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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

Mon, 01 Jun 2009 13:09:38 -0600

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/2009/Jun/01/t1243883421t2lq1ur04mz4mic.htm/, Retrieved Wed, 03 Jun 2026 16:57:40 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=41048, Retrieved Wed, 03 Jun 2026 16:57:40 +0000

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Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords

Estimated Impact

344

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

-       [Exponential Smoothing] [Opgave 10 oefenin...] [2009-06-01 19:09:38] [c0b80eb26a0ae341c828c46b0228b15b] [Current]

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Dataseries X:

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5,29
5,29
5,29
5,31
5,33
5,34
5,34
5,37
5,41
5,41
5,38
5,44
5,44
5,46
5,46
5,45
5,46
5,46
5,48
5,47
5,48
5,51
5,55
5,58
5,59
5,6
5,6
5,67
5,71
5,7
5,73
5,72
5,75
5,75
5,77
5,83
5,85
5,87
5,86
5,87
5,93
5,97
5,98
5,99
5,99
6,03
6,06
6,07
6,08
6,08
6,1
6,13
6,14
6,14
6,16
6,2
6,19
6,32
6,32
6,33
6,32
6,33
6,38
6,42
6,46
6,47
6,42
6,48
6,47
6,49
6,48
6,51
6,51
6,52
6,57
6,59
6,62
6,63
6,61
6,64

Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	2 seconds
R Server	'Sir Ronald Aylmer Fisher' @ 193.190.124.24

\begin{tabular}{lllllllll}
\hline
Summary of computational transaction \tabularnewline
Raw Input & view raw input (R code)  \tabularnewline
Raw Output & view raw output of R engine  \tabularnewline
Computing time & 2 seconds \tabularnewline
R Server & 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=41048&T=0

[TABLE]
[ROW][C]Summary of computational transaction[/C][/ROW]
[ROW][C]Raw Input[/C][C]view raw input (R code) [/C][/ROW]
[ROW][C]Raw Output[/C][C]view raw output of R engine [/C][/ROW]
[ROW][C]Computing time[/C][C]2 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Sir Ronald Aylmer Fisher' @ 193.190.124.24[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=41048&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=41048&T=0

As an alternative you can also use a QR Code:

The GUIDs for individual cells are displayed in the table below:

Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	2 seconds
R Server	'Sir Ronald Aylmer Fisher' @ 193.190.124.24

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.999944487040144
beta	0
gamma	0

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.999944487040144 \tabularnewline
beta & 0 \tabularnewline
gamma & 0 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=41048&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]0.999944487040144[/C][/ROW]
[ROW][C]beta[/C][C]0[/C][/ROW]
[ROW][C]gamma[/C][C]0[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=41048&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=41048&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	0.999944487040144
beta	0
gamma	0

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
2	5.29	5.29	0
3	5.29	5.29	0
4	5.31	5.29	0.0199999999999996
5	5.33	5.3099988897408	0.0200011102591979
6	5.34	5.32999888967917	0.0100011103208306
7	5.34	5.33999944480876	5.55191236095709e-07
8	5.37	5.33999999996918	0.0300000000308209
9	5.41	5.3699983346112	0.0400016653887976
10	5.41	5.40999777938916	2.22061084453884e-06
11	5.38	5.40999999987673	-0.0299999998767273
12	5.44	5.38000166538879	0.0599983346112118
13	5.44	5.43999666931486	3.33068514102308e-06
14	5.46	5.4399999998151	0.0200000001848961
15	5.46	5.45999888974079	1.11025920723762e-06
16	5.45	5.45999999993837	-0.00999999993836642
17	5.46	5.4500005551296	0.00999944487040505
18	5.46	5.45999944490122	5.55098782051289e-07
19	5.48	5.45999999996918	0.0200000000308158
20	5.47	5.4799988897408	-0.00999888974080143
21	5.48	5.47000055506796	0.00999944493203575
22	5.51	5.47999944490122	0.0300005550987841
23	5.55	5.50999833458039	0.0400016654196111
24	5.58	5.54999777938915	0.0300022206108466
25	5.59	5.57999833448793	0.0100016655120685
26	5.6	5.58999944477794	0.0100005552220557
27	5.6	5.59999944483958	5.55160420745437e-07
28	5.67	5.59999999996918	0.0700000000308192
29	5.71	5.66999611409281	0.0400038859071916
30	5.7	5.70999777926589	-0.00999777926588763
31	5.73	5.70000055500632	0.0299994449936811
32	5.72	5.72999833464201	-0.00999833464201494
33	5.75	5.72000055503715	0.0299994449628507
34	5.75	5.74999833464202	1.66535798395984e-06
35	5.77	5.74999999990755	0.0200000000924483
36	5.83	5.7699988897408	0.0600011102592024
37	5.85	5.82999666916078	0.0200033308392245
38	5.87	5.8499988895559	0.0200011104441025
39	5.86	5.86999888967916	-0.0099988896791583
40	5.87	5.86000055506796	0.00999944493203841
41	5.93	5.86999944490122	0.0600005550987843
42	5.97	5.92999666919159	0.0400033308084069
43	5.98	5.9699977792967	0.0100022207032984
44	5.99	5.97999944474712	0.0100005552528764
45	5.99	5.98999944483958	5.55160422521794e-07
46	6.03	5.98999999996918	0.0400000000308189
47	6.06	6.0299977794816	0.0300022205183952
48	6.07	6.05999833448794	0.0100016655120641
49	6.08	6.06999944477794	0.0100005552220557
50	6.08	6.07999944483958	5.55160420745437e-07
51	6.1	6.07999999996918	0.0200000000308185
52	6.13	6.0999988897408	0.0300011102591995
53	6.14	6.12999833454957	0.0100016654504289
54	6.14	6.13999944477795	5.55222052334159e-07
55	6.16	6.13999999996918	0.0200000000308229
56	6.2	6.1599988897408	0.0400011102591993
57	6.19	6.19999777941997	-0.00999777941997149
58	6.32	6.19000055500633	0.129999444993672
59	6.32	6.31999278334603	7.21665397129811e-06
60	6.33	6.31999999959938	0.010000000400618
61	6.32	6.32999944487038	-0.0099994448703793
62	6.33	6.32000055509878	0.00999944490121774
63	6.38	6.32999944490122	0.0500005550987828
64	6.42	6.37999722432119	0.0400027756788077
65	6.46	6.41999777932752	0.0400022206724806
66	6.47	6.45999777935833	0.0100022206416703
67	6.42	6.46999944474713	-0.0499994447471268
68	6.48	6.42000277561717	0.0599972243828315
69	6.47	6.47999666937649	-0.00999666937649213
70	6.49	6.4700005549447	0.0199994450552943
71	6.48	6.48999888977161	-0.00999888977160968
72	6.51	6.48000055506797	0.0299994449320327
73	6.51	6.50999833464202	1.66535798218348e-06
74	6.52	6.50999999990755	0.0100000000924485
75	6.57	6.5199994448704	0.0500005551296043
76	6.59	6.56999722432119	0.0200027756788090
77	6.62	6.58999888958672	0.0300011104132833
78	6.63	6.61999833454956	0.0100016654504378
79	6.61	6.62999944477795	-0.0199994447779472
80	6.64	6.61000111022838	0.0299988897716243

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
2 & 5.29 & 5.29 & 0 \tabularnewline
3 & 5.29 & 5.29 & 0 \tabularnewline
4 & 5.31 & 5.29 & 0.0199999999999996 \tabularnewline
5 & 5.33 & 5.3099988897408 & 0.0200011102591979 \tabularnewline
6 & 5.34 & 5.32999888967917 & 0.0100011103208306 \tabularnewline
7 & 5.34 & 5.33999944480876 & 5.55191236095709e-07 \tabularnewline
8 & 5.37 & 5.33999999996918 & 0.0300000000308209 \tabularnewline
9 & 5.41 & 5.3699983346112 & 0.0400016653887976 \tabularnewline
10 & 5.41 & 5.40999777938916 & 2.22061084453884e-06 \tabularnewline
11 & 5.38 & 5.40999999987673 & -0.0299999998767273 \tabularnewline
12 & 5.44 & 5.38000166538879 & 0.0599983346112118 \tabularnewline
13 & 5.44 & 5.43999666931486 & 3.33068514102308e-06 \tabularnewline
14 & 5.46 & 5.4399999998151 & 0.0200000001848961 \tabularnewline
15 & 5.46 & 5.45999888974079 & 1.11025920723762e-06 \tabularnewline
16 & 5.45 & 5.45999999993837 & -0.00999999993836642 \tabularnewline
17 & 5.46 & 5.4500005551296 & 0.00999944487040505 \tabularnewline
18 & 5.46 & 5.45999944490122 & 5.55098782051289e-07 \tabularnewline
19 & 5.48 & 5.45999999996918 & 0.0200000000308158 \tabularnewline
20 & 5.47 & 5.4799988897408 & -0.00999888974080143 \tabularnewline
21 & 5.48 & 5.47000055506796 & 0.00999944493203575 \tabularnewline
22 & 5.51 & 5.47999944490122 & 0.0300005550987841 \tabularnewline
23 & 5.55 & 5.50999833458039 & 0.0400016654196111 \tabularnewline
24 & 5.58 & 5.54999777938915 & 0.0300022206108466 \tabularnewline
25 & 5.59 & 5.57999833448793 & 0.0100016655120685 \tabularnewline
26 & 5.6 & 5.58999944477794 & 0.0100005552220557 \tabularnewline
27 & 5.6 & 5.59999944483958 & 5.55160420745437e-07 \tabularnewline
28 & 5.67 & 5.59999999996918 & 0.0700000000308192 \tabularnewline
29 & 5.71 & 5.66999611409281 & 0.0400038859071916 \tabularnewline
30 & 5.7 & 5.70999777926589 & -0.00999777926588763 \tabularnewline
31 & 5.73 & 5.70000055500632 & 0.0299994449936811 \tabularnewline
32 & 5.72 & 5.72999833464201 & -0.00999833464201494 \tabularnewline
33 & 5.75 & 5.72000055503715 & 0.0299994449628507 \tabularnewline
34 & 5.75 & 5.74999833464202 & 1.66535798395984e-06 \tabularnewline
35 & 5.77 & 5.74999999990755 & 0.0200000000924483 \tabularnewline
36 & 5.83 & 5.7699988897408 & 0.0600011102592024 \tabularnewline
37 & 5.85 & 5.82999666916078 & 0.0200033308392245 \tabularnewline
38 & 5.87 & 5.8499988895559 & 0.0200011104441025 \tabularnewline
39 & 5.86 & 5.86999888967916 & -0.0099988896791583 \tabularnewline
40 & 5.87 & 5.86000055506796 & 0.00999944493203841 \tabularnewline
41 & 5.93 & 5.86999944490122 & 0.0600005550987843 \tabularnewline
42 & 5.97 & 5.92999666919159 & 0.0400033308084069 \tabularnewline
43 & 5.98 & 5.9699977792967 & 0.0100022207032984 \tabularnewline
44 & 5.99 & 5.97999944474712 & 0.0100005552528764 \tabularnewline
45 & 5.99 & 5.98999944483958 & 5.55160422521794e-07 \tabularnewline
46 & 6.03 & 5.98999999996918 & 0.0400000000308189 \tabularnewline
47 & 6.06 & 6.0299977794816 & 0.0300022205183952 \tabularnewline
48 & 6.07 & 6.05999833448794 & 0.0100016655120641 \tabularnewline
49 & 6.08 & 6.06999944477794 & 0.0100005552220557 \tabularnewline
50 & 6.08 & 6.07999944483958 & 5.55160420745437e-07 \tabularnewline
51 & 6.1 & 6.07999999996918 & 0.0200000000308185 \tabularnewline
52 & 6.13 & 6.0999988897408 & 0.0300011102591995 \tabularnewline
53 & 6.14 & 6.12999833454957 & 0.0100016654504289 \tabularnewline
54 & 6.14 & 6.13999944477795 & 5.55222052334159e-07 \tabularnewline
55 & 6.16 & 6.13999999996918 & 0.0200000000308229 \tabularnewline
56 & 6.2 & 6.1599988897408 & 0.0400011102591993 \tabularnewline
57 & 6.19 & 6.19999777941997 & -0.00999777941997149 \tabularnewline
58 & 6.32 & 6.19000055500633 & 0.129999444993672 \tabularnewline
59 & 6.32 & 6.31999278334603 & 7.21665397129811e-06 \tabularnewline
60 & 6.33 & 6.31999999959938 & 0.010000000400618 \tabularnewline
61 & 6.32 & 6.32999944487038 & -0.0099994448703793 \tabularnewline
62 & 6.33 & 6.32000055509878 & 0.00999944490121774 \tabularnewline
63 & 6.38 & 6.32999944490122 & 0.0500005550987828 \tabularnewline
64 & 6.42 & 6.37999722432119 & 0.0400027756788077 \tabularnewline
65 & 6.46 & 6.41999777932752 & 0.0400022206724806 \tabularnewline
66 & 6.47 & 6.45999777935833 & 0.0100022206416703 \tabularnewline
67 & 6.42 & 6.46999944474713 & -0.0499994447471268 \tabularnewline
68 & 6.48 & 6.42000277561717 & 0.0599972243828315 \tabularnewline
69 & 6.47 & 6.47999666937649 & -0.00999666937649213 \tabularnewline
70 & 6.49 & 6.4700005549447 & 0.0199994450552943 \tabularnewline
71 & 6.48 & 6.48999888977161 & -0.00999888977160968 \tabularnewline
72 & 6.51 & 6.48000055506797 & 0.0299994449320327 \tabularnewline
73 & 6.51 & 6.50999833464202 & 1.66535798218348e-06 \tabularnewline
74 & 6.52 & 6.50999999990755 & 0.0100000000924485 \tabularnewline
75 & 6.57 & 6.5199994448704 & 0.0500005551296043 \tabularnewline
76 & 6.59 & 6.56999722432119 & 0.0200027756788090 \tabularnewline
77 & 6.62 & 6.58999888958672 & 0.0300011104132833 \tabularnewline
78 & 6.63 & 6.61999833454956 & 0.0100016654504378 \tabularnewline
79 & 6.61 & 6.62999944477795 & -0.0199994447779472 \tabularnewline
80 & 6.64 & 6.61000111022838 & 0.0299988897716243 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=41048&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]2[/C][C]5.29[/C][C]5.29[/C][C]0[/C][/ROW]
[ROW][C]3[/C][C]5.29[/C][C]5.29[/C][C]0[/C][/ROW]
[ROW][C]4[/C][C]5.31[/C][C]5.29[/C][C]0.0199999999999996[/C][/ROW]
[ROW][C]5[/C][C]5.33[/C][C]5.3099988897408[/C][C]0.0200011102591979[/C][/ROW]
[ROW][C]6[/C][C]5.34[/C][C]5.32999888967917[/C][C]0.0100011103208306[/C][/ROW]
[ROW][C]7[/C][C]5.34[/C][C]5.33999944480876[/C][C]5.55191236095709e-07[/C][/ROW]
[ROW][C]8[/C][C]5.37[/C][C]5.33999999996918[/C][C]0.0300000000308209[/C][/ROW]
[ROW][C]9[/C][C]5.41[/C][C]5.3699983346112[/C][C]0.0400016653887976[/C][/ROW]
[ROW][C]10[/C][C]5.41[/C][C]5.40999777938916[/C][C]2.22061084453884e-06[/C][/ROW]
[ROW][C]11[/C][C]5.38[/C][C]5.40999999987673[/C][C]-0.0299999998767273[/C][/ROW]
[ROW][C]12[/C][C]5.44[/C][C]5.38000166538879[/C][C]0.0599983346112118[/C][/ROW]
[ROW][C]13[/C][C]5.44[/C][C]5.43999666931486[/C][C]3.33068514102308e-06[/C][/ROW]
[ROW][C]14[/C][C]5.46[/C][C]5.4399999998151[/C][C]0.0200000001848961[/C][/ROW]
[ROW][C]15[/C][C]5.46[/C][C]5.45999888974079[/C][C]1.11025920723762e-06[/C][/ROW]
[ROW][C]16[/C][C]5.45[/C][C]5.45999999993837[/C][C]-0.00999999993836642[/C][/ROW]
[ROW][C]17[/C][C]5.46[/C][C]5.4500005551296[/C][C]0.00999944487040505[/C][/ROW]
[ROW][C]18[/C][C]5.46[/C][C]5.45999944490122[/C][C]5.55098782051289e-07[/C][/ROW]
[ROW][C]19[/C][C]5.48[/C][C]5.45999999996918[/C][C]0.0200000000308158[/C][/ROW]
[ROW][C]20[/C][C]5.47[/C][C]5.4799988897408[/C][C]-0.00999888974080143[/C][/ROW]
[ROW][C]21[/C][C]5.48[/C][C]5.47000055506796[/C][C]0.00999944493203575[/C][/ROW]
[ROW][C]22[/C][C]5.51[/C][C]5.47999944490122[/C][C]0.0300005550987841[/C][/ROW]
[ROW][C]23[/C][C]5.55[/C][C]5.50999833458039[/C][C]0.0400016654196111[/C][/ROW]
[ROW][C]24[/C][C]5.58[/C][C]5.54999777938915[/C][C]0.0300022206108466[/C][/ROW]
[ROW][C]25[/C][C]5.59[/C][C]5.57999833448793[/C][C]0.0100016655120685[/C][/ROW]
[ROW][C]26[/C][C]5.6[/C][C]5.58999944477794[/C][C]0.0100005552220557[/C][/ROW]
[ROW][C]27[/C][C]5.6[/C][C]5.59999944483958[/C][C]5.55160420745437e-07[/C][/ROW]
[ROW][C]28[/C][C]5.67[/C][C]5.59999999996918[/C][C]0.0700000000308192[/C][/ROW]
[ROW][C]29[/C][C]5.71[/C][C]5.66999611409281[/C][C]0.0400038859071916[/C][/ROW]
[ROW][C]30[/C][C]5.7[/C][C]5.70999777926589[/C][C]-0.00999777926588763[/C][/ROW]
[ROW][C]31[/C][C]5.73[/C][C]5.70000055500632[/C][C]0.0299994449936811[/C][/ROW]
[ROW][C]32[/C][C]5.72[/C][C]5.72999833464201[/C][C]-0.00999833464201494[/C][/ROW]
[ROW][C]33[/C][C]5.75[/C][C]5.72000055503715[/C][C]0.0299994449628507[/C][/ROW]
[ROW][C]34[/C][C]5.75[/C][C]5.74999833464202[/C][C]1.66535798395984e-06[/C][/ROW]
[ROW][C]35[/C][C]5.77[/C][C]5.74999999990755[/C][C]0.0200000000924483[/C][/ROW]
[ROW][C]36[/C][C]5.83[/C][C]5.7699988897408[/C][C]0.0600011102592024[/C][/ROW]
[ROW][C]37[/C][C]5.85[/C][C]5.82999666916078[/C][C]0.0200033308392245[/C][/ROW]
[ROW][C]38[/C][C]5.87[/C][C]5.8499988895559[/C][C]0.0200011104441025[/C][/ROW]
[ROW][C]39[/C][C]5.86[/C][C]5.86999888967916[/C][C]-0.0099988896791583[/C][/ROW]
[ROW][C]40[/C][C]5.87[/C][C]5.86000055506796[/C][C]0.00999944493203841[/C][/ROW]
[ROW][C]41[/C][C]5.93[/C][C]5.86999944490122[/C][C]0.0600005550987843[/C][/ROW]
[ROW][C]42[/C][C]5.97[/C][C]5.92999666919159[/C][C]0.0400033308084069[/C][/ROW]
[ROW][C]43[/C][C]5.98[/C][C]5.9699977792967[/C][C]0.0100022207032984[/C][/ROW]
[ROW][C]44[/C][C]5.99[/C][C]5.97999944474712[/C][C]0.0100005552528764[/C][/ROW]
[ROW][C]45[/C][C]5.99[/C][C]5.98999944483958[/C][C]5.55160422521794e-07[/C][/ROW]
[ROW][C]46[/C][C]6.03[/C][C]5.98999999996918[/C][C]0.0400000000308189[/C][/ROW]
[ROW][C]47[/C][C]6.06[/C][C]6.0299977794816[/C][C]0.0300022205183952[/C][/ROW]
[ROW][C]48[/C][C]6.07[/C][C]6.05999833448794[/C][C]0.0100016655120641[/C][/ROW]
[ROW][C]49[/C][C]6.08[/C][C]6.06999944477794[/C][C]0.0100005552220557[/C][/ROW]
[ROW][C]50[/C][C]6.08[/C][C]6.07999944483958[/C][C]5.55160420745437e-07[/C][/ROW]
[ROW][C]51[/C][C]6.1[/C][C]6.07999999996918[/C][C]0.0200000000308185[/C][/ROW]
[ROW][C]52[/C][C]6.13[/C][C]6.0999988897408[/C][C]0.0300011102591995[/C][/ROW]
[ROW][C]53[/C][C]6.14[/C][C]6.12999833454957[/C][C]0.0100016654504289[/C][/ROW]
[ROW][C]54[/C][C]6.14[/C][C]6.13999944477795[/C][C]5.55222052334159e-07[/C][/ROW]
[ROW][C]55[/C][C]6.16[/C][C]6.13999999996918[/C][C]0.0200000000308229[/C][/ROW]
[ROW][C]56[/C][C]6.2[/C][C]6.1599988897408[/C][C]0.0400011102591993[/C][/ROW]
[ROW][C]57[/C][C]6.19[/C][C]6.19999777941997[/C][C]-0.00999777941997149[/C][/ROW]
[ROW][C]58[/C][C]6.32[/C][C]6.19000055500633[/C][C]0.129999444993672[/C][/ROW]
[ROW][C]59[/C][C]6.32[/C][C]6.31999278334603[/C][C]7.21665397129811e-06[/C][/ROW]
[ROW][C]60[/C][C]6.33[/C][C]6.31999999959938[/C][C]0.010000000400618[/C][/ROW]
[ROW][C]61[/C][C]6.32[/C][C]6.32999944487038[/C][C]-0.0099994448703793[/C][/ROW]
[ROW][C]62[/C][C]6.33[/C][C]6.32000055509878[/C][C]0.00999944490121774[/C][/ROW]
[ROW][C]63[/C][C]6.38[/C][C]6.32999944490122[/C][C]0.0500005550987828[/C][/ROW]
[ROW][C]64[/C][C]6.42[/C][C]6.37999722432119[/C][C]0.0400027756788077[/C][/ROW]
[ROW][C]65[/C][C]6.46[/C][C]6.41999777932752[/C][C]0.0400022206724806[/C][/ROW]
[ROW][C]66[/C][C]6.47[/C][C]6.45999777935833[/C][C]0.0100022206416703[/C][/ROW]
[ROW][C]67[/C][C]6.42[/C][C]6.46999944474713[/C][C]-0.0499994447471268[/C][/ROW]
[ROW][C]68[/C][C]6.48[/C][C]6.42000277561717[/C][C]0.0599972243828315[/C][/ROW]
[ROW][C]69[/C][C]6.47[/C][C]6.47999666937649[/C][C]-0.00999666937649213[/C][/ROW]
[ROW][C]70[/C][C]6.49[/C][C]6.4700005549447[/C][C]0.0199994450552943[/C][/ROW]
[ROW][C]71[/C][C]6.48[/C][C]6.48999888977161[/C][C]-0.00999888977160968[/C][/ROW]
[ROW][C]72[/C][C]6.51[/C][C]6.48000055506797[/C][C]0.0299994449320327[/C][/ROW]
[ROW][C]73[/C][C]6.51[/C][C]6.50999833464202[/C][C]1.66535798218348e-06[/C][/ROW]
[ROW][C]74[/C][C]6.52[/C][C]6.50999999990755[/C][C]0.0100000000924485[/C][/ROW]
[ROW][C]75[/C][C]6.57[/C][C]6.5199994448704[/C][C]0.0500005551296043[/C][/ROW]
[ROW][C]76[/C][C]6.59[/C][C]6.56999722432119[/C][C]0.0200027756788090[/C][/ROW]
[ROW][C]77[/C][C]6.62[/C][C]6.58999888958672[/C][C]0.0300011104132833[/C][/ROW]
[ROW][C]78[/C][C]6.63[/C][C]6.61999833454956[/C][C]0.0100016654504378[/C][/ROW]
[ROW][C]79[/C][C]6.61[/C][C]6.62999944477795[/C][C]-0.0199994447779472[/C][/ROW]
[ROW][C]80[/C][C]6.64[/C][C]6.61000111022838[/C][C]0.0299988897716243[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=41048&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=41048&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
2	5.29	5.29	0
3	5.29	5.29	0
4	5.31	5.29	0.0199999999999996
5	5.33	5.3099988897408	0.0200011102591979
6	5.34	5.32999888967917	0.0100011103208306
7	5.34	5.33999944480876	5.55191236095709e-07
8	5.37	5.33999999996918	0.0300000000308209
9	5.41	5.3699983346112	0.0400016653887976
10	5.41	5.40999777938916	2.22061084453884e-06
11	5.38	5.40999999987673	-0.0299999998767273
12	5.44	5.38000166538879	0.0599983346112118
13	5.44	5.43999666931486	3.33068514102308e-06
14	5.46	5.4399999998151	0.0200000001848961
15	5.46	5.45999888974079	1.11025920723762e-06
16	5.45	5.45999999993837	-0.00999999993836642
17	5.46	5.4500005551296	0.00999944487040505
18	5.46	5.45999944490122	5.55098782051289e-07
19	5.48	5.45999999996918	0.0200000000308158
20	5.47	5.4799988897408	-0.00999888974080143
21	5.48	5.47000055506796	0.00999944493203575
22	5.51	5.47999944490122	0.0300005550987841
23	5.55	5.50999833458039	0.0400016654196111
24	5.58	5.54999777938915	0.0300022206108466
25	5.59	5.57999833448793	0.0100016655120685
26	5.6	5.58999944477794	0.0100005552220557
27	5.6	5.59999944483958	5.55160420745437e-07
28	5.67	5.59999999996918	0.0700000000308192
29	5.71	5.66999611409281	0.0400038859071916
30	5.7	5.70999777926589	-0.00999777926588763
31	5.73	5.70000055500632	0.0299994449936811
32	5.72	5.72999833464201	-0.00999833464201494
33	5.75	5.72000055503715	0.0299994449628507
34	5.75	5.74999833464202	1.66535798395984e-06
35	5.77	5.74999999990755	0.0200000000924483
36	5.83	5.7699988897408	0.0600011102592024
37	5.85	5.82999666916078	0.0200033308392245
38	5.87	5.8499988895559	0.0200011104441025
39	5.86	5.86999888967916	-0.0099988896791583
40	5.87	5.86000055506796	0.00999944493203841
41	5.93	5.86999944490122	0.0600005550987843
42	5.97	5.92999666919159	0.0400033308084069
43	5.98	5.9699977792967	0.0100022207032984
44	5.99	5.97999944474712	0.0100005552528764
45	5.99	5.98999944483958	5.55160422521794e-07
46	6.03	5.98999999996918	0.0400000000308189
47	6.06	6.0299977794816	0.0300022205183952
48	6.07	6.05999833448794	0.0100016655120641
49	6.08	6.06999944477794	0.0100005552220557
50	6.08	6.07999944483958	5.55160420745437e-07
51	6.1	6.07999999996918	0.0200000000308185
52	6.13	6.0999988897408	0.0300011102591995
53	6.14	6.12999833454957	0.0100016654504289
54	6.14	6.13999944477795	5.55222052334159e-07
55	6.16	6.13999999996918	0.0200000000308229
56	6.2	6.1599988897408	0.0400011102591993
57	6.19	6.19999777941997	-0.00999777941997149
58	6.32	6.19000055500633	0.129999444993672
59	6.32	6.31999278334603	7.21665397129811e-06
60	6.33	6.31999999959938	0.010000000400618
61	6.32	6.32999944487038	-0.0099994448703793
62	6.33	6.32000055509878	0.00999944490121774
63	6.38	6.32999944490122	0.0500005550987828
64	6.42	6.37999722432119	0.0400027756788077
65	6.46	6.41999777932752	0.0400022206724806
66	6.47	6.45999777935833	0.0100022206416703
67	6.42	6.46999944474713	-0.0499994447471268
68	6.48	6.42000277561717	0.0599972243828315
69	6.47	6.47999666937649	-0.00999666937649213
70	6.49	6.4700005549447	0.0199994450552943
71	6.48	6.48999888977161	-0.00999888977160968
72	6.51	6.48000055506797	0.0299994449320327
73	6.51	6.50999833464202	1.66535798218348e-06
74	6.52	6.50999999990755	0.0100000000924485
75	6.57	6.5199994448704	0.0500005551296043
76	6.59	6.56999722432119	0.0200027756788090
77	6.62	6.58999888958672	0.0300011104132833
78	6.63	6.61999833454956	0.0100016654504378
79	6.61	6.62999944477795	-0.0199994447779472
80	6.64	6.61000111022838	0.0299988897716243

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
81	6.63999833467284	6.59006371065429	6.68993295869139
82	6.63999833467284	6.56938207223987	6.7106145971058
83	6.63999833467284	6.55351222963782	6.72648443970786
84	6.63999833467284	6.54013324463505	6.73986342471062
85	6.63999833467284	6.52834607964687	6.7516505896988
86	6.63999833467284	6.51768964366312	6.76230702568255
87	6.63999833467284	6.50789002403101	6.77210664531466
88	6.63999833467284	6.49876875000743	6.78122791933825
89	6.63999833467284	6.49020185464446	6.78979481470121
90	6.63999833467284	6.48209907795192	6.79789759139375
91	6.63999833467284	6.47439228068584	6.80560438865984
92	6.63999833467284	6.46702852528035	6.81296814406532

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
81 & 6.63999833467284 & 6.59006371065429 & 6.68993295869139 \tabularnewline
82 & 6.63999833467284 & 6.56938207223987 & 6.7106145971058 \tabularnewline
83 & 6.63999833467284 & 6.55351222963782 & 6.72648443970786 \tabularnewline
84 & 6.63999833467284 & 6.54013324463505 & 6.73986342471062 \tabularnewline
85 & 6.63999833467284 & 6.52834607964687 & 6.7516505896988 \tabularnewline
86 & 6.63999833467284 & 6.51768964366312 & 6.76230702568255 \tabularnewline
87 & 6.63999833467284 & 6.50789002403101 & 6.77210664531466 \tabularnewline
88 & 6.63999833467284 & 6.49876875000743 & 6.78122791933825 \tabularnewline
89 & 6.63999833467284 & 6.49020185464446 & 6.78979481470121 \tabularnewline
90 & 6.63999833467284 & 6.48209907795192 & 6.79789759139375 \tabularnewline
91 & 6.63999833467284 & 6.47439228068584 & 6.80560438865984 \tabularnewline
92 & 6.63999833467284 & 6.46702852528035 & 6.81296814406532 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=41048&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]81[/C][C]6.63999833467284[/C][C]6.59006371065429[/C][C]6.68993295869139[/C][/ROW]
[ROW][C]82[/C][C]6.63999833467284[/C][C]6.56938207223987[/C][C]6.7106145971058[/C][/ROW]
[ROW][C]83[/C][C]6.63999833467284[/C][C]6.55351222963782[/C][C]6.72648443970786[/C][/ROW]
[ROW][C]84[/C][C]6.63999833467284[/C][C]6.54013324463505[/C][C]6.73986342471062[/C][/ROW]
[ROW][C]85[/C][C]6.63999833467284[/C][C]6.52834607964687[/C][C]6.7516505896988[/C][/ROW]
[ROW][C]86[/C][C]6.63999833467284[/C][C]6.51768964366312[/C][C]6.76230702568255[/C][/ROW]
[ROW][C]87[/C][C]6.63999833467284[/C][C]6.50789002403101[/C][C]6.77210664531466[/C][/ROW]
[ROW][C]88[/C][C]6.63999833467284[/C][C]6.49876875000743[/C][C]6.78122791933825[/C][/ROW]
[ROW][C]89[/C][C]6.63999833467284[/C][C]6.49020185464446[/C][C]6.78979481470121[/C][/ROW]
[ROW][C]90[/C][C]6.63999833467284[/C][C]6.48209907795192[/C][C]6.79789759139375[/C][/ROW]
[ROW][C]91[/C][C]6.63999833467284[/C][C]6.47439228068584[/C][C]6.80560438865984[/C][/ROW]
[ROW][C]92[/C][C]6.63999833467284[/C][C]6.46702852528035[/C][C]6.81296814406532[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=41048&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=41048&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
81	6.63999833467284	6.59006371065429	6.68993295869139
82	6.63999833467284	6.56938207223987	6.7106145971058
83	6.63999833467284	6.55351222963782	6.72648443970786
84	6.63999833467284	6.54013324463505	6.73986342471062
85	6.63999833467284	6.52834607964687	6.7516505896988
86	6.63999833467284	6.51768964366312	6.76230702568255
87	6.63999833467284	6.50789002403101	6.77210664531466
88	6.63999833467284	6.49876875000743	6.78122791933825
89	6.63999833467284	6.49020185464446	6.78979481470121
90	6.63999833467284	6.48209907795192	6.79789759139375
91	6.63999833467284	6.47439228068584	6.80560438865984
92	6.63999833467284	6.46702852528035	6.81296814406532

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 = Single ; par3 = additive ;

Parameters (R input):

par1 = 12 ; par2 = Single ; 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=0, beta=0)
if (par2 == 'Double') fit <- HoltWinters(x, gamma=0)
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