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Author's title

Author

*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Thu, 16 Dec 2010 18:46:42 +0000

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/2010/Dec/16/t1292525097bt7078a20hymx0p.htm/, Retrieved Fri, 03 May 2024 10:02:23 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=111176, Retrieved Fri, 03 May 2024 10:02:23 +0000

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IsPrivate?

No (this computation is public)

User-defined keywords

KDGP2W102

Estimated Impact

109

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

-       [Exponential Smoothing] [index huishoudcon...] [2010-12-16 18:46:42] [bc974f2989c3f1048b8acb0f98df66e5] [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	3 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 & 3 seconds \tabularnewline
R Server & 'RServer@AstonUniversity' @ vre.aston.ac.uk \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=111176&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]3 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=111176&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=111176&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	3 seconds
R Server	'RServer@AstonUniversity' @ vre.aston.ac.uk

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.264786991916332
beta	0
gamma	0.560459608377657

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.264786991916332 \tabularnewline
beta & 0 \tabularnewline
gamma & 0.560459608377657 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=111176&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]0.264786991916332[/C][/ROW]
[ROW][C]beta[/C][C]0[/C][/ROW]
[ROW][C]gamma[/C][C]0.560459608377657[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=111176&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=111176&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	0.264786991916332
beta	0
gamma	0.560459608377657

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	143.6	142.173980044541	1.42601995545917
14	134.9	134.365028304488	0.534971695512496
15	135.6	135.774958166406	-0.174958166405872
16	105.3	105.601841065423	-0.301841065422948
17	86.4	86.3180700268043	0.0819299731956988
18	74.6	74.0924554423624	0.507544557637559
19	67.6	72.0429399868427	-4.44293998684273
20	73.4	76.4372287961622	-3.03722879616215
21	78.5	81.7397668170721	-3.23976681707212
22	98.2	92.3344957596896	5.86550424031039
23	118.6	108.878760864691	9.72123913530852
24	136.9	127.189707921916	9.71029207808361
25	137.9	147.644612757203	-9.74461275720321
26	115.6	136.394400516438	-20.7944005164376
27	119.3	131.838319286585	-12.5383192865845
28	98.5	99.9301808323502	-1.43018083235017
29	84.3	81.5647557048365	2.73524429516351
30	73.5	70.7894846140615	2.71051538593845
31	60.7	67.4218853677964	-6.72188536779643
32	69.5	71.479012348698	-1.97901234869806
33	77.9	76.6893604148253	1.21063958517473
34	113.9	91.6924469850936	22.2075530149064
35	126.3	114.276444622482	12.0235553775177
36	135.1	133.383369224023	1.71663077597671
37	130.5	143.639214465184	-13.1392144651844
38	113.1	126.662594057663	-13.5625940576634
39	110	127.315401397604	-17.3154013976038
40	90.8	98.8385397337297	-8.03853973372972
41	85.4	80.8013539679064	4.59864603209361
42	72.5	70.6917247241964	1.80827527580355
43	64.7	63.2852170468452	1.41478295315484
44	67.2	71.5421602325398	-4.34216023253977
45	77.9	77.4642649645822	0.435735035417849
46	105.2	100.383397353405	4.8166026465951
47	107.2	113.447698272686	-6.24769827268567
48	120.7	122.46044767874	-1.7604476787396
49	121.3	125.215775187045	-3.91577518704493
50	107.9	111.433937939176	-3.53393793917628
51	105.6	112.788691259977	-7.18869125997661
52	81.3	91.5659419822415	-10.2659419822415
53	71.7	78.6431468994277	-6.94314689942767
54	64.8	65.3835692010754	-0.583569201075392
55	57.3	57.9301120758796	-0.630112075879637
56	61.9	62.6953672047879	-0.795367204787887
57	70.1	70.722838696787	-0.622838696787028
58	88.8	92.8706743370042	-4.07067433700423
59	106.8	98.1694126780139	8.63058732198614
60	110.7	111.969466595427	-1.26946659542683
61	114.1	113.774555576511	0.325444423489216
62	108	102.160153421408	5.83984657859156
63	111.5	104.407400404395	7.09259959560545
64	86.8	85.8838742119237	0.916125788076286
65	78.4	77.0630361235264	1.33696387647359
66	68	68.200525200477	-0.200525200477031
67	57.3	60.4710569561418	-3.17105695614182
68	65.3	64.672786003542	0.627213996457925
69	73.3	73.5039877680701	-0.203987768070078
70	88.6	95.2545830564975	-6.65458305649746
71	101.3	105.451498128919	-4.15149812891936
72	122.9	111.810219912972	11.0897800870277
73	126.6	117.634212576751	8.96578742324911
74	114.1	110.04132122644	4.0586787735599
75	124.7	112.333334253362	12.3666657466375
76	93.3	91.3123297740655	1.98767022593447
77	77.2	82.394821327376	-5.19482132737605
78	66.5	70.782085782487	-4.282085782487
79	57.9	60.5211339888723	-2.62113398887232
80	63.7	66.6020547613356	-2.90205476133561
81	65.8	74.249820023212	-8.44982002321204
82	85	90.7560925869753	-5.75609258697534
83	101	101.988318606604	-0.988318606603556
84	105.3	115.227173674076	-9.92717367407599
85	121	114.432826619369	6.56717338063099
86	117.9	104.901540256754	12.998459743246
87	106	112.665242402336	-6.66524240233562
88	86.6	84.6486404125142	1.95135958748584
89	79.9	73.7351517341059	6.16484826589411
90	65.2	65.9196959853728	-0.719695985372752
91	61.2	57.5433755080228	3.65662449197723
92	67.6	65.1303761272483	2.46962387275165
93	78.9	71.9123773089754	6.98762269102458
94	95.5	95.0842121279258	0.415787872074205
95	113.1	111.320461914416	1.77953808558435
96	124.4	122.510997187222	1.8890028127783
97	122	132.739630759354	-10.7396307593545
98	110.3	120.284440230791	-9.98444023079098
99	114	113.68138832278	0.318611677219891
100	93.3	89.8787498770735	3.42125012292649
101	75.5	80.4869532874741	-4.9869532874741
102	65.4	66.6818333484786	-1.2818333484786
103	59.2	59.8454344490295	-0.645434449029501
104	63.8	65.7592486826884	-1.95924868268835
105	74.2	72.973954566782	1.22604543321805
106	91.7	91.1024516068109	0.597548393189143
107	107	107.230251561379	-0.230251561378807
108	120.7	117.421726190031	3.27827380996926
109	127.4	122.524134959165	4.87586504083508
110	119.7	114.547258798018	5.15274120198201
111	112.7	116.202276472486	-3.50227647248602
112	84.4	92.3779805670601	-7.97798056706013
113	75.6	76.7584937295897	-1.15849372958974
114	66.5	65.5852433361242	0.91475666387582
115	59.9	59.5906836162079	0.309316383792051
116	64.8	65.2339501219855	-0.433950121985546
117	74.3	74.2586214117738	0.041378588226209
118	100.4	91.9244085036175	8.4755914963825
119	105.9	110.249785506968	-4.34978550696826
120	131.1	121.005813457758	10.0941865422421

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 143.6 & 142.173980044541 & 1.42601995545917 \tabularnewline
14 & 134.9 & 134.365028304488 & 0.534971695512496 \tabularnewline
15 & 135.6 & 135.774958166406 & -0.174958166405872 \tabularnewline
16 & 105.3 & 105.601841065423 & -0.301841065422948 \tabularnewline
17 & 86.4 & 86.3180700268043 & 0.0819299731956988 \tabularnewline
18 & 74.6 & 74.0924554423624 & 0.507544557637559 \tabularnewline
19 & 67.6 & 72.0429399868427 & -4.44293998684273 \tabularnewline
20 & 73.4 & 76.4372287961622 & -3.03722879616215 \tabularnewline
21 & 78.5 & 81.7397668170721 & -3.23976681707212 \tabularnewline
22 & 98.2 & 92.3344957596896 & 5.86550424031039 \tabularnewline
23 & 118.6 & 108.878760864691 & 9.72123913530852 \tabularnewline
24 & 136.9 & 127.189707921916 & 9.71029207808361 \tabularnewline
25 & 137.9 & 147.644612757203 & -9.74461275720321 \tabularnewline
26 & 115.6 & 136.394400516438 & -20.7944005164376 \tabularnewline
27 & 119.3 & 131.838319286585 & -12.5383192865845 \tabularnewline
28 & 98.5 & 99.9301808323502 & -1.43018083235017 \tabularnewline
29 & 84.3 & 81.5647557048365 & 2.73524429516351 \tabularnewline
30 & 73.5 & 70.7894846140615 & 2.71051538593845 \tabularnewline
31 & 60.7 & 67.4218853677964 & -6.72188536779643 \tabularnewline
32 & 69.5 & 71.479012348698 & -1.97901234869806 \tabularnewline
33 & 77.9 & 76.6893604148253 & 1.21063958517473 \tabularnewline
34 & 113.9 & 91.6924469850936 & 22.2075530149064 \tabularnewline
35 & 126.3 & 114.276444622482 & 12.0235553775177 \tabularnewline
36 & 135.1 & 133.383369224023 & 1.71663077597671 \tabularnewline
37 & 130.5 & 143.639214465184 & -13.1392144651844 \tabularnewline
38 & 113.1 & 126.662594057663 & -13.5625940576634 \tabularnewline
39 & 110 & 127.315401397604 & -17.3154013976038 \tabularnewline
40 & 90.8 & 98.8385397337297 & -8.03853973372972 \tabularnewline
41 & 85.4 & 80.8013539679064 & 4.59864603209361 \tabularnewline
42 & 72.5 & 70.6917247241964 & 1.80827527580355 \tabularnewline
43 & 64.7 & 63.2852170468452 & 1.41478295315484 \tabularnewline
44 & 67.2 & 71.5421602325398 & -4.34216023253977 \tabularnewline
45 & 77.9 & 77.4642649645822 & 0.435735035417849 \tabularnewline
46 & 105.2 & 100.383397353405 & 4.8166026465951 \tabularnewline
47 & 107.2 & 113.447698272686 & -6.24769827268567 \tabularnewline
48 & 120.7 & 122.46044767874 & -1.7604476787396 \tabularnewline
49 & 121.3 & 125.215775187045 & -3.91577518704493 \tabularnewline
50 & 107.9 & 111.433937939176 & -3.53393793917628 \tabularnewline
51 & 105.6 & 112.788691259977 & -7.18869125997661 \tabularnewline
52 & 81.3 & 91.5659419822415 & -10.2659419822415 \tabularnewline
53 & 71.7 & 78.6431468994277 & -6.94314689942767 \tabularnewline
54 & 64.8 & 65.3835692010754 & -0.583569201075392 \tabularnewline
55 & 57.3 & 57.9301120758796 & -0.630112075879637 \tabularnewline
56 & 61.9 & 62.6953672047879 & -0.795367204787887 \tabularnewline
57 & 70.1 & 70.722838696787 & -0.622838696787028 \tabularnewline
58 & 88.8 & 92.8706743370042 & -4.07067433700423 \tabularnewline
59 & 106.8 & 98.1694126780139 & 8.63058732198614 \tabularnewline
60 & 110.7 & 111.969466595427 & -1.26946659542683 \tabularnewline
61 & 114.1 & 113.774555576511 & 0.325444423489216 \tabularnewline
62 & 108 & 102.160153421408 & 5.83984657859156 \tabularnewline
63 & 111.5 & 104.407400404395 & 7.09259959560545 \tabularnewline
64 & 86.8 & 85.8838742119237 & 0.916125788076286 \tabularnewline
65 & 78.4 & 77.0630361235264 & 1.33696387647359 \tabularnewline
66 & 68 & 68.200525200477 & -0.200525200477031 \tabularnewline
67 & 57.3 & 60.4710569561418 & -3.17105695614182 \tabularnewline
68 & 65.3 & 64.672786003542 & 0.627213996457925 \tabularnewline
69 & 73.3 & 73.5039877680701 & -0.203987768070078 \tabularnewline
70 & 88.6 & 95.2545830564975 & -6.65458305649746 \tabularnewline
71 & 101.3 & 105.451498128919 & -4.15149812891936 \tabularnewline
72 & 122.9 & 111.810219912972 & 11.0897800870277 \tabularnewline
73 & 126.6 & 117.634212576751 & 8.96578742324911 \tabularnewline
74 & 114.1 & 110.04132122644 & 4.0586787735599 \tabularnewline
75 & 124.7 & 112.333334253362 & 12.3666657466375 \tabularnewline
76 & 93.3 & 91.3123297740655 & 1.98767022593447 \tabularnewline
77 & 77.2 & 82.394821327376 & -5.19482132737605 \tabularnewline
78 & 66.5 & 70.782085782487 & -4.282085782487 \tabularnewline
79 & 57.9 & 60.5211339888723 & -2.62113398887232 \tabularnewline
80 & 63.7 & 66.6020547613356 & -2.90205476133561 \tabularnewline
81 & 65.8 & 74.249820023212 & -8.44982002321204 \tabularnewline
82 & 85 & 90.7560925869753 & -5.75609258697534 \tabularnewline
83 & 101 & 101.988318606604 & -0.988318606603556 \tabularnewline
84 & 105.3 & 115.227173674076 & -9.92717367407599 \tabularnewline
85 & 121 & 114.432826619369 & 6.56717338063099 \tabularnewline
86 & 117.9 & 104.901540256754 & 12.998459743246 \tabularnewline
87 & 106 & 112.665242402336 & -6.66524240233562 \tabularnewline
88 & 86.6 & 84.6486404125142 & 1.95135958748584 \tabularnewline
89 & 79.9 & 73.7351517341059 & 6.16484826589411 \tabularnewline
90 & 65.2 & 65.9196959853728 & -0.719695985372752 \tabularnewline
91 & 61.2 & 57.5433755080228 & 3.65662449197723 \tabularnewline
92 & 67.6 & 65.1303761272483 & 2.46962387275165 \tabularnewline
93 & 78.9 & 71.9123773089754 & 6.98762269102458 \tabularnewline
94 & 95.5 & 95.0842121279258 & 0.415787872074205 \tabularnewline
95 & 113.1 & 111.320461914416 & 1.77953808558435 \tabularnewline
96 & 124.4 & 122.510997187222 & 1.8890028127783 \tabularnewline
97 & 122 & 132.739630759354 & -10.7396307593545 \tabularnewline
98 & 110.3 & 120.284440230791 & -9.98444023079098 \tabularnewline
99 & 114 & 113.68138832278 & 0.318611677219891 \tabularnewline
100 & 93.3 & 89.8787498770735 & 3.42125012292649 \tabularnewline
101 & 75.5 & 80.4869532874741 & -4.9869532874741 \tabularnewline
102 & 65.4 & 66.6818333484786 & -1.2818333484786 \tabularnewline
103 & 59.2 & 59.8454344490295 & -0.645434449029501 \tabularnewline
104 & 63.8 & 65.7592486826884 & -1.95924868268835 \tabularnewline
105 & 74.2 & 72.973954566782 & 1.22604543321805 \tabularnewline
106 & 91.7 & 91.1024516068109 & 0.597548393189143 \tabularnewline
107 & 107 & 107.230251561379 & -0.230251561378807 \tabularnewline
108 & 120.7 & 117.421726190031 & 3.27827380996926 \tabularnewline
109 & 127.4 & 122.524134959165 & 4.87586504083508 \tabularnewline
110 & 119.7 & 114.547258798018 & 5.15274120198201 \tabularnewline
111 & 112.7 & 116.202276472486 & -3.50227647248602 \tabularnewline
112 & 84.4 & 92.3779805670601 & -7.97798056706013 \tabularnewline
113 & 75.6 & 76.7584937295897 & -1.15849372958974 \tabularnewline
114 & 66.5 & 65.5852433361242 & 0.91475666387582 \tabularnewline
115 & 59.9 & 59.5906836162079 & 0.309316383792051 \tabularnewline
116 & 64.8 & 65.2339501219855 & -0.433950121985546 \tabularnewline
117 & 74.3 & 74.2586214117738 & 0.041378588226209 \tabularnewline
118 & 100.4 & 91.9244085036175 & 8.4755914963825 \tabularnewline
119 & 105.9 & 110.249785506968 & -4.34978550696826 \tabularnewline
120 & 131.1 & 121.005813457758 & 10.0941865422421 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=111176&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]143.6[/C][C]142.173980044541[/C][C]1.42601995545917[/C][/ROW]
[ROW][C]14[/C][C]134.9[/C][C]134.365028304488[/C][C]0.534971695512496[/C][/ROW]
[ROW][C]15[/C][C]135.6[/C][C]135.774958166406[/C][C]-0.174958166405872[/C][/ROW]
[ROW][C]16[/C][C]105.3[/C][C]105.601841065423[/C][C]-0.301841065422948[/C][/ROW]
[ROW][C]17[/C][C]86.4[/C][C]86.3180700268043[/C][C]0.0819299731956988[/C][/ROW]
[ROW][C]18[/C][C]74.6[/C][C]74.0924554423624[/C][C]0.507544557637559[/C][/ROW]
[ROW][C]19[/C][C]67.6[/C][C]72.0429399868427[/C][C]-4.44293998684273[/C][/ROW]
[ROW][C]20[/C][C]73.4[/C][C]76.4372287961622[/C][C]-3.03722879616215[/C][/ROW]
[ROW][C]21[/C][C]78.5[/C][C]81.7397668170721[/C][C]-3.23976681707212[/C][/ROW]
[ROW][C]22[/C][C]98.2[/C][C]92.3344957596896[/C][C]5.86550424031039[/C][/ROW]
[ROW][C]23[/C][C]118.6[/C][C]108.878760864691[/C][C]9.72123913530852[/C][/ROW]
[ROW][C]24[/C][C]136.9[/C][C]127.189707921916[/C][C]9.71029207808361[/C][/ROW]
[ROW][C]25[/C][C]137.9[/C][C]147.644612757203[/C][C]-9.74461275720321[/C][/ROW]
[ROW][C]26[/C][C]115.6[/C][C]136.394400516438[/C][C]-20.7944005164376[/C][/ROW]
[ROW][C]27[/C][C]119.3[/C][C]131.838319286585[/C][C]-12.5383192865845[/C][/ROW]
[ROW][C]28[/C][C]98.5[/C][C]99.9301808323502[/C][C]-1.43018083235017[/C][/ROW]
[ROW][C]29[/C][C]84.3[/C][C]81.5647557048365[/C][C]2.73524429516351[/C][/ROW]
[ROW][C]30[/C][C]73.5[/C][C]70.7894846140615[/C][C]2.71051538593845[/C][/ROW]
[ROW][C]31[/C][C]60.7[/C][C]67.4218853677964[/C][C]-6.72188536779643[/C][/ROW]
[ROW][C]32[/C][C]69.5[/C][C]71.479012348698[/C][C]-1.97901234869806[/C][/ROW]
[ROW][C]33[/C][C]77.9[/C][C]76.6893604148253[/C][C]1.21063958517473[/C][/ROW]
[ROW][C]34[/C][C]113.9[/C][C]91.6924469850936[/C][C]22.2075530149064[/C][/ROW]
[ROW][C]35[/C][C]126.3[/C][C]114.276444622482[/C][C]12.0235553775177[/C][/ROW]
[ROW][C]36[/C][C]135.1[/C][C]133.383369224023[/C][C]1.71663077597671[/C][/ROW]
[ROW][C]37[/C][C]130.5[/C][C]143.639214465184[/C][C]-13.1392144651844[/C][/ROW]
[ROW][C]38[/C][C]113.1[/C][C]126.662594057663[/C][C]-13.5625940576634[/C][/ROW]
[ROW][C]39[/C][C]110[/C][C]127.315401397604[/C][C]-17.3154013976038[/C][/ROW]
[ROW][C]40[/C][C]90.8[/C][C]98.8385397337297[/C][C]-8.03853973372972[/C][/ROW]
[ROW][C]41[/C][C]85.4[/C][C]80.8013539679064[/C][C]4.59864603209361[/C][/ROW]
[ROW][C]42[/C][C]72.5[/C][C]70.6917247241964[/C][C]1.80827527580355[/C][/ROW]
[ROW][C]43[/C][C]64.7[/C][C]63.2852170468452[/C][C]1.41478295315484[/C][/ROW]
[ROW][C]44[/C][C]67.2[/C][C]71.5421602325398[/C][C]-4.34216023253977[/C][/ROW]
[ROW][C]45[/C][C]77.9[/C][C]77.4642649645822[/C][C]0.435735035417849[/C][/ROW]
[ROW][C]46[/C][C]105.2[/C][C]100.383397353405[/C][C]4.8166026465951[/C][/ROW]
[ROW][C]47[/C][C]107.2[/C][C]113.447698272686[/C][C]-6.24769827268567[/C][/ROW]
[ROW][C]48[/C][C]120.7[/C][C]122.46044767874[/C][C]-1.7604476787396[/C][/ROW]
[ROW][C]49[/C][C]121.3[/C][C]125.215775187045[/C][C]-3.91577518704493[/C][/ROW]
[ROW][C]50[/C][C]107.9[/C][C]111.433937939176[/C][C]-3.53393793917628[/C][/ROW]
[ROW][C]51[/C][C]105.6[/C][C]112.788691259977[/C][C]-7.18869125997661[/C][/ROW]
[ROW][C]52[/C][C]81.3[/C][C]91.5659419822415[/C][C]-10.2659419822415[/C][/ROW]
[ROW][C]53[/C][C]71.7[/C][C]78.6431468994277[/C][C]-6.94314689942767[/C][/ROW]
[ROW][C]54[/C][C]64.8[/C][C]65.3835692010754[/C][C]-0.583569201075392[/C][/ROW]
[ROW][C]55[/C][C]57.3[/C][C]57.9301120758796[/C][C]-0.630112075879637[/C][/ROW]
[ROW][C]56[/C][C]61.9[/C][C]62.6953672047879[/C][C]-0.795367204787887[/C][/ROW]
[ROW][C]57[/C][C]70.1[/C][C]70.722838696787[/C][C]-0.622838696787028[/C][/ROW]
[ROW][C]58[/C][C]88.8[/C][C]92.8706743370042[/C][C]-4.07067433700423[/C][/ROW]
[ROW][C]59[/C][C]106.8[/C][C]98.1694126780139[/C][C]8.63058732198614[/C][/ROW]
[ROW][C]60[/C][C]110.7[/C][C]111.969466595427[/C][C]-1.26946659542683[/C][/ROW]
[ROW][C]61[/C][C]114.1[/C][C]113.774555576511[/C][C]0.325444423489216[/C][/ROW]
[ROW][C]62[/C][C]108[/C][C]102.160153421408[/C][C]5.83984657859156[/C][/ROW]
[ROW][C]63[/C][C]111.5[/C][C]104.407400404395[/C][C]7.09259959560545[/C][/ROW]
[ROW][C]64[/C][C]86.8[/C][C]85.8838742119237[/C][C]0.916125788076286[/C][/ROW]
[ROW][C]65[/C][C]78.4[/C][C]77.0630361235264[/C][C]1.33696387647359[/C][/ROW]
[ROW][C]66[/C][C]68[/C][C]68.200525200477[/C][C]-0.200525200477031[/C][/ROW]
[ROW][C]67[/C][C]57.3[/C][C]60.4710569561418[/C][C]-3.17105695614182[/C][/ROW]
[ROW][C]68[/C][C]65.3[/C][C]64.672786003542[/C][C]0.627213996457925[/C][/ROW]
[ROW][C]69[/C][C]73.3[/C][C]73.5039877680701[/C][C]-0.203987768070078[/C][/ROW]
[ROW][C]70[/C][C]88.6[/C][C]95.2545830564975[/C][C]-6.65458305649746[/C][/ROW]
[ROW][C]71[/C][C]101.3[/C][C]105.451498128919[/C][C]-4.15149812891936[/C][/ROW]
[ROW][C]72[/C][C]122.9[/C][C]111.810219912972[/C][C]11.0897800870277[/C][/ROW]
[ROW][C]73[/C][C]126.6[/C][C]117.634212576751[/C][C]8.96578742324911[/C][/ROW]
[ROW][C]74[/C][C]114.1[/C][C]110.04132122644[/C][C]4.0586787735599[/C][/ROW]
[ROW][C]75[/C][C]124.7[/C][C]112.333334253362[/C][C]12.3666657466375[/C][/ROW]
[ROW][C]76[/C][C]93.3[/C][C]91.3123297740655[/C][C]1.98767022593447[/C][/ROW]
[ROW][C]77[/C][C]77.2[/C][C]82.394821327376[/C][C]-5.19482132737605[/C][/ROW]
[ROW][C]78[/C][C]66.5[/C][C]70.782085782487[/C][C]-4.282085782487[/C][/ROW]
[ROW][C]79[/C][C]57.9[/C][C]60.5211339888723[/C][C]-2.62113398887232[/C][/ROW]
[ROW][C]80[/C][C]63.7[/C][C]66.6020547613356[/C][C]-2.90205476133561[/C][/ROW]
[ROW][C]81[/C][C]65.8[/C][C]74.249820023212[/C][C]-8.44982002321204[/C][/ROW]
[ROW][C]82[/C][C]85[/C][C]90.7560925869753[/C][C]-5.75609258697534[/C][/ROW]
[ROW][C]83[/C][C]101[/C][C]101.988318606604[/C][C]-0.988318606603556[/C][/ROW]
[ROW][C]84[/C][C]105.3[/C][C]115.227173674076[/C][C]-9.92717367407599[/C][/ROW]
[ROW][C]85[/C][C]121[/C][C]114.432826619369[/C][C]6.56717338063099[/C][/ROW]
[ROW][C]86[/C][C]117.9[/C][C]104.901540256754[/C][C]12.998459743246[/C][/ROW]
[ROW][C]87[/C][C]106[/C][C]112.665242402336[/C][C]-6.66524240233562[/C][/ROW]
[ROW][C]88[/C][C]86.6[/C][C]84.6486404125142[/C][C]1.95135958748584[/C][/ROW]
[ROW][C]89[/C][C]79.9[/C][C]73.7351517341059[/C][C]6.16484826589411[/C][/ROW]
[ROW][C]90[/C][C]65.2[/C][C]65.9196959853728[/C][C]-0.719695985372752[/C][/ROW]
[ROW][C]91[/C][C]61.2[/C][C]57.5433755080228[/C][C]3.65662449197723[/C][/ROW]
[ROW][C]92[/C][C]67.6[/C][C]65.1303761272483[/C][C]2.46962387275165[/C][/ROW]
[ROW][C]93[/C][C]78.9[/C][C]71.9123773089754[/C][C]6.98762269102458[/C][/ROW]
[ROW][C]94[/C][C]95.5[/C][C]95.0842121279258[/C][C]0.415787872074205[/C][/ROW]
[ROW][C]95[/C][C]113.1[/C][C]111.320461914416[/C][C]1.77953808558435[/C][/ROW]
[ROW][C]96[/C][C]124.4[/C][C]122.510997187222[/C][C]1.8890028127783[/C][/ROW]
[ROW][C]97[/C][C]122[/C][C]132.739630759354[/C][C]-10.7396307593545[/C][/ROW]
[ROW][C]98[/C][C]110.3[/C][C]120.284440230791[/C][C]-9.98444023079098[/C][/ROW]
[ROW][C]99[/C][C]114[/C][C]113.68138832278[/C][C]0.318611677219891[/C][/ROW]
[ROW][C]100[/C][C]93.3[/C][C]89.8787498770735[/C][C]3.42125012292649[/C][/ROW]
[ROW][C]101[/C][C]75.5[/C][C]80.4869532874741[/C][C]-4.9869532874741[/C][/ROW]
[ROW][C]102[/C][C]65.4[/C][C]66.6818333484786[/C][C]-1.2818333484786[/C][/ROW]
[ROW][C]103[/C][C]59.2[/C][C]59.8454344490295[/C][C]-0.645434449029501[/C][/ROW]
[ROW][C]104[/C][C]63.8[/C][C]65.7592486826884[/C][C]-1.95924868268835[/C][/ROW]
[ROW][C]105[/C][C]74.2[/C][C]72.973954566782[/C][C]1.22604543321805[/C][/ROW]
[ROW][C]106[/C][C]91.7[/C][C]91.1024516068109[/C][C]0.597548393189143[/C][/ROW]
[ROW][C]107[/C][C]107[/C][C]107.230251561379[/C][C]-0.230251561378807[/C][/ROW]
[ROW][C]108[/C][C]120.7[/C][C]117.421726190031[/C][C]3.27827380996926[/C][/ROW]
[ROW][C]109[/C][C]127.4[/C][C]122.524134959165[/C][C]4.87586504083508[/C][/ROW]
[ROW][C]110[/C][C]119.7[/C][C]114.547258798018[/C][C]5.15274120198201[/C][/ROW]
[ROW][C]111[/C][C]112.7[/C][C]116.202276472486[/C][C]-3.50227647248602[/C][/ROW]
[ROW][C]112[/C][C]84.4[/C][C]92.3779805670601[/C][C]-7.97798056706013[/C][/ROW]
[ROW][C]113[/C][C]75.6[/C][C]76.7584937295897[/C][C]-1.15849372958974[/C][/ROW]
[ROW][C]114[/C][C]66.5[/C][C]65.5852433361242[/C][C]0.91475666387582[/C][/ROW]
[ROW][C]115[/C][C]59.9[/C][C]59.5906836162079[/C][C]0.309316383792051[/C][/ROW]
[ROW][C]116[/C][C]64.8[/C][C]65.2339501219855[/C][C]-0.433950121985546[/C][/ROW]
[ROW][C]117[/C][C]74.3[/C][C]74.2586214117738[/C][C]0.041378588226209[/C][/ROW]
[ROW][C]118[/C][C]100.4[/C][C]91.9244085036175[/C][C]8.4755914963825[/C][/ROW]
[ROW][C]119[/C][C]105.9[/C][C]110.249785506968[/C][C]-4.34978550696826[/C][/ROW]
[ROW][C]120[/C][C]131.1[/C][C]121.005813457758[/C][C]10.0941865422421[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=111176&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=111176&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	143.6	142.173980044541	1.42601995545917
14	134.9	134.365028304488	0.534971695512496
15	135.6	135.774958166406	-0.174958166405872
16	105.3	105.601841065423	-0.301841065422948
17	86.4	86.3180700268043	0.0819299731956988
18	74.6	74.0924554423624	0.507544557637559
19	67.6	72.0429399868427	-4.44293998684273
20	73.4	76.4372287961622	-3.03722879616215
21	78.5	81.7397668170721	-3.23976681707212
22	98.2	92.3344957596896	5.86550424031039
23	118.6	108.878760864691	9.72123913530852
24	136.9	127.189707921916	9.71029207808361
25	137.9	147.644612757203	-9.74461275720321
26	115.6	136.394400516438	-20.7944005164376
27	119.3	131.838319286585	-12.5383192865845
28	98.5	99.9301808323502	-1.43018083235017
29	84.3	81.5647557048365	2.73524429516351
30	73.5	70.7894846140615	2.71051538593845
31	60.7	67.4218853677964	-6.72188536779643
32	69.5	71.479012348698	-1.97901234869806
33	77.9	76.6893604148253	1.21063958517473
34	113.9	91.6924469850936	22.2075530149064
35	126.3	114.276444622482	12.0235553775177
36	135.1	133.383369224023	1.71663077597671
37	130.5	143.639214465184	-13.1392144651844
38	113.1	126.662594057663	-13.5625940576634
39	110	127.315401397604	-17.3154013976038
40	90.8	98.8385397337297	-8.03853973372972
41	85.4	80.8013539679064	4.59864603209361
42	72.5	70.6917247241964	1.80827527580355
43	64.7	63.2852170468452	1.41478295315484
44	67.2	71.5421602325398	-4.34216023253977
45	77.9	77.4642649645822	0.435735035417849
46	105.2	100.383397353405	4.8166026465951
47	107.2	113.447698272686	-6.24769827268567
48	120.7	122.46044767874	-1.7604476787396
49	121.3	125.215775187045	-3.91577518704493
50	107.9	111.433937939176	-3.53393793917628
51	105.6	112.788691259977	-7.18869125997661
52	81.3	91.5659419822415	-10.2659419822415
53	71.7	78.6431468994277	-6.94314689942767
54	64.8	65.3835692010754	-0.583569201075392
55	57.3	57.9301120758796	-0.630112075879637
56	61.9	62.6953672047879	-0.795367204787887
57	70.1	70.722838696787	-0.622838696787028
58	88.8	92.8706743370042	-4.07067433700423
59	106.8	98.1694126780139	8.63058732198614
60	110.7	111.969466595427	-1.26946659542683
61	114.1	113.774555576511	0.325444423489216
62	108	102.160153421408	5.83984657859156
63	111.5	104.407400404395	7.09259959560545
64	86.8	85.8838742119237	0.916125788076286
65	78.4	77.0630361235264	1.33696387647359
66	68	68.200525200477	-0.200525200477031
67	57.3	60.4710569561418	-3.17105695614182
68	65.3	64.672786003542	0.627213996457925
69	73.3	73.5039877680701	-0.203987768070078
70	88.6	95.2545830564975	-6.65458305649746
71	101.3	105.451498128919	-4.15149812891936
72	122.9	111.810219912972	11.0897800870277
73	126.6	117.634212576751	8.96578742324911
74	114.1	110.04132122644	4.0586787735599
75	124.7	112.333334253362	12.3666657466375
76	93.3	91.3123297740655	1.98767022593447
77	77.2	82.394821327376	-5.19482132737605
78	66.5	70.782085782487	-4.282085782487
79	57.9	60.5211339888723	-2.62113398887232
80	63.7	66.6020547613356	-2.90205476133561
81	65.8	74.249820023212	-8.44982002321204
82	85	90.7560925869753	-5.75609258697534
83	101	101.988318606604	-0.988318606603556
84	105.3	115.227173674076	-9.92717367407599
85	121	114.432826619369	6.56717338063099
86	117.9	104.901540256754	12.998459743246
87	106	112.665242402336	-6.66524240233562
88	86.6	84.6486404125142	1.95135958748584
89	79.9	73.7351517341059	6.16484826589411
90	65.2	65.9196959853728	-0.719695985372752
91	61.2	57.5433755080228	3.65662449197723
92	67.6	65.1303761272483	2.46962387275165
93	78.9	71.9123773089754	6.98762269102458
94	95.5	95.0842121279258	0.415787872074205
95	113.1	111.320461914416	1.77953808558435
96	124.4	122.510997187222	1.8890028127783
97	122	132.739630759354	-10.7396307593545
98	110.3	120.284440230791	-9.98444023079098
99	114	113.68138832278	0.318611677219891
100	93.3	89.8787498770735	3.42125012292649
101	75.5	80.4869532874741	-4.9869532874741
102	65.4	66.6818333484786	-1.2818333484786
103	59.2	59.8454344490295	-0.645434449029501
104	63.8	65.7592486826884	-1.95924868268835
105	74.2	72.973954566782	1.22604543321805
106	91.7	91.1024516068109	0.597548393189143
107	107	107.230251561379	-0.230251561378807
108	120.7	117.421726190031	3.27827380996926
109	127.4	122.524134959165	4.87586504083508
110	119.7	114.547258798018	5.15274120198201
111	112.7	116.202276472486	-3.50227647248602
112	84.4	92.3779805670601	-7.97798056706013
113	75.6	76.7584937295897	-1.15849372958974
114	66.5	65.5852433361242	0.91475666387582
115	59.9	59.5906836162079	0.309316383792051
116	64.8	65.2339501219855	-0.433950121985546
117	74.3	74.2586214117738	0.041378588226209
118	100.4	91.9244085036175	8.4755914963825
119	105.9	110.249785506968	-4.34978550696826
120	131.1	121.005813457758	10.0941865422421

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
121	128.712263377168	120.049377636982	137.375149117355
122	119.321170256173	110.106796928996	128.535543583349
123	115.997286361633	106.24053656276	125.754036160505
124	90.7060434262522	80.9917962838517	100.420290568653
125	79.5466091430092	69.6112754443992	89.4819428416193
126	69.060332706455	58.9851601022579	79.135505310652
127	62.2925527072407	52.0024794229805	72.582625991501
128	67.7647371947554	56.5809821915007	78.9484921980102
129	77.5042770359619	65.0373954501326	89.9711586217911
130	99.4604167751058	84.4219638472468	114.498869702965
131	110.435261069344	93.8438942830944	127.026627855593
132	128.72102132047	-75.5544682539183	332.996510894858

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
121 & 128.712263377168 & 120.049377636982 & 137.375149117355 \tabularnewline
122 & 119.321170256173 & 110.106796928996 & 128.535543583349 \tabularnewline
123 & 115.997286361633 & 106.24053656276 & 125.754036160505 \tabularnewline
124 & 90.7060434262522 & 80.9917962838517 & 100.420290568653 \tabularnewline
125 & 79.5466091430092 & 69.6112754443992 & 89.4819428416193 \tabularnewline
126 & 69.060332706455 & 58.9851601022579 & 79.135505310652 \tabularnewline
127 & 62.2925527072407 & 52.0024794229805 & 72.582625991501 \tabularnewline
128 & 67.7647371947554 & 56.5809821915007 & 78.9484921980102 \tabularnewline
129 & 77.5042770359619 & 65.0373954501326 & 89.9711586217911 \tabularnewline
130 & 99.4604167751058 & 84.4219638472468 & 114.498869702965 \tabularnewline
131 & 110.435261069344 & 93.8438942830944 & 127.026627855593 \tabularnewline
132 & 128.72102132047 & -75.5544682539183 & 332.996510894858 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=111176&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]121[/C][C]128.712263377168[/C][C]120.049377636982[/C][C]137.375149117355[/C][/ROW]
[ROW][C]122[/C][C]119.321170256173[/C][C]110.106796928996[/C][C]128.535543583349[/C][/ROW]
[ROW][C]123[/C][C]115.997286361633[/C][C]106.24053656276[/C][C]125.754036160505[/C][/ROW]
[ROW][C]124[/C][C]90.7060434262522[/C][C]80.9917962838517[/C][C]100.420290568653[/C][/ROW]
[ROW][C]125[/C][C]79.5466091430092[/C][C]69.6112754443992[/C][C]89.4819428416193[/C][/ROW]
[ROW][C]126[/C][C]69.060332706455[/C][C]58.9851601022579[/C][C]79.135505310652[/C][/ROW]
[ROW][C]127[/C][C]62.2925527072407[/C][C]52.0024794229805[/C][C]72.582625991501[/C][/ROW]
[ROW][C]128[/C][C]67.7647371947554[/C][C]56.5809821915007[/C][C]78.9484921980102[/C][/ROW]
[ROW][C]129[/C][C]77.5042770359619[/C][C]65.0373954501326[/C][C]89.9711586217911[/C][/ROW]
[ROW][C]130[/C][C]99.4604167751058[/C][C]84.4219638472468[/C][C]114.498869702965[/C][/ROW]
[ROW][C]131[/C][C]110.435261069344[/C][C]93.8438942830944[/C][C]127.026627855593[/C][/ROW]
[ROW][C]132[/C][C]128.72102132047[/C][C]-75.5544682539183[/C][C]332.996510894858[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=111176&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=111176&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
121	128.712263377168	120.049377636982	137.375149117355
122	119.321170256173	110.106796928996	128.535543583349
123	115.997286361633	106.24053656276	125.754036160505
124	90.7060434262522	80.9917962838517	100.420290568653
125	79.5466091430092	69.6112754443992	89.4819428416193
126	69.060332706455	58.9851601022579	79.135505310652
127	62.2925527072407	52.0024794229805	72.582625991501
128	67.7647371947554	56.5809821915007	78.9484921980102
129	77.5042770359619	65.0373954501326	89.9711586217911
130	99.4604167751058	84.4219638472468	114.498869702965
131	110.435261069344	93.8438942830944	127.026627855593
132	128.72102132047	-75.5544682539183	332.996510894858

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 = Triple ; par3 = multiplicative ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; 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