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Workshop 8 Regression Analysis of Time Series

*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: Sat, 27 Nov 2010 13:02:33 +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/Nov/27/t1290862881mc63hzbml9zrxhb.htm/, Retrieved Sat, 27 Nov 2010 14:01:25 +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/Nov/27/t1290862881mc63hzbml9zrxhb.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 «

24 25 17 18 18 16 20 16 18 17 23 30 23 18 15 12 21 15 20 31 27 34 21 31 19 16 20 21 22 17 24 25 26 25 17 32 33 13 32 25 29 22 18 17 20 15 20 33 29 23 26 18 20 11 28 26 22 17 12 14 17 21 19 18 10 29 31 19 9 20 28 19 30 29 26 23 13 21 19 28 23 18 21 20 23 21 21 15 28 19 26 10 16 22 19 31 31 29 19 22 23 15 20 18 23 25 21 24 25 17 13 28 21 25 9 16 19 17 25 20 29 14 22 15 19 20 15 20 18 33 22 16 17 16 21 26 18 18 17 22 30 30 24 21 21 29 31 20 16 22 20 28 38 22 20 17 28 22 31

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	'Gwilym Jenkins' @ 72.249.127.135

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.098155242834915
beta	0.0343410839300175
gamma	0.291862308416832

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	23	23.9407051282051	-0.940705128205131
14	18	18.4734042273127	-0.47340422731267
15	15	14.6753756290066	0.324624370993439
16	12	10.8734386182343	1.12656138176569
17	21	19.6123466279767	1.38765337202334
18	15	14.0482262950089	0.951773704991053
19	20	21.5695302857669	-1.56953028576691
20	31	18.004731231842	12.995268768158
21	27	21.9550141313493	5.04498586865068
22	34	22.1002738391955	11.8997261608045
23	21	29.7501400020546	-8.75014000205462
24	31	36.1352856044103	-5.13528560441032
25	19	28.7353310927983	-9.73533109279827
26	16	22.5549834855285	-6.55498348552847
27	20	18.3767607337301	1.6232392662699
28	21	14.9244403853849	6.07555961461514
29	22	24.245592945388	-2.24559294538802
30	17	18.2256250315818	-1.22562503158176
31	24	24.8777271260046	-0.877727126004594
32	25	25.2249952297038	-0.224995229703811
33	26	25.7509447799475	0.249055220052501
34	25	27.1794943734804	-2.17949437348036
35	17	27.9143781778676	-10.9143781778676
36	32	34.9335903984696	-2.93359039846963
37	33	26.4413824311170	6.55861756888296
38	13	22.6548450980675	-9.65484509806753
39	32	20.2718927525035	11.7281072474965
40	25	18.9643057982640	6.03569420173602
41	29	26.0721354148673	2.92786458513271
42	22	20.8267162939575	1.17328370604254
43	18	27.8122182439617	-9.81221824396166
44	17	27.430574310146	-10.430574310146
45	20	27.0214157974156	-7.02141579741562
46	15	27.0144401806347	-12.0144401806347
47	20	24.3690123524528	-4.36901235245283
48	33	34.0375962623899	-1.03759626238988
49	29	28.1426184114701	0.857381588529947
50	23	19.4222806898592	3.57771931014081
51	26	23.9045047988361	2.0954952011639
52	18	20.0586595880739	-2.05865958807387
53	20	25.4322347737117	-5.43223477371167
54	11	18.7544917138134	-7.75449171381337
55	28	21.7921437713945	6.20785622860549
56	26	22.694195038791	3.30580496120899
57	22	24.4509772793287	-2.45097727932865
58	17	23.5140868067799	-6.51408680677995
59	12	23.3751842851588	-11.3751842851588
60	14	33.163585956078	-19.163585956078
61	17	25.8577677766990	-8.85776777669897
62	21	16.7366570692108	4.26334293078918
63	19	20.7351344812155	-1.73513448121549
64	18	15.2460456319187	2.75395436808126
65	10	20.0464388758477	-10.0464388758477
66	29	12.1313910944195	16.8686089055805
67	31	21.1708756869899	9.8291243130101
68	19	21.5865716512114	-2.58657165121142
69	9	21.1519161460759	-12.1519161460759
70	20	18.0628572670495	1.93714273295050
71	28	17.3719618880004	10.6280381119996
72	19	27.2422284888402	-8.24222848884021
73	30	23.7300062409474	6.26999375905261
74	29	19.6073898868591	9.3926101131409
75	26	22.6076938397626	3.39230616023736
76	23	18.8980150381552	4.10198496184484
77	13	20.5605510453933	-7.56055104539328
78	21	20.081391665784	0.918608334216017
79	19	25.7560969619197	-6.75609696191973
80	28	21.2736618807934	6.72633811920658
81	23	19.2645580477712	3.73544195222881
82	18	21.5261016266073	-3.52610162660731
83	21	22.6508449980264	-1.65084499802639
84	20	26.3718732685513	-6.37187326855135
85	23	26.892293994386	-3.89229399438598
86	21	22.5890896770853	-1.58908967708532
87	21	22.890074592955	-1.89007459295501
88	15	18.7888538806877	-3.78885388068774
89	28	16.5206697924175	11.4793302075825
90	19	20.1199466417961	-1.11994664179613
91	26	23.5453263708249	2.45467362917508
92	10	23.5176592493214	-13.5176592493214
93	16	18.6679263427215	-2.66792634272150
94	22	18.3016906625713	3.69830933742869
95	19	20.5655827255514	-1.56558272555139
96	31	22.9890644294635	8.01093557053652
97	31	25.5591029653854	5.4408970346146
98	29	22.7949054740391	6.20509452596095
99	19	23.8246475836695	-4.82464758366946
100	22	18.9686459288086	3.03135407119143
101	23	21.4447466123348	1.5552533876652
102	15	20.7761976475165	-5.77619764751653
103	20	24.6923253165335	-4.69232531653347
104	18	19.7417940100302	-1.74179401003025
105	23	18.9262252706317	4.07377472936827
106	25	20.9426273425743	4.05737265742570
107	21	21.9026599410408	-0.902659941040831
108	24	26.9605532153950	-2.96055321539497
109	25	27.7888928073504	-2.78889280735043
110	17	24.4019871167118	-7.40198711671178
111	13	21.1310096444457	-8.13100964444574
112	28	17.9452145586073	10.0547854413927
113	21	20.6727748948518	0.327225105148205
114	25	17.9004090991798	7.0995909008202
115	9	23.3555197336149	-14.3555197336149
116	16	18.1904114571158	-2.19041145711583
117	19	18.8173319815893	0.182668018410684
118	17	20.3901497325718	-3.39014973257184
119	25	19.2311781730949	5.76882182690513
120	20	24.3422946477942	-4.34229464779417
121	29	25.0155846874287	3.98441531257131
122	14	21.0375047068512	-7.03750470685117
123	22	17.5698656660359	4.43013433396406
124	15	20.4055699810182	-5.40556998101823
125	19	19.0048598459936	-0.00485984599361799
126	20	17.9310429487519	2.06895705124815
127	15	17.1766773394644	-2.17667733946443
128	20	16.3816739262784	3.61832607372156
129	18	18.195616076569	-0.195616076569014
130	33	18.7818454174540	14.2181545825460
131	22	21.8123081876901	0.187691812309911
132	16	23.7457153533617	-7.7457153533617
133	17	26.2966806952366	-9.29668069523663
134	16	18.0891232220929	-2.08912322209294
135	21	18.1175831505994	2.88241684940057
136	26	18.1991945695216	7.80080543047845
137	18	19.5475436969447	-1.54754369694466
138	18	18.8942020398857	-0.894202039885734
139	17	16.7475253234629	0.252474676537052
140	22	17.7405238770503	4.25947612294967
141	30	18.6399136566837	11.3600863433163
142	30	24.2196608145824	5.78033918541763
143	24	22.7657965052121	1.23420349478794
144	21	22.7541739536077	-1.7541739536077
145	21	25.5456373902177	-4.54563739021775
146	29	19.7782013738207	9.22179862617935
147	31	22.3402434183874	8.65975658161259
148	20	24.4177706252788	-4.41777062527876
149	16	22.1992440725596	-6.19924407255963
150	22	21.3386654672080	0.661334532791972
151	20	19.7291209263115	0.270879073688477
152	28	21.8613150328721	6.13868496712787
153	38	24.9031650989477	13.0968349010523
154	22	29.2795896433223	-7.27958964332232
155	20	25.3980709242075	-5.39807092420746
156	17	23.9773706994309	-6.97737069943092
157	28	25.5322893165103	2.46771068348967
158	22	24.1115721224774	-2.11157212247739
159	31	25.4095780065939	5.59042199340615

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
160	23.7296244693920	11.0971215548138	36.3621273839703
161	21.4766920955035	8.77925114622493	34.1741330447821
162	23.0521781224104	10.2857800612796	35.8185761835411
163	21.2944900039497	8.4550392924907	34.1339407154087
164	24.9632178025419	12.0465480184480	37.8798875866358
165	29.2319479837333	16.2338274667531	42.2300685007136
166	26.9132713398171	13.8294088802634	39.9971337993708
167	24.2198553063170	11.0459058502224	37.3938047624115
168	22.9098348438952	9.64140518894856	36.1782644988419
169	27.6557456938485	14.2884001506674	41.0230912370295
170	24.7992025169713	11.3284685109857	38.2699365229570
171	28.3505907056985	14.7719642959105	41.9292171154866

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Nov/27/t1290862881mc63hzbml9zrxhb/14uzi1290862949.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/27/t1290862881mc63hzbml9zrxhb/14uzi1290862949.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/27/t1290862881mc63hzbml9zrxhb/2x3y31290862949.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/27/t1290862881mc63hzbml9zrxhb/2x3y31290862949.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/27/t1290862881mc63hzbml9zrxhb/3x3y31290862949.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/27/t1290862881mc63hzbml9zrxhb/3x3y31290862949.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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