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Author

*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Sun, 12 Jan 2014 14:54:43 -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/2014/Jan/12/t1389556558gi5sfvmose3uf0q.htm/, Retrieved Wed, 29 Apr 2026 03:22:28 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=233052, Retrieved Wed, 29 Apr 2026 03:22:28 +0000

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IsPrivate?

No (this computation is public)

User-defined keywords

Estimated Impact

381

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

-     [Notched Boxplots] [] [2013-09-30 10:24:51] [94c8d1afc76fa1f7d1059d0b56e0a268]
- RMPD    [Exponential Smoothing] [] [2014-01-12 19:54:43] [626ac9e5ebaea65a3a0a76f5178dd51c] [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	'Herman Ole Andreas Wold' @ wold.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 & 3 seconds \tabularnewline
R Server & 'Herman Ole Andreas Wold' @ wold.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=233052&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]'Herman Ole Andreas Wold' @ wold.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=233052&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=233052&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	'Herman Ole Andreas Wold' @ wold.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.891573760043234
beta	0.433728993113943
gamma	FALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.891573760043234 \tabularnewline
beta & 0.433728993113943 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=233052&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]0.891573760043234[/C][/ROW]
[ROW][C]beta[/C][C]0.433728993113943[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=233052&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=233052&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	0.891573760043234
beta	0.433728993113943
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
3	75.79	75.33	0.459999999999994
4	74.88	73.7980065686659	1.08199343133413
5	74.5	73.2389745226635	1.26102547733647
6	74.59	73.3272030550246	1.26279694497538
7	74.59	73.9053363143872	0.684663685612833
8	73.57	74.2327815282759	-0.662781528275872
9	73.3	73.102581408625	0.197418591374955
10	73.23	72.8156551875855	0.414344812414456
11	73	72.8823624076713	0.11763759232872
12	72.31	72.7300238762924	-0.420023876292419
13	72.31	71.9358966712325	0.374103328767546
14	71.24	71.9944587212905	-0.754458721290547
15	70.82	70.7550742252962	0.0649257747037808
16	70.66	70.2713383325915	0.388661667408513
17	69.94	70.2265328737339	-0.286532873733933
18	69.87	69.4689390187363	0.40106098126374
19	69.87	69.4794766410423	0.390523358957736
20	68.88	69.6316351212429	-0.75163512124287
21	68.09	68.4748167251068	-0.384816725106774
22	68.38	67.4962348234689	0.883765176531071
23	66.78	67.9904404793151	-1.2104404793151
24	67.2	66.1494283093429	1.05057169065712
25	67.2	66.7305327935195	0.469467206480488
26	66.67	66.9750833887049	-0.305083388704901
27	65.86	66.4110888271538	-0.551088827153833
28	66.05	65.414655456804	0.635344543195956
29	66.31	65.7217035649916	0.588296435008417
30	66.39	66.2142998631565	0.175700136843545
31	66.39	66.4069796153919	-0.0169796153919179
32	65.72	66.421305115582	-0.701305115581974
33	65.52	65.5543082940042	-0.0343082940042052
34	64.93	65.2687212716238	-0.338721271623839
35	65.27	64.5807436398824	0.689256360117611
36	65.04	65.0758202825063	-0.0358202825063216
37	65.02	64.9105858635387	0.109414136461282
38	64.72	64.9171492401488	-0.197149240148761
39	64.68	64.5741508693318	0.105849130668204
40	64.41	64.5422299011493	-0.13222990114933
41	64.79	64.2469104289096	0.543089571090405
42	64.71	64.7637015693822	-0.0537015693822553
43	64.71	64.7276429172992	-0.0176429172992414
44	64.83	64.7169106726063	0.113089327393737
45	64.77	64.8664676668834	-0.0964676668833988
46	64.19	64.7918849630299	-0.601884963029875
47	64.27	64.0339357086996	0.236064291300352
48	64.23	64.1143664111716	0.11563358882843
49	64.23	64.1321399288576	0.0978600711424065
50	63.03	64.1719096700121	-1.14190967001205
51	62.85	62.6647551856884	0.185244814311645
52	62.15	62.4124912421534	-0.262491242153416
53	61.69	61.6595318512475	0.0304681487525471
54	62.1	61.1795494414814	0.920450558518617
55	62.1	61.8489915048598	0.251008495140198
56	61.81	62.0186419244402	-0.208641924440158
57	61.28	61.6977979690971	-0.417797969097101
58	61.05	61.028912917516	0.0210870824839873
59	61.08	60.7594806656912	0.320519334308791
60	60.98	60.8809596243565	0.0990403756435327
61	60.98	60.8432728059253	0.136727194074673
62	61.11	60.8920591618011	0.217940838198849
63	60.58	61.0975314966371	-0.517531496637112
64	60.37	60.4471458477756	-0.0771458477755544
65	59.44	60.1595640812304	-0.719564081230409
66	59.29	59.0209626449115	0.269037355088486
67	59.29	58.8678094273232	0.422190572676818
68	59.33	59.0144652807769	0.315534719223109
69	59.06	59.1880472882102	-0.12804728821024
70	58.75	58.9166271531506	-0.166627153150642
71	58.92	58.5463752712373	0.373624728762692
72	58.73	58.8022789927152	-0.0722789927152405
73	58.73	58.6326762697227	0.0973237302773455
74	58.46	58.6519221058855	-0.191922105885475
75	58.18	58.3390673993538	-0.15906739935383
76	58.02	57.9939935027483	0.0260064972516716
77	56.97	57.8239833846415	-0.853983384641495
78	57.22	56.5391608175146	0.680839182485357
79	57.19	56.8860272352698	0.303972764730155
80	57.06	57.0144361343224	0.0455638656776429
81	57.08	56.9300740497742	0.14992595022575
82	56.59	56.9967350345866	-0.406735034586617
83	56.91	56.4098066891772	0.500193310822844
84	56.54	56.8248973069531	-0.28489730695312

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 75.79 & 75.33 & 0.459999999999994 \tabularnewline
4 & 74.88 & 73.7980065686659 & 1.08199343133413 \tabularnewline
5 & 74.5 & 73.2389745226635 & 1.26102547733647 \tabularnewline
6 & 74.59 & 73.3272030550246 & 1.26279694497538 \tabularnewline
7 & 74.59 & 73.9053363143872 & 0.684663685612833 \tabularnewline
8 & 73.57 & 74.2327815282759 & -0.662781528275872 \tabularnewline
9 & 73.3 & 73.102581408625 & 0.197418591374955 \tabularnewline
10 & 73.23 & 72.8156551875855 & 0.414344812414456 \tabularnewline
11 & 73 & 72.8823624076713 & 0.11763759232872 \tabularnewline
12 & 72.31 & 72.7300238762924 & -0.420023876292419 \tabularnewline
13 & 72.31 & 71.9358966712325 & 0.374103328767546 \tabularnewline
14 & 71.24 & 71.9944587212905 & -0.754458721290547 \tabularnewline
15 & 70.82 & 70.7550742252962 & 0.0649257747037808 \tabularnewline
16 & 70.66 & 70.2713383325915 & 0.388661667408513 \tabularnewline
17 & 69.94 & 70.2265328737339 & -0.286532873733933 \tabularnewline
18 & 69.87 & 69.4689390187363 & 0.40106098126374 \tabularnewline
19 & 69.87 & 69.4794766410423 & 0.390523358957736 \tabularnewline
20 & 68.88 & 69.6316351212429 & -0.75163512124287 \tabularnewline
21 & 68.09 & 68.4748167251068 & -0.384816725106774 \tabularnewline
22 & 68.38 & 67.4962348234689 & 0.883765176531071 \tabularnewline
23 & 66.78 & 67.9904404793151 & -1.2104404793151 \tabularnewline
24 & 67.2 & 66.1494283093429 & 1.05057169065712 \tabularnewline
25 & 67.2 & 66.7305327935195 & 0.469467206480488 \tabularnewline
26 & 66.67 & 66.9750833887049 & -0.305083388704901 \tabularnewline
27 & 65.86 & 66.4110888271538 & -0.551088827153833 \tabularnewline
28 & 66.05 & 65.414655456804 & 0.635344543195956 \tabularnewline
29 & 66.31 & 65.7217035649916 & 0.588296435008417 \tabularnewline
30 & 66.39 & 66.2142998631565 & 0.175700136843545 \tabularnewline
31 & 66.39 & 66.4069796153919 & -0.0169796153919179 \tabularnewline
32 & 65.72 & 66.421305115582 & -0.701305115581974 \tabularnewline
33 & 65.52 & 65.5543082940042 & -0.0343082940042052 \tabularnewline
34 & 64.93 & 65.2687212716238 & -0.338721271623839 \tabularnewline
35 & 65.27 & 64.5807436398824 & 0.689256360117611 \tabularnewline
36 & 65.04 & 65.0758202825063 & -0.0358202825063216 \tabularnewline
37 & 65.02 & 64.9105858635387 & 0.109414136461282 \tabularnewline
38 & 64.72 & 64.9171492401488 & -0.197149240148761 \tabularnewline
39 & 64.68 & 64.5741508693318 & 0.105849130668204 \tabularnewline
40 & 64.41 & 64.5422299011493 & -0.13222990114933 \tabularnewline
41 & 64.79 & 64.2469104289096 & 0.543089571090405 \tabularnewline
42 & 64.71 & 64.7637015693822 & -0.0537015693822553 \tabularnewline
43 & 64.71 & 64.7276429172992 & -0.0176429172992414 \tabularnewline
44 & 64.83 & 64.7169106726063 & 0.113089327393737 \tabularnewline
45 & 64.77 & 64.8664676668834 & -0.0964676668833988 \tabularnewline
46 & 64.19 & 64.7918849630299 & -0.601884963029875 \tabularnewline
47 & 64.27 & 64.0339357086996 & 0.236064291300352 \tabularnewline
48 & 64.23 & 64.1143664111716 & 0.11563358882843 \tabularnewline
49 & 64.23 & 64.1321399288576 & 0.0978600711424065 \tabularnewline
50 & 63.03 & 64.1719096700121 & -1.14190967001205 \tabularnewline
51 & 62.85 & 62.6647551856884 & 0.185244814311645 \tabularnewline
52 & 62.15 & 62.4124912421534 & -0.262491242153416 \tabularnewline
53 & 61.69 & 61.6595318512475 & 0.0304681487525471 \tabularnewline
54 & 62.1 & 61.1795494414814 & 0.920450558518617 \tabularnewline
55 & 62.1 & 61.8489915048598 & 0.251008495140198 \tabularnewline
56 & 61.81 & 62.0186419244402 & -0.208641924440158 \tabularnewline
57 & 61.28 & 61.6977979690971 & -0.417797969097101 \tabularnewline
58 & 61.05 & 61.028912917516 & 0.0210870824839873 \tabularnewline
59 & 61.08 & 60.7594806656912 & 0.320519334308791 \tabularnewline
60 & 60.98 & 60.8809596243565 & 0.0990403756435327 \tabularnewline
61 & 60.98 & 60.8432728059253 & 0.136727194074673 \tabularnewline
62 & 61.11 & 60.8920591618011 & 0.217940838198849 \tabularnewline
63 & 60.58 & 61.0975314966371 & -0.517531496637112 \tabularnewline
64 & 60.37 & 60.4471458477756 & -0.0771458477755544 \tabularnewline
65 & 59.44 & 60.1595640812304 & -0.719564081230409 \tabularnewline
66 & 59.29 & 59.0209626449115 & 0.269037355088486 \tabularnewline
67 & 59.29 & 58.8678094273232 & 0.422190572676818 \tabularnewline
68 & 59.33 & 59.0144652807769 & 0.315534719223109 \tabularnewline
69 & 59.06 & 59.1880472882102 & -0.12804728821024 \tabularnewline
70 & 58.75 & 58.9166271531506 & -0.166627153150642 \tabularnewline
71 & 58.92 & 58.5463752712373 & 0.373624728762692 \tabularnewline
72 & 58.73 & 58.8022789927152 & -0.0722789927152405 \tabularnewline
73 & 58.73 & 58.6326762697227 & 0.0973237302773455 \tabularnewline
74 & 58.46 & 58.6519221058855 & -0.191922105885475 \tabularnewline
75 & 58.18 & 58.3390673993538 & -0.15906739935383 \tabularnewline
76 & 58.02 & 57.9939935027483 & 0.0260064972516716 \tabularnewline
77 & 56.97 & 57.8239833846415 & -0.853983384641495 \tabularnewline
78 & 57.22 & 56.5391608175146 & 0.680839182485357 \tabularnewline
79 & 57.19 & 56.8860272352698 & 0.303972764730155 \tabularnewline
80 & 57.06 & 57.0144361343224 & 0.0455638656776429 \tabularnewline
81 & 57.08 & 56.9300740497742 & 0.14992595022575 \tabularnewline
82 & 56.59 & 56.9967350345866 & -0.406735034586617 \tabularnewline
83 & 56.91 & 56.4098066891772 & 0.500193310822844 \tabularnewline
84 & 56.54 & 56.8248973069531 & -0.28489730695312 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=233052&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]75.79[/C][C]75.33[/C][C]0.459999999999994[/C][/ROW]
[ROW][C]4[/C][C]74.88[/C][C]73.7980065686659[/C][C]1.08199343133413[/C][/ROW]
[ROW][C]5[/C][C]74.5[/C][C]73.2389745226635[/C][C]1.26102547733647[/C][/ROW]
[ROW][C]6[/C][C]74.59[/C][C]73.3272030550246[/C][C]1.26279694497538[/C][/ROW]
[ROW][C]7[/C][C]74.59[/C][C]73.9053363143872[/C][C]0.684663685612833[/C][/ROW]
[ROW][C]8[/C][C]73.57[/C][C]74.2327815282759[/C][C]-0.662781528275872[/C][/ROW]
[ROW][C]9[/C][C]73.3[/C][C]73.102581408625[/C][C]0.197418591374955[/C][/ROW]
[ROW][C]10[/C][C]73.23[/C][C]72.8156551875855[/C][C]0.414344812414456[/C][/ROW]
[ROW][C]11[/C][C]73[/C][C]72.8823624076713[/C][C]0.11763759232872[/C][/ROW]
[ROW][C]12[/C][C]72.31[/C][C]72.7300238762924[/C][C]-0.420023876292419[/C][/ROW]
[ROW][C]13[/C][C]72.31[/C][C]71.9358966712325[/C][C]0.374103328767546[/C][/ROW]
[ROW][C]14[/C][C]71.24[/C][C]71.9944587212905[/C][C]-0.754458721290547[/C][/ROW]
[ROW][C]15[/C][C]70.82[/C][C]70.7550742252962[/C][C]0.0649257747037808[/C][/ROW]
[ROW][C]16[/C][C]70.66[/C][C]70.2713383325915[/C][C]0.388661667408513[/C][/ROW]
[ROW][C]17[/C][C]69.94[/C][C]70.2265328737339[/C][C]-0.286532873733933[/C][/ROW]
[ROW][C]18[/C][C]69.87[/C][C]69.4689390187363[/C][C]0.40106098126374[/C][/ROW]
[ROW][C]19[/C][C]69.87[/C][C]69.4794766410423[/C][C]0.390523358957736[/C][/ROW]
[ROW][C]20[/C][C]68.88[/C][C]69.6316351212429[/C][C]-0.75163512124287[/C][/ROW]
[ROW][C]21[/C][C]68.09[/C][C]68.4748167251068[/C][C]-0.384816725106774[/C][/ROW]
[ROW][C]22[/C][C]68.38[/C][C]67.4962348234689[/C][C]0.883765176531071[/C][/ROW]
[ROW][C]23[/C][C]66.78[/C][C]67.9904404793151[/C][C]-1.2104404793151[/C][/ROW]
[ROW][C]24[/C][C]67.2[/C][C]66.1494283093429[/C][C]1.05057169065712[/C][/ROW]
[ROW][C]25[/C][C]67.2[/C][C]66.7305327935195[/C][C]0.469467206480488[/C][/ROW]
[ROW][C]26[/C][C]66.67[/C][C]66.9750833887049[/C][C]-0.305083388704901[/C][/ROW]
[ROW][C]27[/C][C]65.86[/C][C]66.4110888271538[/C][C]-0.551088827153833[/C][/ROW]
[ROW][C]28[/C][C]66.05[/C][C]65.414655456804[/C][C]0.635344543195956[/C][/ROW]
[ROW][C]29[/C][C]66.31[/C][C]65.7217035649916[/C][C]0.588296435008417[/C][/ROW]
[ROW][C]30[/C][C]66.39[/C][C]66.2142998631565[/C][C]0.175700136843545[/C][/ROW]
[ROW][C]31[/C][C]66.39[/C][C]66.4069796153919[/C][C]-0.0169796153919179[/C][/ROW]
[ROW][C]32[/C][C]65.72[/C][C]66.421305115582[/C][C]-0.701305115581974[/C][/ROW]
[ROW][C]33[/C][C]65.52[/C][C]65.5543082940042[/C][C]-0.0343082940042052[/C][/ROW]
[ROW][C]34[/C][C]64.93[/C][C]65.2687212716238[/C][C]-0.338721271623839[/C][/ROW]
[ROW][C]35[/C][C]65.27[/C][C]64.5807436398824[/C][C]0.689256360117611[/C][/ROW]
[ROW][C]36[/C][C]65.04[/C][C]65.0758202825063[/C][C]-0.0358202825063216[/C][/ROW]
[ROW][C]37[/C][C]65.02[/C][C]64.9105858635387[/C][C]0.109414136461282[/C][/ROW]
[ROW][C]38[/C][C]64.72[/C][C]64.9171492401488[/C][C]-0.197149240148761[/C][/ROW]
[ROW][C]39[/C][C]64.68[/C][C]64.5741508693318[/C][C]0.105849130668204[/C][/ROW]
[ROW][C]40[/C][C]64.41[/C][C]64.5422299011493[/C][C]-0.13222990114933[/C][/ROW]
[ROW][C]41[/C][C]64.79[/C][C]64.2469104289096[/C][C]0.543089571090405[/C][/ROW]
[ROW][C]42[/C][C]64.71[/C][C]64.7637015693822[/C][C]-0.0537015693822553[/C][/ROW]
[ROW][C]43[/C][C]64.71[/C][C]64.7276429172992[/C][C]-0.0176429172992414[/C][/ROW]
[ROW][C]44[/C][C]64.83[/C][C]64.7169106726063[/C][C]0.113089327393737[/C][/ROW]
[ROW][C]45[/C][C]64.77[/C][C]64.8664676668834[/C][C]-0.0964676668833988[/C][/ROW]
[ROW][C]46[/C][C]64.19[/C][C]64.7918849630299[/C][C]-0.601884963029875[/C][/ROW]
[ROW][C]47[/C][C]64.27[/C][C]64.0339357086996[/C][C]0.236064291300352[/C][/ROW]
[ROW][C]48[/C][C]64.23[/C][C]64.1143664111716[/C][C]0.11563358882843[/C][/ROW]
[ROW][C]49[/C][C]64.23[/C][C]64.1321399288576[/C][C]0.0978600711424065[/C][/ROW]
[ROW][C]50[/C][C]63.03[/C][C]64.1719096700121[/C][C]-1.14190967001205[/C][/ROW]
[ROW][C]51[/C][C]62.85[/C][C]62.6647551856884[/C][C]0.185244814311645[/C][/ROW]
[ROW][C]52[/C][C]62.15[/C][C]62.4124912421534[/C][C]-0.262491242153416[/C][/ROW]
[ROW][C]53[/C][C]61.69[/C][C]61.6595318512475[/C][C]0.0304681487525471[/C][/ROW]
[ROW][C]54[/C][C]62.1[/C][C]61.1795494414814[/C][C]0.920450558518617[/C][/ROW]
[ROW][C]55[/C][C]62.1[/C][C]61.8489915048598[/C][C]0.251008495140198[/C][/ROW]
[ROW][C]56[/C][C]61.81[/C][C]62.0186419244402[/C][C]-0.208641924440158[/C][/ROW]
[ROW][C]57[/C][C]61.28[/C][C]61.6977979690971[/C][C]-0.417797969097101[/C][/ROW]
[ROW][C]58[/C][C]61.05[/C][C]61.028912917516[/C][C]0.0210870824839873[/C][/ROW]
[ROW][C]59[/C][C]61.08[/C][C]60.7594806656912[/C][C]0.320519334308791[/C][/ROW]
[ROW][C]60[/C][C]60.98[/C][C]60.8809596243565[/C][C]0.0990403756435327[/C][/ROW]
[ROW][C]61[/C][C]60.98[/C][C]60.8432728059253[/C][C]0.136727194074673[/C][/ROW]
[ROW][C]62[/C][C]61.11[/C][C]60.8920591618011[/C][C]0.217940838198849[/C][/ROW]
[ROW][C]63[/C][C]60.58[/C][C]61.0975314966371[/C][C]-0.517531496637112[/C][/ROW]
[ROW][C]64[/C][C]60.37[/C][C]60.4471458477756[/C][C]-0.0771458477755544[/C][/ROW]
[ROW][C]65[/C][C]59.44[/C][C]60.1595640812304[/C][C]-0.719564081230409[/C][/ROW]
[ROW][C]66[/C][C]59.29[/C][C]59.0209626449115[/C][C]0.269037355088486[/C][/ROW]
[ROW][C]67[/C][C]59.29[/C][C]58.8678094273232[/C][C]0.422190572676818[/C][/ROW]
[ROW][C]68[/C][C]59.33[/C][C]59.0144652807769[/C][C]0.315534719223109[/C][/ROW]
[ROW][C]69[/C][C]59.06[/C][C]59.1880472882102[/C][C]-0.12804728821024[/C][/ROW]
[ROW][C]70[/C][C]58.75[/C][C]58.9166271531506[/C][C]-0.166627153150642[/C][/ROW]
[ROW][C]71[/C][C]58.92[/C][C]58.5463752712373[/C][C]0.373624728762692[/C][/ROW]
[ROW][C]72[/C][C]58.73[/C][C]58.8022789927152[/C][C]-0.0722789927152405[/C][/ROW]
[ROW][C]73[/C][C]58.73[/C][C]58.6326762697227[/C][C]0.0973237302773455[/C][/ROW]
[ROW][C]74[/C][C]58.46[/C][C]58.6519221058855[/C][C]-0.191922105885475[/C][/ROW]
[ROW][C]75[/C][C]58.18[/C][C]58.3390673993538[/C][C]-0.15906739935383[/C][/ROW]
[ROW][C]76[/C][C]58.02[/C][C]57.9939935027483[/C][C]0.0260064972516716[/C][/ROW]
[ROW][C]77[/C][C]56.97[/C][C]57.8239833846415[/C][C]-0.853983384641495[/C][/ROW]
[ROW][C]78[/C][C]57.22[/C][C]56.5391608175146[/C][C]0.680839182485357[/C][/ROW]
[ROW][C]79[/C][C]57.19[/C][C]56.8860272352698[/C][C]0.303972764730155[/C][/ROW]
[ROW][C]80[/C][C]57.06[/C][C]57.0144361343224[/C][C]0.0455638656776429[/C][/ROW]
[ROW][C]81[/C][C]57.08[/C][C]56.9300740497742[/C][C]0.14992595022575[/C][/ROW]
[ROW][C]82[/C][C]56.59[/C][C]56.9967350345866[/C][C]-0.406735034586617[/C][/ROW]
[ROW][C]83[/C][C]56.91[/C][C]56.4098066891772[/C][C]0.500193310822844[/C][/ROW]
[ROW][C]84[/C][C]56.54[/C][C]56.8248973069531[/C][C]-0.28489730695312[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=233052&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=233052&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	75.79	75.33	0.459999999999994
4	74.88	73.7980065686659	1.08199343133413
5	74.5	73.2389745226635	1.26102547733647
6	74.59	73.3272030550246	1.26279694497538
7	74.59	73.9053363143872	0.684663685612833
8	73.57	74.2327815282759	-0.662781528275872
9	73.3	73.102581408625	0.197418591374955
10	73.23	72.8156551875855	0.414344812414456
11	73	72.8823624076713	0.11763759232872
12	72.31	72.7300238762924	-0.420023876292419
13	72.31	71.9358966712325	0.374103328767546
14	71.24	71.9944587212905	-0.754458721290547
15	70.82	70.7550742252962	0.0649257747037808
16	70.66	70.2713383325915	0.388661667408513
17	69.94	70.2265328737339	-0.286532873733933
18	69.87	69.4689390187363	0.40106098126374
19	69.87	69.4794766410423	0.390523358957736
20	68.88	69.6316351212429	-0.75163512124287
21	68.09	68.4748167251068	-0.384816725106774
22	68.38	67.4962348234689	0.883765176531071
23	66.78	67.9904404793151	-1.2104404793151
24	67.2	66.1494283093429	1.05057169065712
25	67.2	66.7305327935195	0.469467206480488
26	66.67	66.9750833887049	-0.305083388704901
27	65.86	66.4110888271538	-0.551088827153833
28	66.05	65.414655456804	0.635344543195956
29	66.31	65.7217035649916	0.588296435008417
30	66.39	66.2142998631565	0.175700136843545
31	66.39	66.4069796153919	-0.0169796153919179
32	65.72	66.421305115582	-0.701305115581974
33	65.52	65.5543082940042	-0.0343082940042052
34	64.93	65.2687212716238	-0.338721271623839
35	65.27	64.5807436398824	0.689256360117611
36	65.04	65.0758202825063	-0.0358202825063216
37	65.02	64.9105858635387	0.109414136461282
38	64.72	64.9171492401488	-0.197149240148761
39	64.68	64.5741508693318	0.105849130668204
40	64.41	64.5422299011493	-0.13222990114933
41	64.79	64.2469104289096	0.543089571090405
42	64.71	64.7637015693822	-0.0537015693822553
43	64.71	64.7276429172992	-0.0176429172992414
44	64.83	64.7169106726063	0.113089327393737
45	64.77	64.8664676668834	-0.0964676668833988
46	64.19	64.7918849630299	-0.601884963029875
47	64.27	64.0339357086996	0.236064291300352
48	64.23	64.1143664111716	0.11563358882843
49	64.23	64.1321399288576	0.0978600711424065
50	63.03	64.1719096700121	-1.14190967001205
51	62.85	62.6647551856884	0.185244814311645
52	62.15	62.4124912421534	-0.262491242153416
53	61.69	61.6595318512475	0.0304681487525471
54	62.1	61.1795494414814	0.920450558518617
55	62.1	61.8489915048598	0.251008495140198
56	61.81	62.0186419244402	-0.208641924440158
57	61.28	61.6977979690971	-0.417797969097101
58	61.05	61.028912917516	0.0210870824839873
59	61.08	60.7594806656912	0.320519334308791
60	60.98	60.8809596243565	0.0990403756435327
61	60.98	60.8432728059253	0.136727194074673
62	61.11	60.8920591618011	0.217940838198849
63	60.58	61.0975314966371	-0.517531496637112
64	60.37	60.4471458477756	-0.0771458477755544
65	59.44	60.1595640812304	-0.719564081230409
66	59.29	59.0209626449115	0.269037355088486
67	59.29	58.8678094273232	0.422190572676818
68	59.33	59.0144652807769	0.315534719223109
69	59.06	59.1880472882102	-0.12804728821024
70	58.75	58.9166271531506	-0.166627153150642
71	58.92	58.5463752712373	0.373624728762692
72	58.73	58.8022789927152	-0.0722789927152405
73	58.73	58.6326762697227	0.0973237302773455
74	58.46	58.6519221058855	-0.191922105885475
75	58.18	58.3390673993538	-0.15906739935383
76	58.02	57.9939935027483	0.0260064972516716
77	56.97	57.8239833846415	-0.853983384641495
78	57.22	56.5391608175146	0.680839182485357
79	57.19	56.8860272352698	0.303972764730155
80	57.06	57.0144361343224	0.0455638656776429
81	57.08	56.9300740497742	0.14992595022575
82	56.59	56.9967350345866	-0.406735034586617
83	56.91	56.4098066891772	0.500193310822844
84	56.54	56.8248973069531	-0.28489730695312

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
85	56.4298515462771	55.4528344809225	57.4068686116318
86	56.2888127487875	54.7031566854951	57.87446881208
87	56.147773951298	53.8761022817293	58.4194456208666
88	56.0067351538084	52.9771130592102	59.0363572484065
89	55.8656963563188	52.0115887609586	59.719803951679
90	55.7246575588292	50.9841317698247	60.4651833478336
91	55.5836187613396	49.8986020634595	61.2686354592197
92	55.44257996385	48.7582608307013	62.1268990969987
93	55.3015411663604	47.5658968084344	63.0371855242865
94	55.1605023688708	46.3239240972168	63.9970806405248
95	55.0194635713812	45.0344560776268	65.0044710651356
96	54.8784247738916	43.6993614164189	66.0574881313644

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
85 & 56.4298515462771 & 55.4528344809225 & 57.4068686116318 \tabularnewline
86 & 56.2888127487875 & 54.7031566854951 & 57.87446881208 \tabularnewline
87 & 56.147773951298 & 53.8761022817293 & 58.4194456208666 \tabularnewline
88 & 56.0067351538084 & 52.9771130592102 & 59.0363572484065 \tabularnewline
89 & 55.8656963563188 & 52.0115887609586 & 59.719803951679 \tabularnewline
90 & 55.7246575588292 & 50.9841317698247 & 60.4651833478336 \tabularnewline
91 & 55.5836187613396 & 49.8986020634595 & 61.2686354592197 \tabularnewline
92 & 55.44257996385 & 48.7582608307013 & 62.1268990969987 \tabularnewline
93 & 55.3015411663604 & 47.5658968084344 & 63.0371855242865 \tabularnewline
94 & 55.1605023688708 & 46.3239240972168 & 63.9970806405248 \tabularnewline
95 & 55.0194635713812 & 45.0344560776268 & 65.0044710651356 \tabularnewline
96 & 54.8784247738916 & 43.6993614164189 & 66.0574881313644 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=233052&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]56.4298515462771[/C][C]55.4528344809225[/C][C]57.4068686116318[/C][/ROW]
[ROW][C]86[/C][C]56.2888127487875[/C][C]54.7031566854951[/C][C]57.87446881208[/C][/ROW]
[ROW][C]87[/C][C]56.147773951298[/C][C]53.8761022817293[/C][C]58.4194456208666[/C][/ROW]
[ROW][C]88[/C][C]56.0067351538084[/C][C]52.9771130592102[/C][C]59.0363572484065[/C][/ROW]
[ROW][C]89[/C][C]55.8656963563188[/C][C]52.0115887609586[/C][C]59.719803951679[/C][/ROW]
[ROW][C]90[/C][C]55.7246575588292[/C][C]50.9841317698247[/C][C]60.4651833478336[/C][/ROW]
[ROW][C]91[/C][C]55.5836187613396[/C][C]49.8986020634595[/C][C]61.2686354592197[/C][/ROW]
[ROW][C]92[/C][C]55.44257996385[/C][C]48.7582608307013[/C][C]62.1268990969987[/C][/ROW]
[ROW][C]93[/C][C]55.3015411663604[/C][C]47.5658968084344[/C][C]63.0371855242865[/C][/ROW]
[ROW][C]94[/C][C]55.1605023688708[/C][C]46.3239240972168[/C][C]63.9970806405248[/C][/ROW]
[ROW][C]95[/C][C]55.0194635713812[/C][C]45.0344560776268[/C][C]65.0044710651356[/C][/ROW]
[ROW][C]96[/C][C]54.8784247738916[/C][C]43.6993614164189[/C][C]66.0574881313644[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=233052&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=233052&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	56.4298515462771	55.4528344809225	57.4068686116318
86	56.2888127487875	54.7031566854951	57.87446881208
87	56.147773951298	53.8761022817293	58.4194456208666
88	56.0067351538084	52.9771130592102	59.0363572484065
89	55.8656963563188	52.0115887609586	59.719803951679
90	55.7246575588292	50.9841317698247	60.4651833478336
91	55.5836187613396	49.8986020634595	61.2686354592197
92	55.44257996385	48.7582608307013	62.1268990969987
93	55.3015411663604	47.5658968084344	63.0371855242865
94	55.1605023688708	46.3239240972168	63.9970806405248
95	55.0194635713812	45.0344560776268	65.0044710651356
96	54.8784247738916	43.6993614164189	66.0574881313644

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 = additive ;

Parameters (R input):

par1 = 12 ; par2 = Double ; 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