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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 computationMon, 19 Dec 2016 22:10:20 +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/19/t1482181838wpy9xpvledzpvyc.htm/, Retrieved Fri, 17 May 2024 16:33:08 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=301513, Retrieved Fri, 17 May 2024 16:33:08 +0000
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Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact58
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [] [2016-12-19 21:10:20] [9412b5b3b31fe4708efb1e5c8c74b28f] [Current]
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Dataseries X:
588.55
930.75
3228.65
2268.55
2414.5
3305.25
4342.05
3198.75
3091.35
3993.05
5331.5
3814.65
3707.6
4513.6
5634.2
4344.4
4060
4530.35
5348.75
4504.9
4281.35
4423.45
5197.9
4883.9
4155.25
4415.75
5384.05
5153.8
4564.1
5545
7585.4
6252.2
5785.65
6664.95
8639.85
6841.35




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=301513&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=301513&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=301513&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
alpha0.196060031469733
beta0.641380989938513
gammaFALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.196060031469733 \tabularnewline
beta & 0.641380989938513 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=301513&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]0.196060031469733[/C][/ROW]
[ROW][C]beta[/C][C]0.641380989938513[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=301513&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=301513&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
alpha0.196060031469733
beta0.641380989938513
gammaFALSE







Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
33228.651272.951955.7
42268.552244.5122691439624.0377308560392
52414.52840.37549788447-425.875497884471
63305.253294.4752314456110.7747685543914
74342.053835.53954826487506.510451735133
83198.754537.49109121299-1338.74109121299
93091.354709.31696804937-1617.96696804937
103993.054622.93979592509-629.889795925085
115331.54651.07694177874680.423058221262
123814.655021.67670668327-1207.02670668327
133707.64870.44039621442-1162.84039621442
144513.64581.64103227723-68.0410322772332
155634.24498.931962270671135.26803772933
164344.44794.90272789737-450.502727897368
1740604723.31688000998-663.316880009984
184530.354526.595130963353.75486903665114
195348.754461.13166171874887.618338281261
204504.94680.57576766156-175.675767661556
214281.354669.45931451681-388.109314516809
224423.454567.88870657615-144.438706576148
235197.94495.92911720737701.970882792631
244883.94678.1888794058205.711120594201
254155.254789.01994113961-633.769941139605
264415.754655.56627094072-239.81627094072
275384.054569.19447099951814.855529000491
285153.84792.06906954893361.730930451067
294564.14971.59141187255-407.491411872552
3055454949.05828784633595.94171215367
317585.45198.197473580332387.20252641967
326252.26098.72006412994153.479935870062
335785.656580.59890891437-794.948908914374
346664.956776.56459343375-111.614593433754
358639.857092.469382077161547.38061792284
366841.357928.21866338735-1086.86866338735

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 3228.65 & 1272.95 & 1955.7 \tabularnewline
4 & 2268.55 & 2244.51226914396 & 24.0377308560392 \tabularnewline
5 & 2414.5 & 2840.37549788447 & -425.875497884471 \tabularnewline
6 & 3305.25 & 3294.47523144561 & 10.7747685543914 \tabularnewline
7 & 4342.05 & 3835.53954826487 & 506.510451735133 \tabularnewline
8 & 3198.75 & 4537.49109121299 & -1338.74109121299 \tabularnewline
9 & 3091.35 & 4709.31696804937 & -1617.96696804937 \tabularnewline
10 & 3993.05 & 4622.93979592509 & -629.889795925085 \tabularnewline
11 & 5331.5 & 4651.07694177874 & 680.423058221262 \tabularnewline
12 & 3814.65 & 5021.67670668327 & -1207.02670668327 \tabularnewline
13 & 3707.6 & 4870.44039621442 & -1162.84039621442 \tabularnewline
14 & 4513.6 & 4581.64103227723 & -68.0410322772332 \tabularnewline
15 & 5634.2 & 4498.93196227067 & 1135.26803772933 \tabularnewline
16 & 4344.4 & 4794.90272789737 & -450.502727897368 \tabularnewline
17 & 4060 & 4723.31688000998 & -663.316880009984 \tabularnewline
18 & 4530.35 & 4526.59513096335 & 3.75486903665114 \tabularnewline
19 & 5348.75 & 4461.13166171874 & 887.618338281261 \tabularnewline
20 & 4504.9 & 4680.57576766156 & -175.675767661556 \tabularnewline
21 & 4281.35 & 4669.45931451681 & -388.109314516809 \tabularnewline
22 & 4423.45 & 4567.88870657615 & -144.438706576148 \tabularnewline
23 & 5197.9 & 4495.92911720737 & 701.970882792631 \tabularnewline
24 & 4883.9 & 4678.1888794058 & 205.711120594201 \tabularnewline
25 & 4155.25 & 4789.01994113961 & -633.769941139605 \tabularnewline
26 & 4415.75 & 4655.56627094072 & -239.81627094072 \tabularnewline
27 & 5384.05 & 4569.19447099951 & 814.855529000491 \tabularnewline
28 & 5153.8 & 4792.06906954893 & 361.730930451067 \tabularnewline
29 & 4564.1 & 4971.59141187255 & -407.491411872552 \tabularnewline
30 & 5545 & 4949.05828784633 & 595.94171215367 \tabularnewline
31 & 7585.4 & 5198.19747358033 & 2387.20252641967 \tabularnewline
32 & 6252.2 & 6098.72006412994 & 153.479935870062 \tabularnewline
33 & 5785.65 & 6580.59890891437 & -794.948908914374 \tabularnewline
34 & 6664.95 & 6776.56459343375 & -111.614593433754 \tabularnewline
35 & 8639.85 & 7092.46938207716 & 1547.38061792284 \tabularnewline
36 & 6841.35 & 7928.21866338735 & -1086.86866338735 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=301513&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]3228.65[/C][C]1272.95[/C][C]1955.7[/C][/ROW]
[ROW][C]4[/C][C]2268.55[/C][C]2244.51226914396[/C][C]24.0377308560392[/C][/ROW]
[ROW][C]5[/C][C]2414.5[/C][C]2840.37549788447[/C][C]-425.875497884471[/C][/ROW]
[ROW][C]6[/C][C]3305.25[/C][C]3294.47523144561[/C][C]10.7747685543914[/C][/ROW]
[ROW][C]7[/C][C]4342.05[/C][C]3835.53954826487[/C][C]506.510451735133[/C][/ROW]
[ROW][C]8[/C][C]3198.75[/C][C]4537.49109121299[/C][C]-1338.74109121299[/C][/ROW]
[ROW][C]9[/C][C]3091.35[/C][C]4709.31696804937[/C][C]-1617.96696804937[/C][/ROW]
[ROW][C]10[/C][C]3993.05[/C][C]4622.93979592509[/C][C]-629.889795925085[/C][/ROW]
[ROW][C]11[/C][C]5331.5[/C][C]4651.07694177874[/C][C]680.423058221262[/C][/ROW]
[ROW][C]12[/C][C]3814.65[/C][C]5021.67670668327[/C][C]-1207.02670668327[/C][/ROW]
[ROW][C]13[/C][C]3707.6[/C][C]4870.44039621442[/C][C]-1162.84039621442[/C][/ROW]
[ROW][C]14[/C][C]4513.6[/C][C]4581.64103227723[/C][C]-68.0410322772332[/C][/ROW]
[ROW][C]15[/C][C]5634.2[/C][C]4498.93196227067[/C][C]1135.26803772933[/C][/ROW]
[ROW][C]16[/C][C]4344.4[/C][C]4794.90272789737[/C][C]-450.502727897368[/C][/ROW]
[ROW][C]17[/C][C]4060[/C][C]4723.31688000998[/C][C]-663.316880009984[/C][/ROW]
[ROW][C]18[/C][C]4530.35[/C][C]4526.59513096335[/C][C]3.75486903665114[/C][/ROW]
[ROW][C]19[/C][C]5348.75[/C][C]4461.13166171874[/C][C]887.618338281261[/C][/ROW]
[ROW][C]20[/C][C]4504.9[/C][C]4680.57576766156[/C][C]-175.675767661556[/C][/ROW]
[ROW][C]21[/C][C]4281.35[/C][C]4669.45931451681[/C][C]-388.109314516809[/C][/ROW]
[ROW][C]22[/C][C]4423.45[/C][C]4567.88870657615[/C][C]-144.438706576148[/C][/ROW]
[ROW][C]23[/C][C]5197.9[/C][C]4495.92911720737[/C][C]701.970882792631[/C][/ROW]
[ROW][C]24[/C][C]4883.9[/C][C]4678.1888794058[/C][C]205.711120594201[/C][/ROW]
[ROW][C]25[/C][C]4155.25[/C][C]4789.01994113961[/C][C]-633.769941139605[/C][/ROW]
[ROW][C]26[/C][C]4415.75[/C][C]4655.56627094072[/C][C]-239.81627094072[/C][/ROW]
[ROW][C]27[/C][C]5384.05[/C][C]4569.19447099951[/C][C]814.855529000491[/C][/ROW]
[ROW][C]28[/C][C]5153.8[/C][C]4792.06906954893[/C][C]361.730930451067[/C][/ROW]
[ROW][C]29[/C][C]4564.1[/C][C]4971.59141187255[/C][C]-407.491411872552[/C][/ROW]
[ROW][C]30[/C][C]5545[/C][C]4949.05828784633[/C][C]595.94171215367[/C][/ROW]
[ROW][C]31[/C][C]7585.4[/C][C]5198.19747358033[/C][C]2387.20252641967[/C][/ROW]
[ROW][C]32[/C][C]6252.2[/C][C]6098.72006412994[/C][C]153.479935870062[/C][/ROW]
[ROW][C]33[/C][C]5785.65[/C][C]6580.59890891437[/C][C]-794.948908914374[/C][/ROW]
[ROW][C]34[/C][C]6664.95[/C][C]6776.56459343375[/C][C]-111.614593433754[/C][/ROW]
[ROW][C]35[/C][C]8639.85[/C][C]7092.46938207716[/C][C]1547.38061792284[/C][/ROW]
[ROW][C]36[/C][C]6841.35[/C][C]7928.21866338735[/C][C]-1086.86866338735[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=301513&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=301513&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
33228.651272.951955.7
42268.552244.5122691439624.0377308560392
52414.52840.37549788447-425.875497884471
63305.253294.4752314456110.7747685543914
74342.053835.53954826487506.510451735133
83198.754537.49109121299-1338.74109121299
93091.354709.31696804937-1617.96696804937
103993.054622.93979592509-629.889795925085
115331.54651.07694177874680.423058221262
123814.655021.67670668327-1207.02670668327
133707.64870.44039621442-1162.84039621442
144513.64581.64103227723-68.0410322772332
155634.24498.931962270671135.26803772933
164344.44794.90272789737-450.502727897368
1740604723.31688000998-663.316880009984
184530.354526.595130963353.75486903665114
195348.754461.13166171874887.618338281261
204504.94680.57576766156-175.675767661556
214281.354669.45931451681-388.109314516809
224423.454567.88870657615-144.438706576148
235197.94495.92911720737701.970882792631
244883.94678.1888794058205.711120594201
254155.254789.01994113961-633.769941139605
264415.754655.56627094072-239.81627094072
275384.054569.19447099951814.855529000491
285153.84792.06906954893361.730930451067
294564.14971.59141187255-407.491411872552
3055454949.05828784633595.94171215367
317585.45198.197473580332387.20252641967
326252.26098.72006412994153.479935870062
335785.656580.59890891437-794.948908914374
346664.956776.56459343375-111.614593433754
358639.857092.469382077161547.38061792284
366841.357928.21866338735-1086.86866338735







Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
378110.824107699066321.10166231089900.54655308732
388506.521056357966626.4083281829310386.633784533
398902.218005016866858.5856157123210945.8503943214
409297.914953675767011.1628101417311584.6670972098
419693.611902334667086.9798877996512300.2439168697
4210089.30885099367093.7088875700313084.9088144171
4310485.00579965257040.0381757879613929.973423517
4410880.70274831146933.7571769159414827.6483197068
4511276.39969697036781.2187326889115771.5806612516
4611672.09664562926587.421619411916756.7716718464
4712067.79359428816356.2715095997217779.3156789764
4812463.4905429476090.8423002540418836.1387856399

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
37 & 8110.82410769906 & 6321.1016623108 & 9900.54655308732 \tabularnewline
38 & 8506.52105635796 & 6626.40832818293 & 10386.633784533 \tabularnewline
39 & 8902.21800501686 & 6858.58561571232 & 10945.8503943214 \tabularnewline
40 & 9297.91495367576 & 7011.16281014173 & 11584.6670972098 \tabularnewline
41 & 9693.61190233466 & 7086.97988779965 & 12300.2439168697 \tabularnewline
42 & 10089.3088509936 & 7093.70888757003 & 13084.9088144171 \tabularnewline
43 & 10485.0057996525 & 7040.03817578796 & 13929.973423517 \tabularnewline
44 & 10880.7027483114 & 6933.75717691594 & 14827.6483197068 \tabularnewline
45 & 11276.3996969703 & 6781.21873268891 & 15771.5806612516 \tabularnewline
46 & 11672.0966456292 & 6587.4216194119 & 16756.7716718464 \tabularnewline
47 & 12067.7935942881 & 6356.27150959972 & 17779.3156789764 \tabularnewline
48 & 12463.490542947 & 6090.84230025404 & 18836.1387856399 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=301513&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]8110.82410769906[/C][C]6321.1016623108[/C][C]9900.54655308732[/C][/ROW]
[ROW][C]38[/C][C]8506.52105635796[/C][C]6626.40832818293[/C][C]10386.633784533[/C][/ROW]
[ROW][C]39[/C][C]8902.21800501686[/C][C]6858.58561571232[/C][C]10945.8503943214[/C][/ROW]
[ROW][C]40[/C][C]9297.91495367576[/C][C]7011.16281014173[/C][C]11584.6670972098[/C][/ROW]
[ROW][C]41[/C][C]9693.61190233466[/C][C]7086.97988779965[/C][C]12300.2439168697[/C][/ROW]
[ROW][C]42[/C][C]10089.3088509936[/C][C]7093.70888757003[/C][C]13084.9088144171[/C][/ROW]
[ROW][C]43[/C][C]10485.0057996525[/C][C]7040.03817578796[/C][C]13929.973423517[/C][/ROW]
[ROW][C]44[/C][C]10880.7027483114[/C][C]6933.75717691594[/C][C]14827.6483197068[/C][/ROW]
[ROW][C]45[/C][C]11276.3996969703[/C][C]6781.21873268891[/C][C]15771.5806612516[/C][/ROW]
[ROW][C]46[/C][C]11672.0966456292[/C][C]6587.4216194119[/C][C]16756.7716718464[/C][/ROW]
[ROW][C]47[/C][C]12067.7935942881[/C][C]6356.27150959972[/C][C]17779.3156789764[/C][/ROW]
[ROW][C]48[/C][C]12463.490542947[/C][C]6090.84230025404[/C][C]18836.1387856399[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=301513&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=301513&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
378110.824107699066321.10166231089900.54655308732
388506.521056357966626.4083281829310386.633784533
398902.218005016866858.5856157123210945.8503943214
409297.914953675767011.1628101417311584.6670972098
419693.611902334667086.9798877996512300.2439168697
4210089.30885099367093.7088875700313084.9088144171
4310485.00579965257040.0381757879613929.973423517
4410880.70274831146933.7571769159414827.6483197068
4511276.39969697036781.2187326889115771.5806612516
4611672.09664562926587.421619411916756.7716718464
4712067.79359428816356.2715095997217779.3156789764
4812463.4905429476090.8423002540418836.1387856399



Parameters (Session):
par1 = 12 ; par2 = Double ; par3 = multiplicative ; par4 = 12 ;
Parameters (R input):
par1 = 12 ; par2 = Double ; 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')