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Description of Statistical Computation

Author's title

Author*Unverified author*
R Software Modulerwasp_exponentialsmoothing.wasp
Title produced by softwareExponential Smoothing
Date of computationWed, 18 Dec 2013 09:36:51 -0500
Cite this page as followsStatistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2013/Dec/18/t1387377439lc81sv8behvucu4.htm/, Retrieved Thu, 25 Apr 2024 01:43:24 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=232438, Retrieved Thu, 25 Apr 2024 01:43:24 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact161

Tree of Dependent Computations

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-     [Harrell-Davis Quantiles] [] [2013-10-03 12:28:46] [14940b3acdb362d975a3925771725f1c]
- R P   [Harrell-Davis Quantiles] [] [2013-10-07 06:42:43] [bc3b96987d5d985d411360ed2ed90621]
- RMPD      [Exponential Smoothing] [] [2013-12-18 14:36:51] [c1a61ab38fcc49939d713dd54579d7ce] [Current]
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Dataset

Dataseries X:
Download CSV
Histogram
Boxplots 
107,2
107,56
107,72
108,14
108,16
108,16
108,16
108,1
108,95
110,49
110,72
110,82
110,82
110,75
110,71
110,86
110,84
110,84
110,84
110,92
111,46
112,46
113,04
113,15
113,15
113,21
113,37
113,47
113,71
113,71
113,71
113,8
115,46
117
117,94
118,08
118,08
118,47
118,49
118,45
118,54
118,55
118,55
118,55
119,04
121,37
122
122,14
122,14
122,03
121,91
122,23
121,73
121,83
121,83
122,49
123,02
125,98
126,13
126,39
126,39
126,57
126,87
127,26
126,82
126,7
126,7
126,7
128,53
130,37
130,39
130,65
130,65
130,65
130,85
130,89
130,85
131,6
131,6
131,6
132,53
133,05
133,49
133,46

Tables (Output of Computation)





Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time4 seconds
R Server'Gwilym Jenkins' @ jenkins.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 & 'Gwilym Jenkins' @ jenkins.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232438&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]'Gwilym Jenkins' @ jenkins.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232438&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232438&T=0

As an alternative you can also use a QR Code:  
QR Code


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

Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time4 seconds
R Server'Gwilym Jenkins' @ jenkins.wessa.net






Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.881401691360718
beta0.0275706998028344
gamma0

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.881401691360718 \tabularnewline
beta & 0.0275706998028344 \tabularnewline
gamma & 0 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232438&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]0.881401691360718[/C][/ROW]
[ROW][C]beta[/C][C]0.0275706998028344[/C][/ROW]
[ROW][C]gamma[/C][C]0[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232438&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232438&T=1

As an alternative you can also use a QR Code:  
QR Code


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

Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.881401691360718
beta0.0275706998028344
gamma0






Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
13110.82109.4149743548481.40502564515249
14110.75110.6110898749230.138910125077288
15110.71110.731222314566-0.0212223145656054
16110.86110.93504306376-0.0750430637604893
17110.84110.927359060936-0.0873590609363504
18110.84110.911622937775-0.0716229377748476
19110.84110.866484762678-0.026484762678237
20110.92110.744102574650.175897425349888
21111.46111.764208654783-0.304208654783096
22112.46113.078420500256-0.618420500256263
23113.04112.7733526584880.266647341511586
24113.15113.1238845670560.0261154329443798
25113.15113.160876718317-0.0108767183174336
26113.21113.0875122612960.122487738704308
27113.37113.1722349698580.197765030141667
28113.47113.558416368476-0.0884163684758192
29113.71113.5241807751080.185819224892356
30113.71113.740836851976-0.0308368519762325
31113.71113.723148546556-0.0131485465563657
32113.8113.6013541654320.198645834568012
33115.46114.6557315382530.804268461747
34117117.018786202513-0.0187862025126009
35117.94117.2826021300370.65739786996285
36118.08118.0217435455910.0582564544085642
37118.08118.12744329084-0.0474432908395812
38118.47118.0578385113020.412161488697663
39118.49118.4417542927080.0482457072921534
40118.45118.747039468415-0.297039468414695
41118.54118.567401937069-0.0274019370691718
42118.55118.62984943606-0.0798494360596465
43118.55118.599632790064-0.0496327900644502
44118.55118.4702905198620.0797094801382059
45119.04119.483193335694-0.443193335694431
46121.37120.7981889886380.571811011361774
47122121.6040028195960.395997180404052
48122.14122.1230824016330.0169175983667884
49122.14122.198032626574-0.0580326265741604
50122.03122.121736812503-0.0917368125027167
51121.91122.054327142026-0.144327142025887
52122.23122.1854383434090.0445616565913269
53121.73122.305187941655-0.575187941654519
54121.83121.870841580591-0.0408415805907367
55121.83121.860748582307-0.0307485823070408
56122.49121.7308499788560.759150021143896
57123.02123.373502289999-0.353502289998758
58125.98124.8269425337771.15305746622344
59126.13126.171346202153-0.0413462021531217
60126.39126.315735845620.074264154379577
61126.39126.449499709342-0.0594997093424894
62126.57126.377198249150.192801750849924
63126.87126.5735580282340.296441971766043
64127.26127.125881205790.134118794210337
65126.82127.352035977604-0.532035977603826
66126.7126.984304140211-0.28430414021085
67126.7126.780732043927-0.0807320439273411
68126.7126.6215402329470.0784597670534026
69128.53127.7014611933730.828538806626881
70130.37130.3037397263330.0662602736674955
71130.39130.706862896249-0.316862896249461
72130.65130.6133577201840.036642279815851
73130.65130.714181034863-0.0641810348625995
74130.65130.6349507758670.0150492241332358
75130.85130.669324033630.180675966370416
76130.89131.119866134047-0.229866134046944
77130.85131.011374696511-0.161374696511103
78131.6130.9651515544550.634848445544691
79131.6131.5856412195810.0143587804185188
80131.6131.5211044618920.0788955381077869
81132.53132.654725890124-0.124725890124409
82133.05134.469167274037-1.41916727403708
83133.49133.530304434787-0.040304434786492
84133.46133.65103891631-0.191038916310134

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 110.82 & 109.414974354848 & 1.40502564515249 \tabularnewline
14 & 110.75 & 110.611089874923 & 0.138910125077288 \tabularnewline
15 & 110.71 & 110.731222314566 & -0.0212223145656054 \tabularnewline
16 & 110.86 & 110.93504306376 & -0.0750430637604893 \tabularnewline
17 & 110.84 & 110.927359060936 & -0.0873590609363504 \tabularnewline
18 & 110.84 & 110.911622937775 & -0.0716229377748476 \tabularnewline
19 & 110.84 & 110.866484762678 & -0.026484762678237 \tabularnewline
20 & 110.92 & 110.74410257465 & 0.175897425349888 \tabularnewline
21 & 111.46 & 111.764208654783 & -0.304208654783096 \tabularnewline
22 & 112.46 & 113.078420500256 & -0.618420500256263 \tabularnewline
23 & 113.04 & 112.773352658488 & 0.266647341511586 \tabularnewline
24 & 113.15 & 113.123884567056 & 0.0261154329443798 \tabularnewline
25 & 113.15 & 113.160876718317 & -0.0108767183174336 \tabularnewline
26 & 113.21 & 113.087512261296 & 0.122487738704308 \tabularnewline
27 & 113.37 & 113.172234969858 & 0.197765030141667 \tabularnewline
28 & 113.47 & 113.558416368476 & -0.0884163684758192 \tabularnewline
29 & 113.71 & 113.524180775108 & 0.185819224892356 \tabularnewline
30 & 113.71 & 113.740836851976 & -0.0308368519762325 \tabularnewline
31 & 113.71 & 113.723148546556 & -0.0131485465563657 \tabularnewline
32 & 113.8 & 113.601354165432 & 0.198645834568012 \tabularnewline
33 & 115.46 & 114.655731538253 & 0.804268461747 \tabularnewline
34 & 117 & 117.018786202513 & -0.0187862025126009 \tabularnewline
35 & 117.94 & 117.282602130037 & 0.65739786996285 \tabularnewline
36 & 118.08 & 118.021743545591 & 0.0582564544085642 \tabularnewline
37 & 118.08 & 118.12744329084 & -0.0474432908395812 \tabularnewline
38 & 118.47 & 118.057838511302 & 0.412161488697663 \tabularnewline
39 & 118.49 & 118.441754292708 & 0.0482457072921534 \tabularnewline
40 & 118.45 & 118.747039468415 & -0.297039468414695 \tabularnewline
41 & 118.54 & 118.567401937069 & -0.0274019370691718 \tabularnewline
42 & 118.55 & 118.62984943606 & -0.0798494360596465 \tabularnewline
43 & 118.55 & 118.599632790064 & -0.0496327900644502 \tabularnewline
44 & 118.55 & 118.470290519862 & 0.0797094801382059 \tabularnewline
45 & 119.04 & 119.483193335694 & -0.443193335694431 \tabularnewline
46 & 121.37 & 120.798188988638 & 0.571811011361774 \tabularnewline
47 & 122 & 121.604002819596 & 0.395997180404052 \tabularnewline
48 & 122.14 & 122.123082401633 & 0.0169175983667884 \tabularnewline
49 & 122.14 & 122.198032626574 & -0.0580326265741604 \tabularnewline
50 & 122.03 & 122.121736812503 & -0.0917368125027167 \tabularnewline
51 & 121.91 & 122.054327142026 & -0.144327142025887 \tabularnewline
52 & 122.23 & 122.185438343409 & 0.0445616565913269 \tabularnewline
53 & 121.73 & 122.305187941655 & -0.575187941654519 \tabularnewline
54 & 121.83 & 121.870841580591 & -0.0408415805907367 \tabularnewline
55 & 121.83 & 121.860748582307 & -0.0307485823070408 \tabularnewline
56 & 122.49 & 121.730849978856 & 0.759150021143896 \tabularnewline
57 & 123.02 & 123.373502289999 & -0.353502289998758 \tabularnewline
58 & 125.98 & 124.826942533777 & 1.15305746622344 \tabularnewline
59 & 126.13 & 126.171346202153 & -0.0413462021531217 \tabularnewline
60 & 126.39 & 126.31573584562 & 0.074264154379577 \tabularnewline
61 & 126.39 & 126.449499709342 & -0.0594997093424894 \tabularnewline
62 & 126.57 & 126.37719824915 & 0.192801750849924 \tabularnewline
63 & 126.87 & 126.573558028234 & 0.296441971766043 \tabularnewline
64 & 127.26 & 127.12588120579 & 0.134118794210337 \tabularnewline
65 & 126.82 & 127.352035977604 & -0.532035977603826 \tabularnewline
66 & 126.7 & 126.984304140211 & -0.28430414021085 \tabularnewline
67 & 126.7 & 126.780732043927 & -0.0807320439273411 \tabularnewline
68 & 126.7 & 126.621540232947 & 0.0784597670534026 \tabularnewline
69 & 128.53 & 127.701461193373 & 0.828538806626881 \tabularnewline
70 & 130.37 & 130.303739726333 & 0.0662602736674955 \tabularnewline
71 & 130.39 & 130.706862896249 & -0.316862896249461 \tabularnewline
72 & 130.65 & 130.613357720184 & 0.036642279815851 \tabularnewline
73 & 130.65 & 130.714181034863 & -0.0641810348625995 \tabularnewline
74 & 130.65 & 130.634950775867 & 0.0150492241332358 \tabularnewline
75 & 130.85 & 130.66932403363 & 0.180675966370416 \tabularnewline
76 & 130.89 & 131.119866134047 & -0.229866134046944 \tabularnewline
77 & 130.85 & 131.011374696511 & -0.161374696511103 \tabularnewline
78 & 131.6 & 130.965151554455 & 0.634848445544691 \tabularnewline
79 & 131.6 & 131.585641219581 & 0.0143587804185188 \tabularnewline
80 & 131.6 & 131.521104461892 & 0.0788955381077869 \tabularnewline
81 & 132.53 & 132.654725890124 & -0.124725890124409 \tabularnewline
82 & 133.05 & 134.469167274037 & -1.41916727403708 \tabularnewline
83 & 133.49 & 133.530304434787 & -0.040304434786492 \tabularnewline
84 & 133.46 & 133.65103891631 & -0.191038916310134 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232438&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]110.82[/C][C]109.414974354848[/C][C]1.40502564515249[/C][/ROW]
[ROW][C]14[/C][C]110.75[/C][C]110.611089874923[/C][C]0.138910125077288[/C][/ROW]
[ROW][C]15[/C][C]110.71[/C][C]110.731222314566[/C][C]-0.0212223145656054[/C][/ROW]
[ROW][C]16[/C][C]110.86[/C][C]110.93504306376[/C][C]-0.0750430637604893[/C][/ROW]
[ROW][C]17[/C][C]110.84[/C][C]110.927359060936[/C][C]-0.0873590609363504[/C][/ROW]
[ROW][C]18[/C][C]110.84[/C][C]110.911622937775[/C][C]-0.0716229377748476[/C][/ROW]
[ROW][C]19[/C][C]110.84[/C][C]110.866484762678[/C][C]-0.026484762678237[/C][/ROW]
[ROW][C]20[/C][C]110.92[/C][C]110.74410257465[/C][C]0.175897425349888[/C][/ROW]
[ROW][C]21[/C][C]111.46[/C][C]111.764208654783[/C][C]-0.304208654783096[/C][/ROW]
[ROW][C]22[/C][C]112.46[/C][C]113.078420500256[/C][C]-0.618420500256263[/C][/ROW]
[ROW][C]23[/C][C]113.04[/C][C]112.773352658488[/C][C]0.266647341511586[/C][/ROW]
[ROW][C]24[/C][C]113.15[/C][C]113.123884567056[/C][C]0.0261154329443798[/C][/ROW]
[ROW][C]25[/C][C]113.15[/C][C]113.160876718317[/C][C]-0.0108767183174336[/C][/ROW]
[ROW][C]26[/C][C]113.21[/C][C]113.087512261296[/C][C]0.122487738704308[/C][/ROW]
[ROW][C]27[/C][C]113.37[/C][C]113.172234969858[/C][C]0.197765030141667[/C][/ROW]
[ROW][C]28[/C][C]113.47[/C][C]113.558416368476[/C][C]-0.0884163684758192[/C][/ROW]
[ROW][C]29[/C][C]113.71[/C][C]113.524180775108[/C][C]0.185819224892356[/C][/ROW]
[ROW][C]30[/C][C]113.71[/C][C]113.740836851976[/C][C]-0.0308368519762325[/C][/ROW]
[ROW][C]31[/C][C]113.71[/C][C]113.723148546556[/C][C]-0.0131485465563657[/C][/ROW]
[ROW][C]32[/C][C]113.8[/C][C]113.601354165432[/C][C]0.198645834568012[/C][/ROW]
[ROW][C]33[/C][C]115.46[/C][C]114.655731538253[/C][C]0.804268461747[/C][/ROW]
[ROW][C]34[/C][C]117[/C][C]117.018786202513[/C][C]-0.0187862025126009[/C][/ROW]
[ROW][C]35[/C][C]117.94[/C][C]117.282602130037[/C][C]0.65739786996285[/C][/ROW]
[ROW][C]36[/C][C]118.08[/C][C]118.021743545591[/C][C]0.0582564544085642[/C][/ROW]
[ROW][C]37[/C][C]118.08[/C][C]118.12744329084[/C][C]-0.0474432908395812[/C][/ROW]
[ROW][C]38[/C][C]118.47[/C][C]118.057838511302[/C][C]0.412161488697663[/C][/ROW]
[ROW][C]39[/C][C]118.49[/C][C]118.441754292708[/C][C]0.0482457072921534[/C][/ROW]
[ROW][C]40[/C][C]118.45[/C][C]118.747039468415[/C][C]-0.297039468414695[/C][/ROW]
[ROW][C]41[/C][C]118.54[/C][C]118.567401937069[/C][C]-0.0274019370691718[/C][/ROW]
[ROW][C]42[/C][C]118.55[/C][C]118.62984943606[/C][C]-0.0798494360596465[/C][/ROW]
[ROW][C]43[/C][C]118.55[/C][C]118.599632790064[/C][C]-0.0496327900644502[/C][/ROW]
[ROW][C]44[/C][C]118.55[/C][C]118.470290519862[/C][C]0.0797094801382059[/C][/ROW]
[ROW][C]45[/C][C]119.04[/C][C]119.483193335694[/C][C]-0.443193335694431[/C][/ROW]
[ROW][C]46[/C][C]121.37[/C][C]120.798188988638[/C][C]0.571811011361774[/C][/ROW]
[ROW][C]47[/C][C]122[/C][C]121.604002819596[/C][C]0.395997180404052[/C][/ROW]
[ROW][C]48[/C][C]122.14[/C][C]122.123082401633[/C][C]0.0169175983667884[/C][/ROW]
[ROW][C]49[/C][C]122.14[/C][C]122.198032626574[/C][C]-0.0580326265741604[/C][/ROW]
[ROW][C]50[/C][C]122.03[/C][C]122.121736812503[/C][C]-0.0917368125027167[/C][/ROW]
[ROW][C]51[/C][C]121.91[/C][C]122.054327142026[/C][C]-0.144327142025887[/C][/ROW]
[ROW][C]52[/C][C]122.23[/C][C]122.185438343409[/C][C]0.0445616565913269[/C][/ROW]
[ROW][C]53[/C][C]121.73[/C][C]122.305187941655[/C][C]-0.575187941654519[/C][/ROW]
[ROW][C]54[/C][C]121.83[/C][C]121.870841580591[/C][C]-0.0408415805907367[/C][/ROW]
[ROW][C]55[/C][C]121.83[/C][C]121.860748582307[/C][C]-0.0307485823070408[/C][/ROW]
[ROW][C]56[/C][C]122.49[/C][C]121.730849978856[/C][C]0.759150021143896[/C][/ROW]
[ROW][C]57[/C][C]123.02[/C][C]123.373502289999[/C][C]-0.353502289998758[/C][/ROW]
[ROW][C]58[/C][C]125.98[/C][C]124.826942533777[/C][C]1.15305746622344[/C][/ROW]
[ROW][C]59[/C][C]126.13[/C][C]126.171346202153[/C][C]-0.0413462021531217[/C][/ROW]
[ROW][C]60[/C][C]126.39[/C][C]126.31573584562[/C][C]0.074264154379577[/C][/ROW]
[ROW][C]61[/C][C]126.39[/C][C]126.449499709342[/C][C]-0.0594997093424894[/C][/ROW]
[ROW][C]62[/C][C]126.57[/C][C]126.37719824915[/C][C]0.192801750849924[/C][/ROW]
[ROW][C]63[/C][C]126.87[/C][C]126.573558028234[/C][C]0.296441971766043[/C][/ROW]
[ROW][C]64[/C][C]127.26[/C][C]127.12588120579[/C][C]0.134118794210337[/C][/ROW]
[ROW][C]65[/C][C]126.82[/C][C]127.352035977604[/C][C]-0.532035977603826[/C][/ROW]
[ROW][C]66[/C][C]126.7[/C][C]126.984304140211[/C][C]-0.28430414021085[/C][/ROW]
[ROW][C]67[/C][C]126.7[/C][C]126.780732043927[/C][C]-0.0807320439273411[/C][/ROW]
[ROW][C]68[/C][C]126.7[/C][C]126.621540232947[/C][C]0.0784597670534026[/C][/ROW]
[ROW][C]69[/C][C]128.53[/C][C]127.701461193373[/C][C]0.828538806626881[/C][/ROW]
[ROW][C]70[/C][C]130.37[/C][C]130.303739726333[/C][C]0.0662602736674955[/C][/ROW]
[ROW][C]71[/C][C]130.39[/C][C]130.706862896249[/C][C]-0.316862896249461[/C][/ROW]
[ROW][C]72[/C][C]130.65[/C][C]130.613357720184[/C][C]0.036642279815851[/C][/ROW]
[ROW][C]73[/C][C]130.65[/C][C]130.714181034863[/C][C]-0.0641810348625995[/C][/ROW]
[ROW][C]74[/C][C]130.65[/C][C]130.634950775867[/C][C]0.0150492241332358[/C][/ROW]
[ROW][C]75[/C][C]130.85[/C][C]130.66932403363[/C][C]0.180675966370416[/C][/ROW]
[ROW][C]76[/C][C]130.89[/C][C]131.119866134047[/C][C]-0.229866134046944[/C][/ROW]
[ROW][C]77[/C][C]130.85[/C][C]131.011374696511[/C][C]-0.161374696511103[/C][/ROW]
[ROW][C]78[/C][C]131.6[/C][C]130.965151554455[/C][C]0.634848445544691[/C][/ROW]
[ROW][C]79[/C][C]131.6[/C][C]131.585641219581[/C][C]0.0143587804185188[/C][/ROW]
[ROW][C]80[/C][C]131.6[/C][C]131.521104461892[/C][C]0.0788955381077869[/C][/ROW]
[ROW][C]81[/C][C]132.53[/C][C]132.654725890124[/C][C]-0.124725890124409[/C][/ROW]
[ROW][C]82[/C][C]133.05[/C][C]134.469167274037[/C][C]-1.41916727403708[/C][/ROW]
[ROW][C]83[/C][C]133.49[/C][C]133.530304434787[/C][C]-0.040304434786492[/C][/ROW]
[ROW][C]84[/C][C]133.46[/C][C]133.65103891631[/C][C]-0.191038916310134[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232438&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232438&T=2

As an alternative you can also use a QR Code:  
QR Code


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

Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
13110.82109.4149743548481.40502564515249
14110.75110.6110898749230.138910125077288
15110.71110.731222314566-0.0212223145656054
16110.86110.93504306376-0.0750430637604893
17110.84110.927359060936-0.0873590609363504
18110.84110.911622937775-0.0716229377748476
19110.84110.866484762678-0.026484762678237
20110.92110.744102574650.175897425349888
21111.46111.764208654783-0.304208654783096
22112.46113.078420500256-0.618420500256263
23113.04112.7733526584880.266647341511586
24113.15113.1238845670560.0261154329443798
25113.15113.160876718317-0.0108767183174336
26113.21113.0875122612960.122487738704308
27113.37113.1722349698580.197765030141667
28113.47113.558416368476-0.0884163684758192
29113.71113.5241807751080.185819224892356
30113.71113.740836851976-0.0308368519762325
31113.71113.723148546556-0.0131485465563657
32113.8113.6013541654320.198645834568012
33115.46114.6557315382530.804268461747
34117117.018786202513-0.0187862025126009
35117.94117.2826021300370.65739786996285
36118.08118.0217435455910.0582564544085642
37118.08118.12744329084-0.0474432908395812
38118.47118.0578385113020.412161488697663
39118.49118.4417542927080.0482457072921534
40118.45118.747039468415-0.297039468414695
41118.54118.567401937069-0.0274019370691718
42118.55118.62984943606-0.0798494360596465
43118.55118.599632790064-0.0496327900644502
44118.55118.4702905198620.0797094801382059
45119.04119.483193335694-0.443193335694431
46121.37120.7981889886380.571811011361774
47122121.6040028195960.395997180404052
48122.14122.1230824016330.0169175983667884
49122.14122.198032626574-0.0580326265741604
50122.03122.121736812503-0.0917368125027167
51121.91122.054327142026-0.144327142025887
52122.23122.1854383434090.0445616565913269
53121.73122.305187941655-0.575187941654519
54121.83121.870841580591-0.0408415805907367
55121.83121.860748582307-0.0307485823070408
56122.49121.7308499788560.759150021143896
57123.02123.373502289999-0.353502289998758
58125.98124.8269425337771.15305746622344
59126.13126.171346202153-0.0413462021531217
60126.39126.315735845620.074264154379577
61126.39126.449499709342-0.0594997093424894
62126.57126.377198249150.192801750849924
63126.87126.5735580282340.296441971766043
64127.26127.125881205790.134118794210337
65126.82127.352035977604-0.532035977603826
66126.7126.984304140211-0.28430414021085
67126.7126.780732043927-0.0807320439273411
68126.7126.6215402329470.0784597670534026
69128.53127.7014611933730.828538806626881
70130.37130.3037397263330.0662602736674955
71130.39130.706862896249-0.316862896249461
72130.65130.6133577201840.036642279815851
73130.65130.714181034863-0.0641810348625995
74130.65130.6349507758670.0150492241332358
75130.85130.669324033630.180675966370416
76130.89131.119866134047-0.229866134046944
77130.85131.011374696511-0.161374696511103
78131.6130.9651515544550.634848445544691
79131.6131.5856412195810.0143587804185188
80131.6131.5211044618920.0788955381077869
81132.53132.654725890124-0.124725890124409
82133.05134.469167274037-1.41916727403708
83133.49133.530304434787-0.040304434786492
84133.46133.65103891631-0.191038916310134






Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
85133.513909495403132.835722424196134.192096566609
86133.453636756006132.482573534865134.424699977147
87133.437925582859132.233692196843134.642158968876
88133.693796236773132.283853444039135.103739029507
89133.754018115746132.158096239976135.349939991516
90133.820049937469132.050231055023135.589868819915
91133.835837320633131.901493266692135.770181374575
92133.711066550399131.620754758016135.801378342782
93134.74411921043132.485411573215137.002826847646
94136.654626408511134.216200692771139.093052124252
95136.961092146953134.372640566507139.549543727399
96137.108799136045133.618435982867140.599162289224

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
85 & 133.513909495403 & 132.835722424196 & 134.192096566609 \tabularnewline
86 & 133.453636756006 & 132.482573534865 & 134.424699977147 \tabularnewline
87 & 133.437925582859 & 132.233692196843 & 134.642158968876 \tabularnewline
88 & 133.693796236773 & 132.283853444039 & 135.103739029507 \tabularnewline
89 & 133.754018115746 & 132.158096239976 & 135.349939991516 \tabularnewline
90 & 133.820049937469 & 132.050231055023 & 135.589868819915 \tabularnewline
91 & 133.835837320633 & 131.901493266692 & 135.770181374575 \tabularnewline
92 & 133.711066550399 & 131.620754758016 & 135.801378342782 \tabularnewline
93 & 134.74411921043 & 132.485411573215 & 137.002826847646 \tabularnewline
94 & 136.654626408511 & 134.216200692771 & 139.093052124252 \tabularnewline
95 & 136.961092146953 & 134.372640566507 & 139.549543727399 \tabularnewline
96 & 137.108799136045 & 133.618435982867 & 140.599162289224 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=232438&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]133.513909495403[/C][C]132.835722424196[/C][C]134.192096566609[/C][/ROW]
[ROW][C]86[/C][C]133.453636756006[/C][C]132.482573534865[/C][C]134.424699977147[/C][/ROW]
[ROW][C]87[/C][C]133.437925582859[/C][C]132.233692196843[/C][C]134.642158968876[/C][/ROW]
[ROW][C]88[/C][C]133.693796236773[/C][C]132.283853444039[/C][C]135.103739029507[/C][/ROW]
[ROW][C]89[/C][C]133.754018115746[/C][C]132.158096239976[/C][C]135.349939991516[/C][/ROW]
[ROW][C]90[/C][C]133.820049937469[/C][C]132.050231055023[/C][C]135.589868819915[/C][/ROW]
[ROW][C]91[/C][C]133.835837320633[/C][C]131.901493266692[/C][C]135.770181374575[/C][/ROW]
[ROW][C]92[/C][C]133.711066550399[/C][C]131.620754758016[/C][C]135.801378342782[/C][/ROW]
[ROW][C]93[/C][C]134.74411921043[/C][C]132.485411573215[/C][C]137.002826847646[/C][/ROW]
[ROW][C]94[/C][C]136.654626408511[/C][C]134.216200692771[/C][C]139.093052124252[/C][/ROW]
[ROW][C]95[/C][C]136.961092146953[/C][C]134.372640566507[/C][C]139.549543727399[/C][/ROW]
[ROW][C]96[/C][C]137.108799136045[/C][C]133.618435982867[/C][C]140.599162289224[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=232438&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=232438&T=3

As an alternative you can also use a QR Code:  
QR Code


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

Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
85133.513909495403132.835722424196134.192096566609
86133.453636756006132.482573534865134.424699977147
87133.437925582859132.233692196843134.642158968876
88133.693796236773132.283853444039135.103739029507
89133.754018115746132.158096239976135.349939991516
90133.820049937469132.050231055023135.589868819915
91133.835837320633131.901493266692135.770181374575
92133.711066550399131.620754758016135.801378342782
93134.74411921043132.485411573215137.002826847646
94136.654626408511134.216200692771139.093052124252
95136.961092146953134.372640566507139.549543727399
96137.108799136045133.618435982867140.599162289224


Figures (Output of Computation)

Input Parameters & R Code

Parameters (Session):
par1 = 12 ; par2 = Triple ; par3 = multiplicative ;
Parameters (R input):
par1 = 12 ; par2 = Triple ; par3 = multiplicative ;
R code (references can be found in the software module):
par1 <- as.numeric(par1)
if (par2 == 'Single') K <- 1
if (par2 == 'Double') K <- 2
if (par2 == 'Triple') K <- par1
nx <- length(x)
nxmK <- nx - K
x <- ts(x, frequency = par1)
if (par2 == 'Single') fit <- HoltWinters(x, gamma=F, beta=F)
if (par2 == 'Double') fit <- HoltWinters(x, gamma=F)
if (par2 == 'Triple') fit <- HoltWinters(x, seasonal=par3)
fit
myresid <- x - fit$fitted[,'xhat']
bitmap(file='test1.png')
op <- par(mfrow=c(2,1))
plot(fit,ylab='Observed (black) / Fitted (red)',main='Interpolation Fit of Exponential Smoothing')
plot(myresid,ylab='Residuals',main='Interpolation Prediction Errors')
par(op)
dev.off()
bitmap(file='test2.png')
p <- predict(fit, par1, prediction.interval=TRUE)
np <- length(p[,1])
plot(fit,p,ylab='Observed (black) / Fitted (red)',main='Extrapolation Fit of Exponential Smoothing')
dev.off()
bitmap(file='test3.png')
op <- par(mfrow = c(2,2))
acf(as.numeric(myresid),lag.max = nx/2,main='Residual ACF')
spectrum(myresid,main='Residals Periodogram')
cpgram(myresid,main='Residal Cumulative Periodogram')
qqnorm(myresid,main='Residual Normal QQ Plot')
qqline(myresid)
par(op)
dev.off()
load(file='createtable')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Estimated Parameters of Exponential Smoothing',2,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Parameter',header=TRUE)
a<-table.element(a,'Value',header=TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'alpha',header=TRUE)
a<-table.element(a,fit$alpha)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'beta',header=TRUE)
a<-table.element(a,fit$beta)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'gamma',header=TRUE)
a<-table.element(a,fit$gamma)
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file='mytable.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Interpolation Forecasts of Exponential Smoothing',4,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'t',header=TRUE)
a<-table.element(a,'Observed',header=TRUE)
a<-table.element(a,'Fitted',header=TRUE)
a<-table.element(a,'Residuals',header=TRUE)
a<-table.row.end(a)
for (i in 1:nxmK) {
a<-table.row.start(a)
a<-table.element(a,i+K,header=TRUE)
a<-table.element(a,x[i+K])
a<-table.element(a,fit$fitted[i,'xhat'])
a<-table.element(a,myresid[i])
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable1.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Extrapolation Forecasts of Exponential Smoothing',4,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'t',header=TRUE)
a<-table.element(a,'Forecast',header=TRUE)
a<-table.element(a,'95% Lower Bound',header=TRUE)
a<-table.element(a,'95% Upper Bound',header=TRUE)
a<-table.row.end(a)
for (i in 1:np) {
a<-table.row.start(a)
a<-table.element(a,nx+i,header=TRUE)
a<-table.element(a,p[i,'fit'])
a<-table.element(a,p[i,'lwr'])
a<-table.element(a,p[i,'upr'])
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable2.tab')