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Author's title

Author*The author of this computation has been verified*
R Software Modulerwasp_exponentialsmoothing.wasp
Title produced by softwareExponential Smoothing
Date of computationWed, 07 Dec 2016 18:39:35 +0100
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/Dec/07/t1481132551nk06q3ikx6v9qj8.htm/, Retrieved Wed, 08 May 2024 00:52:56 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=298263, Retrieved Wed, 08 May 2024 00:52:56 +0000
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Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact64
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [N776] [2016-12-07 17:39:35] [40b26b3aac7c05a245868a452a1f2cfc] [Current]
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Dataseries X:
1575.17
1184.88
1227.35
1524.42
2040.63
1556.98
1684.28
1813.09
2265
1705.56
1794.49
1887.58
2609.02
1961.94
1967.67
2069.48
2653.73
2039.95
2662.62
3110.71
3661.27
2740.05
2766.81
2877.17
3568.61
2680.25
2757.06
2926.84
3855.35
3009.72
2962.93
3076.82
3905.86
2955.93
2926.8
3104.93




Summary of computational transaction
Raw Input view raw input (R code)
Raw Outputview raw output of R engine
Computing time1 seconds
R ServerBig Analytics Cloud Computing Center

\begin{tabular}{lllllllll}
\hline
Summary of computational transaction \tabularnewline
Raw Input view raw input (R code)  \tabularnewline
Raw Outputview raw output of R engine  \tabularnewline
Computing time1 seconds \tabularnewline
R ServerBig Analytics Cloud Computing Center \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=298263&T=0

[TABLE]
[ROW]
Summary of computational transaction[/C][/ROW] [ROW]Raw Input[/C] view raw input (R code) [/C][/ROW] [ROW]Raw Output[/C]view raw output of R engine [/C][/ROW] [ROW]Computing time[/C]1 seconds[/C][/ROW] [ROW]R Server[/C]Big Analytics Cloud Computing Center[/C][/ROW] [/TABLE] Source: https://freestatistics.org/blog/index.php?pk=298263&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=298263&T=0

As an alternative you can also use a QR Code:  

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

Summary of computational transaction
Raw Input view raw input (R code)
Raw Outputview raw output of R engine
Computing time1 seconds
R ServerBig Analytics Cloud Computing Center







Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.848926333732298
beta0.0257616816117401
gamma1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.848926333732298 \tabularnewline
beta & 0.0257616816117401 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=298263&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]0.848926333732298[/C][/ROW]
[ROW][C]beta[/C][C]0.0257616816117401[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=298263&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=298263&T=1

As an alternative you can also use a QR Code:  

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

Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.848926333732298
beta0.0257616816117401
gamma1







Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
52040.631793.66941832876246.96058167124
61556.981541.2012418220115.7787581779908
71684.281573.38912368562110.890876314379
81813.092038.13015968626-225.040159686258
922652491.23251917572-226.232519175718
101705.561722.03635557136-16.476355571363
111794.491728.939712080365.5502879197008
121887.582106.05612413072-218.476124130723
132609.022582.6027052594526.41729474055
141961.941961.352022192440.587977807562311
151967.671984.37652963015-16.7065296301537
162069.482258.42604657018-188.94604657018
172653.732856.21100276786-202.481002767857
182039.952008.1211039708931.8288960291059
192662.622046.12439825137616.495601748627
203110.712889.24347747734221.466522522659
213661.274165.52586240512-504.25586240512
222740.052813.66060694239-73.6106069423899
232766.812840.5065127352-73.6965127352028
242877.173035.80584962265-158.635849622653
253568.613795.11329113232-226.503291132316
262680.252748.86118806111-68.6111880611134
272757.062769.64552863888-12.5855286388819
282926.842993.26290530165-66.4229053016502
293855.353825.6224509986929.7275490013071
303009.722945.8683125322263.8516874677816
312962.933088.84439479583-125.914394795835
323076.823217.27070409797-140.450704097968
333905.864043.00557093115-137.145570931154
342955.933001.99793897019-46.0679389701895
352926.83013.74263333946-86.9426333394608
363104.933162.61319840818-57.6831984081832

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
5 & 2040.63 & 1793.66941832876 & 246.96058167124 \tabularnewline
6 & 1556.98 & 1541.20124182201 & 15.7787581779908 \tabularnewline
7 & 1684.28 & 1573.38912368562 & 110.890876314379 \tabularnewline
8 & 1813.09 & 2038.13015968626 & -225.040159686258 \tabularnewline
9 & 2265 & 2491.23251917572 & -226.232519175718 \tabularnewline
10 & 1705.56 & 1722.03635557136 & -16.476355571363 \tabularnewline
11 & 1794.49 & 1728.9397120803 & 65.5502879197008 \tabularnewline
12 & 1887.58 & 2106.05612413072 & -218.476124130723 \tabularnewline
13 & 2609.02 & 2582.60270525945 & 26.41729474055 \tabularnewline
14 & 1961.94 & 1961.35202219244 & 0.587977807562311 \tabularnewline
15 & 1967.67 & 1984.37652963015 & -16.7065296301537 \tabularnewline
16 & 2069.48 & 2258.42604657018 & -188.94604657018 \tabularnewline
17 & 2653.73 & 2856.21100276786 & -202.481002767857 \tabularnewline
18 & 2039.95 & 2008.12110397089 & 31.8288960291059 \tabularnewline
19 & 2662.62 & 2046.12439825137 & 616.495601748627 \tabularnewline
20 & 3110.71 & 2889.24347747734 & 221.466522522659 \tabularnewline
21 & 3661.27 & 4165.52586240512 & -504.25586240512 \tabularnewline
22 & 2740.05 & 2813.66060694239 & -73.6106069423899 \tabularnewline
23 & 2766.81 & 2840.5065127352 & -73.6965127352028 \tabularnewline
24 & 2877.17 & 3035.80584962265 & -158.635849622653 \tabularnewline
25 & 3568.61 & 3795.11329113232 & -226.503291132316 \tabularnewline
26 & 2680.25 & 2748.86118806111 & -68.6111880611134 \tabularnewline
27 & 2757.06 & 2769.64552863888 & -12.5855286388819 \tabularnewline
28 & 2926.84 & 2993.26290530165 & -66.4229053016502 \tabularnewline
29 & 3855.35 & 3825.62245099869 & 29.7275490013071 \tabularnewline
30 & 3009.72 & 2945.86831253222 & 63.8516874677816 \tabularnewline
31 & 2962.93 & 3088.84439479583 & -125.914394795835 \tabularnewline
32 & 3076.82 & 3217.27070409797 & -140.450704097968 \tabularnewline
33 & 3905.86 & 4043.00557093115 & -137.145570931154 \tabularnewline
34 & 2955.93 & 3001.99793897019 & -46.0679389701895 \tabularnewline
35 & 2926.8 & 3013.74263333946 & -86.9426333394608 \tabularnewline
36 & 3104.93 & 3162.61319840818 & -57.6831984081832 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=298263&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]5[/C][C]2040.63[/C][C]1793.66941832876[/C][C]246.96058167124[/C][/ROW]
[ROW][C]6[/C][C]1556.98[/C][C]1541.20124182201[/C][C]15.7787581779908[/C][/ROW]
[ROW][C]7[/C][C]1684.28[/C][C]1573.38912368562[/C][C]110.890876314379[/C][/ROW]
[ROW][C]8[/C][C]1813.09[/C][C]2038.13015968626[/C][C]-225.040159686258[/C][/ROW]
[ROW][C]9[/C][C]2265[/C][C]2491.23251917572[/C][C]-226.232519175718[/C][/ROW]
[ROW][C]10[/C][C]1705.56[/C][C]1722.03635557136[/C][C]-16.476355571363[/C][/ROW]
[ROW][C]11[/C][C]1794.49[/C][C]1728.9397120803[/C][C]65.5502879197008[/C][/ROW]
[ROW][C]12[/C][C]1887.58[/C][C]2106.05612413072[/C][C]-218.476124130723[/C][/ROW]
[ROW][C]13[/C][C]2609.02[/C][C]2582.60270525945[/C][C]26.41729474055[/C][/ROW]
[ROW][C]14[/C][C]1961.94[/C][C]1961.35202219244[/C][C]0.587977807562311[/C][/ROW]
[ROW][C]15[/C][C]1967.67[/C][C]1984.37652963015[/C][C]-16.7065296301537[/C][/ROW]
[ROW][C]16[/C][C]2069.48[/C][C]2258.42604657018[/C][C]-188.94604657018[/C][/ROW]
[ROW][C]17[/C][C]2653.73[/C][C]2856.21100276786[/C][C]-202.481002767857[/C][/ROW]
[ROW][C]18[/C][C]2039.95[/C][C]2008.12110397089[/C][C]31.8288960291059[/C][/ROW]
[ROW][C]19[/C][C]2662.62[/C][C]2046.12439825137[/C][C]616.495601748627[/C][/ROW]
[ROW][C]20[/C][C]3110.71[/C][C]2889.24347747734[/C][C]221.466522522659[/C][/ROW]
[ROW][C]21[/C][C]3661.27[/C][C]4165.52586240512[/C][C]-504.25586240512[/C][/ROW]
[ROW][C]22[/C][C]2740.05[/C][C]2813.66060694239[/C][C]-73.6106069423899[/C][/ROW]
[ROW][C]23[/C][C]2766.81[/C][C]2840.5065127352[/C][C]-73.6965127352028[/C][/ROW]
[ROW][C]24[/C][C]2877.17[/C][C]3035.80584962265[/C][C]-158.635849622653[/C][/ROW]
[ROW][C]25[/C][C]3568.61[/C][C]3795.11329113232[/C][C]-226.503291132316[/C][/ROW]
[ROW][C]26[/C][C]2680.25[/C][C]2748.86118806111[/C][C]-68.6111880611134[/C][/ROW]
[ROW][C]27[/C][C]2757.06[/C][C]2769.64552863888[/C][C]-12.5855286388819[/C][/ROW]
[ROW][C]28[/C][C]2926.84[/C][C]2993.26290530165[/C][C]-66.4229053016502[/C][/ROW]
[ROW][C]29[/C][C]3855.35[/C][C]3825.62245099869[/C][C]29.7275490013071[/C][/ROW]
[ROW][C]30[/C][C]3009.72[/C][C]2945.86831253222[/C][C]63.8516874677816[/C][/ROW]
[ROW][C]31[/C][C]2962.93[/C][C]3088.84439479583[/C][C]-125.914394795835[/C][/ROW]
[ROW][C]32[/C][C]3076.82[/C][C]3217.27070409797[/C][C]-140.450704097968[/C][/ROW]
[ROW][C]33[/C][C]3905.86[/C][C]4043.00557093115[/C][C]-137.145570931154[/C][/ROW]
[ROW][C]34[/C][C]2955.93[/C][C]3001.99793897019[/C][C]-46.0679389701895[/C][/ROW]
[ROW][C]35[/C][C]2926.8[/C][C]3013.74263333946[/C][C]-86.9426333394608[/C][/ROW]
[ROW][C]36[/C][C]3104.93[/C][C]3162.61319840818[/C][C]-57.6831984081832[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=298263&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=298263&T=2

As an alternative you can also use a QR Code:  

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

Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
52040.631793.66941832876246.96058167124
61556.981541.2012418220115.7787581779908
71684.281573.38912368562110.890876314379
81813.092038.13015968626-225.040159686258
922652491.23251917572-226.232519175718
101705.561722.03635557136-16.476355571363
111794.491728.939712080365.5502879197008
121887.582106.05612413072-218.476124130723
132609.022582.6027052594526.41729474055
141961.941961.352022192440.587977807562311
151967.671984.37652963015-16.7065296301537
162069.482258.42604657018-188.94604657018
172653.732856.21100276786-202.481002767857
182039.952008.1211039708931.8288960291059
192662.622046.12439825137616.495601748627
203110.712889.24347747734221.466522522659
213661.274165.52586240512-504.25586240512
222740.052813.66060694239-73.6106069423899
232766.812840.5065127352-73.6965127352028
242877.173035.80584962265-158.635849622653
253568.613795.11329113232-226.503291132316
262680.252748.86118806111-68.6111880611134
272757.062769.64552863888-12.5855286388819
282926.842993.26290530165-66.4229053016502
293855.353825.6224509986929.7275490013071
303009.722945.8683125322263.8516874677816
312962.933088.84439479583-125.914394795835
323076.823217.27070409797-140.450704097968
333905.864043.00557093115-137.145570931154
342955.933001.99793897019-46.0679389701895
352926.83013.74263333946-86.9426333394608
363104.933162.61319840818-57.6831984081832







Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
374060.151449276243692.026576939014428.27632161346
383106.336879384592666.894965474473545.77879329471
393146.330108639392602.218497937273690.44171934151
403383.735111162032834.995677588883932.47454473518
414416.725450190763583.661411856275249.78948852525
423373.283187793612639.518522922844107.04785266437
433411.026535953542604.533203852564217.51986805452
443662.540222324062847.433534980694477.64690966744
454773.299451105283607.053477154555939.54542505602
463640.229496202622667.111134380414613.34785802484
473675.722963267692642.105255819434709.34067071596
483941.34533348612894.596195094384988.09447187782

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
37 & 4060.15144927624 & 3692.02657693901 & 4428.27632161346 \tabularnewline
38 & 3106.33687938459 & 2666.89496547447 & 3545.77879329471 \tabularnewline
39 & 3146.33010863939 & 2602.21849793727 & 3690.44171934151 \tabularnewline
40 & 3383.73511116203 & 2834.99567758888 & 3932.47454473518 \tabularnewline
41 & 4416.72545019076 & 3583.66141185627 & 5249.78948852525 \tabularnewline
42 & 3373.28318779361 & 2639.51852292284 & 4107.04785266437 \tabularnewline
43 & 3411.02653595354 & 2604.53320385256 & 4217.51986805452 \tabularnewline
44 & 3662.54022232406 & 2847.43353498069 & 4477.64690966744 \tabularnewline
45 & 4773.29945110528 & 3607.05347715455 & 5939.54542505602 \tabularnewline
46 & 3640.22949620262 & 2667.11113438041 & 4613.34785802484 \tabularnewline
47 & 3675.72296326769 & 2642.10525581943 & 4709.34067071596 \tabularnewline
48 & 3941.3453334861 & 2894.59619509438 & 4988.09447187782 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=298263&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]37[/C][C]4060.15144927624[/C][C]3692.02657693901[/C][C]4428.27632161346[/C][/ROW]
[ROW][C]38[/C][C]3106.33687938459[/C][C]2666.89496547447[/C][C]3545.77879329471[/C][/ROW]
[ROW][C]39[/C][C]3146.33010863939[/C][C]2602.21849793727[/C][C]3690.44171934151[/C][/ROW]
[ROW][C]40[/C][C]3383.73511116203[/C][C]2834.99567758888[/C][C]3932.47454473518[/C][/ROW]
[ROW][C]41[/C][C]4416.72545019076[/C][C]3583.66141185627[/C][C]5249.78948852525[/C][/ROW]
[ROW][C]42[/C][C]3373.28318779361[/C][C]2639.51852292284[/C][C]4107.04785266437[/C][/ROW]
[ROW][C]43[/C][C]3411.02653595354[/C][C]2604.53320385256[/C][C]4217.51986805452[/C][/ROW]
[ROW][C]44[/C][C]3662.54022232406[/C][C]2847.43353498069[/C][C]4477.64690966744[/C][/ROW]
[ROW][C]45[/C][C]4773.29945110528[/C][C]3607.05347715455[/C][C]5939.54542505602[/C][/ROW]
[ROW][C]46[/C][C]3640.22949620262[/C][C]2667.11113438041[/C][C]4613.34785802484[/C][/ROW]
[ROW][C]47[/C][C]3675.72296326769[/C][C]2642.10525581943[/C][C]4709.34067071596[/C][/ROW]
[ROW][C]48[/C][C]3941.3453334861[/C][C]2894.59619509438[/C][C]4988.09447187782[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=298263&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=298263&T=3

As an alternative you can also use a QR Code:  

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

Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
374060.151449276243692.026576939014428.27632161346
383106.336879384592666.894965474473545.77879329471
393146.330108639392602.218497937273690.44171934151
403383.735111162032834.995677588883932.47454473518
414416.725450190763583.661411856275249.78948852525
423373.283187793612639.518522922844107.04785266437
433411.026535953542604.533203852564217.51986805452
443662.540222324062847.433534980694477.64690966744
454773.299451105283607.053477154555939.54542505602
463640.229496202622667.111134380414613.34785802484
473675.722963267692642.105255819434709.34067071596
483941.34533348612894.596195094384988.09447187782



Parameters (Session):
par1 = 4 ; par2 = Triple ; par3 = multiplicative ; par4 = 12 ;
Parameters (R input):
par1 = 4 ; par2 = Triple ; par3 = multiplicative ; par4 = 12 ;
R code (references can be found in the software module):
par1 <- as.numeric(par1)
par4 <- as.numeric(par4)
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$fitted[,'xhat']
bitmap(file='test1.png')
op <- par(mfrow=c(2,1))
plot(fit,ylab='Observed (black) / Fitted (red)',main='Interpolation Fit of Exponential Smoothing')
plot(myresid,ylab='Residuals',main='Interpolation Prediction Errors')
par(op)
dev.off()
bitmap(file='test2.png')
p <- predict(fit, par4, prediction.interval=TRUE)
np <- length(p[,1])
plot(fit,p,ylab='Observed (black) / Fitted (red)',main='Extrapolation Fit of Exponential Smoothing')
dev.off()
bitmap(file='test3.png')
op <- par(mfrow = c(2,2))
acf(as.numeric(myresid),lag.max = nx/2,main='Residual ACF')
spectrum(myresid,main='Residals Periodogram')
cpgram(myresid,main='Residal Cumulative Periodogram')
qqnorm(myresid,main='Residual Normal QQ Plot')
qqline(myresid)
par(op)
dev.off()
load(file='createtable')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Estimated Parameters of Exponential Smoothing',2,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Parameter',header=TRUE)
a<-table.element(a,'Value',header=TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'alpha',header=TRUE)
a<-table.element(a,fit$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')