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*Unverified author*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Wed, 25 Nov 2015 18:48:04 +0000

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Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2015/Nov/25/t1448477334gyjc46lw4yqlqo6.htm/, Retrieved Thu, 16 May 2024 01:07:14 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=284152, Retrieved Thu, 16 May 2024 01:07:14 +0000

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Estimated Impact

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

-       [Exponential Smoothing] [] [2015-11-25 18:48:04] [45f7fcffe569af381f5269bedb4e9f14] [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	2 seconds
R Server	'Sir Ronald Aylmer Fisher' @ fisher.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 & 2 seconds \tabularnewline
R Server & 'Sir Ronald Aylmer Fisher' @ fisher.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284152&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' @ fisher.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284152&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284152&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' @ fisher.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.77370614769999
beta	0
gamma	1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.77370614769999 \tabularnewline
beta & 0 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284152&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]0.77370614769999[/C][/ROW]
[ROW][C]beta[/C][C]0[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284152&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284152&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	0.77370614769999
beta	0
gamma	1

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	83.09	81.6837419871795	1.40625801282047
14	83.21	82.9172284592821	0.292771540717936
15	83.13	83.1208702692048	0.00912973079519475
16	84.31	84.3246400003789	-0.014640000378904
17	83.62	83.5821022777478	0.0378977222522394
18	83.75	83.6468799807691	0.103120019230857
19	84.1	84.2112872425967	-0.111287242596688
20	83.71	84.29313962117	-0.583139621170019
21	84.2	84.4532502469677	-0.253250246967653
22	85.13	84.0181816429799	1.11181835702007
23	86.16	85.627608343263	0.532391656736962
24	86.65	85.5545623767289	1.09543762327107
25	87.44	87.5182927456556	-0.0782927456555598
26	87.62	87.3511980260965	0.268801973903535
27	88.03	87.4721080369764	0.557891963023565
28	89.1	89.0950795368157	0.0049204631843196
29	89.68	88.3795648287405	1.30043517125948
30	89.47	89.4359349225994	0.0340650774005837
31	90.13	89.8983949061638	0.231605093836237
32	89.49	90.1387679009702	-0.648767900970185
33	89.52	90.3227534605445	-0.802753460544551
34	89.86	89.7714374750816	0.088562524918359
35	89.77	90.4580441472653	-0.688044147265259
36	89.8	89.5681533370904	0.231846662909561
37	90.89	90.5981101041413	0.291889895858688
38	90.82	90.7959733712956	0.0240266287043625
39	90.68	90.7929184800889	-0.112918480088936
40	90.92	91.7717457652399	-0.85174576523994
41	90.82	90.6865901437076	0.13340985629236
42	90.09	90.5534538098781	-0.463453809878089
43	89.71	90.6756824630607	-0.965682463060702
44	89.34	89.7904837180756	-0.450483718075617
45	89.2	90.0930369834725	-0.893036983472541
46	89.48	89.6735674092512	-0.193567409251216
47	89.72	89.9661471013473	-0.246147101347319
48	89.58	89.6263203873795	-0.0463203873795095
49	90.65	90.4546450120228	0.19535498797724
50	90.93	90.5172028168675	0.412797183132469
51	91.42	90.7839522574441	0.63604774255586
52	91.52	92.1750672409339	-0.655067240933889
53	91.76	91.4650176634893	0.294982336510714
54	91.47	91.3218243725882	0.148175627411788
55	91.37	91.8036232248521	-0.433623224852127
56	91.35	91.4466682921124	-0.0966682921124118
57	91.74	91.9228236444535	-0.182823644453521
58	91.78	92.2111361613269	-0.431136161326918
59	91.88	92.3080089883635	-0.428008988363459
60	91.99	91.8726941712752	0.117305828724838
61	92.55	92.8823070569388	-0.332307056938774
62	92.94	92.5858153257183	0.354184674281683
63	92.81	92.857736136985	-0.0477361369849802
64	93.35	93.4276319457997	-0.0776319457996664
65	93.72	93.3793379848553	0.340662015144673
66	93.94	93.2382658863928	0.701734113607188
67	94.03	94.0166988389951	0.0133011610049181
68	93.66	94.0817828809312	-0.421782880931175
69	93.78	94.2868986506187	-0.506898650618723
70	94.1	94.2682807468886	-0.168280746888612
71	94.85	94.569234084049	0.280765915950994
72	94.83	94.8057041584394	0.0242958415605585
73	95.06	95.641610013296	-0.581610013296
74	95.87	95.3075799105322	0.562420089467835
75	95.97	95.6496615339962	0.320338466003847
76	95.96	96.4975736882112	-0.537573688211211
77	96.3	96.1880773253951	0.111922674604855
78	96.17	95.9517365890553	0.218263410944743
79	96.18	96.2003171418801	-0.0203171418801134
80	96.55	96.1409336522749	0.409066347725144
81	96.76	96.9696214025715	-0.209621402571528
82	97.63	97.2576358831197	0.372364116880291
83	97.86	98.078505974297	-0.218505974296974
84	97.82	97.8706487166953	-0.0506487166953065
85	98.62	98.511456736066	0.108543263933981
86	99.24	98.9702894458519	0.269710554148062
87	99.63	99.0311183192039	0.598881680796097
88	100.27	99.9004008247915	0.369599175208492
89	100.84	100.439766717426	0.400233282573652
90	101.05	100.450557925802	0.59944207419818
91	100.38	100.940069441375	-0.560069441375319
92	100.02	100.560243123392	-0.540243123392187
93	99.97	100.51443906543	-0.544439065430112
94	99.95	100.675102807046	-0.725102807045616
95	100	100.513145723143	-0.513145723142657
96	100.04	100.115308945961	-0.0753089459614813
97	100.51	100.773061360897	-0.26306136089714
98	100.29	100.980852454905	-0.690852454904757
99	100.22	100.372977225214	-0.152977225214485
100	101.29	100.608656651564	0.681343348435703
101	100.29	101.396153237702	-1.10615323770212
102	100.26	100.286523659397	-0.0265236593965739
103	100.39	100.029331310993	0.36066868900707
104	99.3	100.366372318782	-1.06637231878177
105	98.9	99.9125493519745	-1.01254935197454
106	98.76	99.6701501930279	-0.910150193027903
107	99.12	99.4129853940133	-0.292985394013314
108	99.28	99.2845677879461	-0.00456778794611523

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 83.09 & 81.6837419871795 & 1.40625801282047 \tabularnewline
14 & 83.21 & 82.9172284592821 & 0.292771540717936 \tabularnewline
15 & 83.13 & 83.1208702692048 & 0.00912973079519475 \tabularnewline
16 & 84.31 & 84.3246400003789 & -0.014640000378904 \tabularnewline
17 & 83.62 & 83.5821022777478 & 0.0378977222522394 \tabularnewline
18 & 83.75 & 83.6468799807691 & 0.103120019230857 \tabularnewline
19 & 84.1 & 84.2112872425967 & -0.111287242596688 \tabularnewline
20 & 83.71 & 84.29313962117 & -0.583139621170019 \tabularnewline
21 & 84.2 & 84.4532502469677 & -0.253250246967653 \tabularnewline
22 & 85.13 & 84.0181816429799 & 1.11181835702007 \tabularnewline
23 & 86.16 & 85.627608343263 & 0.532391656736962 \tabularnewline
24 & 86.65 & 85.5545623767289 & 1.09543762327107 \tabularnewline
25 & 87.44 & 87.5182927456556 & -0.0782927456555598 \tabularnewline
26 & 87.62 & 87.3511980260965 & 0.268801973903535 \tabularnewline
27 & 88.03 & 87.4721080369764 & 0.557891963023565 \tabularnewline
28 & 89.1 & 89.0950795368157 & 0.0049204631843196 \tabularnewline
29 & 89.68 & 88.3795648287405 & 1.30043517125948 \tabularnewline
30 & 89.47 & 89.4359349225994 & 0.0340650774005837 \tabularnewline
31 & 90.13 & 89.8983949061638 & 0.231605093836237 \tabularnewline
32 & 89.49 & 90.1387679009702 & -0.648767900970185 \tabularnewline
33 & 89.52 & 90.3227534605445 & -0.802753460544551 \tabularnewline
34 & 89.86 & 89.7714374750816 & 0.088562524918359 \tabularnewline
35 & 89.77 & 90.4580441472653 & -0.688044147265259 \tabularnewline
36 & 89.8 & 89.5681533370904 & 0.231846662909561 \tabularnewline
37 & 90.89 & 90.5981101041413 & 0.291889895858688 \tabularnewline
38 & 90.82 & 90.7959733712956 & 0.0240266287043625 \tabularnewline
39 & 90.68 & 90.7929184800889 & -0.112918480088936 \tabularnewline
40 & 90.92 & 91.7717457652399 & -0.85174576523994 \tabularnewline
41 & 90.82 & 90.6865901437076 & 0.13340985629236 \tabularnewline
42 & 90.09 & 90.5534538098781 & -0.463453809878089 \tabularnewline
43 & 89.71 & 90.6756824630607 & -0.965682463060702 \tabularnewline
44 & 89.34 & 89.7904837180756 & -0.450483718075617 \tabularnewline
45 & 89.2 & 90.0930369834725 & -0.893036983472541 \tabularnewline
46 & 89.48 & 89.6735674092512 & -0.193567409251216 \tabularnewline
47 & 89.72 & 89.9661471013473 & -0.246147101347319 \tabularnewline
48 & 89.58 & 89.6263203873795 & -0.0463203873795095 \tabularnewline
49 & 90.65 & 90.4546450120228 & 0.19535498797724 \tabularnewline
50 & 90.93 & 90.5172028168675 & 0.412797183132469 \tabularnewline
51 & 91.42 & 90.7839522574441 & 0.63604774255586 \tabularnewline
52 & 91.52 & 92.1750672409339 & -0.655067240933889 \tabularnewline
53 & 91.76 & 91.4650176634893 & 0.294982336510714 \tabularnewline
54 & 91.47 & 91.3218243725882 & 0.148175627411788 \tabularnewline
55 & 91.37 & 91.8036232248521 & -0.433623224852127 \tabularnewline
56 & 91.35 & 91.4466682921124 & -0.0966682921124118 \tabularnewline
57 & 91.74 & 91.9228236444535 & -0.182823644453521 \tabularnewline
58 & 91.78 & 92.2111361613269 & -0.431136161326918 \tabularnewline
59 & 91.88 & 92.3080089883635 & -0.428008988363459 \tabularnewline
60 & 91.99 & 91.8726941712752 & 0.117305828724838 \tabularnewline
61 & 92.55 & 92.8823070569388 & -0.332307056938774 \tabularnewline
62 & 92.94 & 92.5858153257183 & 0.354184674281683 \tabularnewline
63 & 92.81 & 92.857736136985 & -0.0477361369849802 \tabularnewline
64 & 93.35 & 93.4276319457997 & -0.0776319457996664 \tabularnewline
65 & 93.72 & 93.3793379848553 & 0.340662015144673 \tabularnewline
66 & 93.94 & 93.2382658863928 & 0.701734113607188 \tabularnewline
67 & 94.03 & 94.0166988389951 & 0.0133011610049181 \tabularnewline
68 & 93.66 & 94.0817828809312 & -0.421782880931175 \tabularnewline
69 & 93.78 & 94.2868986506187 & -0.506898650618723 \tabularnewline
70 & 94.1 & 94.2682807468886 & -0.168280746888612 \tabularnewline
71 & 94.85 & 94.569234084049 & 0.280765915950994 \tabularnewline
72 & 94.83 & 94.8057041584394 & 0.0242958415605585 \tabularnewline
73 & 95.06 & 95.641610013296 & -0.581610013296 \tabularnewline
74 & 95.87 & 95.3075799105322 & 0.562420089467835 \tabularnewline
75 & 95.97 & 95.6496615339962 & 0.320338466003847 \tabularnewline
76 & 95.96 & 96.4975736882112 & -0.537573688211211 \tabularnewline
77 & 96.3 & 96.1880773253951 & 0.111922674604855 \tabularnewline
78 & 96.17 & 95.9517365890553 & 0.218263410944743 \tabularnewline
79 & 96.18 & 96.2003171418801 & -0.0203171418801134 \tabularnewline
80 & 96.55 & 96.1409336522749 & 0.409066347725144 \tabularnewline
81 & 96.76 & 96.9696214025715 & -0.209621402571528 \tabularnewline
82 & 97.63 & 97.2576358831197 & 0.372364116880291 \tabularnewline
83 & 97.86 & 98.078505974297 & -0.218505974296974 \tabularnewline
84 & 97.82 & 97.8706487166953 & -0.0506487166953065 \tabularnewline
85 & 98.62 & 98.511456736066 & 0.108543263933981 \tabularnewline
86 & 99.24 & 98.9702894458519 & 0.269710554148062 \tabularnewline
87 & 99.63 & 99.0311183192039 & 0.598881680796097 \tabularnewline
88 & 100.27 & 99.9004008247915 & 0.369599175208492 \tabularnewline
89 & 100.84 & 100.439766717426 & 0.400233282573652 \tabularnewline
90 & 101.05 & 100.450557925802 & 0.59944207419818 \tabularnewline
91 & 100.38 & 100.940069441375 & -0.560069441375319 \tabularnewline
92 & 100.02 & 100.560243123392 & -0.540243123392187 \tabularnewline
93 & 99.97 & 100.51443906543 & -0.544439065430112 \tabularnewline
94 & 99.95 & 100.675102807046 & -0.725102807045616 \tabularnewline
95 & 100 & 100.513145723143 & -0.513145723142657 \tabularnewline
96 & 100.04 & 100.115308945961 & -0.0753089459614813 \tabularnewline
97 & 100.51 & 100.773061360897 & -0.26306136089714 \tabularnewline
98 & 100.29 & 100.980852454905 & -0.690852454904757 \tabularnewline
99 & 100.22 & 100.372977225214 & -0.152977225214485 \tabularnewline
100 & 101.29 & 100.608656651564 & 0.681343348435703 \tabularnewline
101 & 100.29 & 101.396153237702 & -1.10615323770212 \tabularnewline
102 & 100.26 & 100.286523659397 & -0.0265236593965739 \tabularnewline
103 & 100.39 & 100.029331310993 & 0.36066868900707 \tabularnewline
104 & 99.3 & 100.366372318782 & -1.06637231878177 \tabularnewline
105 & 98.9 & 99.9125493519745 & -1.01254935197454 \tabularnewline
106 & 98.76 & 99.6701501930279 & -0.910150193027903 \tabularnewline
107 & 99.12 & 99.4129853940133 & -0.292985394013314 \tabularnewline
108 & 99.28 & 99.2845677879461 & -0.00456778794611523 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284152&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]83.09[/C][C]81.6837419871795[/C][C]1.40625801282047[/C][/ROW]
[ROW][C]14[/C][C]83.21[/C][C]82.9172284592821[/C][C]0.292771540717936[/C][/ROW]
[ROW][C]15[/C][C]83.13[/C][C]83.1208702692048[/C][C]0.00912973079519475[/C][/ROW]
[ROW][C]16[/C][C]84.31[/C][C]84.3246400003789[/C][C]-0.014640000378904[/C][/ROW]
[ROW][C]17[/C][C]83.62[/C][C]83.5821022777478[/C][C]0.0378977222522394[/C][/ROW]
[ROW][C]18[/C][C]83.75[/C][C]83.6468799807691[/C][C]0.103120019230857[/C][/ROW]
[ROW][C]19[/C][C]84.1[/C][C]84.2112872425967[/C][C]-0.111287242596688[/C][/ROW]
[ROW][C]20[/C][C]83.71[/C][C]84.29313962117[/C][C]-0.583139621170019[/C][/ROW]
[ROW][C]21[/C][C]84.2[/C][C]84.4532502469677[/C][C]-0.253250246967653[/C][/ROW]
[ROW][C]22[/C][C]85.13[/C][C]84.0181816429799[/C][C]1.11181835702007[/C][/ROW]
[ROW][C]23[/C][C]86.16[/C][C]85.627608343263[/C][C]0.532391656736962[/C][/ROW]
[ROW][C]24[/C][C]86.65[/C][C]85.5545623767289[/C][C]1.09543762327107[/C][/ROW]
[ROW][C]25[/C][C]87.44[/C][C]87.5182927456556[/C][C]-0.0782927456555598[/C][/ROW]
[ROW][C]26[/C][C]87.62[/C][C]87.3511980260965[/C][C]0.268801973903535[/C][/ROW]
[ROW][C]27[/C][C]88.03[/C][C]87.4721080369764[/C][C]0.557891963023565[/C][/ROW]
[ROW][C]28[/C][C]89.1[/C][C]89.0950795368157[/C][C]0.0049204631843196[/C][/ROW]
[ROW][C]29[/C][C]89.68[/C][C]88.3795648287405[/C][C]1.30043517125948[/C][/ROW]
[ROW][C]30[/C][C]89.47[/C][C]89.4359349225994[/C][C]0.0340650774005837[/C][/ROW]
[ROW][C]31[/C][C]90.13[/C][C]89.8983949061638[/C][C]0.231605093836237[/C][/ROW]
[ROW][C]32[/C][C]89.49[/C][C]90.1387679009702[/C][C]-0.648767900970185[/C][/ROW]
[ROW][C]33[/C][C]89.52[/C][C]90.3227534605445[/C][C]-0.802753460544551[/C][/ROW]
[ROW][C]34[/C][C]89.86[/C][C]89.7714374750816[/C][C]0.088562524918359[/C][/ROW]
[ROW][C]35[/C][C]89.77[/C][C]90.4580441472653[/C][C]-0.688044147265259[/C][/ROW]
[ROW][C]36[/C][C]89.8[/C][C]89.5681533370904[/C][C]0.231846662909561[/C][/ROW]
[ROW][C]37[/C][C]90.89[/C][C]90.5981101041413[/C][C]0.291889895858688[/C][/ROW]
[ROW][C]38[/C][C]90.82[/C][C]90.7959733712956[/C][C]0.0240266287043625[/C][/ROW]
[ROW][C]39[/C][C]90.68[/C][C]90.7929184800889[/C][C]-0.112918480088936[/C][/ROW]
[ROW][C]40[/C][C]90.92[/C][C]91.7717457652399[/C][C]-0.85174576523994[/C][/ROW]
[ROW][C]41[/C][C]90.82[/C][C]90.6865901437076[/C][C]0.13340985629236[/C][/ROW]
[ROW][C]42[/C][C]90.09[/C][C]90.5534538098781[/C][C]-0.463453809878089[/C][/ROW]
[ROW][C]43[/C][C]89.71[/C][C]90.6756824630607[/C][C]-0.965682463060702[/C][/ROW]
[ROW][C]44[/C][C]89.34[/C][C]89.7904837180756[/C][C]-0.450483718075617[/C][/ROW]
[ROW][C]45[/C][C]89.2[/C][C]90.0930369834725[/C][C]-0.893036983472541[/C][/ROW]
[ROW][C]46[/C][C]89.48[/C][C]89.6735674092512[/C][C]-0.193567409251216[/C][/ROW]
[ROW][C]47[/C][C]89.72[/C][C]89.9661471013473[/C][C]-0.246147101347319[/C][/ROW]
[ROW][C]48[/C][C]89.58[/C][C]89.6263203873795[/C][C]-0.0463203873795095[/C][/ROW]
[ROW][C]49[/C][C]90.65[/C][C]90.4546450120228[/C][C]0.19535498797724[/C][/ROW]
[ROW][C]50[/C][C]90.93[/C][C]90.5172028168675[/C][C]0.412797183132469[/C][/ROW]
[ROW][C]51[/C][C]91.42[/C][C]90.7839522574441[/C][C]0.63604774255586[/C][/ROW]
[ROW][C]52[/C][C]91.52[/C][C]92.1750672409339[/C][C]-0.655067240933889[/C][/ROW]
[ROW][C]53[/C][C]91.76[/C][C]91.4650176634893[/C][C]0.294982336510714[/C][/ROW]
[ROW][C]54[/C][C]91.47[/C][C]91.3218243725882[/C][C]0.148175627411788[/C][/ROW]
[ROW][C]55[/C][C]91.37[/C][C]91.8036232248521[/C][C]-0.433623224852127[/C][/ROW]
[ROW][C]56[/C][C]91.35[/C][C]91.4466682921124[/C][C]-0.0966682921124118[/C][/ROW]
[ROW][C]57[/C][C]91.74[/C][C]91.9228236444535[/C][C]-0.182823644453521[/C][/ROW]
[ROW][C]58[/C][C]91.78[/C][C]92.2111361613269[/C][C]-0.431136161326918[/C][/ROW]
[ROW][C]59[/C][C]91.88[/C][C]92.3080089883635[/C][C]-0.428008988363459[/C][/ROW]
[ROW][C]60[/C][C]91.99[/C][C]91.8726941712752[/C][C]0.117305828724838[/C][/ROW]
[ROW][C]61[/C][C]92.55[/C][C]92.8823070569388[/C][C]-0.332307056938774[/C][/ROW]
[ROW][C]62[/C][C]92.94[/C][C]92.5858153257183[/C][C]0.354184674281683[/C][/ROW]
[ROW][C]63[/C][C]92.81[/C][C]92.857736136985[/C][C]-0.0477361369849802[/C][/ROW]
[ROW][C]64[/C][C]93.35[/C][C]93.4276319457997[/C][C]-0.0776319457996664[/C][/ROW]
[ROW][C]65[/C][C]93.72[/C][C]93.3793379848553[/C][C]0.340662015144673[/C][/ROW]
[ROW][C]66[/C][C]93.94[/C][C]93.2382658863928[/C][C]0.701734113607188[/C][/ROW]
[ROW][C]67[/C][C]94.03[/C][C]94.0166988389951[/C][C]0.0133011610049181[/C][/ROW]
[ROW][C]68[/C][C]93.66[/C][C]94.0817828809312[/C][C]-0.421782880931175[/C][/ROW]
[ROW][C]69[/C][C]93.78[/C][C]94.2868986506187[/C][C]-0.506898650618723[/C][/ROW]
[ROW][C]70[/C][C]94.1[/C][C]94.2682807468886[/C][C]-0.168280746888612[/C][/ROW]
[ROW][C]71[/C][C]94.85[/C][C]94.569234084049[/C][C]0.280765915950994[/C][/ROW]
[ROW][C]72[/C][C]94.83[/C][C]94.8057041584394[/C][C]0.0242958415605585[/C][/ROW]
[ROW][C]73[/C][C]95.06[/C][C]95.641610013296[/C][C]-0.581610013296[/C][/ROW]
[ROW][C]74[/C][C]95.87[/C][C]95.3075799105322[/C][C]0.562420089467835[/C][/ROW]
[ROW][C]75[/C][C]95.97[/C][C]95.6496615339962[/C][C]0.320338466003847[/C][/ROW]
[ROW][C]76[/C][C]95.96[/C][C]96.4975736882112[/C][C]-0.537573688211211[/C][/ROW]
[ROW][C]77[/C][C]96.3[/C][C]96.1880773253951[/C][C]0.111922674604855[/C][/ROW]
[ROW][C]78[/C][C]96.17[/C][C]95.9517365890553[/C][C]0.218263410944743[/C][/ROW]
[ROW][C]79[/C][C]96.18[/C][C]96.2003171418801[/C][C]-0.0203171418801134[/C][/ROW]
[ROW][C]80[/C][C]96.55[/C][C]96.1409336522749[/C][C]0.409066347725144[/C][/ROW]
[ROW][C]81[/C][C]96.76[/C][C]96.9696214025715[/C][C]-0.209621402571528[/C][/ROW]
[ROW][C]82[/C][C]97.63[/C][C]97.2576358831197[/C][C]0.372364116880291[/C][/ROW]
[ROW][C]83[/C][C]97.86[/C][C]98.078505974297[/C][C]-0.218505974296974[/C][/ROW]
[ROW][C]84[/C][C]97.82[/C][C]97.8706487166953[/C][C]-0.0506487166953065[/C][/ROW]
[ROW][C]85[/C][C]98.62[/C][C]98.511456736066[/C][C]0.108543263933981[/C][/ROW]
[ROW][C]86[/C][C]99.24[/C][C]98.9702894458519[/C][C]0.269710554148062[/C][/ROW]
[ROW][C]87[/C][C]99.63[/C][C]99.0311183192039[/C][C]0.598881680796097[/C][/ROW]
[ROW][C]88[/C][C]100.27[/C][C]99.9004008247915[/C][C]0.369599175208492[/C][/ROW]
[ROW][C]89[/C][C]100.84[/C][C]100.439766717426[/C][C]0.400233282573652[/C][/ROW]
[ROW][C]90[/C][C]101.05[/C][C]100.450557925802[/C][C]0.59944207419818[/C][/ROW]
[ROW][C]91[/C][C]100.38[/C][C]100.940069441375[/C][C]-0.560069441375319[/C][/ROW]
[ROW][C]92[/C][C]100.02[/C][C]100.560243123392[/C][C]-0.540243123392187[/C][/ROW]
[ROW][C]93[/C][C]99.97[/C][C]100.51443906543[/C][C]-0.544439065430112[/C][/ROW]
[ROW][C]94[/C][C]99.95[/C][C]100.675102807046[/C][C]-0.725102807045616[/C][/ROW]
[ROW][C]95[/C][C]100[/C][C]100.513145723143[/C][C]-0.513145723142657[/C][/ROW]
[ROW][C]96[/C][C]100.04[/C][C]100.115308945961[/C][C]-0.0753089459614813[/C][/ROW]
[ROW][C]97[/C][C]100.51[/C][C]100.773061360897[/C][C]-0.26306136089714[/C][/ROW]
[ROW][C]98[/C][C]100.29[/C][C]100.980852454905[/C][C]-0.690852454904757[/C][/ROW]
[ROW][C]99[/C][C]100.22[/C][C]100.372977225214[/C][C]-0.152977225214485[/C][/ROW]
[ROW][C]100[/C][C]101.29[/C][C]100.608656651564[/C][C]0.681343348435703[/C][/ROW]
[ROW][C]101[/C][C]100.29[/C][C]101.396153237702[/C][C]-1.10615323770212[/C][/ROW]
[ROW][C]102[/C][C]100.26[/C][C]100.286523659397[/C][C]-0.0265236593965739[/C][/ROW]
[ROW][C]103[/C][C]100.39[/C][C]100.029331310993[/C][C]0.36066868900707[/C][/ROW]
[ROW][C]104[/C][C]99.3[/C][C]100.366372318782[/C][C]-1.06637231878177[/C][/ROW]
[ROW][C]105[/C][C]98.9[/C][C]99.9125493519745[/C][C]-1.01254935197454[/C][/ROW]
[ROW][C]106[/C][C]98.76[/C][C]99.6701501930279[/C][C]-0.910150193027903[/C][/ROW]
[ROW][C]107[/C][C]99.12[/C][C]99.4129853940133[/C][C]-0.292985394013314[/C][/ROW]
[ROW][C]108[/C][C]99.28[/C][C]99.2845677879461[/C][C]-0.00456778794611523[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284152&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284152&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	83.09	81.6837419871795	1.40625801282047
14	83.21	82.9172284592821	0.292771540717936
15	83.13	83.1208702692048	0.00912973079519475
16	84.31	84.3246400003789	-0.014640000378904
17	83.62	83.5821022777478	0.0378977222522394
18	83.75	83.6468799807691	0.103120019230857
19	84.1	84.2112872425967	-0.111287242596688
20	83.71	84.29313962117	-0.583139621170019
21	84.2	84.4532502469677	-0.253250246967653
22	85.13	84.0181816429799	1.11181835702007
23	86.16	85.627608343263	0.532391656736962
24	86.65	85.5545623767289	1.09543762327107
25	87.44	87.5182927456556	-0.0782927456555598
26	87.62	87.3511980260965	0.268801973903535
27	88.03	87.4721080369764	0.557891963023565
28	89.1	89.0950795368157	0.0049204631843196
29	89.68	88.3795648287405	1.30043517125948
30	89.47	89.4359349225994	0.0340650774005837
31	90.13	89.8983949061638	0.231605093836237
32	89.49	90.1387679009702	-0.648767900970185
33	89.52	90.3227534605445	-0.802753460544551
34	89.86	89.7714374750816	0.088562524918359
35	89.77	90.4580441472653	-0.688044147265259
36	89.8	89.5681533370904	0.231846662909561
37	90.89	90.5981101041413	0.291889895858688
38	90.82	90.7959733712956	0.0240266287043625
39	90.68	90.7929184800889	-0.112918480088936
40	90.92	91.7717457652399	-0.85174576523994
41	90.82	90.6865901437076	0.13340985629236
42	90.09	90.5534538098781	-0.463453809878089
43	89.71	90.6756824630607	-0.965682463060702
44	89.34	89.7904837180756	-0.450483718075617
45	89.2	90.0930369834725	-0.893036983472541
46	89.48	89.6735674092512	-0.193567409251216
47	89.72	89.9661471013473	-0.246147101347319
48	89.58	89.6263203873795	-0.0463203873795095
49	90.65	90.4546450120228	0.19535498797724
50	90.93	90.5172028168675	0.412797183132469
51	91.42	90.7839522574441	0.63604774255586
52	91.52	92.1750672409339	-0.655067240933889
53	91.76	91.4650176634893	0.294982336510714
54	91.47	91.3218243725882	0.148175627411788
55	91.37	91.8036232248521	-0.433623224852127
56	91.35	91.4466682921124	-0.0966682921124118
57	91.74	91.9228236444535	-0.182823644453521
58	91.78	92.2111361613269	-0.431136161326918
59	91.88	92.3080089883635	-0.428008988363459
60	91.99	91.8726941712752	0.117305828724838
61	92.55	92.8823070569388	-0.332307056938774
62	92.94	92.5858153257183	0.354184674281683
63	92.81	92.857736136985	-0.0477361369849802
64	93.35	93.4276319457997	-0.0776319457996664
65	93.72	93.3793379848553	0.340662015144673
66	93.94	93.2382658863928	0.701734113607188
67	94.03	94.0166988389951	0.0133011610049181
68	93.66	94.0817828809312	-0.421782880931175
69	93.78	94.2868986506187	-0.506898650618723
70	94.1	94.2682807468886	-0.168280746888612
71	94.85	94.569234084049	0.280765915950994
72	94.83	94.8057041584394	0.0242958415605585
73	95.06	95.641610013296	-0.581610013296
74	95.87	95.3075799105322	0.562420089467835
75	95.97	95.6496615339962	0.320338466003847
76	95.96	96.4975736882112	-0.537573688211211
77	96.3	96.1880773253951	0.111922674604855
78	96.17	95.9517365890553	0.218263410944743
79	96.18	96.2003171418801	-0.0203171418801134
80	96.55	96.1409336522749	0.409066347725144
81	96.76	96.9696214025715	-0.209621402571528
82	97.63	97.2576358831197	0.372364116880291
83	97.86	98.078505974297	-0.218505974296974
84	97.82	97.8706487166953	-0.0506487166953065
85	98.62	98.511456736066	0.108543263933981
86	99.24	98.9702894458519	0.269710554148062
87	99.63	99.0311183192039	0.598881680796097
88	100.27	99.9004008247915	0.369599175208492
89	100.84	100.439766717426	0.400233282573652
90	101.05	100.450557925802	0.59944207419818
91	100.38	100.940069441375	-0.560069441375319
92	100.02	100.560243123392	-0.540243123392187
93	99.97	100.51443906543	-0.544439065430112
94	99.95	100.675102807046	-0.725102807045616
95	100	100.513145723143	-0.513145723142657
96	100.04	100.115308945961	-0.0753089459614813
97	100.51	100.773061360897	-0.26306136089714
98	100.29	100.980852454905	-0.690852454904757
99	100.22	100.372977225214	-0.152977225214485
100	101.29	100.608656651564	0.681343348435703
101	100.29	101.396153237702	-1.10615323770212
102	100.26	100.286523659397	-0.0265236593965739
103	100.39	100.029331310993	0.36066868900707
104	99.3	100.366372318782	-1.06637231878177
105	98.9	99.9125493519745	-1.01254935197454
106	98.76	99.6701501930279	-0.910150193027903
107	99.12	99.4129853940133	-0.292985394013314
108	99.28	99.2845677879461	-0.00456778794611523

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
109	99.9545658544793	98.9543149107891	100.954816798169
110	100.269082645993	99.0043994321142	101.533765859871
111	100.317442065599	98.834760268119	101.800123863079
112	100.86028252822	99.187779316357	102.532785740083
113	100.716120088528	98.8732449775116	102.558995199545
114	100.706641606863	98.7078643916842	102.705418822042
115	100.557590024895	98.4142207060352	102.700959343756
116	100.292648843674	98.0138435167221	102.571454170626
117	100.676064502146	98.2694329559272	103.082696048365
118	101.240253301822	98.7122507147251	103.768255888919
119	101.826937902357	99.1831302766995	104.470745528014
120	101.990472027972	99.2357233202682	104.745220735676

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
109 & 99.9545658544793 & 98.9543149107891 & 100.954816798169 \tabularnewline
110 & 100.269082645993 & 99.0043994321142 & 101.533765859871 \tabularnewline
111 & 100.317442065599 & 98.834760268119 & 101.800123863079 \tabularnewline
112 & 100.86028252822 & 99.187779316357 & 102.532785740083 \tabularnewline
113 & 100.716120088528 & 98.8732449775116 & 102.558995199545 \tabularnewline
114 & 100.706641606863 & 98.7078643916842 & 102.705418822042 \tabularnewline
115 & 100.557590024895 & 98.4142207060352 & 102.700959343756 \tabularnewline
116 & 100.292648843674 & 98.0138435167221 & 102.571454170626 \tabularnewline
117 & 100.676064502146 & 98.2694329559272 & 103.082696048365 \tabularnewline
118 & 101.240253301822 & 98.7122507147251 & 103.768255888919 \tabularnewline
119 & 101.826937902357 & 99.1831302766995 & 104.470745528014 \tabularnewline
120 & 101.990472027972 & 99.2357233202682 & 104.745220735676 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=284152&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]109[/C][C]99.9545658544793[/C][C]98.9543149107891[/C][C]100.954816798169[/C][/ROW]
[ROW][C]110[/C][C]100.269082645993[/C][C]99.0043994321142[/C][C]101.533765859871[/C][/ROW]
[ROW][C]111[/C][C]100.317442065599[/C][C]98.834760268119[/C][C]101.800123863079[/C][/ROW]
[ROW][C]112[/C][C]100.86028252822[/C][C]99.187779316357[/C][C]102.532785740083[/C][/ROW]
[ROW][C]113[/C][C]100.716120088528[/C][C]98.8732449775116[/C][C]102.558995199545[/C][/ROW]
[ROW][C]114[/C][C]100.706641606863[/C][C]98.7078643916842[/C][C]102.705418822042[/C][/ROW]
[ROW][C]115[/C][C]100.557590024895[/C][C]98.4142207060352[/C][C]102.700959343756[/C][/ROW]
[ROW][C]116[/C][C]100.292648843674[/C][C]98.0138435167221[/C][C]102.571454170626[/C][/ROW]
[ROW][C]117[/C][C]100.676064502146[/C][C]98.2694329559272[/C][C]103.082696048365[/C][/ROW]
[ROW][C]118[/C][C]101.240253301822[/C][C]98.7122507147251[/C][C]103.768255888919[/C][/ROW]
[ROW][C]119[/C][C]101.826937902357[/C][C]99.1831302766995[/C][C]104.470745528014[/C][/ROW]
[ROW][C]120[/C][C]101.990472027972[/C][C]99.2357233202682[/C][C]104.745220735676[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=284152&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=284152&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
109	99.9545658544793	98.9543149107891	100.954816798169
110	100.269082645993	99.0043994321142	101.533765859871
111	100.317442065599	98.834760268119	101.800123863079
112	100.86028252822	99.187779316357	102.532785740083
113	100.716120088528	98.8732449775116	102.558995199545
114	100.706641606863	98.7078643916842	102.705418822042
115	100.557590024895	98.4142207060352	102.700959343756
116	100.292648843674	98.0138435167221	102.571454170626
117	100.676064502146	98.2694329559272	103.082696048365
118	101.240253301822	98.7122507147251	103.768255888919
119	101.826937902357	99.1831302766995	104.470745528014
120	101.990472027972	99.2357233202682	104.745220735676

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