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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, 22 May 2008 10:03:14 -0600
Cite this page as followsStatistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2008/May/22/t12114722620i5ssi095nnt6w8.htm/, Retrieved Tue, 14 May 2024 22:51:07 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=13026, Retrieved Tue, 14 May 2024 22:51:07 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact253

Tree of Dependent Computations

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-     [] [Clélia Comes - op...] [-0001-11-30 00:00:00] [74be16979710d4c4e7c6647856088456]
- RMPD  [Standard Deviation Plot] [Opgave 8 - oefeni...] [2008-05-09 12:11:18] [74be16979710d4c4e7c6647856088456]
- RMPD    [Classical Decomposition] [Clélia Comes - in...] [2008-05-17 16:16:28] [74be16979710d4c4e7c6647856088456]
- RMPD        [Exponential Smoothing] [Clélia Comes- exp...] [2008-05-22 16:03:14] [d41d8cd98f00b204e9800998ecf8427e] [Current]
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Dataset

Dataseries X:
Download CSV
Histogram
Boxplots 
1.48
1.57
1.58
1.58
1.58
1.58
1.59
1.6
1.6
1.61
1.61
1.61
1.62
1.63
1.63
1.64
1.64
1.64
1.64
1.64
1.65
1.65
1.65
1.65
1.65
1.66
1.66
1.67
1.68
1.68
1.68
1.68
1.69
1.7
1.7
1.71
1.72
1.73
1.74
1.74
1.75
1.75
1.75
1.76
1.79
1.83

Tables (Output of Computation)





Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time3 seconds
R Server'Herman Ole Andreas Wold' @ 193.190.124.10:1001

\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 & 3 seconds \tabularnewline
R Server & 'Herman Ole Andreas Wold' @ 193.190.124.10:1001 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=13026&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]3 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Herman Ole Andreas Wold' @ 193.190.124.10:1001[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=13026&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=13026&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 time3 seconds
R Server'Herman Ole Andreas Wold' @ 193.190.124.10:1001






Estimated Parameters of Exponential Smoothing
ParameterValue
alpha1
beta1
gamma0

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 1 \tabularnewline
beta & 1 \tabularnewline
gamma & 0 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=13026&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]1[/C][/ROW]
[ROW][C]gamma[/C][C]0[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=13026&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=13026&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
alpha1
beta1
gamma0






Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
31.581.66-0.08
41.581.59-0.01
51.581.580
61.581.580
71.591.580.01
81.61.60
91.61.61-0.01
101.611.60.01
111.611.62-0.01
121.611.610
131.621.610.01
141.631.63-2.22044604925031e-16
151.631.64-0.00999999999999979
161.641.630.01
171.641.65-0.01
181.641.640
191.641.640
201.641.640
211.651.640.01
221.651.66-0.01
231.651.650
241.651.650
251.651.650
261.661.650.01
271.661.67-0.01
281.671.660.01
291.681.680
301.681.69-0.01
311.681.680
321.681.680
331.691.680.01
341.71.70
351.71.71-0.01
361.711.70.01
371.721.720
381.731.730
391.741.740
401.741.75-0.01
411.751.740.01
421.751.76-0.01
431.751.750
441.761.750.01
451.791.770.02
461.831.820.01

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 1.58 & 1.66 & -0.08 \tabularnewline
4 & 1.58 & 1.59 & -0.01 \tabularnewline
5 & 1.58 & 1.58 & 0 \tabularnewline
6 & 1.58 & 1.58 & 0 \tabularnewline
7 & 1.59 & 1.58 & 0.01 \tabularnewline
8 & 1.6 & 1.6 & 0 \tabularnewline
9 & 1.6 & 1.61 & -0.01 \tabularnewline
10 & 1.61 & 1.6 & 0.01 \tabularnewline
11 & 1.61 & 1.62 & -0.01 \tabularnewline
12 & 1.61 & 1.61 & 0 \tabularnewline
13 & 1.62 & 1.61 & 0.01 \tabularnewline
14 & 1.63 & 1.63 & -2.22044604925031e-16 \tabularnewline
15 & 1.63 & 1.64 & -0.00999999999999979 \tabularnewline
16 & 1.64 & 1.63 & 0.01 \tabularnewline
17 & 1.64 & 1.65 & -0.01 \tabularnewline
18 & 1.64 & 1.64 & 0 \tabularnewline
19 & 1.64 & 1.64 & 0 \tabularnewline
20 & 1.64 & 1.64 & 0 \tabularnewline
21 & 1.65 & 1.64 & 0.01 \tabularnewline
22 & 1.65 & 1.66 & -0.01 \tabularnewline
23 & 1.65 & 1.65 & 0 \tabularnewline
24 & 1.65 & 1.65 & 0 \tabularnewline
25 & 1.65 & 1.65 & 0 \tabularnewline
26 & 1.66 & 1.65 & 0.01 \tabularnewline
27 & 1.66 & 1.67 & -0.01 \tabularnewline
28 & 1.67 & 1.66 & 0.01 \tabularnewline
29 & 1.68 & 1.68 & 0 \tabularnewline
30 & 1.68 & 1.69 & -0.01 \tabularnewline
31 & 1.68 & 1.68 & 0 \tabularnewline
32 & 1.68 & 1.68 & 0 \tabularnewline
33 & 1.69 & 1.68 & 0.01 \tabularnewline
34 & 1.7 & 1.7 & 0 \tabularnewline
35 & 1.7 & 1.71 & -0.01 \tabularnewline
36 & 1.71 & 1.7 & 0.01 \tabularnewline
37 & 1.72 & 1.72 & 0 \tabularnewline
38 & 1.73 & 1.73 & 0 \tabularnewline
39 & 1.74 & 1.74 & 0 \tabularnewline
40 & 1.74 & 1.75 & -0.01 \tabularnewline
41 & 1.75 & 1.74 & 0.01 \tabularnewline
42 & 1.75 & 1.76 & -0.01 \tabularnewline
43 & 1.75 & 1.75 & 0 \tabularnewline
44 & 1.76 & 1.75 & 0.01 \tabularnewline
45 & 1.79 & 1.77 & 0.02 \tabularnewline
46 & 1.83 & 1.82 & 0.01 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=13026&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]3[/C][C]1.58[/C][C]1.66[/C][C]-0.08[/C][/ROW]
[ROW][C]4[/C][C]1.58[/C][C]1.59[/C][C]-0.01[/C][/ROW]
[ROW][C]5[/C][C]1.58[/C][C]1.58[/C][C]0[/C][/ROW]
[ROW][C]6[/C][C]1.58[/C][C]1.58[/C][C]0[/C][/ROW]
[ROW][C]7[/C][C]1.59[/C][C]1.58[/C][C]0.01[/C][/ROW]
[ROW][C]8[/C][C]1.6[/C][C]1.6[/C][C]0[/C][/ROW]
[ROW][C]9[/C][C]1.6[/C][C]1.61[/C][C]-0.01[/C][/ROW]
[ROW][C]10[/C][C]1.61[/C][C]1.6[/C][C]0.01[/C][/ROW]
[ROW][C]11[/C][C]1.61[/C][C]1.62[/C][C]-0.01[/C][/ROW]
[ROW][C]12[/C][C]1.61[/C][C]1.61[/C][C]0[/C][/ROW]
[ROW][C]13[/C][C]1.62[/C][C]1.61[/C][C]0.01[/C][/ROW]
[ROW][C]14[/C][C]1.63[/C][C]1.63[/C][C]-2.22044604925031e-16[/C][/ROW]
[ROW][C]15[/C][C]1.63[/C][C]1.64[/C][C]-0.00999999999999979[/C][/ROW]
[ROW][C]16[/C][C]1.64[/C][C]1.63[/C][C]0.01[/C][/ROW]
[ROW][C]17[/C][C]1.64[/C][C]1.65[/C][C]-0.01[/C][/ROW]
[ROW][C]18[/C][C]1.64[/C][C]1.64[/C][C]0[/C][/ROW]
[ROW][C]19[/C][C]1.64[/C][C]1.64[/C][C]0[/C][/ROW]
[ROW][C]20[/C][C]1.64[/C][C]1.64[/C][C]0[/C][/ROW]
[ROW][C]21[/C][C]1.65[/C][C]1.64[/C][C]0.01[/C][/ROW]
[ROW][C]22[/C][C]1.65[/C][C]1.66[/C][C]-0.01[/C][/ROW]
[ROW][C]23[/C][C]1.65[/C][C]1.65[/C][C]0[/C][/ROW]
[ROW][C]24[/C][C]1.65[/C][C]1.65[/C][C]0[/C][/ROW]
[ROW][C]25[/C][C]1.65[/C][C]1.65[/C][C]0[/C][/ROW]
[ROW][C]26[/C][C]1.66[/C][C]1.65[/C][C]0.01[/C][/ROW]
[ROW][C]27[/C][C]1.66[/C][C]1.67[/C][C]-0.01[/C][/ROW]
[ROW][C]28[/C][C]1.67[/C][C]1.66[/C][C]0.01[/C][/ROW]
[ROW][C]29[/C][C]1.68[/C][C]1.68[/C][C]0[/C][/ROW]
[ROW][C]30[/C][C]1.68[/C][C]1.69[/C][C]-0.01[/C][/ROW]
[ROW][C]31[/C][C]1.68[/C][C]1.68[/C][C]0[/C][/ROW]
[ROW][C]32[/C][C]1.68[/C][C]1.68[/C][C]0[/C][/ROW]
[ROW][C]33[/C][C]1.69[/C][C]1.68[/C][C]0.01[/C][/ROW]
[ROW][C]34[/C][C]1.7[/C][C]1.7[/C][C]0[/C][/ROW]
[ROW][C]35[/C][C]1.7[/C][C]1.71[/C][C]-0.01[/C][/ROW]
[ROW][C]36[/C][C]1.71[/C][C]1.7[/C][C]0.01[/C][/ROW]
[ROW][C]37[/C][C]1.72[/C][C]1.72[/C][C]0[/C][/ROW]
[ROW][C]38[/C][C]1.73[/C][C]1.73[/C][C]0[/C][/ROW]
[ROW][C]39[/C][C]1.74[/C][C]1.74[/C][C]0[/C][/ROW]
[ROW][C]40[/C][C]1.74[/C][C]1.75[/C][C]-0.01[/C][/ROW]
[ROW][C]41[/C][C]1.75[/C][C]1.74[/C][C]0.01[/C][/ROW]
[ROW][C]42[/C][C]1.75[/C][C]1.76[/C][C]-0.01[/C][/ROW]
[ROW][C]43[/C][C]1.75[/C][C]1.75[/C][C]0[/C][/ROW]
[ROW][C]44[/C][C]1.76[/C][C]1.75[/C][C]0.01[/C][/ROW]
[ROW][C]45[/C][C]1.79[/C][C]1.77[/C][C]0.02[/C][/ROW]
[ROW][C]46[/C][C]1.83[/C][C]1.82[/C][C]0.01[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=13026&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=13026&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
31.581.66-0.08
41.581.59-0.01
51.581.580
61.581.580
71.591.580.01
81.61.60
91.61.61-0.01
101.611.60.01
111.611.62-0.01
121.611.610
131.621.610.01
141.631.63-2.22044604925031e-16
151.631.64-0.00999999999999979
161.641.630.01
171.641.65-0.01
181.641.640
191.641.640
201.641.640
211.651.640.01
221.651.66-0.01
231.651.650
241.651.650
251.651.650
261.661.650.01
271.661.67-0.01
281.671.660.01
291.681.680
301.681.69-0.01
311.681.680
321.681.680
331.691.680.01
341.71.70
351.71.71-0.01
361.711.70.01
371.721.720
381.731.730
391.741.740
401.741.75-0.01
411.751.740.01
421.751.76-0.01
431.751.750
441.761.750.01
451.791.770.02
461.831.820.01






Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
471.871.841576700287051.89842329971295
481.911.8464435696971.973556430303
491.951.843649750672562.05635024932744
501.991.834319175884892.14568082411511
512.031.819207167670582.24079283232942
522.071.798859001701972.34114099829803
532.111.773690982398512.44630901760149
542.151.744034078793942.55596592120606
552.191.710159473915542.66984052608446
562.231.672294587501422.78770541249858
572.271.630633683178872.90936631682114
582.311.585345200614953.03465479938505

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
47 & 1.87 & 1.84157670028705 & 1.89842329971295 \tabularnewline
48 & 1.91 & 1.846443569697 & 1.973556430303 \tabularnewline
49 & 1.95 & 1.84364975067256 & 2.05635024932744 \tabularnewline
50 & 1.99 & 1.83431917588489 & 2.14568082411511 \tabularnewline
51 & 2.03 & 1.81920716767058 & 2.24079283232942 \tabularnewline
52 & 2.07 & 1.79885900170197 & 2.34114099829803 \tabularnewline
53 & 2.11 & 1.77369098239851 & 2.44630901760149 \tabularnewline
54 & 2.15 & 1.74403407879394 & 2.55596592120606 \tabularnewline
55 & 2.19 & 1.71015947391554 & 2.66984052608446 \tabularnewline
56 & 2.23 & 1.67229458750142 & 2.78770541249858 \tabularnewline
57 & 2.27 & 1.63063368317887 & 2.90936631682114 \tabularnewline
58 & 2.31 & 1.58534520061495 & 3.03465479938505 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=13026&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]47[/C][C]1.87[/C][C]1.84157670028705[/C][C]1.89842329971295[/C][/ROW]
[ROW][C]48[/C][C]1.91[/C][C]1.846443569697[/C][C]1.973556430303[/C][/ROW]
[ROW][C]49[/C][C]1.95[/C][C]1.84364975067256[/C][C]2.05635024932744[/C][/ROW]
[ROW][C]50[/C][C]1.99[/C][C]1.83431917588489[/C][C]2.14568082411511[/C][/ROW]
[ROW][C]51[/C][C]2.03[/C][C]1.81920716767058[/C][C]2.24079283232942[/C][/ROW]
[ROW][C]52[/C][C]2.07[/C][C]1.79885900170197[/C][C]2.34114099829803[/C][/ROW]
[ROW][C]53[/C][C]2.11[/C][C]1.77369098239851[/C][C]2.44630901760149[/C][/ROW]
[ROW][C]54[/C][C]2.15[/C][C]1.74403407879394[/C][C]2.55596592120606[/C][/ROW]
[ROW][C]55[/C][C]2.19[/C][C]1.71015947391554[/C][C]2.66984052608446[/C][/ROW]
[ROW][C]56[/C][C]2.23[/C][C]1.67229458750142[/C][C]2.78770541249858[/C][/ROW]
[ROW][C]57[/C][C]2.27[/C][C]1.63063368317887[/C][C]2.90936631682114[/C][/ROW]
[ROW][C]58[/C][C]2.31[/C][C]1.58534520061495[/C][C]3.03465479938505[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=13026&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=13026&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
471.871.841576700287051.89842329971295
481.911.8464435696971.973556430303
491.951.843649750672562.05635024932744
501.991.834319175884892.14568082411511
512.031.819207167670582.24079283232942
522.071.798859001701972.34114099829803
532.111.773690982398512.44630901760149
542.151.744034078793942.55596592120606
552.191.710159473915542.66984052608446
562.231.672294587501422.78770541249858
572.271.630633683178872.90936631682114
582.311.585345200614953.03465479938505


Figures (Output of Computation)

Input Parameters & R Code

Parameters (Session):
par1 = 12 ; par2 = Double ; par3 = additive ;
Parameters (R input):
par1 = 12 ; par2 = Double ; 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=0, beta=0)
if (par2 == 'Double') fit <- HoltWinters(x, gamma=0)
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')