2.89996313768162[/C][/ROW]
[ROW][C]40[/C][C]101.16[/C][C]101.316284467844[/C][C]-0.156284467844301[/C][/ROW]
[ROW][C]41[/C][C]99.34[/C][C]100.331353032498[/C][C]-0.991353032497557[/C][/ROW]
[ROW][C]42[/C][C]99.35[/C][C]98.7877872916304[/C][C]0.562212708369572[/C][/ROW]
[ROW][C]43[/C][C]99.88[/C][C]96.4371622842092[/C][C]3.44283771579082[/C][/ROW]
[ROW][C]44[/C][C]99.31[/C][C]98.6550089396787[/C][C]0.654991060321279[/C][/ROW]
[ROW][C]45[/C][C]99.91[/C][C]99.4108890038495[/C][C]0.499110996150478[/C][/ROW]
[ROW][C]46[/C][C]98.39[/C][C]98.9412394114001[/C][C]-0.55123941140009[/C][/ROW]
[ROW][C]47[/C][C]98.02[/C][C]97.8572537891402[/C][C]0.162746210859765[/C][/ROW]
[ROW][C]48[/C][C]98.7[/C][C]99.3447126496292[/C][C]-0.644712649629241[/C][/ROW]
[ROW][C]49[/C][C]98.01[/C][C]99.3854930362606[/C][C]-1.37549303626058[/C][/ROW]
[ROW][C]50[/C][C]98.42[/C][C]98.1222244618845[/C][C]0.297775538115459[/C][/ROW]
[ROW][C]51[/C][C]98.2[/C][C]100.47768087204[/C][C]-2.2776808720398[/C][/ROW]
[ROW][C]52[/C][C]93.5[/C][C]98.1692596270892[/C][C]-4.66925962708916[/C][/ROW]
[ROW][C]53[/C][C]93.17[/C][C]92.2146300940159[/C][C]0.955369905984099[/C][/ROW]
[ROW][C]54[/C][C]93.42[/C][C]92.2284599380951[/C][C]1.19154006190486[/C][/ROW]
[ROW][C]55[/C][C]93.13[/C][C]90.4389299959518[/C][C]2.69107000404821[/C][/ROW]
[ROW][C]56[/C][C]92.31[/C][C]91.5315092049629[/C][C]0.778490795037058[/C][/ROW]
[ROW][C]57[/C][C]92.09[/C][C]92.0320562242188[/C][C]0.0579437757811689[/C][/ROW]
[ROW][C]58[/C][C]92.62[/C][C]90.7688180025323[/C][C]1.85118199746773[/C][/ROW]
[ROW][C]59[/C][C]91.43[/C][C]91.9526685751149[/C][C]-0.522668575114878[/C][/ROW]
[ROW][C]60[/C][C]89.38[/C][C]92.4696973145884[/C][C]-3.08969731458845[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284172&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284172&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
1390.3783.43453508082826.93546491917181
1492.2192.8083404384583-0.59834043845828
1595.7596.5626461243587-0.81264612435865
1697.2998.0192383452437-0.729238345243687
1798.2998.9221041734142-0.63210417341422
1899.51100.148000054915-0.638000054914542
1999.0498.60776152025790.432238479742153
2098.999.7654426634382-0.865442663438188
21100.74100.741032654764-0.00103265476410286
22100.3101.376925319696-1.07692531969587
23101.68101.2195269574980.46047304250169
24101.3104.729632767722-3.42963276772221
25103.13103.0913525927050.0386474072948459
26104.17104.585258292681-0.415258292681074
27105.98107.795291702497-1.81529170249664
28106.25107.181075248621-0.93107524862134
29104.01106.708942717444-2.6989427174444
30101.68104.520631576916-2.84063157691622
31101.9399.17542444115742.75457555884256
32104.41101.1422507191183.26774928088165
33105.51105.3206730275820.189326972418016
34104.71105.26189000101-0.551890001009696
35103.14104.923605697648-1.78360569764793
36102.66105.123889958828-2.46388995882815
37102.68103.647671318669-0.967671318669176
38101.89103.116121409581-1.22612140958078
39101.37104.269963137682-2.89996313768162
40101.16101.316284467844-0.156284467844301
4199.34100.331353032498-0.991353032497557
4299.3598.78778729163040.562212708369572
4399.8896.43716228420923.44283771579082
4499.3198.65500893967870.654991060321279
4599.9199.41088900384950.499110996150478
4698.3998.9412394114001-0.55123941140009
4798.0297.85725378914020.162746210859765
4898.799.3447126496292-0.644712649629241
4998.0199.3854930362606-1.37549303626058
5098.4298.12222446188450.297775538115459
5198.2100.47768087204-2.2776808720398
5293.598.1692596270892-4.66925962708916
5393.1792.21463009401590.955369905984099
5493.4292.22845993809511.19154006190486
5593.1390.43892999595182.69107000404821
5692.3191.53150920496290.778490795037058
5792.0992.03205622421880.0579437757811689
5892.6290.76881800253231.85118199746773
5991.4391.9526685751149-0.522668575114878
6089.3892.4696973145884-3.08969731458845






Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
6189.550276211595285.655590459720193.4449619634702
6289.304908243636983.526926122657395.0828903646166
6390.691777306420783.086860759061798.2966938537798
6490.367803634606881.07722941728899.6583778519256
6589.628007128957578.6879874938811100.568026764034
6689.110162182072376.4753574305605101.744966933584
6786.546288625224472.4904813905087100.60209585994
6884.936004625832369.3207544599729100.551254791692
6984.434701037156867.0380849890118101.831317085302
7083.0312379374864.0063253706675102.056150504293
7181.945308372372761.2034189672524102.687197777493
7282.357021762792458.89753934842105.816504177165

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
61 & 89.5502762115952 & 85.6555904597201 & 93.4449619634702 \tabularnewline
62 & 89.3049082436369 & 83.5269261226573 & 95.0828903646166 \tabularnewline
63 & 90.6917773064207 & 83.0868607590617 & 98.2966938537798 \tabularnewline
64 & 90.3678036346068 & 81.077229417288 & 99.6583778519256 \tabularnewline
65 & 89.6280071289575 & 78.6879874938811 & 100.568026764034 \tabularnewline
66 & 89.1101621820723 & 76.4753574305605 & 101.744966933584 \tabularnewline
67 & 86.5462886252244 & 72.4904813905087 & 100.60209585994 \tabularnewline
68 & 84.9360046258323 & 69.3207544599729 & 100.551254791692 \tabularnewline
69 & 84.4347010371568 & 67.0380849890118 & 101.831317085302 \tabularnewline
70 & 83.03123793748 & 64.0063253706675 & 102.056150504293 \tabularnewline
71 & 81.9453083723727 & 61.2034189672524 & 102.687197777493 \tabularnewline
72 & 82.3570217627924 & 58.89753934842 & 105.816504177165 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284172&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]61[/C][C]89.5502762115952[/C][C]85.6555904597201[/C][C]93.4449619634702[/C][/ROW]
[ROW][C]62[/C][C]89.3049082436369[/C][C]83.5269261226573[/C][C]95.0828903646166[/C][/ROW]
[ROW][C]63[/C][C]90.6917773064207[/C][C]83.0868607590617[/C][C]98.2966938537798[/C][/ROW]
[ROW][C]64[/C][C]90.3678036346068[/C][C]81.077229417288[/C][C]99.6583778519256[/C][/ROW]
[ROW][C]65[/C][C]89.6280071289575[/C][C]78.6879874938811[/C][C]100.568026764034[/C][/ROW]
[ROW][C]66[/C][C]89.1101621820723[/C][C]76.4753574305605[/C][C]101.744966933584[/C][/ROW]
[ROW][C]67[/C][C]86.5462886252244[/C][C]72.4904813905087[/C][C]100.60209585994[/C][/ROW]
[ROW][C]68[/C][C]84.9360046258323[/C][C]69.3207544599729[/C][C]100.551254791692[/C][/ROW]
[ROW][C]69[/C][C]84.4347010371568[/C][C]67.0380849890118[/C][C]101.831317085302[/C][/ROW]
[ROW][C]70[/C][C]83.03123793748[/C][C]64.0063253706675[/C][C]102.056150504293[/C][/ROW]
[ROW][C]71[/C][C]81.9453083723727[/C][C]61.2034189672524[/C][C]102.687197777493[/C][/ROW]
[ROW][C]72[/C][C]82.3570217627924[/C][C]58.89753934842[/C][C]105.816504177165[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284172&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284172&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
6189.550276211595285.655590459720193.4449619634702
6289.304908243636983.526926122657395.0828903646166
6390.691777306420783.086860759061798.2966938537798
6490.367803634606881.07722941728899.6583778519256
6589.628007128957578.6879874938811100.568026764034
6689.110162182072376.4753574305605101.744966933584
6786.546288625224472.4904813905087100.60209585994
6884.936004625832369.3207544599729100.551254791692
6984.434701037156867.0380849890118101.831317085302
7083.0312379374864.0063253706675102.056150504293
7181.945308372372761.2034189672524102.687197777493
7282.357021762792458.89753934842105.816504177165


Figures (Output of Computation)

Input Parameters & R Code

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')