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*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Fri, 13 Dec 2013 03:41:18 -0500

Cite this page as follows

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2013/Dec/13/t1386924158jxxbjwkbyqvirr0.htm/, Retrieved Wed, 11 Feb 2026 00:37:23 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=232284, Retrieved Wed, 11 Feb 2026 00:37:23 +0000

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IsPrivate?

No (this computation is public)

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Estimated Impact

245

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

-       [Exponential Smoothing] [] [2013-12-13 08:41:18] [2ad58ca14453c04e73fc838d0bf536d8] [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	5 seconds
R Server	'Gertrude Mary Cox' @ cox.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 & 5 seconds \tabularnewline
R Server & 'Gertrude Mary Cox' @ cox.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232284&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]5 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Gertrude Mary Cox' @ cox.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232284&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232284&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	5 seconds
R Server	'Gertrude Mary Cox' @ cox.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	1
beta	0.106401285899114
gamma	FALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 1 \tabularnewline
beta & 0.106401285899114 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232284&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]0.106401285899114[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232284&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232284&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	1
beta	0.106401285899114
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
3	123.07	124.79	-1.72
4	119.52	121.876989788254	-2.35698978825351
5	118.03	118.076203043932	-0.0462030439322518
6	117.27	116.581286980645	0.68871301935458
7	117.27	115.89456693152	1.37543306847979
8	116.69	116.040914778675	0.649085221325379
9	115.38	115.529978280882	-0.149978280881754
10	114.31	114.204020398939	0.105979601061009
11	113.33	113.145296764771	0.184703235229037
12	111.79	112.184949426509	-0.394949426509044
13	111.79	110.602926299663	1.18707370033663
14	110.92	110.729232467836	0.190767532163775
15	109.37	109.879530378566	-0.50953037856624
16	107.04	108.275315691082	-1.23531569108214
17	104.72	105.81387651306	-1.09387651305964
18	104.14	103.377486645455	0.762513354544751
19	104.14	102.878619046894	1.26138095310594
20	102.95	103.012831602313	-0.0628316023131816
21	102.13	101.816146239032	0.31385376096803
22	101.01	101.029540682783	-0.0195406827832159
23	100.07	99.9074615290078	0.162538470992232
24	99.4	98.9847558313294	0.415244168670611
25	99.4	98.3589383448381	1.04106165516194
26	99.34	98.4697086436476	0.870291356352439
27	97.72	98.5023087630703	-0.782308763070347
28	96.26	96.7990701047095	-0.539070104709495
29	95.77	95.2817123523786	0.488287647621348
30	95.04	94.8436667859742	0.196333214025799
31	95.04	94.1345568924113	0.905443107588738
32	94.55	94.2308972033672	0.319102796632791
33	94	93.7748501512629	0.225149848737075
34	93.14	93.2488063846885	-0.108806384688549
35	91.21	92.3772292454436	-1.16722924544365
36	90.3	90.3230345527894	-0.0230345527893832
37	90.3	89.4105836467525	0.889416353247512
38	89.74	89.5052186904377	0.23478130956228
39	89.07	88.9701997236802	0.099800276319769
40	89.06	88.3108186014137	0.749181398586273
41	88.97	88.380532465595	0.589467534404974
42	88.78	88.3532525692515	0.42674743074852
43	88.78	88.2086590446373	0.571340955362729
44	88.23	88.2694504569747	-0.0394504569746914
45	87.91	87.7152528776233	0.194747122376711
46	87.79	87.4159742218693	0.374025778130701
47	87.89	87.3357710456218	0.554228954378161
48	88	87.4947417190502	0.505258280949803
49	88	87.6585018498544	0.341498150145569
50	87.08	87.6948376921621	-0.614837692162084
51	85.75	86.7094181710968	-0.9594181710968
52	84.29	85.2773348439771	-0.987334843977109
53	84.39	83.712281146965	0.677718853035046
54	83.72	83.884391304406	-0.164391304405953
55	83.72	83.1968998582265	0.523100141773469
56	81.76	83.2525583859652	-1.49255838596522
57	81.53	81.133748254419	0.396251745580969
58	80.55	80.9459099496886	-0.395909949688615
59	79.83	79.9237846219415	-0.0937846219414951
60	78.98	79.1938058175694	-0.213805817569352
61	78.98	78.3210566036473	0.65894339635274
62	78.27	78.3911690283539	-0.121169028353933
63	77.41	77.6682764879259	-0.258276487925926
64	76.75	76.7807955374931	-0.0307955374930913
65	76.38	76.1175188527039	0.262481147296114
66	74.96	75.7754471843004	-0.815447184300453
67	74.96	74.2686825553081	0.691317444691919
68	74.46	74.3422396203878	0.117760379612207
69	74.04	73.8547694762065	0.185230523793521
70	73.22	73.4544782421259	-0.234478242125903
71	72.97	72.6095294556483	0.360470544351671
72	72.91	72.3978839850961	0.512116014903896
73	72.91	72.3923737876114	0.517626212388592
74	73.27	72.4474498822246	0.822550117775364
75	72.93	72.8949702724724	0.0350297275276006
76	72.67	72.558697480526	0.111302519473952
77	71.94	72.3105402117219	-0.370540211721888
78	71.9	71.5411142567173	0.35888574328267
79	71.89	71.5393001612935	0.350699838706518
80	71.72	71.5666150750965	0.153384924903534
81	70.85	71.4129354283437	-0.562935428343735
82	69.82	70.4830383748898	-0.663038374889794
83	69.61	69.3824902392011	0.227509760798938
84	69.48	69.1966975703047	0.28330242969534

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 123.07 & 124.79 & -1.72 \tabularnewline
4 & 119.52 & 121.876989788254 & -2.35698978825351 \tabularnewline
5 & 118.03 & 118.076203043932 & -0.0462030439322518 \tabularnewline
6 & 117.27 & 116.581286980645 & 0.68871301935458 \tabularnewline
7 & 117.27 & 115.89456693152 & 1.37543306847979 \tabularnewline
8 & 116.69 & 116.040914778675 & 0.649085221325379 \tabularnewline
9 & 115.38 & 115.529978280882 & -0.149978280881754 \tabularnewline
10 & 114.31 & 114.204020398939 & 0.105979601061009 \tabularnewline
11 & 113.33 & 113.145296764771 & 0.184703235229037 \tabularnewline
12 & 111.79 & 112.184949426509 & -0.394949426509044 \tabularnewline
13 & 111.79 & 110.602926299663 & 1.18707370033663 \tabularnewline
14 & 110.92 & 110.729232467836 & 0.190767532163775 \tabularnewline
15 & 109.37 & 109.879530378566 & -0.50953037856624 \tabularnewline
16 & 107.04 & 108.275315691082 & -1.23531569108214 \tabularnewline
17 & 104.72 & 105.81387651306 & -1.09387651305964 \tabularnewline
18 & 104.14 & 103.377486645455 & 0.762513354544751 \tabularnewline
19 & 104.14 & 102.878619046894 & 1.26138095310594 \tabularnewline
20 & 102.95 & 103.012831602313 & -0.0628316023131816 \tabularnewline
21 & 102.13 & 101.816146239032 & 0.31385376096803 \tabularnewline
22 & 101.01 & 101.029540682783 & -0.0195406827832159 \tabularnewline
23 & 100.07 & 99.9074615290078 & 0.162538470992232 \tabularnewline
24 & 99.4 & 98.9847558313294 & 0.415244168670611 \tabularnewline
25 & 99.4 & 98.3589383448381 & 1.04106165516194 \tabularnewline
26 & 99.34 & 98.4697086436476 & 0.870291356352439 \tabularnewline
27 & 97.72 & 98.5023087630703 & -0.782308763070347 \tabularnewline
28 & 96.26 & 96.7990701047095 & -0.539070104709495 \tabularnewline
29 & 95.77 & 95.2817123523786 & 0.488287647621348 \tabularnewline
30 & 95.04 & 94.8436667859742 & 0.196333214025799 \tabularnewline
31 & 95.04 & 94.1345568924113 & 0.905443107588738 \tabularnewline
32 & 94.55 & 94.2308972033672 & 0.319102796632791 \tabularnewline
33 & 94 & 93.7748501512629 & 0.225149848737075 \tabularnewline
34 & 93.14 & 93.2488063846885 & -0.108806384688549 \tabularnewline
35 & 91.21 & 92.3772292454436 & -1.16722924544365 \tabularnewline
36 & 90.3 & 90.3230345527894 & -0.0230345527893832 \tabularnewline
37 & 90.3 & 89.4105836467525 & 0.889416353247512 \tabularnewline
38 & 89.74 & 89.5052186904377 & 0.23478130956228 \tabularnewline
39 & 89.07 & 88.9701997236802 & 0.099800276319769 \tabularnewline
40 & 89.06 & 88.3108186014137 & 0.749181398586273 \tabularnewline
41 & 88.97 & 88.380532465595 & 0.589467534404974 \tabularnewline
42 & 88.78 & 88.3532525692515 & 0.42674743074852 \tabularnewline
43 & 88.78 & 88.2086590446373 & 0.571340955362729 \tabularnewline
44 & 88.23 & 88.2694504569747 & -0.0394504569746914 \tabularnewline
45 & 87.91 & 87.7152528776233 & 0.194747122376711 \tabularnewline
46 & 87.79 & 87.4159742218693 & 0.374025778130701 \tabularnewline
47 & 87.89 & 87.3357710456218 & 0.554228954378161 \tabularnewline
48 & 88 & 87.4947417190502 & 0.505258280949803 \tabularnewline
49 & 88 & 87.6585018498544 & 0.341498150145569 \tabularnewline
50 & 87.08 & 87.6948376921621 & -0.614837692162084 \tabularnewline
51 & 85.75 & 86.7094181710968 & -0.9594181710968 \tabularnewline
52 & 84.29 & 85.2773348439771 & -0.987334843977109 \tabularnewline
53 & 84.39 & 83.712281146965 & 0.677718853035046 \tabularnewline
54 & 83.72 & 83.884391304406 & -0.164391304405953 \tabularnewline
55 & 83.72 & 83.1968998582265 & 0.523100141773469 \tabularnewline
56 & 81.76 & 83.2525583859652 & -1.49255838596522 \tabularnewline
57 & 81.53 & 81.133748254419 & 0.396251745580969 \tabularnewline
58 & 80.55 & 80.9459099496886 & -0.395909949688615 \tabularnewline
59 & 79.83 & 79.9237846219415 & -0.0937846219414951 \tabularnewline
60 & 78.98 & 79.1938058175694 & -0.213805817569352 \tabularnewline
61 & 78.98 & 78.3210566036473 & 0.65894339635274 \tabularnewline
62 & 78.27 & 78.3911690283539 & -0.121169028353933 \tabularnewline
63 & 77.41 & 77.6682764879259 & -0.258276487925926 \tabularnewline
64 & 76.75 & 76.7807955374931 & -0.0307955374930913 \tabularnewline
65 & 76.38 & 76.1175188527039 & 0.262481147296114 \tabularnewline
66 & 74.96 & 75.7754471843004 & -0.815447184300453 \tabularnewline
67 & 74.96 & 74.2686825553081 & 0.691317444691919 \tabularnewline
68 & 74.46 & 74.3422396203878 & 0.117760379612207 \tabularnewline
69 & 74.04 & 73.8547694762065 & 0.185230523793521 \tabularnewline
70 & 73.22 & 73.4544782421259 & -0.234478242125903 \tabularnewline
71 & 72.97 & 72.6095294556483 & 0.360470544351671 \tabularnewline
72 & 72.91 & 72.3978839850961 & 0.512116014903896 \tabularnewline
73 & 72.91 & 72.3923737876114 & 0.517626212388592 \tabularnewline
74 & 73.27 & 72.4474498822246 & 0.822550117775364 \tabularnewline
75 & 72.93 & 72.8949702724724 & 0.0350297275276006 \tabularnewline
76 & 72.67 & 72.558697480526 & 0.111302519473952 \tabularnewline
77 & 71.94 & 72.3105402117219 & -0.370540211721888 \tabularnewline
78 & 71.9 & 71.5411142567173 & 0.35888574328267 \tabularnewline
79 & 71.89 & 71.5393001612935 & 0.350699838706518 \tabularnewline
80 & 71.72 & 71.5666150750965 & 0.153384924903534 \tabularnewline
81 & 70.85 & 71.4129354283437 & -0.562935428343735 \tabularnewline
82 & 69.82 & 70.4830383748898 & -0.663038374889794 \tabularnewline
83 & 69.61 & 69.3824902392011 & 0.227509760798938 \tabularnewline
84 & 69.48 & 69.1966975703047 & 0.28330242969534 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232284&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]123.07[/C][C]124.79[/C][C]-1.72[/C][/ROW]
[ROW][C]4[/C][C]119.52[/C][C]121.876989788254[/C][C]-2.35698978825351[/C][/ROW]
[ROW][C]5[/C][C]118.03[/C][C]118.076203043932[/C][C]-0.0462030439322518[/C][/ROW]
[ROW][C]6[/C][C]117.27[/C][C]116.581286980645[/C][C]0.68871301935458[/C][/ROW]
[ROW][C]7[/C][C]117.27[/C][C]115.89456693152[/C][C]1.37543306847979[/C][/ROW]
[ROW][C]8[/C][C]116.69[/C][C]116.040914778675[/C][C]0.649085221325379[/C][/ROW]
[ROW][C]9[/C][C]115.38[/C][C]115.529978280882[/C][C]-0.149978280881754[/C][/ROW]
[ROW][C]10[/C][C]114.31[/C][C]114.204020398939[/C][C]0.105979601061009[/C][/ROW]
[ROW][C]11[/C][C]113.33[/C][C]113.145296764771[/C][C]0.184703235229037[/C][/ROW]
[ROW][C]12[/C][C]111.79[/C][C]112.184949426509[/C][C]-0.394949426509044[/C][/ROW]
[ROW][C]13[/C][C]111.79[/C][C]110.602926299663[/C][C]1.18707370033663[/C][/ROW]
[ROW][C]14[/C][C]110.92[/C][C]110.729232467836[/C][C]0.190767532163775[/C][/ROW]
[ROW][C]15[/C][C]109.37[/C][C]109.879530378566[/C][C]-0.50953037856624[/C][/ROW]
[ROW][C]16[/C][C]107.04[/C][C]108.275315691082[/C][C]-1.23531569108214[/C][/ROW]
[ROW][C]17[/C][C]104.72[/C][C]105.81387651306[/C][C]-1.09387651305964[/C][/ROW]
[ROW][C]18[/C][C]104.14[/C][C]103.377486645455[/C][C]0.762513354544751[/C][/ROW]
[ROW][C]19[/C][C]104.14[/C][C]102.878619046894[/C][C]1.26138095310594[/C][/ROW]
[ROW][C]20[/C][C]102.95[/C][C]103.012831602313[/C][C]-0.0628316023131816[/C][/ROW]
[ROW][C]21[/C][C]102.13[/C][C]101.816146239032[/C][C]0.31385376096803[/C][/ROW]
[ROW][C]22[/C][C]101.01[/C][C]101.029540682783[/C][C]-0.0195406827832159[/C][/ROW]
[ROW][C]23[/C][C]100.07[/C][C]99.9074615290078[/C][C]0.162538470992232[/C][/ROW]
[ROW][C]24[/C][C]99.4[/C][C]98.9847558313294[/C][C]0.415244168670611[/C][/ROW]
[ROW][C]25[/C][C]99.4[/C][C]98.3589383448381[/C][C]1.04106165516194[/C][/ROW]
[ROW][C]26[/C][C]99.34[/C][C]98.4697086436476[/C][C]0.870291356352439[/C][/ROW]
[ROW][C]27[/C][C]97.72[/C][C]98.5023087630703[/C][C]-0.782308763070347[/C][/ROW]
[ROW][C]28[/C][C]96.26[/C][C]96.7990701047095[/C][C]-0.539070104709495[/C][/ROW]
[ROW][C]29[/C][C]95.77[/C][C]95.2817123523786[/C][C]0.488287647621348[/C][/ROW]
[ROW][C]30[/C][C]95.04[/C][C]94.8436667859742[/C][C]0.196333214025799[/C][/ROW]
[ROW][C]31[/C][C]95.04[/C][C]94.1345568924113[/C][C]0.905443107588738[/C][/ROW]
[ROW][C]32[/C][C]94.55[/C][C]94.2308972033672[/C][C]0.319102796632791[/C][/ROW]
[ROW][C]33[/C][C]94[/C][C]93.7748501512629[/C][C]0.225149848737075[/C][/ROW]
[ROW][C]34[/C][C]93.14[/C][C]93.2488063846885[/C][C]-0.108806384688549[/C][/ROW]
[ROW][C]35[/C][C]91.21[/C][C]92.3772292454436[/C][C]-1.16722924544365[/C][/ROW]
[ROW][C]36[/C][C]90.3[/C][C]90.3230345527894[/C][C]-0.0230345527893832[/C][/ROW]
[ROW][C]37[/C][C]90.3[/C][C]89.4105836467525[/C][C]0.889416353247512[/C][/ROW]
[ROW][C]38[/C][C]89.74[/C][C]89.5052186904377[/C][C]0.23478130956228[/C][/ROW]
[ROW][C]39[/C][C]89.07[/C][C]88.9701997236802[/C][C]0.099800276319769[/C][/ROW]
[ROW][C]40[/C][C]89.06[/C][C]88.3108186014137[/C][C]0.749181398586273[/C][/ROW]
[ROW][C]41[/C][C]88.97[/C][C]88.380532465595[/C][C]0.589467534404974[/C][/ROW]
[ROW][C]42[/C][C]88.78[/C][C]88.3532525692515[/C][C]0.42674743074852[/C][/ROW]
[ROW][C]43[/C][C]88.78[/C][C]88.2086590446373[/C][C]0.571340955362729[/C][/ROW]
[ROW][C]44[/C][C]88.23[/C][C]88.2694504569747[/C][C]-0.0394504569746914[/C][/ROW]
[ROW][C]45[/C][C]87.91[/C][C]87.7152528776233[/C][C]0.194747122376711[/C][/ROW]
[ROW][C]46[/C][C]87.79[/C][C]87.4159742218693[/C][C]0.374025778130701[/C][/ROW]
[ROW][C]47[/C][C]87.89[/C][C]87.3357710456218[/C][C]0.554228954378161[/C][/ROW]
[ROW][C]48[/C][C]88[/C][C]87.4947417190502[/C][C]0.505258280949803[/C][/ROW]
[ROW][C]49[/C][C]88[/C][C]87.6585018498544[/C][C]0.341498150145569[/C][/ROW]
[ROW][C]50[/C][C]87.08[/C][C]87.6948376921621[/C][C]-0.614837692162084[/C][/ROW]
[ROW][C]51[/C][C]85.75[/C][C]86.7094181710968[/C][C]-0.9594181710968[/C][/ROW]
[ROW][C]52[/C][C]84.29[/C][C]85.2773348439771[/C][C]-0.987334843977109[/C][/ROW]
[ROW][C]53[/C][C]84.39[/C][C]83.712281146965[/C][C]0.677718853035046[/C][/ROW]
[ROW][C]54[/C][C]83.72[/C][C]83.884391304406[/C][C]-0.164391304405953[/C][/ROW]
[ROW][C]55[/C][C]83.72[/C][C]83.1968998582265[/C][C]0.523100141773469[/C][/ROW]
[ROW][C]56[/C][C]81.76[/C][C]83.2525583859652[/C][C]-1.49255838596522[/C][/ROW]
[ROW][C]57[/C][C]81.53[/C][C]81.133748254419[/C][C]0.396251745580969[/C][/ROW]
[ROW][C]58[/C][C]80.55[/C][C]80.9459099496886[/C][C]-0.395909949688615[/C][/ROW]
[ROW][C]59[/C][C]79.83[/C][C]79.9237846219415[/C][C]-0.0937846219414951[/C][/ROW]
[ROW][C]60[/C][C]78.98[/C][C]79.1938058175694[/C][C]-0.213805817569352[/C][/ROW]
[ROW][C]61[/C][C]78.98[/C][C]78.3210566036473[/C][C]0.65894339635274[/C][/ROW]
[ROW][C]62[/C][C]78.27[/C][C]78.3911690283539[/C][C]-0.121169028353933[/C][/ROW]
[ROW][C]63[/C][C]77.41[/C][C]77.6682764879259[/C][C]-0.258276487925926[/C][/ROW]
[ROW][C]64[/C][C]76.75[/C][C]76.7807955374931[/C][C]-0.0307955374930913[/C][/ROW]
[ROW][C]65[/C][C]76.38[/C][C]76.1175188527039[/C][C]0.262481147296114[/C][/ROW]
[ROW][C]66[/C][C]74.96[/C][C]75.7754471843004[/C][C]-0.815447184300453[/C][/ROW]
[ROW][C]67[/C][C]74.96[/C][C]74.2686825553081[/C][C]0.691317444691919[/C][/ROW]
[ROW][C]68[/C][C]74.46[/C][C]74.3422396203878[/C][C]0.117760379612207[/C][/ROW]
[ROW][C]69[/C][C]74.04[/C][C]73.8547694762065[/C][C]0.185230523793521[/C][/ROW]
[ROW][C]70[/C][C]73.22[/C][C]73.4544782421259[/C][C]-0.234478242125903[/C][/ROW]
[ROW][C]71[/C][C]72.97[/C][C]72.6095294556483[/C][C]0.360470544351671[/C][/ROW]
[ROW][C]72[/C][C]72.91[/C][C]72.3978839850961[/C][C]0.512116014903896[/C][/ROW]
[ROW][C]73[/C][C]72.91[/C][C]72.3923737876114[/C][C]0.517626212388592[/C][/ROW]
[ROW][C]74[/C][C]73.27[/C][C]72.4474498822246[/C][C]0.822550117775364[/C][/ROW]
[ROW][C]75[/C][C]72.93[/C][C]72.8949702724724[/C][C]0.0350297275276006[/C][/ROW]
[ROW][C]76[/C][C]72.67[/C][C]72.558697480526[/C][C]0.111302519473952[/C][/ROW]
[ROW][C]77[/C][C]71.94[/C][C]72.3105402117219[/C][C]-0.370540211721888[/C][/ROW]
[ROW][C]78[/C][C]71.9[/C][C]71.5411142567173[/C][C]0.35888574328267[/C][/ROW]
[ROW][C]79[/C][C]71.89[/C][C]71.5393001612935[/C][C]0.350699838706518[/C][/ROW]
[ROW][C]80[/C][C]71.72[/C][C]71.5666150750965[/C][C]0.153384924903534[/C][/ROW]
[ROW][C]81[/C][C]70.85[/C][C]71.4129354283437[/C][C]-0.562935428343735[/C][/ROW]
[ROW][C]82[/C][C]69.82[/C][C]70.4830383748898[/C][C]-0.663038374889794[/C][/ROW]
[ROW][C]83[/C][C]69.61[/C][C]69.3824902392011[/C][C]0.227509760798938[/C][/ROW]
[ROW][C]84[/C][C]69.48[/C][C]69.1966975703047[/C][C]0.28330242969534[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232284&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232284&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
3	123.07	124.79	-1.72
4	119.52	121.876989788254	-2.35698978825351
5	118.03	118.076203043932	-0.0462030439322518
6	117.27	116.581286980645	0.68871301935458
7	117.27	115.89456693152	1.37543306847979
8	116.69	116.040914778675	0.649085221325379
9	115.38	115.529978280882	-0.149978280881754
10	114.31	114.204020398939	0.105979601061009
11	113.33	113.145296764771	0.184703235229037
12	111.79	112.184949426509	-0.394949426509044
13	111.79	110.602926299663	1.18707370033663
14	110.92	110.729232467836	0.190767532163775
15	109.37	109.879530378566	-0.50953037856624
16	107.04	108.275315691082	-1.23531569108214
17	104.72	105.81387651306	-1.09387651305964
18	104.14	103.377486645455	0.762513354544751
19	104.14	102.878619046894	1.26138095310594
20	102.95	103.012831602313	-0.0628316023131816
21	102.13	101.816146239032	0.31385376096803
22	101.01	101.029540682783	-0.0195406827832159
23	100.07	99.9074615290078	0.162538470992232
24	99.4	98.9847558313294	0.415244168670611
25	99.4	98.3589383448381	1.04106165516194
26	99.34	98.4697086436476	0.870291356352439
27	97.72	98.5023087630703	-0.782308763070347
28	96.26	96.7990701047095	-0.539070104709495
29	95.77	95.2817123523786	0.488287647621348
30	95.04	94.8436667859742	0.196333214025799
31	95.04	94.1345568924113	0.905443107588738
32	94.55	94.2308972033672	0.319102796632791
33	94	93.7748501512629	0.225149848737075
34	93.14	93.2488063846885	-0.108806384688549
35	91.21	92.3772292454436	-1.16722924544365
36	90.3	90.3230345527894	-0.0230345527893832
37	90.3	89.4105836467525	0.889416353247512
38	89.74	89.5052186904377	0.23478130956228
39	89.07	88.9701997236802	0.099800276319769
40	89.06	88.3108186014137	0.749181398586273
41	88.97	88.380532465595	0.589467534404974
42	88.78	88.3532525692515	0.42674743074852
43	88.78	88.2086590446373	0.571340955362729
44	88.23	88.2694504569747	-0.0394504569746914
45	87.91	87.7152528776233	0.194747122376711
46	87.79	87.4159742218693	0.374025778130701
47	87.89	87.3357710456218	0.554228954378161
48	88	87.4947417190502	0.505258280949803
49	88	87.6585018498544	0.341498150145569
50	87.08	87.6948376921621	-0.614837692162084
51	85.75	86.7094181710968	-0.9594181710968
52	84.29	85.2773348439771	-0.987334843977109
53	84.39	83.712281146965	0.677718853035046
54	83.72	83.884391304406	-0.164391304405953
55	83.72	83.1968998582265	0.523100141773469
56	81.76	83.2525583859652	-1.49255838596522
57	81.53	81.133748254419	0.396251745580969
58	80.55	80.9459099496886	-0.395909949688615
59	79.83	79.9237846219415	-0.0937846219414951
60	78.98	79.1938058175694	-0.213805817569352
61	78.98	78.3210566036473	0.65894339635274
62	78.27	78.3911690283539	-0.121169028353933
63	77.41	77.6682764879259	-0.258276487925926
64	76.75	76.7807955374931	-0.0307955374930913
65	76.38	76.1175188527039	0.262481147296114
66	74.96	75.7754471843004	-0.815447184300453
67	74.96	74.2686825553081	0.691317444691919
68	74.46	74.3422396203878	0.117760379612207
69	74.04	73.8547694762065	0.185230523793521
70	73.22	73.4544782421259	-0.234478242125903
71	72.97	72.6095294556483	0.360470544351671
72	72.91	72.3978839850961	0.512116014903896
73	72.91	72.3923737876114	0.517626212388592
74	73.27	72.4474498822246	0.822550117775364
75	72.93	72.8949702724724	0.0350297275276006
76	72.67	72.558697480526	0.111302519473952
77	71.94	72.3105402117219	-0.370540211721888
78	71.9	71.5411142567173	0.35888574328267
79	71.89	71.5393001612935	0.350699838706518
80	71.72	71.5666150750965	0.153384924903534
81	70.85	71.4129354283437	-0.562935428343735
82	69.82	70.4830383748898	-0.663038374889794
83	69.61	69.3824902392011	0.227509760798938
84	69.48	69.1966975703047	0.28330242969534

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
85	69.0968413131226	67.7790318175173	70.4146508087279
86	68.7136826262452	66.7483678843297	70.6789973681607
87	68.3305239393678	65.7973753239	70.8636725548356
88	67.9473652524904	64.8750562038635	71.0196743011172
89	67.564206565613	63.9630253170125	71.1653878142135
90	67.1810478787356	63.052557380159	71.3095383773122
91	66.7978891918582	62.1388993304431	71.4568790532732
92	66.4147305049808	61.219243426288	71.6102175836735
93	66.0315718181034	60.2918459040096	71.7712977321972
94	65.648413131226	59.3555909754093	71.9412352870426
95	65.2652544443486	58.4097545547627	72.1207543339344
96	64.8820957574712	57.4538672252541	72.3103242896882

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
85 & 69.0968413131226 & 67.7790318175173 & 70.4146508087279 \tabularnewline
86 & 68.7136826262452 & 66.7483678843297 & 70.6789973681607 \tabularnewline
87 & 68.3305239393678 & 65.7973753239 & 70.8636725548356 \tabularnewline
88 & 67.9473652524904 & 64.8750562038635 & 71.0196743011172 \tabularnewline
89 & 67.564206565613 & 63.9630253170125 & 71.1653878142135 \tabularnewline
90 & 67.1810478787356 & 63.052557380159 & 71.3095383773122 \tabularnewline
91 & 66.7978891918582 & 62.1388993304431 & 71.4568790532732 \tabularnewline
92 & 66.4147305049808 & 61.219243426288 & 71.6102175836735 \tabularnewline
93 & 66.0315718181034 & 60.2918459040096 & 71.7712977321972 \tabularnewline
94 & 65.648413131226 & 59.3555909754093 & 71.9412352870426 \tabularnewline
95 & 65.2652544443486 & 58.4097545547627 & 72.1207543339344 \tabularnewline
96 & 64.8820957574712 & 57.4538672252541 & 72.3103242896882 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232284&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]85[/C][C]69.0968413131226[/C][C]67.7790318175173[/C][C]70.4146508087279[/C][/ROW]
[ROW][C]86[/C][C]68.7136826262452[/C][C]66.7483678843297[/C][C]70.6789973681607[/C][/ROW]
[ROW][C]87[/C][C]68.3305239393678[/C][C]65.7973753239[/C][C]70.8636725548356[/C][/ROW]
[ROW][C]88[/C][C]67.9473652524904[/C][C]64.8750562038635[/C][C]71.0196743011172[/C][/ROW]
[ROW][C]89[/C][C]67.564206565613[/C][C]63.9630253170125[/C][C]71.1653878142135[/C][/ROW]
[ROW][C]90[/C][C]67.1810478787356[/C][C]63.052557380159[/C][C]71.3095383773122[/C][/ROW]
[ROW][C]91[/C][C]66.7978891918582[/C][C]62.1388993304431[/C][C]71.4568790532732[/C][/ROW]
[ROW][C]92[/C][C]66.4147305049808[/C][C]61.219243426288[/C][C]71.6102175836735[/C][/ROW]
[ROW][C]93[/C][C]66.0315718181034[/C][C]60.2918459040096[/C][C]71.7712977321972[/C][/ROW]
[ROW][C]94[/C][C]65.648413131226[/C][C]59.3555909754093[/C][C]71.9412352870426[/C][/ROW]
[ROW][C]95[/C][C]65.2652544443486[/C][C]58.4097545547627[/C][C]72.1207543339344[/C][/ROW]
[ROW][C]96[/C][C]64.8820957574712[/C][C]57.4538672252541[/C][C]72.3103242896882[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232284&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232284&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
85	69.0968413131226	67.7790318175173	70.4146508087279
86	68.7136826262452	66.7483678843297	70.6789973681607
87	68.3305239393678	65.7973753239	70.8636725548356
88	67.9473652524904	64.8750562038635	71.0196743011172
89	67.564206565613	63.9630253170125	71.1653878142135
90	67.1810478787356	63.052557380159	71.3095383773122
91	66.7978891918582	62.1388993304431	71.4568790532732
92	66.4147305049808	61.219243426288	71.6102175836735
93	66.0315718181034	60.2918459040096	71.7712977321972
94	65.648413131226	59.3555909754093	71.9412352870426
95	65.2652544443486	58.4097545547627	72.1207543339344
96	64.8820957574712	57.4538672252541	72.3103242896882

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 = 12 ; par2 = Double ; par3 = multiplicative ;

Parameters (R input):

par1 = 12 ; par2 = Double ; par3 = multiplicative ;

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