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Exponential Smoothing

*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: Sun, 12 Dec 2010 11:16:02 +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/12/t1292152457zozx1t8ecrqzh0w.htm/, Retrieved Sun, 12 Dec 2010 12:14:18 +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/12/t1292152457zozx1t8ecrqzh0w.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 «

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

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	0.859273893725624
beta	0
gamma	1

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	33508	35364.6626602564	-1856.66266025641
14	36080	36276.0354231599	-196.035423159919
15	34560	34389.3418181102	170.658181889841
16	38737	38296.4051215425	440.594878457545
17	38144	37441.7929812939	702.207018706089
18	37594	36733.5189901093	860.48100989069
19	36424	36951.8290409387	-527.829040938726
20	36843	34780.1171753602	2062.8828246398
21	37246	37188.3697153718	57.6302846281615
22	38661	34382.7694307578	4278.23056924218
23	40454	39570.7791198969	883.220880103108
24	44928	42404.7956142132	2523.2043857868
25	48441	42312.3928809279	6128.6071190721
26	48140	50318.9931046143	-2178.99310461434
27	45998	46779.9990947625	-781.999094762476
28	47369	49906.4560109482	-2537.45601094823
29	49554	46529.6981450982	3024.30185490179
30	47510	47839.0129079156	-329.012907915567
31	44873	46839.8504206739	-1966.85042067385
32	45344	43796.2058442976	1547.79415570242
33	42413	45479.6647560848	-3066.66475608481
34	36912	40583.3879508839	-3671.38795088395
35	43452	38462.7314862846	4989.26851371542
36	42142	45055.7560116669	-2913.7560116669
37	44382	40798.8894358358	3583.11056416423
38	43636	45449.1146913578	-1813.11469135783
39	44167	42421.1039777901	1745.89602220988
40	44423	47472.6765575195	-3049.67655751946
41	42868	44438.4654766729	-1570.46547667287
42	43908	41327.7176940411	2580.2823059589
43	42013	42597.950137342	-584.950137342043
44	38846	41236.3386443366	-2390.33864433663
45	35087	38886.4880158068	-3799.4880158068
46	33026	33275.414974234	-249.414974234031
47	34646	35313.9510155476	-667.951015547573
48	37135	35933.7126191115	1201.28738088851
49	37985	36127.0741382522	1857.9258617478
50	43121	38535.503448344	4585.49655165605
51	43722	41506.4980519065	2215.50194809349
52	43630	46286.728487585	-2656.72848758502
53	42234	43798.3310405884	-1564.33104058842
54	39351	41276.9729923135	-1925.97299231351
55	39327	38229.6670621471	1097.33293785288
56	35704	38059.5322026113	-2355.53220261131
57	30466	35541.2857365836	-5075.28573658355
58	28155	29333.5409760028	-1178.54097600283
59	29257	30514.8043525852	-1257.80435258523
60	29998	30890.7710237347	-892.771023734749
61	32529	29377.1690004872	3151.83099951279
62	34787	33281.2576191968	1505.74238080323
63	33855	33272.3797522023	582.62024779775
64	34556	35963.8675531898	-1407.86755318978
65	31348	34702.3125432327	-3354.3125432327
66	30805	30591.9776557521	213.02234424794
67	28353	29808.1126487223	-1455.11264872232
68	24514	26958.8196647792	-2444.8196647792
69	21106	23981.1104886117	-2875.11048861168
70	21346	20212.2925975361	1133.70740246391
71	23335	23369.2562151878	-34.2562151877719
72	24379	24847.955577549	-468.955577549048
73	26290	24267.7081971276	2022.29180287235
74	30084	26969.5656303307	3114.4343696693
75	29429	28213.0874090208	1215.91259097921
76	30632	31168.6331897809	-536.633189780856
77	27349	30381.7914990917	-3032.79149909172
78	27264	27049.7483996167	214.251600383279
79	27474	26032.189517992	1441.81048200796
80	24482	25532.8693376932	-1050.86933769324
81	21453	23692.3921345374	-2239.39213453744
82	18788	21033.9757614542	-2245.97576145418
83	19282	21122.5028951048	-1840.50289510477
84	19713	20987.9680911197	-1274.96809111973
85	21917	20065.7187433838	1851.28125661619
86	23812	22774.324249559	1037.67575044097
87	23785	21966.1699855844	1818.83001441562
88	24696	25193.1580245817	-497.158024581724
89	24562	24088.961673285	473.038326715036
90	23580	24226.3303512646	-646.330351264558
91	24939	22642.045446811	2296.95455318899
92	23899	22526.7431170371	1372.25688296295
93	21454	22601.1388310749	-1147.13883107493
94	19761	20880.3407188114	-1119.34071881143
95	19815	21994.0165500426	-2179.01655004265
96	20780	21648.1913106273	-868.191310627306
97	23462	21515.419528892	1946.58047110803
98	25005	24191.4176272452	813.582372754849
99	24725	23300.6345720368	1424.36542796319
100	26198	25862.7495109902	335.250489009817
101	27543	25610.3520191772	1932.64798082282
102	26471	26844.4007724239	-373.40077242392
103	26558	25908.8341541536	649.165845846434
104	25317	24247.5009031724	1069.49909682764
105	22896	23707.2000064611	-811.200006461102
106	22248	22278.9772761778	-30.9772761777567
107	23406	24178.7313469072	-772.731346907207
108	25073	25225.7576016279	-152.757601627884
109	27691	26103.8512016217	1587.14879837835
110	30599	28311.5566362226	2287.44336377743
111	31948	28773.1769747183	3174.82302528166
112	32946	32686.147524477	259.85247552301
113	34012	32593.7580172313	1418.2419827687
114	32936	33061.269863651	-125.269863650959
115	32974	32482.817476111	491.182523889034
116	30951	30744.885142676	206.11485732405
117	29812	29198.0372468256	613.96275317443
118	29010	29104.2173770665	-94.2173770665431
119	31068	30845.2467178788	222.753282121241
120	32447	32834.9134170942	-387.913417094212
121	34844	33755.7940168548	1088.20598314525
122	35676	35633.3206432976	42.6793567023851
123	35387	34290.9513574894	1096.04864251056
124	36488	36007.4728938153	480.527106184723
125	35652	36267.7189806085	-615.718980608523
126	33488	34770.2888582061	-1282.28885820611
127	32914	33284.3911983022	-370.391198302248
128	29781	30766.0145951278	-985.01459512784
129	27951	28253.0551030731	-302.055103073071

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
130	27272.4655709844	23431.3188675874	31113.6122743814
131	29139.0594909159	24074.6395725795	34203.4794092523
132	30851.3833632508	24806.3614260312	36896.4053004705
133	32313.3163709381	25425.9198139151	39200.7129279611
134	33108.6431139227	25471.2238871213	40746.0623407242
135	31877.83712916	23557.7341515436	40197.9401067764
136	32565.9327315879	23615.0637977712	41516.8016654047
137	32259.0039774962	22718.9832341964	41799.024720796
138	31196.841317568	21101.9942709399	41291.688364196
139	30941.1088047348	20320.3801100058	41561.8374994638
140	28654.5061312669	17532.7338300882	39776.2784324456
141	27084.0541958042	15482.8576958095	38685.2506957989

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/12/t1292152457zozx1t8ecrqzh0w/1ex9a1292152557.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/12/t1292152457zozx1t8ecrqzh0w/1ex9a1292152557.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/12/t1292152457zozx1t8ecrqzh0w/2o68u1292152557.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/12/t1292152457zozx1t8ecrqzh0w/2o68u1292152557.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/12/t1292152457zozx1t8ecrqzh0w/3o68u1292152557.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/12/t1292152457zozx1t8ecrqzh0w/3o68u1292152557.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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