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Exponential Smoothing - NWWZ

*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: Wed, 29 Dec 2010 21:45:07 +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/29/t12936590471x22j3m6ly7gfgg.htm/, Retrieved Wed, 29 Dec 2010 22:44:08 +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/29/t12936590471x22j3m6ly7gfgg.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 «

206010 198112 194519 185705 180173 176142 203401 221902 197378 185001 176356 180449 180144 173666 165688 161570 156145 153730 182698 200765 176512 166618 158644 159585 163095 159044 155511 153745 150569 150605 179612 194690 189917 184128 175335 179566 181140 177876 175041 169292 166070 166972 206348 215706 202108 195411 193111 195198 198770 194163 190420 189733 186029 191531 232571 243477 227247 217859 208679 213188 216234 213586 209465 204045 200237 203666 241476 260307 243324 244460 233575 237217 235243 230354 227184 221678 217142 219452 256446 265845 248624 241114 229245 231805 219277 219313 212610 214771 211142 211457 240048 240636 230580 208795 197922 194596 194581 185686 178106 172608 167302 168053 202300 202388 182516 173476 166444 171297 169701 164182 161914 159612 151001 158114 186530 187069 174330 169362 166827 178037 186413 189226 191563 188906 186005 195309 223532 226899 etc...

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	0.786721671208237
beta	0.155737227444934
gamma	1

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	180144	192256.77590812	-12112.7759081197
14	173666	174646.428106397	-980.428106396546
15	165688	164180.807371187	1507.19262881341
16	161570	159602.165810082	1967.83418991757
17	156145	154188.606970164	1956.39302983636
18	153730	152119.123018617	1610.87698138296
19	182698	179952.348401522	2745.65159847814
20	200765	200442.561219332	322.438780667988
21	176512	176164.427462602	347.572537397995
22	166618	164082.693874569	2535.30612543056
23	158644	157564.603112553	1079.39688744652
24	159585	162699.575040562	-3114.57504056211
25	163095	157033.520593308	6061.47940669156
26	159044	158060.686855116	983.313144883898
27	155511	151876.285741275	3634.71425872456
28	153745	151536.070533916	2208.92946608434
29	150569	148805.699979993	1763.30002000669
30	150605	148982.909933439	1622.09006656095
31	179612	179540.649021074	71.3509789262025
32	194690	199556.1218355	-4866.12183550018
33	189917	172711.692459615	17205.3075403846
34	184128	177934.638374221	6193.36162577895
35	175335	178007.834878522	-2672.83487852153
36	179566	182860.560671384	-3294.56067138413
37	181140	182552.108246818	-1412.10824681763
38	177876	179243.047588606	-1367.04758860619
39	175041	174113.551861834	927.448138166044
40	169292	173346.18218413	-4054.18218412981
41	166070	166832.87427879	-762.874278790143
42	166972	165922.491006595	1049.50899340509
43	206348	196558.795284389	9789.20471561147
44	215706	225216.873886494	-9510.87388649434
45	202108	200907.006969103	1200.99303089728
46	195411	190710.856136706	4700.14386329436
47	193111	187055.840220638	6055.1597793616
48	195198	199049.339913743	-3851.339913743
49	198770	199042.997871197	-272.997871196945
50	194163	197117.93077686	-2954.93077686024
51	190420	191512.249043427	-1092.2490434272
52	189733	188129.678889893	1603.32111010668
53	186029	187498.596717085	-1469.59671708496
54	191531	187061.553357131	4469.44664286866
55	232571	223314.193184279	9256.80681572098
56	243477	248433.712722975	-4956.712722975
57	227247	231545.874832781	-4298.87483278071
58	217859	218649.860480711	-790.860480711417
59	208679	211171.888635607	-2492.88863560685
60	213188	213488.229628962	-300.229628961562
61	216234	216634.512356744	-400.51235674409
62	213586	213617.211916038	-31.211916037777
63	209465	210647.255498068	-1182.25549806794
64	204045	207696.056936062	-3651.05693606203
65	200237	201559.353700861	-1322.35370086058
66	203666	201806.358136103	1859.64186389704
67	241476	236008.629329493	5467.37067050653
68	260307	253632.97407016	6674.02592983964
69	243324	245978.10496673	-2654.1049667302
70	244460	235268.282471266	9191.71752873395
71	233575	236647.931624635	-3072.93162463498
72	237217	240271.63520002	-3054.63520001969
73	235243	242188.152693052	-6945.15269305155
74	230354	234257.517648547	-3903.51764854742
75	227184	227677.911670693	-493.911670693051
76	221678	224508.313359514	-2830.31335951379
77	217142	219381.134929034	-2239.13492903442
78	219452	219340.378779068	111.621220931673
79	256446	252477.564273766	3968.4357262344
80	265845	268537.035194253	-2692.03519425297
81	248624	249733.664875924	-1109.66487592363
82	241114	241164.041968422	-50.0419684220105
83	229245	229923.595156386	-678.595156385709
84	231805	232994.61627415	-1189.61627415035
85	219277	233336.866000631	-14059.8660006305
86	219313	217374.18329822	1938.81670178048
87	212610	213750.413699219	-1140.41369921915
88	214771	207127.034028596	7643.96597140448
89	211142	209202.751430528	1939.24856947243
90	211457	212298.996501291	-841.996501291404
91	240048	244740.097239514	-4692.09723951353
92	240636	250736.072567315	-10100.0725673146
93	230580	223704.945028586	6875.05497141436
94	208795	219884.192553616	-11089.1925536162
95	197922	196713.536485743	1208.46351425743
96	194596	198279.950262188	-3683.95026218813
97	194581	190729.090050536	3851.90994946379
98	185686	191278.927945051	-5592.92794505064
99	178106	179159.001172259	-1053.00117225942
100	172608	172574.581266998	33.4187330016284
101	167302	164610.437912255	2691.56208774468
102	168053	164961.754155404	3091.2458445956
103	202300	197414.376094765	4885.62390523526
104	202388	208703.725659619	-6315.72565961938
105	182516	187645.695259936	-5129.6952599361
106	173476	168453.759284006	5022.24071599363
107	166444	160459.741451464	5984.25854853578
108	171297	165203.67108995	6093.32891004978
109	169701	168613.71455601	1087.28544399026
110	164182	166297.126878907	-2115.12687890694
111	161914	159630.579876355	2283.42012364493
112	159612	158060.539554747	1551.46044525341
113	151001	154201.424991105	-3200.42499110513
114	158114	151624.562134491	6489.43786550936
115	186530	189171.601050891	-2641.60105089133
116	187069	193266.148679687	-6197.14867968677
117	174330	173684.922388921	645.077611079323
118	169362	163039.411629569	6322.58837043142
119	166827	158271.002348831	8555.99765116919
120	178037	167373.950770154	10663.0492298462
121	186413	176183.815808529	10229.1841914707
122	189226	184368.839357322	4857.16064267812
123	191563	188972.400892042	2590.59910795817
124	188906	192372.294027718	-3466.29402771773
125	186005	187821.724755116	-1816.72475511601
126	195309	192839.215792406	2469.78420759359
127	223532	229223.087110724	-5691.08711072433
128	226899	233733.221800341	-6834.22180034057
129	214126	218605.044575766	-4479.04457576631
130	206903	208006.298620989	-1103.29862098902
131	204442	199829.420257715	4612.57974228458
132	220375	207753.529268726	12621.4707312738
133	214320	219725.686840236	-5405.68684023627
134	212588	214263.16287187	-1675.16287187024
135	205816	212242.323659019	-6426.32365901885
136	202196	205149.9623062	-2953.96230619974
137	195722	199310.402816274	-3588.40281627362
138	198563	201587.356456262	-3024.35645626215
139	229139	228974.239624869	164.760375131242
140	229527	235630.868228266	-6103.86822826631
141	211868	219452.446170659	-7584.44617065854
142	203555	204622.969306613	-1067.96930661306
143	195770	195189.669246083	580.330753916613

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
144	198652.31584729	188946.127403192	208358.504291388
145	192306.353232713	179187.823603671	205424.882861755
146	188010.82132942	171506.694169197	204514.948489644
147	182618.374637975	162670.69897198	202566.05030397
148	178433.511095258	154949.581220176	201917.440970341
149	172255.700582945	145126.928102198	199384.473063691
150	175388.800333859	144499.251040227	206278.34962749
151	204118.502515161	169349.186596989	238887.818433332
152	207571.683921239	168802.856144804	246340.511697675
153	194890.525319255	152002.965606995	237778.085031515
154	187357.973538446	140233.710876374	234482.236200517
155	179187.517956523	127710.236010429	230664.799902617

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/29/t12936590471x22j3m6ly7gfgg/1sz161293659102.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/29/t12936590471x22j3m6ly7gfgg/1sz161293659102.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/29/t12936590471x22j3m6ly7gfgg/2sz161293659102.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/29/t12936590471x22j3m6ly7gfgg/2sz161293659102.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/29/t12936590471x22j3m6ly7gfgg/338091293659102.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/29/t12936590471x22j3m6ly7gfgg/338091293659102.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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