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Description of Statistical Computation

Author's title

Author*The author of this computation has been verified*
R Software Modulerwasp_exponentialsmoothing.wasp
Title produced by softwareExponential Smoothing
Date of computationSun, 10 Jan 2016 18:05:52 +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/2016/Jan/10/t1452449173uvld79e848p37fd.htm/, Retrieved Sun, 05 May 2024 06:29:59 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=288011, Retrieved Sun, 05 May 2024 06:29:59 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact86

Tree of Dependent Computations

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [] [2016-01-10 18:05:52] [2faf0b243486f2b1b1e754e881f6604f] [Current]
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Dataset

Dataseries X:
Download CSV
Histogram
Boxplots 
19.6427
13.7242
14.8027
8.42832
6.80835
8.82499
8.76937
7.69536
7.65933
1.86731
3.72965
3.26542
-7.7517
0.957393
-3.04433
-12.5655
-6.9441
-6.8168
-8.56987
-4.46034
-11.229
-3.03305
-9.87928
-4.30931
-2.82241
-2.62636
-4.92814
-2.61029
-6.89788
-7.6867

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 Maurice George Kendall' @ kendall.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 Maurice George Kendall' @ kendall.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=288011&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 Maurice George Kendall' @ kendall.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=288011&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=288011&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 Maurice George Kendall' @ kendall.wessa.net






Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.596160807019071
betaFALSE
gammaFALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.596160807019071 \tabularnewline
beta & FALSE \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=288011&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]0.596160807019071[/C][/ROW]
[ROW][C]beta[/C][C]FALSE[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=288011&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=288011&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
alpha0.596160807019071
betaFALSE
gammaFALSE






Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
213.724219.6427-5.9185
314.802716.1143222636576-1.31162226365763
48.4283215.3323844764513-6.90406447645131
56.8083511.2164518264584-4.4081018264584
68.824998.588514284174720.236475715825282
78.769378.729491837761530.0398781622384696
87.695368.75326563514405-1.05790563514405
97.659338.12258375794655-0.463253757946551
101.867317.84641002375452-5.97910002375452
113.729654.28190492834528-0.552254928345276
123.265423.9526721845827-0.687252184582697
13-7.75173.54295936759626-11.2946593675963
140.957393-3.190473875995444.14786687599544
15-3.04433-0.717678211794325-2.32665178820568
16-12.5655-2.10473681950339-10.4607631804966
17-6.9441-8.341033839223631.39693383922363
18-6.8168-7.508236634279820.691436634279824
19-8.56987-7.09602921238501-1.47384078761499
20-4.46034-7.974675325747193.51433532574719
21-11.229-5.87956634181411-5.34943365818589
22-3.03305-9.068689028573196.03563902857319
23-9.87928-5.47047759442319-4.4088024055768
24-4.30931-8.098832794519483.78952279451948
25-2.82241-5.839667827121583.01725782712158
26-2.62636-4.040896965920171.41453696592017
27-4.92814-3.19760546675889-1.73053453324111
28-2.61029-4.229282330670281.61899233067028
29-6.89788-3.2641025562602-3.6337774437398
30-7.6867-5.43041824964782-2.25628175035218

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
2 & 13.7242 & 19.6427 & -5.9185 \tabularnewline
3 & 14.8027 & 16.1143222636576 & -1.31162226365763 \tabularnewline
4 & 8.42832 & 15.3323844764513 & -6.90406447645131 \tabularnewline
5 & 6.80835 & 11.2164518264584 & -4.4081018264584 \tabularnewline
6 & 8.82499 & 8.58851428417472 & 0.236475715825282 \tabularnewline
7 & 8.76937 & 8.72949183776153 & 0.0398781622384696 \tabularnewline
8 & 7.69536 & 8.75326563514405 & -1.05790563514405 \tabularnewline
9 & 7.65933 & 8.12258375794655 & -0.463253757946551 \tabularnewline
10 & 1.86731 & 7.84641002375452 & -5.97910002375452 \tabularnewline
11 & 3.72965 & 4.28190492834528 & -0.552254928345276 \tabularnewline
12 & 3.26542 & 3.9526721845827 & -0.687252184582697 \tabularnewline
13 & -7.7517 & 3.54295936759626 & -11.2946593675963 \tabularnewline
14 & 0.957393 & -3.19047387599544 & 4.14786687599544 \tabularnewline
15 & -3.04433 & -0.717678211794325 & -2.32665178820568 \tabularnewline
16 & -12.5655 & -2.10473681950339 & -10.4607631804966 \tabularnewline
17 & -6.9441 & -8.34103383922363 & 1.39693383922363 \tabularnewline
18 & -6.8168 & -7.50823663427982 & 0.691436634279824 \tabularnewline
19 & -8.56987 & -7.09602921238501 & -1.47384078761499 \tabularnewline
20 & -4.46034 & -7.97467532574719 & 3.51433532574719 \tabularnewline
21 & -11.229 & -5.87956634181411 & -5.34943365818589 \tabularnewline
22 & -3.03305 & -9.06868902857319 & 6.03563902857319 \tabularnewline
23 & -9.87928 & -5.47047759442319 & -4.4088024055768 \tabularnewline
24 & -4.30931 & -8.09883279451948 & 3.78952279451948 \tabularnewline
25 & -2.82241 & -5.83966782712158 & 3.01725782712158 \tabularnewline
26 & -2.62636 & -4.04089696592017 & 1.41453696592017 \tabularnewline
27 & -4.92814 & -3.19760546675889 & -1.73053453324111 \tabularnewline
28 & -2.61029 & -4.22928233067028 & 1.61899233067028 \tabularnewline
29 & -6.89788 & -3.2641025562602 & -3.6337774437398 \tabularnewline
30 & -7.6867 & -5.43041824964782 & -2.25628175035218 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=288011&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]2[/C][C]13.7242[/C][C]19.6427[/C][C]-5.9185[/C][/ROW]
[ROW][C]3[/C][C]14.8027[/C][C]16.1143222636576[/C][C]-1.31162226365763[/C][/ROW]
[ROW][C]4[/C][C]8.42832[/C][C]15.3323844764513[/C][C]-6.90406447645131[/C][/ROW]
[ROW][C]5[/C][C]6.80835[/C][C]11.2164518264584[/C][C]-4.4081018264584[/C][/ROW]
[ROW][C]6[/C][C]8.82499[/C][C]8.58851428417472[/C][C]0.236475715825282[/C][/ROW]
[ROW][C]7[/C][C]8.76937[/C][C]8.72949183776153[/C][C]0.0398781622384696[/C][/ROW]
[ROW][C]8[/C][C]7.69536[/C][C]8.75326563514405[/C][C]-1.05790563514405[/C][/ROW]
[ROW][C]9[/C][C]7.65933[/C][C]8.12258375794655[/C][C]-0.463253757946551[/C][/ROW]
[ROW][C]10[/C][C]1.86731[/C][C]7.84641002375452[/C][C]-5.97910002375452[/C][/ROW]
[ROW][C]11[/C][C]3.72965[/C][C]4.28190492834528[/C][C]-0.552254928345276[/C][/ROW]
[ROW][C]12[/C][C]3.26542[/C][C]3.9526721845827[/C][C]-0.687252184582697[/C][/ROW]
[ROW][C]13[/C][C]-7.7517[/C][C]3.54295936759626[/C][C]-11.2946593675963[/C][/ROW]
[ROW][C]14[/C][C]0.957393[/C][C]-3.19047387599544[/C][C]4.14786687599544[/C][/ROW]
[ROW][C]15[/C][C]-3.04433[/C][C]-0.717678211794325[/C][C]-2.32665178820568[/C][/ROW]
[ROW][C]16[/C][C]-12.5655[/C][C]-2.10473681950339[/C][C]-10.4607631804966[/C][/ROW]
[ROW][C]17[/C][C]-6.9441[/C][C]-8.34103383922363[/C][C]1.39693383922363[/C][/ROW]
[ROW][C]18[/C][C]-6.8168[/C][C]-7.50823663427982[/C][C]0.691436634279824[/C][/ROW]
[ROW][C]19[/C][C]-8.56987[/C][C]-7.09602921238501[/C][C]-1.47384078761499[/C][/ROW]
[ROW][C]20[/C][C]-4.46034[/C][C]-7.97467532574719[/C][C]3.51433532574719[/C][/ROW]
[ROW][C]21[/C][C]-11.229[/C][C]-5.87956634181411[/C][C]-5.34943365818589[/C][/ROW]
[ROW][C]22[/C][C]-3.03305[/C][C]-9.06868902857319[/C][C]6.03563902857319[/C][/ROW]
[ROW][C]23[/C][C]-9.87928[/C][C]-5.47047759442319[/C][C]-4.4088024055768[/C][/ROW]
[ROW][C]24[/C][C]-4.30931[/C][C]-8.09883279451948[/C][C]3.78952279451948[/C][/ROW]
[ROW][C]25[/C][C]-2.82241[/C][C]-5.83966782712158[/C][C]3.01725782712158[/C][/ROW]
[ROW][C]26[/C][C]-2.62636[/C][C]-4.04089696592017[/C][C]1.41453696592017[/C][/ROW]
[ROW][C]27[/C][C]-4.92814[/C][C]-3.19760546675889[/C][C]-1.73053453324111[/C][/ROW]
[ROW][C]28[/C][C]-2.61029[/C][C]-4.22928233067028[/C][C]1.61899233067028[/C][/ROW]
[ROW][C]29[/C][C]-6.89788[/C][C]-3.2641025562602[/C][C]-3.6337774437398[/C][/ROW]
[ROW][C]30[/C][C]-7.6867[/C][C]-5.43041824964782[/C][C]-2.25628175035218[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=288011&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=288011&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
213.724219.6427-5.9185
314.802716.1143222636576-1.31162226365763
48.4283215.3323844764513-6.90406447645131
56.8083511.2164518264584-4.4081018264584
68.824998.588514284174720.236475715825282
78.769378.729491837761530.0398781622384696
87.695368.75326563514405-1.05790563514405
97.659338.12258375794655-0.463253757946551
101.867317.84641002375452-5.97910002375452
113.729654.28190492834528-0.552254928345276
123.265423.9526721845827-0.687252184582697
13-7.75173.54295936759626-11.2946593675963
140.957393-3.190473875995444.14786687599544
15-3.04433-0.717678211794325-2.32665178820568
16-12.5655-2.10473681950339-10.4607631804966
17-6.9441-8.341033839223631.39693383922363
18-6.8168-7.508236634279820.691436634279824
19-8.56987-7.09602921238501-1.47384078761499
20-4.46034-7.974675325747193.51433532574719
21-11.229-5.87956634181411-5.34943365818589
22-3.03305-9.068689028573196.03563902857319
23-9.87928-5.47047759442319-4.4088024055768
24-4.30931-8.098832794519483.78952279451948
25-2.82241-5.839667827121583.01725782712158
26-2.62636-4.040896965920171.41453696592017
27-4.92814-3.19760546675889-1.73053453324111
28-2.61029-4.229282330670281.61899233067028
29-6.89788-3.2641025562602-3.6337774437398
30-7.6867-5.43041824964782-2.25628175035218






Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
31-6.77552499880018-14.92829979614131.37724979854097
32-6.77552499880018-16.26714672065482.71609672305446
33-6.77552499880018-17.43920301932413.88815302172372
34-6.77552499880018-18.49461945784554.94356946024512
35-6.77552499880018-19.46253886947315.91148887187277
36-6.77552499880018-20.36167482019076.81062482259034
37-6.77552499880018-21.20489133914557.65384134154517
38-6.77552499880018-22.0014817502998.45043175269866
39-6.77552499880018-22.75841913638169.20736913878124
40-6.77552499880018-23.48109446014329.93004446254284
41-6.77552499880018-24.173777699642810.6227277020425
42-6.77552499880018-24.839919343947711.2888693463474

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
31 & -6.77552499880018 & -14.9282997961413 & 1.37724979854097 \tabularnewline
32 & -6.77552499880018 & -16.2671467206548 & 2.71609672305446 \tabularnewline
33 & -6.77552499880018 & -17.4392030193241 & 3.88815302172372 \tabularnewline
34 & -6.77552499880018 & -18.4946194578455 & 4.94356946024512 \tabularnewline
35 & -6.77552499880018 & -19.4625388694731 & 5.91148887187277 \tabularnewline
36 & -6.77552499880018 & -20.3616748201907 & 6.81062482259034 \tabularnewline
37 & -6.77552499880018 & -21.2048913391455 & 7.65384134154517 \tabularnewline
38 & -6.77552499880018 & -22.001481750299 & 8.45043175269866 \tabularnewline
39 & -6.77552499880018 & -22.7584191363816 & 9.20736913878124 \tabularnewline
40 & -6.77552499880018 & -23.4810944601432 & 9.93004446254284 \tabularnewline
41 & -6.77552499880018 & -24.1737776996428 & 10.6227277020425 \tabularnewline
42 & -6.77552499880018 & -24.8399193439477 & 11.2888693463474 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=288011&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]31[/C][C]-6.77552499880018[/C][C]-14.9282997961413[/C][C]1.37724979854097[/C][/ROW]
[ROW][C]32[/C][C]-6.77552499880018[/C][C]-16.2671467206548[/C][C]2.71609672305446[/C][/ROW]
[ROW][C]33[/C][C]-6.77552499880018[/C][C]-17.4392030193241[/C][C]3.88815302172372[/C][/ROW]
[ROW][C]34[/C][C]-6.77552499880018[/C][C]-18.4946194578455[/C][C]4.94356946024512[/C][/ROW]
[ROW][C]35[/C][C]-6.77552499880018[/C][C]-19.4625388694731[/C][C]5.91148887187277[/C][/ROW]
[ROW][C]36[/C][C]-6.77552499880018[/C][C]-20.3616748201907[/C][C]6.81062482259034[/C][/ROW]
[ROW][C]37[/C][C]-6.77552499880018[/C][C]-21.2048913391455[/C][C]7.65384134154517[/C][/ROW]
[ROW][C]38[/C][C]-6.77552499880018[/C][C]-22.001481750299[/C][C]8.45043175269866[/C][/ROW]
[ROW][C]39[/C][C]-6.77552499880018[/C][C]-22.7584191363816[/C][C]9.20736913878124[/C][/ROW]
[ROW][C]40[/C][C]-6.77552499880018[/C][C]-23.4810944601432[/C][C]9.93004446254284[/C][/ROW]
[ROW][C]41[/C][C]-6.77552499880018[/C][C]-24.1737776996428[/C][C]10.6227277020425[/C][/ROW]
[ROW][C]42[/C][C]-6.77552499880018[/C][C]-24.8399193439477[/C][C]11.2888693463474[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=288011&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=288011&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
31-6.77552499880018-14.92829979614131.37724979854097
32-6.77552499880018-16.26714672065482.71609672305446
33-6.77552499880018-17.43920301932413.88815302172372
34-6.77552499880018-18.49461945784554.94356946024512
35-6.77552499880018-19.46253886947315.91148887187277
36-6.77552499880018-20.36167482019076.81062482259034
37-6.77552499880018-21.20489133914557.65384134154517
38-6.77552499880018-22.0014817502998.45043175269866
39-6.77552499880018-22.75841913638169.20736913878124
40-6.77552499880018-23.48109446014329.93004446254284
41-6.77552499880018-24.173777699642810.6227277020425
42-6.77552499880018-24.839919343947711.2888693463474


Figures (Output of Computation)

Input Parameters & R Code

Parameters (Session):
par1 = 1 ; par2 = 2 ; par3 = 3 ; par4 = TRUE ;
Parameters (R input):
par1 = 12 ; par2 = Single ; 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')