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*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Mon, 23 Dec 2013 06:54:27 -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/23/t138779969949icpji5ub5iy2b.htm/, Retrieved Wed, 29 Jul 2026 03:29:20 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=232584, Retrieved Wed, 29 Jul 2026 03:29:20 +0000

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IsPrivate?

No (this computation is public)

User-defined keywords

Estimated Impact

511

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

-       [Exponential Smoothing] [Korte sigaretten] [2013-12-23 11:54:27] [478e7c199ef13b68c565592d49c085e5] [Current]

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Dataseries X:

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4,69
4,69
4,69
4,69
4,69
4,69
4,69
4,73
4,78
4,79
4,79
4,8
4,8
4,81
5,16
5,26
5,29
5,29
5,29
5,3
5,3
5,3
5,3
5,3
5,3
5,3
5,3
5,35
5,44
5,47
5,47
5,48
5,48
5,48
5,48
5,48
5,48
5,48
5,5
5,55
5,57
5,58
5,58
5,58
5,59
5,59
5,59
5,55
5,61
5,61
5,61
5,63
5,69
5,7
5,7
5,7
5,7
5,7
5,7
5,7
5,7
5,7
5,7
5,71
5,74
5,77
5,79
5,79
5,8
5,8
5,8
5,8
5,8
5,81
5,81
5,83
5,94
5,98
5,99
6
6,02
6,02
6,02
6,02

Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	4 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 & 4 seconds \tabularnewline
R Server & 'Gertrude Mary Cox' @ cox.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232584&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]4 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=232584&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232584&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	4 seconds
R Server	'Gertrude Mary Cox' @ cox.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.58822905647819
beta	0.0608846697362494
gamma	1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.58822905647819 \tabularnewline
beta & 0.0608846697362494 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232584&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]0.58822905647819[/C][/ROW]
[ROW][C]beta[/C][C]0.0608846697362494[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232584&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232584&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	0.58822905647819
beta	0.0608846697362494
gamma	1

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	4.8	4.54237179487179	0.257628205128206
14	4.81	4.7081283526343	0.101871647365695
15	5.16	5.12924615532601	0.0307538446739901
16	5.26	5.2621318222891	-0.00213182228909581
17	5.29	5.30601350170499	-0.0160135017049905
18	5.29	5.31157273094124	-0.0215727309412381
19	5.29	5.20308925137918	0.0869107486208236
20	5.3	5.34403153965218	-0.0440315396521829
21	5.3	5.40137281787535	-0.101372817875351
22	5.3	5.36260371064839	-0.0626037106483883
23	5.3	5.32898095457298	-0.0289809545729849
24	5.3	5.32284815285229	-0.0228481528522897
25	5.3	5.41558836564641	-0.115588365646408
26	5.3	5.29066655975239	0.00933344024760974
27	5.3	5.6177467722998	-0.317746772299796
28	5.35	5.50929195783695	-0.159291957836946
29	5.44	5.42658192275176	0.013418077248244
30	5.47	5.41978911526611	0.0502108847338896
31	5.47	5.3733966392619	0.0966033607380989
32	5.48	5.44166475681743	0.0383352431825674
33	5.48	5.50233757536608	-0.0223375753660804
34	5.48	5.50734634233604	-0.0273463423360427
35	5.48	5.49089363699153	-0.0108936369915344
36	5.48	5.48115918039497	-0.00115918039497398
37	5.48	5.53248007360592	-0.0524800736059152
38	5.48	5.48239005890207	-0.00239005890206645
39	5.5	5.6537426641618	-0.153742664161799
40	5.55	5.69873120797817	-0.148731207978171
41	5.57	5.68545279904224	-0.115452799042242
42	5.58	5.60549172021043	-0.0254917202104288
43	5.58	5.5184477364384	0.0615522635616017
44	5.58	5.52562522830998	0.054374771690024
45	5.59	5.5548446675073	0.0351553324926979
46	5.59	5.57776403496992	0.0122359650300812
47	5.59	5.57894121116787	0.0110587888321287
48	5.55	5.57448605497497	-0.0244860549749726
49	5.61	5.57847539767639	0.0315246023236115
50	5.61	5.58895598892337	0.0210440110766257
51	5.61	5.70314086292886	-0.093140862928859
52	5.63	5.77938139299021	-0.149381392990211
53	5.69	5.77294099621653	-0.0829409962165251
54	5.7	5.74382943282059	-0.0438294328205915
55	5.7	5.67586587775518	0.0241341222448241
56	5.7	5.65076237158623	0.0492376284137723
57	5.7	5.66154692742388	0.0384530725761234
58	5.7	5.66958763775232	0.03041236224768
59	5.7	5.67424198976128	0.0257580102387216
60	5.7	5.6575934664681	0.0424065335318975
61	5.7	5.72018669221139	-0.0201866922113902
62	5.7	5.6902737570146	0.00972624298540481
63	5.7	5.74471800412696	-0.0447180041269588
64	5.71	5.82195309971243	-0.111953099712433
65	5.74	5.86189684838295	-0.121896848382953
66	5.77	5.82158966709623	-0.0515896670962261
67	5.79	5.772383148797	0.0176168512030026
68	5.79	5.74888589330673	0.0411141066932732
69	5.8	5.7452632584704	0.0547367415296041
70	5.8	5.75496681818712	0.0450331818128751
71	5.8	5.76222391917736	0.0377760808226366
72	5.8	5.75584945411512	0.0441505458848832
73	5.8	5.79010624903122	0.00989375096878309
74	5.81	5.7876938491062	0.0223061508938027
75	5.81	5.82505901006755	-0.0150590100675503
76	5.83	5.89105674561319	-0.0610567456131914
77	5.94	5.95766928721253	-0.0176692872125281
78	5.98	6.01217968523432	-0.0321796852343246
79	5.99	6.00814051220662	-0.0181405122066201
80	6	5.97725720132872	0.0227427986712829
81	6.02	5.97175145975026	0.0482485402497383
82	6.02	5.97672448295593	0.0432755170440675
83	6.02	5.982978117844	0.0370218821560044
84	6.02	5.9817765264427	0.0382234735573004

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 4.8 & 4.54237179487179 & 0.257628205128206 \tabularnewline
14 & 4.81 & 4.7081283526343 & 0.101871647365695 \tabularnewline
15 & 5.16 & 5.12924615532601 & 0.0307538446739901 \tabularnewline
16 & 5.26 & 5.2621318222891 & -0.00213182228909581 \tabularnewline
17 & 5.29 & 5.30601350170499 & -0.0160135017049905 \tabularnewline
18 & 5.29 & 5.31157273094124 & -0.0215727309412381 \tabularnewline
19 & 5.29 & 5.20308925137918 & 0.0869107486208236 \tabularnewline
20 & 5.3 & 5.34403153965218 & -0.0440315396521829 \tabularnewline
21 & 5.3 & 5.40137281787535 & -0.101372817875351 \tabularnewline
22 & 5.3 & 5.36260371064839 & -0.0626037106483883 \tabularnewline
23 & 5.3 & 5.32898095457298 & -0.0289809545729849 \tabularnewline
24 & 5.3 & 5.32284815285229 & -0.0228481528522897 \tabularnewline
25 & 5.3 & 5.41558836564641 & -0.115588365646408 \tabularnewline
26 & 5.3 & 5.29066655975239 & 0.00933344024760974 \tabularnewline
27 & 5.3 & 5.6177467722998 & -0.317746772299796 \tabularnewline
28 & 5.35 & 5.50929195783695 & -0.159291957836946 \tabularnewline
29 & 5.44 & 5.42658192275176 & 0.013418077248244 \tabularnewline
30 & 5.47 & 5.41978911526611 & 0.0502108847338896 \tabularnewline
31 & 5.47 & 5.3733966392619 & 0.0966033607380989 \tabularnewline
32 & 5.48 & 5.44166475681743 & 0.0383352431825674 \tabularnewline
33 & 5.48 & 5.50233757536608 & -0.0223375753660804 \tabularnewline
34 & 5.48 & 5.50734634233604 & -0.0273463423360427 \tabularnewline
35 & 5.48 & 5.49089363699153 & -0.0108936369915344 \tabularnewline
36 & 5.48 & 5.48115918039497 & -0.00115918039497398 \tabularnewline
37 & 5.48 & 5.53248007360592 & -0.0524800736059152 \tabularnewline
38 & 5.48 & 5.48239005890207 & -0.00239005890206645 \tabularnewline
39 & 5.5 & 5.6537426641618 & -0.153742664161799 \tabularnewline
40 & 5.55 & 5.69873120797817 & -0.148731207978171 \tabularnewline
41 & 5.57 & 5.68545279904224 & -0.115452799042242 \tabularnewline
42 & 5.58 & 5.60549172021043 & -0.0254917202104288 \tabularnewline
43 & 5.58 & 5.5184477364384 & 0.0615522635616017 \tabularnewline
44 & 5.58 & 5.52562522830998 & 0.054374771690024 \tabularnewline
45 & 5.59 & 5.5548446675073 & 0.0351553324926979 \tabularnewline
46 & 5.59 & 5.57776403496992 & 0.0122359650300812 \tabularnewline
47 & 5.59 & 5.57894121116787 & 0.0110587888321287 \tabularnewline
48 & 5.55 & 5.57448605497497 & -0.0244860549749726 \tabularnewline
49 & 5.61 & 5.57847539767639 & 0.0315246023236115 \tabularnewline
50 & 5.61 & 5.58895598892337 & 0.0210440110766257 \tabularnewline
51 & 5.61 & 5.70314086292886 & -0.093140862928859 \tabularnewline
52 & 5.63 & 5.77938139299021 & -0.149381392990211 \tabularnewline
53 & 5.69 & 5.77294099621653 & -0.0829409962165251 \tabularnewline
54 & 5.7 & 5.74382943282059 & -0.0438294328205915 \tabularnewline
55 & 5.7 & 5.67586587775518 & 0.0241341222448241 \tabularnewline
56 & 5.7 & 5.65076237158623 & 0.0492376284137723 \tabularnewline
57 & 5.7 & 5.66154692742388 & 0.0384530725761234 \tabularnewline
58 & 5.7 & 5.66958763775232 & 0.03041236224768 \tabularnewline
59 & 5.7 & 5.67424198976128 & 0.0257580102387216 \tabularnewline
60 & 5.7 & 5.6575934664681 & 0.0424065335318975 \tabularnewline
61 & 5.7 & 5.72018669221139 & -0.0201866922113902 \tabularnewline
62 & 5.7 & 5.6902737570146 & 0.00972624298540481 \tabularnewline
63 & 5.7 & 5.74471800412696 & -0.0447180041269588 \tabularnewline
64 & 5.71 & 5.82195309971243 & -0.111953099712433 \tabularnewline
65 & 5.74 & 5.86189684838295 & -0.121896848382953 \tabularnewline
66 & 5.77 & 5.82158966709623 & -0.0515896670962261 \tabularnewline
67 & 5.79 & 5.772383148797 & 0.0176168512030026 \tabularnewline
68 & 5.79 & 5.74888589330673 & 0.0411141066932732 \tabularnewline
69 & 5.8 & 5.7452632584704 & 0.0547367415296041 \tabularnewline
70 & 5.8 & 5.75496681818712 & 0.0450331818128751 \tabularnewline
71 & 5.8 & 5.76222391917736 & 0.0377760808226366 \tabularnewline
72 & 5.8 & 5.75584945411512 & 0.0441505458848832 \tabularnewline
73 & 5.8 & 5.79010624903122 & 0.00989375096878309 \tabularnewline
74 & 5.81 & 5.7876938491062 & 0.0223061508938027 \tabularnewline
75 & 5.81 & 5.82505901006755 & -0.0150590100675503 \tabularnewline
76 & 5.83 & 5.89105674561319 & -0.0610567456131914 \tabularnewline
77 & 5.94 & 5.95766928721253 & -0.0176692872125281 \tabularnewline
78 & 5.98 & 6.01217968523432 & -0.0321796852343246 \tabularnewline
79 & 5.99 & 6.00814051220662 & -0.0181405122066201 \tabularnewline
80 & 6 & 5.97725720132872 & 0.0227427986712829 \tabularnewline
81 & 6.02 & 5.97175145975026 & 0.0482485402497383 \tabularnewline
82 & 6.02 & 5.97672448295593 & 0.0432755170440675 \tabularnewline
83 & 6.02 & 5.982978117844 & 0.0370218821560044 \tabularnewline
84 & 6.02 & 5.9817765264427 & 0.0382234735573004 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232584&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]4.8[/C][C]4.54237179487179[/C][C]0.257628205128206[/C][/ROW]
[ROW][C]14[/C][C]4.81[/C][C]4.7081283526343[/C][C]0.101871647365695[/C][/ROW]
[ROW][C]15[/C][C]5.16[/C][C]5.12924615532601[/C][C]0.0307538446739901[/C][/ROW]
[ROW][C]16[/C][C]5.26[/C][C]5.2621318222891[/C][C]-0.00213182228909581[/C][/ROW]
[ROW][C]17[/C][C]5.29[/C][C]5.30601350170499[/C][C]-0.0160135017049905[/C][/ROW]
[ROW][C]18[/C][C]5.29[/C][C]5.31157273094124[/C][C]-0.0215727309412381[/C][/ROW]
[ROW][C]19[/C][C]5.29[/C][C]5.20308925137918[/C][C]0.0869107486208236[/C][/ROW]
[ROW][C]20[/C][C]5.3[/C][C]5.34403153965218[/C][C]-0.0440315396521829[/C][/ROW]
[ROW][C]21[/C][C]5.3[/C][C]5.40137281787535[/C][C]-0.101372817875351[/C][/ROW]
[ROW][C]22[/C][C]5.3[/C][C]5.36260371064839[/C][C]-0.0626037106483883[/C][/ROW]
[ROW][C]23[/C][C]5.3[/C][C]5.32898095457298[/C][C]-0.0289809545729849[/C][/ROW]
[ROW][C]24[/C][C]5.3[/C][C]5.32284815285229[/C][C]-0.0228481528522897[/C][/ROW]
[ROW][C]25[/C][C]5.3[/C][C]5.41558836564641[/C][C]-0.115588365646408[/C][/ROW]
[ROW][C]26[/C][C]5.3[/C][C]5.29066655975239[/C][C]0.00933344024760974[/C][/ROW]
[ROW][C]27[/C][C]5.3[/C][C]5.6177467722998[/C][C]-0.317746772299796[/C][/ROW]
[ROW][C]28[/C][C]5.35[/C][C]5.50929195783695[/C][C]-0.159291957836946[/C][/ROW]
[ROW][C]29[/C][C]5.44[/C][C]5.42658192275176[/C][C]0.013418077248244[/C][/ROW]
[ROW][C]30[/C][C]5.47[/C][C]5.41978911526611[/C][C]0.0502108847338896[/C][/ROW]
[ROW][C]31[/C][C]5.47[/C][C]5.3733966392619[/C][C]0.0966033607380989[/C][/ROW]
[ROW][C]32[/C][C]5.48[/C][C]5.44166475681743[/C][C]0.0383352431825674[/C][/ROW]
[ROW][C]33[/C][C]5.48[/C][C]5.50233757536608[/C][C]-0.0223375753660804[/C][/ROW]
[ROW][C]34[/C][C]5.48[/C][C]5.50734634233604[/C][C]-0.0273463423360427[/C][/ROW]
[ROW][C]35[/C][C]5.48[/C][C]5.49089363699153[/C][C]-0.0108936369915344[/C][/ROW]
[ROW][C]36[/C][C]5.48[/C][C]5.48115918039497[/C][C]-0.00115918039497398[/C][/ROW]
[ROW][C]37[/C][C]5.48[/C][C]5.53248007360592[/C][C]-0.0524800736059152[/C][/ROW]
[ROW][C]38[/C][C]5.48[/C][C]5.48239005890207[/C][C]-0.00239005890206645[/C][/ROW]
[ROW][C]39[/C][C]5.5[/C][C]5.6537426641618[/C][C]-0.153742664161799[/C][/ROW]
[ROW][C]40[/C][C]5.55[/C][C]5.69873120797817[/C][C]-0.148731207978171[/C][/ROW]
[ROW][C]41[/C][C]5.57[/C][C]5.68545279904224[/C][C]-0.115452799042242[/C][/ROW]
[ROW][C]42[/C][C]5.58[/C][C]5.60549172021043[/C][C]-0.0254917202104288[/C][/ROW]
[ROW][C]43[/C][C]5.58[/C][C]5.5184477364384[/C][C]0.0615522635616017[/C][/ROW]
[ROW][C]44[/C][C]5.58[/C][C]5.52562522830998[/C][C]0.054374771690024[/C][/ROW]
[ROW][C]45[/C][C]5.59[/C][C]5.5548446675073[/C][C]0.0351553324926979[/C][/ROW]
[ROW][C]46[/C][C]5.59[/C][C]5.57776403496992[/C][C]0.0122359650300812[/C][/ROW]
[ROW][C]47[/C][C]5.59[/C][C]5.57894121116787[/C][C]0.0110587888321287[/C][/ROW]
[ROW][C]48[/C][C]5.55[/C][C]5.57448605497497[/C][C]-0.0244860549749726[/C][/ROW]
[ROW][C]49[/C][C]5.61[/C][C]5.57847539767639[/C][C]0.0315246023236115[/C][/ROW]
[ROW][C]50[/C][C]5.61[/C][C]5.58895598892337[/C][C]0.0210440110766257[/C][/ROW]
[ROW][C]51[/C][C]5.61[/C][C]5.70314086292886[/C][C]-0.093140862928859[/C][/ROW]
[ROW][C]52[/C][C]5.63[/C][C]5.77938139299021[/C][C]-0.149381392990211[/C][/ROW]
[ROW][C]53[/C][C]5.69[/C][C]5.77294099621653[/C][C]-0.0829409962165251[/C][/ROW]
[ROW][C]54[/C][C]5.7[/C][C]5.74382943282059[/C][C]-0.0438294328205915[/C][/ROW]
[ROW][C]55[/C][C]5.7[/C][C]5.67586587775518[/C][C]0.0241341222448241[/C][/ROW]
[ROW][C]56[/C][C]5.7[/C][C]5.65076237158623[/C][C]0.0492376284137723[/C][/ROW]
[ROW][C]57[/C][C]5.7[/C][C]5.66154692742388[/C][C]0.0384530725761234[/C][/ROW]
[ROW][C]58[/C][C]5.7[/C][C]5.66958763775232[/C][C]0.03041236224768[/C][/ROW]
[ROW][C]59[/C][C]5.7[/C][C]5.67424198976128[/C][C]0.0257580102387216[/C][/ROW]
[ROW][C]60[/C][C]5.7[/C][C]5.6575934664681[/C][C]0.0424065335318975[/C][/ROW]
[ROW][C]61[/C][C]5.7[/C][C]5.72018669221139[/C][C]-0.0201866922113902[/C][/ROW]
[ROW][C]62[/C][C]5.7[/C][C]5.6902737570146[/C][C]0.00972624298540481[/C][/ROW]
[ROW][C]63[/C][C]5.7[/C][C]5.74471800412696[/C][C]-0.0447180041269588[/C][/ROW]
[ROW][C]64[/C][C]5.71[/C][C]5.82195309971243[/C][C]-0.111953099712433[/C][/ROW]
[ROW][C]65[/C][C]5.74[/C][C]5.86189684838295[/C][C]-0.121896848382953[/C][/ROW]
[ROW][C]66[/C][C]5.77[/C][C]5.82158966709623[/C][C]-0.0515896670962261[/C][/ROW]
[ROW][C]67[/C][C]5.79[/C][C]5.772383148797[/C][C]0.0176168512030026[/C][/ROW]
[ROW][C]68[/C][C]5.79[/C][C]5.74888589330673[/C][C]0.0411141066932732[/C][/ROW]
[ROW][C]69[/C][C]5.8[/C][C]5.7452632584704[/C][C]0.0547367415296041[/C][/ROW]
[ROW][C]70[/C][C]5.8[/C][C]5.75496681818712[/C][C]0.0450331818128751[/C][/ROW]
[ROW][C]71[/C][C]5.8[/C][C]5.76222391917736[/C][C]0.0377760808226366[/C][/ROW]
[ROW][C]72[/C][C]5.8[/C][C]5.75584945411512[/C][C]0.0441505458848832[/C][/ROW]
[ROW][C]73[/C][C]5.8[/C][C]5.79010624903122[/C][C]0.00989375096878309[/C][/ROW]
[ROW][C]74[/C][C]5.81[/C][C]5.7876938491062[/C][C]0.0223061508938027[/C][/ROW]
[ROW][C]75[/C][C]5.81[/C][C]5.82505901006755[/C][C]-0.0150590100675503[/C][/ROW]
[ROW][C]76[/C][C]5.83[/C][C]5.89105674561319[/C][C]-0.0610567456131914[/C][/ROW]
[ROW][C]77[/C][C]5.94[/C][C]5.95766928721253[/C][C]-0.0176692872125281[/C][/ROW]
[ROW][C]78[/C][C]5.98[/C][C]6.01217968523432[/C][C]-0.0321796852343246[/C][/ROW]
[ROW][C]79[/C][C]5.99[/C][C]6.00814051220662[/C][C]-0.0181405122066201[/C][/ROW]
[ROW][C]80[/C][C]6[/C][C]5.97725720132872[/C][C]0.0227427986712829[/C][/ROW]
[ROW][C]81[/C][C]6.02[/C][C]5.97175145975026[/C][C]0.0482485402497383[/C][/ROW]
[ROW][C]82[/C][C]6.02[/C][C]5.97672448295593[/C][C]0.0432755170440675[/C][/ROW]
[ROW][C]83[/C][C]6.02[/C][C]5.982978117844[/C][C]0.0370218821560044[/C][/ROW]
[ROW][C]84[/C][C]6.02[/C][C]5.9817765264427[/C][C]0.0382234735573004[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232584&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232584&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	4.8	4.54237179487179	0.257628205128206
14	4.81	4.7081283526343	0.101871647365695
15	5.16	5.12924615532601	0.0307538446739901
16	5.26	5.2621318222891	-0.00213182228909581
17	5.29	5.30601350170499	-0.0160135017049905
18	5.29	5.31157273094124	-0.0215727309412381
19	5.29	5.20308925137918	0.0869107486208236
20	5.3	5.34403153965218	-0.0440315396521829
21	5.3	5.40137281787535	-0.101372817875351
22	5.3	5.36260371064839	-0.0626037106483883
23	5.3	5.32898095457298	-0.0289809545729849
24	5.3	5.32284815285229	-0.0228481528522897
25	5.3	5.41558836564641	-0.115588365646408
26	5.3	5.29066655975239	0.00933344024760974
27	5.3	5.6177467722998	-0.317746772299796
28	5.35	5.50929195783695	-0.159291957836946
29	5.44	5.42658192275176	0.013418077248244
30	5.47	5.41978911526611	0.0502108847338896
31	5.47	5.3733966392619	0.0966033607380989
32	5.48	5.44166475681743	0.0383352431825674
33	5.48	5.50233757536608	-0.0223375753660804
34	5.48	5.50734634233604	-0.0273463423360427
35	5.48	5.49089363699153	-0.0108936369915344
36	5.48	5.48115918039497	-0.00115918039497398
37	5.48	5.53248007360592	-0.0524800736059152
38	5.48	5.48239005890207	-0.00239005890206645
39	5.5	5.6537426641618	-0.153742664161799
40	5.55	5.69873120797817	-0.148731207978171
41	5.57	5.68545279904224	-0.115452799042242
42	5.58	5.60549172021043	-0.0254917202104288
43	5.58	5.5184477364384	0.0615522635616017
44	5.58	5.52562522830998	0.054374771690024
45	5.59	5.5548446675073	0.0351553324926979
46	5.59	5.57776403496992	0.0122359650300812
47	5.59	5.57894121116787	0.0110587888321287
48	5.55	5.57448605497497	-0.0244860549749726
49	5.61	5.57847539767639	0.0315246023236115
50	5.61	5.58895598892337	0.0210440110766257
51	5.61	5.70314086292886	-0.093140862928859
52	5.63	5.77938139299021	-0.149381392990211
53	5.69	5.77294099621653	-0.0829409962165251
54	5.7	5.74382943282059	-0.0438294328205915
55	5.7	5.67586587775518	0.0241341222448241
56	5.7	5.65076237158623	0.0492376284137723
57	5.7	5.66154692742388	0.0384530725761234
58	5.7	5.66958763775232	0.03041236224768
59	5.7	5.67424198976128	0.0257580102387216
60	5.7	5.6575934664681	0.0424065335318975
61	5.7	5.72018669221139	-0.0201866922113902
62	5.7	5.6902737570146	0.00972624298540481
63	5.7	5.74471800412696	-0.0447180041269588
64	5.71	5.82195309971243	-0.111953099712433
65	5.74	5.86189684838295	-0.121896848382953
66	5.77	5.82158966709623	-0.0515896670962261
67	5.79	5.772383148797	0.0176168512030026
68	5.79	5.74888589330673	0.0411141066932732
69	5.8	5.7452632584704	0.0547367415296041
70	5.8	5.75496681818712	0.0450331818128751
71	5.8	5.76222391917736	0.0377760808226366
72	5.8	5.75584945411512	0.0441505458848832
73	5.8	5.79010624903122	0.00989375096878309
74	5.81	5.7876938491062	0.0223061508938027
75	5.81	5.82505901006755	-0.0150590100675503
76	5.83	5.89105674561319	-0.0610567456131914
77	5.94	5.95766928721253	-0.0176692872125281
78	5.98	6.01217968523432	-0.0321796852343246
79	5.99	6.00814051220662	-0.0181405122066201
80	6	5.97725720132872	0.0227427986712829
81	6.02	5.97175145975026	0.0482485402497383
82	6.02	5.97672448295593	0.0432755170440675
83	6.02	5.982978117844	0.0370218821560044
84	6.02	5.9817765264427	0.0382234735573004

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
85	6.0012203152194	5.84912159769775	6.15331903274105
86	6.00052427581213	5.82123931008082	6.17980924154344
87	6.01100863435351	5.8055434999317	6.21647376877533
88	6.06908952284688	5.83797952037115	6.30019952532261
89	6.19383534195892	5.93733937692852	6.45033130698931
90	6.25774940899006	5.97595106308273	6.53954775489739
91	6.28455771400671	5.97742327751749	6.59169215049593
92	6.28796695433563	5.95538119270907	6.62055271596219
93	6.28555846277121	5.92734786294322	6.6437690625992
94	6.26434726837366	5.88029578373874	6.64839875300859
95	6.24526476865332	5.83512483924419	6.65540469806245
96	6.22414955138849	5.78765002543119	6.66064907734578

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
85 & 6.0012203152194 & 5.84912159769775 & 6.15331903274105 \tabularnewline
86 & 6.00052427581213 & 5.82123931008082 & 6.17980924154344 \tabularnewline
87 & 6.01100863435351 & 5.8055434999317 & 6.21647376877533 \tabularnewline
88 & 6.06908952284688 & 5.83797952037115 & 6.30019952532261 \tabularnewline
89 & 6.19383534195892 & 5.93733937692852 & 6.45033130698931 \tabularnewline
90 & 6.25774940899006 & 5.97595106308273 & 6.53954775489739 \tabularnewline
91 & 6.28455771400671 & 5.97742327751749 & 6.59169215049593 \tabularnewline
92 & 6.28796695433563 & 5.95538119270907 & 6.62055271596219 \tabularnewline
93 & 6.28555846277121 & 5.92734786294322 & 6.6437690625992 \tabularnewline
94 & 6.26434726837366 & 5.88029578373874 & 6.64839875300859 \tabularnewline
95 & 6.24526476865332 & 5.83512483924419 & 6.65540469806245 \tabularnewline
96 & 6.22414955138849 & 5.78765002543119 & 6.66064907734578 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232584&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]6.0012203152194[/C][C]5.84912159769775[/C][C]6.15331903274105[/C][/ROW]
[ROW][C]86[/C][C]6.00052427581213[/C][C]5.82123931008082[/C][C]6.17980924154344[/C][/ROW]
[ROW][C]87[/C][C]6.01100863435351[/C][C]5.8055434999317[/C][C]6.21647376877533[/C][/ROW]
[ROW][C]88[/C][C]6.06908952284688[/C][C]5.83797952037115[/C][C]6.30019952532261[/C][/ROW]
[ROW][C]89[/C][C]6.19383534195892[/C][C]5.93733937692852[/C][C]6.45033130698931[/C][/ROW]
[ROW][C]90[/C][C]6.25774940899006[/C][C]5.97595106308273[/C][C]6.53954775489739[/C][/ROW]
[ROW][C]91[/C][C]6.28455771400671[/C][C]5.97742327751749[/C][C]6.59169215049593[/C][/ROW]
[ROW][C]92[/C][C]6.28796695433563[/C][C]5.95538119270907[/C][C]6.62055271596219[/C][/ROW]
[ROW][C]93[/C][C]6.28555846277121[/C][C]5.92734786294322[/C][C]6.6437690625992[/C][/ROW]
[ROW][C]94[/C][C]6.26434726837366[/C][C]5.88029578373874[/C][C]6.64839875300859[/C][/ROW]
[ROW][C]95[/C][C]6.24526476865332[/C][C]5.83512483924419[/C][C]6.65540469806245[/C][/ROW]
[ROW][C]96[/C][C]6.22414955138849[/C][C]5.78765002543119[/C][C]6.66064907734578[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232584&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232584&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	6.0012203152194	5.84912159769775	6.15331903274105
86	6.00052427581213	5.82123931008082	6.17980924154344
87	6.01100863435351	5.8055434999317	6.21647376877533
88	6.06908952284688	5.83797952037115	6.30019952532261
89	6.19383534195892	5.93733937692852	6.45033130698931
90	6.25774940899006	5.97595106308273	6.53954775489739
91	6.28455771400671	5.97742327751749	6.59169215049593
92	6.28796695433563	5.95538119270907	6.62055271596219
93	6.28555846277121	5.92734786294322	6.6437690625992
94	6.26434726837366	5.88029578373874	6.64839875300859
95	6.24526476865332	5.83512483924419	6.65540469806245
96	6.22414955138849	5.78765002543119	6.66064907734578

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 = additive ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; par3 = additive ;

R code (references can be found in the software module):

par1 <- as.numeric(par1)
if (par2 == 'Single') K <- 1
if (par2 == 'Double') K <- 2
if (par2 == 'Triple') K <- par1
nx <- length(x)
nxmK <- nx - K
x <- ts(x, frequency = par1)
if (par2 == 'Single') fit <- HoltWinters(x, gamma=F, beta=F)
if (par2 == 'Double') fit <- HoltWinters(x, gamma=F)
if (par2 == 'Triple') fit <- HoltWinters(x, seasonal=par3)
fit
myresid <- x - fit$fitted[,'xhat']
bitmap(file='test1.png')
op <- par(mfrow=c(2,1))
plot(fit,ylab='Observed (black) / Fitted (red)',main='Interpolation Fit of Exponential Smoothing')
plot(myresid,ylab='Residuals',main='Interpolation Prediction Errors')
par(op)
dev.off()
bitmap(file='test2.png')
p <- predict(fit, par1, prediction.interval=TRUE)
np <- length(p[,1])
plot(fit,p,ylab='Observed (black) / Fitted (red)',main='Extrapolation Fit of Exponential Smoothing')
dev.off()
bitmap(file='test3.png')
op <- par(mfrow = c(2,2))
acf(as.numeric(myresid),lag.max = nx/2,main='Residual ACF')
spectrum(myresid,main='Residals Periodogram')
cpgram(myresid,main='Residal Cumulative Periodogram')
qqnorm(myresid,main='Residual Normal QQ Plot')
qqline(myresid)
par(op)
dev.off()
load(file='createtable')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Estimated Parameters of Exponential Smoothing',2,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Parameter',header=TRUE)
a<-table.element(a,'Value',header=TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'alpha',header=TRUE)
a<-table.element(a,fit$alpha)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'beta',header=TRUE)
a<-table.element(a,fit$beta)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'gamma',header=TRUE)
a<-table.element(a,fit$gamma)
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file='mytable.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Interpolation Forecasts of Exponential Smoothing',4,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'t',header=TRUE)
a<-table.element(a,'Observed',header=TRUE)
a<-table.element(a,'Fitted',header=TRUE)
a<-table.element(a,'Residuals',header=TRUE)
a<-table.row.end(a)
for (i in 1:nxmK) {
a<-table.row.start(a)
a<-table.element(a,i+K,header=TRUE)
a<-table.element(a,x[i+K])
a<-table.element(a,fit$fitted[i,'xhat'])
a<-table.element(a,myresid[i])
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable1.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Extrapolation Forecasts of Exponential Smoothing',4,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'t',header=TRUE)
a<-table.element(a,'Forecast',header=TRUE)
a<-table.element(a,'95% Lower Bound',header=TRUE)
a<-table.element(a,'95% Upper Bound',header=TRUE)
a<-table.row.end(a)
for (i in 1:np) {
a<-table.row.start(a)
a<-table.element(a,nx+i,header=TRUE)
a<-table.element(a,p[i,'fit'])
a<-table.element(a,p[i,'lwr'])
a<-table.element(a,p[i,'upr'])
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable2.tab')

Free Statistics

Description of Statistical Computation

Tree of Dependent Computations

Dataset

Tables (Output of Computation)

Figures (Output of Computation)

Input Parameters & R Code