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Exponential Smoothning - OPJV

*The author of this computation has been verified*

R Software Module: /rwasp_exponentialsmoothing.wasp (opens new window with default values)

Title produced by software: Exponential Smoothing

Date of computation: Mon, 27 Dec 2010 21:07:25 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/27/t1293484124gsln8kkodhe3403.htm/, Retrieved Mon, 27 Dec 2010 22:08:45 +0100

BibTeX entries for LaTeX users:

@Manual{KEY,
    author = {{YOUR NAME}},
    publisher = {Office for Research Development and Education},
    title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2010/Dec/27/t1293484124gsln8kkodhe3403.htm/},
    year = {2010},
}
@Manual{R,
    title = {R: A Language and Environment for Statistical Computing},
    author = {{R Development Core Team}},
    organization = {R Foundation for Statistical Computing},
    address = {Vienna, Austria},
    year = {2010},
    note = {{ISBN} 3-900051-07-0},
    url = {http://www.R-project.org},
}

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

» Textbox « » Textfile « » CSV «

20503 22885 26217 26583 27751 28158 27373 28367 26851 26733 26849 26733 27951 29781 32914 33488 35652 36488 35387 35676 34844 32447 31068 29010 29812 30951 32974 32936 34012 32946 31948 30599 27691 25073 23406 22248 22896 25317 26558 26471 27543 26198 24725 25005 23462 20780 19815 19761 21454 23899 24939 23580 24562 24696 23785 23812 21917 19713 19282 18788 21453 24482 27474 27264 27349 30632 29429 30084 26290 24379 23335 21346 21106 24514 28353 30805 31348 34556 33855 34787 32529 29998 29257 28155 30466 35704 39327 39351 42234 43630 43722 43121 37985 37135 34646 33026 35087 38846 42013 43908 42868 44423 44167 43636 44382 42142 43452 36912 42413 45344 44873 47510 49554 47369 45998 48140 48441 44928 40454 38661 37246 36843 36424 37594 38144 38737 34560 36080 33508 35462 33374 32110 35533 35532 37903 36763 40399 44164 44496 43110 43880 43930 44327

Output produced by software:

Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	3 seconds
R Server	'RServer@AstonUniversity' @ vre.aston.ac.uk

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.957106985347539
beta	0.0213231419809721
gamma	1

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	27951	24203.1065705128	3747.89342948717
14	29781	29654.6202551851	126.379744814931
15	32914	32946.4121193041	-32.412119304121
16	33488	33593.0200307429	-105.020030742926
17	35652	35915.2410986995	-263.241098699524
18	36488	36895.8636472487	-407.863647248749
19	35387	34286.1180476208	1100.88195237923
20	35676	36417.4957808125	-741.495780812547
21	34844	34291.6797450763	552.320254923688
22	32447	34813.0811183625	-2366.08111836253
23	31068	32676.8052514775	-1608.80525147745
24	29010	30941.1150593774	-1931.11505937737
25	29812	30347.5787971138	-535.578797113765
26	30951	31417.2352441557	-466.235244155694
27	32974	33996.147292809	-1022.14729280899
28	32936	33533.2865375478	-597.286537547756
29	34012	35208.4510334612	-1196.45103346121
30	32946	35101.5248152052	-2155.52481520519
31	31948	30659.9642363374	1288.0357636626
32	30599	32671.431673882	-2072.43167388201
33	27691	29080.0894447363	-1389.08944473627
34	25073	27331.3798624231	-2258.37986242307
35	23406	25046.0703511278	-1640.07035112784
36	22248	22981.3960861211	-733.39608612112
37	22896	23333.2722700379	-437.272270037862
38	25317	24241.2077194548	1075.7922805452
39	26558	28044.8456366513	-1486.84563665127
40	26471	26918.6438964953	-447.643896495269
41	27543	28477.5879114897	-934.587911489689
42	26198	28351.7548651241	-2153.75486512409
43	24725	23831.2288487044	893.771151295619
44	25005	25084.7917686993	-79.7917686992987
45	23462	23234.1860439703	227.813956029706
46	20780	22832.9944567295	-2052.99445672949
47	19815	20612.2284668615	-797.228466861485
48	19761	19251.7817676975	509.218232302523
49	21454	20689.6820866707	764.31791332933
50	23899	22721.0981287392	1177.90187126078
51	24939	26423.1608238923	-1484.16082389229
52	23580	25254.7722664694	-1674.77226646938
53	24562	25503.9618002236	-941.961800223642
54	24696	25204.2520675701	-508.252067570131
55	23785	22308.4228014564	1476.57719854357
56	23812	24008.98557568	-196.985575680013
57	21917	21987.966387298	-70.9663872980345
58	19713	21125.4410414512	-1412.44104145119
59	19282	19507.1512849837	-225.151284983735
60	18788	18697.4908395731	90.5091604269
61	21453	19684.2482889166	1768.75171108344
62	24482	22653.9183268628	1828.08167313724
63	27474	26836.5214996775	637.47850032249
64	27264	27706.3251380422	-442.32513804223
65	27349	29207.4155870458	-1858.41558704582
66	30632	28071.3458877913	2560.6541122087
67	29429	28282.7365692389	1146.26343076108
68	30084	29673.4414532649	410.558546735116
69	26290	28329.7832894654	-2039.7832894654
70	24379	25575.6399447563	-1196.63994475625
71	23335	24269.5158481254	-934.515848125386
72	21346	22834.6746514593	-1488.67465145932
73	21106	22389.9577075273	-1283.95770752727
74	24514	22386.0903534579	2127.90964654207
75	28353	26756.3987381578	1596.60126184223
76	30805	28469.250045438	2335.74995456201
77	31348	32596.5922140573	-1248.59221405726
78	34556	32274.2585747889	2281.74142521108
79	33855	32192.8629260746	1662.13707392537
80	34787	34091.1161300116	695.883869988422
81	32529	32966.6240006514	-437.624000651398
82	29998	31865.9628984148	-1867.96289841482
83	29257	29998.7329284108	-741.732928410791
84	28155	28798.7492071099	-643.749207109926
85	30466	29262.8540575669	1203.14594243308
86	35704	31927.8712040232	3776.12879597679
87	39327	38028.6649060469	1298.33509395306
88	39351	39657.4133971206	-306.413397120574
89	42234	41217.9221600109	1016.07783998913
90	43630	43376.5081197734	253.49188022656
91	43722	41447.8517953737	2274.14820462626
92	43121	44023.4776915288	-902.477691528788
93	37985	41421.0008466684	-3436.00084666835
94	37135	37428.4661847129	-293.466184712874
95	34646	37187.8839885004	-2541.88398850035
96	33026	34303.8060686304	-1277.80606863038
97	35087	34261.9695397591	825.030460240945
98	38846	36689.43589895	2156.56410105001
99	42013	41114.7831357913	898.216864208669
100	43908	42264.5076086407	1643.49239135932
101	42868	45760.5696005895	-2892.56960058953
102	44423	44078.2415818128	344.758418187237
103	44167	42258.2611563071	1908.73884369292
104	43636	44275.0906888205	-639.090688820528
105	44382	41748.6028286708	2633.39717132922
106	42142	43756.3615625214	-1614.36156252142
107	43452	42184.57959563	1267.42040437003
108	36912	43107.8557682617	-6195.85576826174
109	42413	38455.9684959765	3957.03150402353
110	45344	44008.9799360331	1335.02006396692
111	44873	47648.0523369253	-2775.05233692529
112	47510	45293.0713017395	2216.92869826049
113	49554	49134.1497955218	419.850204478178
114	47369	50819.3642618564	-3450.36426185641
115	45998	45415.0199694621	582.980030537932
116	48140	46007.5063306288	2132.49366937119
117	48441	46284.4859863651	2156.51401363491
118	44928	47654.282754283	-2726.28275428301
119	40454	45119.8543019029	-4665.85430190292
120	38661	39901.1127273156	-1240.11272731556
121	37246	40385.9124353144	-3139.91243531439
122	36843	38847.1078568749	-2004.10785687491
123	36424	38859.0220934936	-2435.02209349359
124	37594	36795.5849014515	798.415098548547
125	38144	38924.9396374962	-780.939637496209
126	38737	38993.3858705731	-256.385870573133
127	34560	36582.7285794758	-2022.72857947585
128	36080	34458.2633380606	1621.73666193943
129	33508	33947.5273990163	-439.52739901626
130	35462	32270.3187389705	3191.68126102947
131	33374	35084.7196646527	-1710.71966465272
132	32110	32669.5071767577	-559.507176757666
133	35533	33566.3299314637	1966.67006853631
134	35532	36910.1059105792	-1378.10591057919
135	37903	37461.7802399504	441.219760049586
136	36763	38307.6984169206	-1544.69841692065
137	40399	38096.6723864796	2302.32761352043
138	44164	41171.5327145291	2992.46728547088
139	44496	41893.8137997328	2602.18620026722
140	43110	44545.7986950495	-1435.79869504952
141	43880	41151.4504753465	2728.54952465348
142	43930	42858.0296394016	1071.97036059845
143	44327	43585.9473051438	741.05269485616

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
144	43769.3449128364	40271.3686085046	47267.3212171683
145	45524.0726400358	40632.4850422983	50415.6602377733
146	47015.9719798344	41006.5861541469	53025.3578055218
147	49166.7071249073	42180.9145999252	56152.4996498895
148	49698.1737835985	41823.939885864	57572.407681333
149	51355.1499768495	42652.868811896	60057.431141803
150	52433.6015513244	42947.3006998522	61919.9024027965
151	50391.5220258944	40154.6098333451	60628.4342184437
152	50443.1192602075	39481.6735387983	61404.5649816167
153	48694.2922620938	37029.093605814	60359.4909183737
154	47755.3030739244	35403.1669813812	60107.4391664677
155	47458.1601579429	34432.8500262172	60483.4702896685

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/27/t1293484124gsln8kkodhe3403/1m43k1293484040.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/27/t1293484124gsln8kkodhe3403/1m43k1293484040.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/27/t1293484124gsln8kkodhe3403/2xw351293484040.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/27/t1293484124gsln8kkodhe3403/2xw351293484040.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/27/t1293484124gsln8kkodhe3403/3xw351293484040.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/27/t1293484124gsln8kkodhe3403/3xw351293484040.ps (open in new window)

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')

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

Software written by Ed van Stee & Patrick Wessa

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