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R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Tue, 21 Dec 2010 00:16:41 +0000

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Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2010/Dec/21/t12928904901rmaqyx4td16uy3.htm/, Retrieved Wed, 05 Nov 2025 00:59:36 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=113178, Retrieved Wed, 05 Nov 2025 00:59:36 +0000

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User-defined keywords

Estimated Impact

430

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)

-     [Classical Decomposition] [HPC Retail Sales] [2008-03-02 16:19:32] [74be16979710d4c4e7c6647856088456]
-  M D  [Classical Decomposition] [] [2010-11-26 09:51:05] [7789b9488494790f41ddb7f073cada1b]
- RMPD    [Exponential Smoothing] [] [2010-12-17 11:57:35] [7789b9488494790f41ddb7f073cada1b]
-   P         [Exponential Smoothing] [] [2010-12-21 00:16:41] [0bf4568947c4284a0258563e64d5d827] [Current]

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Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	2 seconds
R Server	'RServer@AstonUniversity' @ vre.aston.ac.uk

\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 & 'RServer@AstonUniversity' @ vre.aston.ac.uk \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=113178&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]'RServer@AstonUniversity' @ vre.aston.ac.uk[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=113178&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=113178&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 Output	view raw output of R engine
Computing time	2 seconds
R Server	'RServer@AstonUniversity' @ vre.aston.ac.uk

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.800425855140645
beta	0.0331010367938548
gamma	0.155499080750997

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.800425855140645 \tabularnewline
beta & 0.0331010367938548 \tabularnewline
gamma & 0.155499080750997 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=113178&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]0.800425855140645[/C][/ROW]
[ROW][C]beta[/C][C]0.0331010367938548[/C][/ROW]
[ROW][C]gamma[/C][C]0.155499080750997[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=113178&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=113178&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
Parameter	Value
alpha	0.800425855140645
beta	0.0331010367938548
gamma	0.155499080750997

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	104.73	103.368149038462	1.36185096153842
14	104.86	104.608075407099	0.251924592900835
15	105.03	104.98042940301	0.049570596990165
16	105.62	105.610877397986	0.00912260201413062
17	105.63	105.625858141324	0.0041418586764479
18	105.63	105.60487857382	0.0251214261796378
19	105.94	106.117190518229	-0.177190518229409
20	106.61	106.116205435251	0.493794564749024
21	107.69	107.668710544731	0.0212894552693683
22	107.78	107.852324410432	-0.0723244104318752
23	107.93	107.954507754413	-0.0245077544134489
24	108.48	108.10639878838	0.373601211619913
25	108.14	108.929524867571	-0.789524867570535
26	108.48	108.417361551599	0.0626384484012732
27	108.48	108.631283288652	-0.151283288652067
28	108.89	109.093742822335	-0.203742822335499
29	108.93	108.926581581002	0.00341841899826534
30	109.21	108.894050432639	0.315949567361073
31	109.47	109.628952151016	-0.158952151016308
32	109.8	109.659953710454	0.14004628954639
33	111.73	110.901838588991	0.828161411008736
34	111.85	111.736959034911	0.113040965089311
35	112.12	112.002479362695	0.117520637305205
36	112.15	112.297653199882	-0.14765319988237
37	112.17	112.670892042672	-0.500892042671921
38	112.67	112.427285156356	0.242714843643924
39	112.8	112.794558607827	0.00544139217296902
40	113.44	113.400841701517	0.0391582984831871
41	113.53	113.460974484271	0.0690255157290238
42	114.53	113.518834901439	1.01116509856136
43	114.51	114.842065865226	-0.332065865225957
44	115.05	114.785793761209	0.264206238791175
45	116.67	116.193715801099	0.476284198900672
46	117.07	116.760970550048	0.309029449952277
47	116.92	117.224675529598	-0.304675529598185
48	117	117.203668665873	-0.20366866587301
49	117.02	117.549610208994	-0.52961020899393
50	117.35	117.333833669063	0.016166330937466
51	117.36	117.53414631254	-0.174146312539676
52	117.82	118.014708885544	-0.194708885544003
53	117.88	117.89935868561	-0.0193586856101575
54	118.24	117.924153820167	0.31584617983269
55	118.5	118.639167571678	-0.139167571678371
56	118.8	118.750930857801	0.0490691421989595
57	119.76	119.982663244492	-0.222663244491997
58	120.09	119.956183411041	0.133816588958965

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 104.73 & 103.368149038462 & 1.36185096153842 \tabularnewline
14 & 104.86 & 104.608075407099 & 0.251924592900835 \tabularnewline
15 & 105.03 & 104.98042940301 & 0.049570596990165 \tabularnewline
16 & 105.62 & 105.610877397986 & 0.00912260201413062 \tabularnewline
17 & 105.63 & 105.625858141324 & 0.0041418586764479 \tabularnewline
18 & 105.63 & 105.60487857382 & 0.0251214261796378 \tabularnewline
19 & 105.94 & 106.117190518229 & -0.177190518229409 \tabularnewline
20 & 106.61 & 106.116205435251 & 0.493794564749024 \tabularnewline
21 & 107.69 & 107.668710544731 & 0.0212894552693683 \tabularnewline
22 & 107.78 & 107.852324410432 & -0.0723244104318752 \tabularnewline
23 & 107.93 & 107.954507754413 & -0.0245077544134489 \tabularnewline
24 & 108.48 & 108.10639878838 & 0.373601211619913 \tabularnewline
25 & 108.14 & 108.929524867571 & -0.789524867570535 \tabularnewline
26 & 108.48 & 108.417361551599 & 0.0626384484012732 \tabularnewline
27 & 108.48 & 108.631283288652 & -0.151283288652067 \tabularnewline
28 & 108.89 & 109.093742822335 & -0.203742822335499 \tabularnewline
29 & 108.93 & 108.926581581002 & 0.00341841899826534 \tabularnewline
30 & 109.21 & 108.894050432639 & 0.315949567361073 \tabularnewline
31 & 109.47 & 109.628952151016 & -0.158952151016308 \tabularnewline
32 & 109.8 & 109.659953710454 & 0.14004628954639 \tabularnewline
33 & 111.73 & 110.901838588991 & 0.828161411008736 \tabularnewline
34 & 111.85 & 111.736959034911 & 0.113040965089311 \tabularnewline
35 & 112.12 & 112.002479362695 & 0.117520637305205 \tabularnewline
36 & 112.15 & 112.297653199882 & -0.14765319988237 \tabularnewline
37 & 112.17 & 112.670892042672 & -0.500892042671921 \tabularnewline
38 & 112.67 & 112.427285156356 & 0.242714843643924 \tabularnewline
39 & 112.8 & 112.794558607827 & 0.00544139217296902 \tabularnewline
40 & 113.44 & 113.400841701517 & 0.0391582984831871 \tabularnewline
41 & 113.53 & 113.460974484271 & 0.0690255157290238 \tabularnewline
42 & 114.53 & 113.518834901439 & 1.01116509856136 \tabularnewline
43 & 114.51 & 114.842065865226 & -0.332065865225957 \tabularnewline
44 & 115.05 & 114.785793761209 & 0.264206238791175 \tabularnewline
45 & 116.67 & 116.193715801099 & 0.476284198900672 \tabularnewline
46 & 117.07 & 116.760970550048 & 0.309029449952277 \tabularnewline
47 & 116.92 & 117.224675529598 & -0.304675529598185 \tabularnewline
48 & 117 & 117.203668665873 & -0.20366866587301 \tabularnewline
49 & 117.02 & 117.549610208994 & -0.52961020899393 \tabularnewline
50 & 117.35 & 117.333833669063 & 0.016166330937466 \tabularnewline
51 & 117.36 & 117.53414631254 & -0.174146312539676 \tabularnewline
52 & 117.82 & 118.014708885544 & -0.194708885544003 \tabularnewline
53 & 117.88 & 117.89935868561 & -0.0193586856101575 \tabularnewline
54 & 118.24 & 117.924153820167 & 0.31584617983269 \tabularnewline
55 & 118.5 & 118.639167571678 & -0.139167571678371 \tabularnewline
56 & 118.8 & 118.750930857801 & 0.0490691421989595 \tabularnewline
57 & 119.76 & 119.982663244492 & -0.222663244491997 \tabularnewline
58 & 120.09 & 119.956183411041 & 0.133816588958965 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=113178&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]104.73[/C][C]103.368149038462[/C][C]1.36185096153842[/C][/ROW]
[ROW][C]14[/C][C]104.86[/C][C]104.608075407099[/C][C]0.251924592900835[/C][/ROW]
[ROW][C]15[/C][C]105.03[/C][C]104.98042940301[/C][C]0.049570596990165[/C][/ROW]
[ROW][C]16[/C][C]105.62[/C][C]105.610877397986[/C][C]0.00912260201413062[/C][/ROW]
[ROW][C]17[/C][C]105.63[/C][C]105.625858141324[/C][C]0.0041418586764479[/C][/ROW]
[ROW][C]18[/C][C]105.63[/C][C]105.60487857382[/C][C]0.0251214261796378[/C][/ROW]
[ROW][C]19[/C][C]105.94[/C][C]106.117190518229[/C][C]-0.177190518229409[/C][/ROW]
[ROW][C]20[/C][C]106.61[/C][C]106.116205435251[/C][C]0.493794564749024[/C][/ROW]
[ROW][C]21[/C][C]107.69[/C][C]107.668710544731[/C][C]0.0212894552693683[/C][/ROW]
[ROW][C]22[/C][C]107.78[/C][C]107.852324410432[/C][C]-0.0723244104318752[/C][/ROW]
[ROW][C]23[/C][C]107.93[/C][C]107.954507754413[/C][C]-0.0245077544134489[/C][/ROW]
[ROW][C]24[/C][C]108.48[/C][C]108.10639878838[/C][C]0.373601211619913[/C][/ROW]
[ROW][C]25[/C][C]108.14[/C][C]108.929524867571[/C][C]-0.789524867570535[/C][/ROW]
[ROW][C]26[/C][C]108.48[/C][C]108.417361551599[/C][C]0.0626384484012732[/C][/ROW]
[ROW][C]27[/C][C]108.48[/C][C]108.631283288652[/C][C]-0.151283288652067[/C][/ROW]
[ROW][C]28[/C][C]108.89[/C][C]109.093742822335[/C][C]-0.203742822335499[/C][/ROW]
[ROW][C]29[/C][C]108.93[/C][C]108.926581581002[/C][C]0.00341841899826534[/C][/ROW]
[ROW][C]30[/C][C]109.21[/C][C]108.894050432639[/C][C]0.315949567361073[/C][/ROW]
[ROW][C]31[/C][C]109.47[/C][C]109.628952151016[/C][C]-0.158952151016308[/C][/ROW]
[ROW][C]32[/C][C]109.8[/C][C]109.659953710454[/C][C]0.14004628954639[/C][/ROW]
[ROW][C]33[/C][C]111.73[/C][C]110.901838588991[/C][C]0.828161411008736[/C][/ROW]
[ROW][C]34[/C][C]111.85[/C][C]111.736959034911[/C][C]0.113040965089311[/C][/ROW]
[ROW][C]35[/C][C]112.12[/C][C]112.002479362695[/C][C]0.117520637305205[/C][/ROW]
[ROW][C]36[/C][C]112.15[/C][C]112.297653199882[/C][C]-0.14765319988237[/C][/ROW]
[ROW][C]37[/C][C]112.17[/C][C]112.670892042672[/C][C]-0.500892042671921[/C][/ROW]
[ROW][C]38[/C][C]112.67[/C][C]112.427285156356[/C][C]0.242714843643924[/C][/ROW]
[ROW][C]39[/C][C]112.8[/C][C]112.794558607827[/C][C]0.00544139217296902[/C][/ROW]
[ROW][C]40[/C][C]113.44[/C][C]113.400841701517[/C][C]0.0391582984831871[/C][/ROW]
[ROW][C]41[/C][C]113.53[/C][C]113.460974484271[/C][C]0.0690255157290238[/C][/ROW]
[ROW][C]42[/C][C]114.53[/C][C]113.518834901439[/C][C]1.01116509856136[/C][/ROW]
[ROW][C]43[/C][C]114.51[/C][C]114.842065865226[/C][C]-0.332065865225957[/C][/ROW]
[ROW][C]44[/C][C]115.05[/C][C]114.785793761209[/C][C]0.264206238791175[/C][/ROW]
[ROW][C]45[/C][C]116.67[/C][C]116.193715801099[/C][C]0.476284198900672[/C][/ROW]
[ROW][C]46[/C][C]117.07[/C][C]116.760970550048[/C][C]0.309029449952277[/C][/ROW]
[ROW][C]47[/C][C]116.92[/C][C]117.224675529598[/C][C]-0.304675529598185[/C][/ROW]
[ROW][C]48[/C][C]117[/C][C]117.203668665873[/C][C]-0.20366866587301[/C][/ROW]
[ROW][C]49[/C][C]117.02[/C][C]117.549610208994[/C][C]-0.52961020899393[/C][/ROW]
[ROW][C]50[/C][C]117.35[/C][C]117.333833669063[/C][C]0.016166330937466[/C][/ROW]
[ROW][C]51[/C][C]117.36[/C][C]117.53414631254[/C][C]-0.174146312539676[/C][/ROW]
[ROW][C]52[/C][C]117.82[/C][C]118.014708885544[/C][C]-0.194708885544003[/C][/ROW]
[ROW][C]53[/C][C]117.88[/C][C]117.89935868561[/C][C]-0.0193586856101575[/C][/ROW]
[ROW][C]54[/C][C]118.24[/C][C]117.924153820167[/C][C]0.31584617983269[/C][/ROW]
[ROW][C]55[/C][C]118.5[/C][C]118.639167571678[/C][C]-0.139167571678371[/C][/ROW]
[ROW][C]56[/C][C]118.8[/C][C]118.750930857801[/C][C]0.0490691421989595[/C][/ROW]
[ROW][C]57[/C][C]119.76[/C][C]119.982663244492[/C][C]-0.222663244491997[/C][/ROW]
[ROW][C]58[/C][C]120.09[/C][C]119.956183411041[/C][C]0.133816588958965[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=113178&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=113178&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
t	Observed	Fitted	Residuals
13	104.73	103.368149038462	1.36185096153842
14	104.86	104.608075407099	0.251924592900835
15	105.03	104.98042940301	0.049570596990165
16	105.62	105.610877397986	0.00912260201413062
17	105.63	105.625858141324	0.0041418586764479
18	105.63	105.60487857382	0.0251214261796378
19	105.94	106.117190518229	-0.177190518229409
20	106.61	106.116205435251	0.493794564749024
21	107.69	107.668710544731	0.0212894552693683
22	107.78	107.852324410432	-0.0723244104318752
23	107.93	107.954507754413	-0.0245077544134489
24	108.48	108.10639878838	0.373601211619913
25	108.14	108.929524867571	-0.789524867570535
26	108.48	108.417361551599	0.0626384484012732
27	108.48	108.631283288652	-0.151283288652067
28	108.89	109.093742822335	-0.203742822335499
29	108.93	108.926581581002	0.00341841899826534
30	109.21	108.894050432639	0.315949567361073
31	109.47	109.628952151016	-0.158952151016308
32	109.8	109.659953710454	0.14004628954639
33	111.73	110.901838588991	0.828161411008736
34	111.85	111.736959034911	0.113040965089311
35	112.12	112.002479362695	0.117520637305205
36	112.15	112.297653199882	-0.14765319988237
37	112.17	112.670892042672	-0.500892042671921
38	112.67	112.427285156356	0.242714843643924
39	112.8	112.794558607827	0.00544139217296902
40	113.44	113.400841701517	0.0391582984831871
41	113.53	113.460974484271	0.0690255157290238
42	114.53	113.518834901439	1.01116509856136
43	114.51	114.842065865226	-0.332065865225957
44	115.05	114.785793761209	0.264206238791175
45	116.67	116.193715801099	0.476284198900672
46	117.07	116.760970550048	0.309029449952277
47	116.92	117.224675529598	-0.304675529598185
48	117	117.203668665873	-0.20366866587301
49	117.02	117.549610208994	-0.52961020899393
50	117.35	117.333833669063	0.016166330937466
51	117.36	117.53414631254	-0.174146312539676
52	117.82	118.014708885544	-0.194708885544003
53	117.88	117.89935868561	-0.0193586856101575
54	118.24	117.924153820167	0.31584617983269
55	118.5	118.639167571678	-0.139167571678371
56	118.8	118.750930857801	0.0490691421989595
57	119.76	119.982663244492	-0.222663244491997
58	120.09	119.956183411041	0.133816588958965

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
59	120.226867301325	119.495761998643	120.957972604006
60	120.427206857907	119.478515540984	121.375898174829
61	120.905792757828	119.770313880513	122.041271635144
62	121.124637143078	119.819586722415	122.429687563741
63	121.299445237922	119.835785915359	122.763104560485
64	121.91671637142	120.301944326895	123.531488415945
65	121.965772222373	120.205238931287	123.726305513459
66	122.020092346141	120.117721406749	123.922463285533
67	122.463432799368	120.422145862008	124.504719736729
68	122.691377167944	120.513364446082	124.869389889805
69	123.873046525867	121.559947566095	126.186145485638
70	124.039400422033	121.592429857695	126.486370986372

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
59 & 120.226867301325 & 119.495761998643 & 120.957972604006 \tabularnewline
60 & 120.427206857907 & 119.478515540984 & 121.375898174829 \tabularnewline
61 & 120.905792757828 & 119.770313880513 & 122.041271635144 \tabularnewline
62 & 121.124637143078 & 119.819586722415 & 122.429687563741 \tabularnewline
63 & 121.299445237922 & 119.835785915359 & 122.763104560485 \tabularnewline
64 & 121.91671637142 & 120.301944326895 & 123.531488415945 \tabularnewline
65 & 121.965772222373 & 120.205238931287 & 123.726305513459 \tabularnewline
66 & 122.020092346141 & 120.117721406749 & 123.922463285533 \tabularnewline
67 & 122.463432799368 & 120.422145862008 & 124.504719736729 \tabularnewline
68 & 122.691377167944 & 120.513364446082 & 124.869389889805 \tabularnewline
69 & 123.873046525867 & 121.559947566095 & 126.186145485638 \tabularnewline
70 & 124.039400422033 & 121.592429857695 & 126.486370986372 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=113178&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]59[/C][C]120.226867301325[/C][C]119.495761998643[/C][C]120.957972604006[/C][/ROW]
[ROW][C]60[/C][C]120.427206857907[/C][C]119.478515540984[/C][C]121.375898174829[/C][/ROW]
[ROW][C]61[/C][C]120.905792757828[/C][C]119.770313880513[/C][C]122.041271635144[/C][/ROW]
[ROW][C]62[/C][C]121.124637143078[/C][C]119.819586722415[/C][C]122.429687563741[/C][/ROW]
[ROW][C]63[/C][C]121.299445237922[/C][C]119.835785915359[/C][C]122.763104560485[/C][/ROW]
[ROW][C]64[/C][C]121.91671637142[/C][C]120.301944326895[/C][C]123.531488415945[/C][/ROW]
[ROW][C]65[/C][C]121.965772222373[/C][C]120.205238931287[/C][C]123.726305513459[/C][/ROW]
[ROW][C]66[/C][C]122.020092346141[/C][C]120.117721406749[/C][C]123.922463285533[/C][/ROW]
[ROW][C]67[/C][C]122.463432799368[/C][C]120.422145862008[/C][C]124.504719736729[/C][/ROW]
[ROW][C]68[/C][C]122.691377167944[/C][C]120.513364446082[/C][C]124.869389889805[/C][/ROW]
[ROW][C]69[/C][C]123.873046525867[/C][C]121.559947566095[/C][C]126.186145485638[/C][/ROW]
[ROW][C]70[/C][C]124.039400422033[/C][C]121.592429857695[/C][C]126.486370986372[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=113178&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=113178&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
t	Forecast	95% Lower Bound	95% Upper Bound
59	120.226867301325	119.495761998643	120.957972604006
60	120.427206857907	119.478515540984	121.375898174829
61	120.905792757828	119.770313880513	122.041271635144
62	121.124637143078	119.819586722415	122.429687563741
63	121.299445237922	119.835785915359	122.763104560485
64	121.91671637142	120.301944326895	123.531488415945
65	121.965772222373	120.205238931287	123.726305513459
66	122.020092346141	120.117721406749	123.922463285533
67	122.463432799368	120.422145862008	124.504719736729
68	122.691377167944	120.513364446082	124.869389889805
69	123.873046525867	121.559947566095	126.186145485638
70	124.039400422033	121.592429857695	126.486370986372

Figure 1

PNG link

Postscript link

PDF link

Figure 2

PNG link

Postscript link

PDF link

Figure 3

PNG link

Postscript link

PDF link

Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ; par4 = no ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; 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')

Free Statistics

Description of Statistical Computation

Tree of Dependent Computations

Dataset

Tables (Output of Computation)

Figures (Output of Computation)

Input Parameters & R Code