Home » date » 2010 » Nov » 30 »

Personal Standards (Yt) - Single 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: Tue, 30 Nov 2010 12:15:29 +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/30/t1291119290sl31sp861zsq1fz.htm/, Retrieved Tue, 30 Nov 2010 13:14:54 +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/30/t1291119290sl31sp861zsq1fz.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 30 19 22 22 25 23 17 21 19 19 15 16 23 27 22 14 22 23 23 21 19 18 20 23 25 19 24 22 25 26 29 32 25 29 28 17 28 29 26 25 14 25 26 20 18 32 25 25 23 21 20 15 30 24 26 24 22 14 24 24 24 24 19 31 22 27 19 25 20 21 27 23 25 20 21 22 23 25 25 17 19 25 19 20 26 23 27 17 17 19 17 22 21 32 21 21 18 18 23 19 20 21 20 17 18 19 22 15 14 18 24 35 29 21 25 20 22 13 26 17 25 20 19 21 22 24 21 26 24 16 23 18 16 26 19 21 21 22 23 29 21 21 23 27 25 21 10 20 26 24 29 19 24 19 24 22 17

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	'George Udny Yule' @ 72.249.76.132

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.0482511366093053
beta	FALSE
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
2	25	24	1
3	30	24.0482511366093	5.9517488633907
4	19	24.3354297840810	-5.33542978408105
5	22	24.0779892327	-2.07798923269999
6	22	23.9777238903603	-1.97772389036032
7	25	23.8822964647511	1.11770353524894
8	23	23.9362269307191	-0.936226930719062
9	17	23.8910529171876	-6.89105291718763
10	21	23.5585517814985	-2.55855178149845
11	19	23.4350987499674	-4.43509874996739
12	19	23.2211001943070	-4.22110019430695
13	15	23.0174273121899	-8.01742731218988
14	16	22.6305773316942	-6.63057733169423
15	23	22.3106444390641	0.689355560935908
16	27	22.3439066284072	4.65609337159281
17	22	22.5685684257456	-0.568568425745596
18	14	22.5411343529632	-8.54113435296321
19	22	22.1290149125000	-0.129014912499951
20	23	22.1227897963323	0.877210203667723
21	23	22.1651161857045	0.834883814295473
22	21	22.205400278681	-1.20540027868100
23	19	22.1472383451655	-3.14723834516547
24	18	21.9953805178308	-3.99538051783084
25	20	21.8025988666588	-1.80259886665883
26	23	21.7156214224919	1.28437857750811
27	25	21.7775941486933	3.22240585130670
28	19	21.9330788936353	-2.93307889363533
29	24	21.7915545032527	2.20844549674734
30	22	21.8981145086104	0.101885491389581
31	25	21.9030305993740	3.09696940062604
32	26	22.0524628929984	3.94753710700159
33	29	22.2429360452186	6.75706395478135
34	32	22.5689720611786	9.43102793882139
35	25	23.0240298786209	1.97597012137914
36	29	23.1193726828834	5.88062731711657
37	28	23.4031196349100	4.59688036508997
38	17	23.6249243373826	-6.62492433738262
39	28	23.3052642081533	4.69473579184674
40	29	23.5317905461903	5.46820945380975
41	26	23.7956378675543	2.20436213244568
42	25	23.9020008459433	1.09799915405667
43	14	23.9549805531226	-9.95498055312262
44	25	23.4746414265109	1.52535857348907
45	26	23.5482417114185	2.45175828858147
46	20	23.6665418355339	-3.66654183553387
47	18	23.4896270245438	-5.48962702454379
48	32	23.2247462810484	8.7752537189516
49	25	23.6481622470228	1.35183775297716
50	25	23.7133899551154	1.28661004488464
51	23	23.775470352154	-0.775470352153995
52	21	23.7380530262557	-2.73805302625575
53	20	23.6059388556424	-3.60593885564236
54	15	23.4319482073140	-8.43194820731396
55	30	23.0250971224803	6.97490287751974
56	24	23.3616441140601	0.638355885939895
57	26	23.3924455111179	2.60755448888206
58	24	23.5182629789772	0.481737021022802
59	22	23.5415073377883	-1.54150733778833
60	14	23.4671278566485	-9.46712785664846
61	24	23.0103281771396	0.989671822860448
62	24	23.0580809674628	0.941919032537228
63	24	23.1035296313766	0.896470368623369
64	24	23.1467853455993	0.853214654400727
65	19	23.1879539224458	-4.18795392244582
66	31	22.9858803856204	8.01411961437958
67	22	23.3725707659372	-1.37257076593716
68	27	23.306342666404	3.69365733359601
69	19	23.4845658309953	-4.48456583099529
70	25	23.2681804324505	1.73181956754949
71	20	23.351742694987	-3.35174269498701
72	21	23.1900173003319	-2.19001730033195
73	27	23.0843464763969	3.91565352360311
74	23	23.2732812094790	-0.273281209478967
75	25	23.2600950805076	1.73990491949236
76	20	23.3440474704653	-3.34404747046527
77	21	23.1826933791399	-2.18269337913985
78	22	23.0773759427267	-1.07737594272675
79	23	23.0253913289347	-0.0253913289346634
80	25	23.0241661684535	1.97583383154645
81	25	23.1195023965768	1.88049760342322
82	17	23.2102385433330	-6.21023854333303
83	19	22.9105874750023	-3.91058747500229
84	25	22.7218971845233	2.27810281547668
85	19	22.8318182346829	-3.83181823468293
86	20	22.6469286495792	-2.64692864957922
87	26	22.5192113337133	3.48078866628672
88	23	22.6871633431584	0.312836656841597
89	27	22.7022580674241	4.29774193257593
90	17	22.9096290005243	-5.90962900052433
91	17	22.6244826843097	-5.62448268430971
92	19	22.3530950019524	-3.35309500195241
93	17	22.1913043569492	-5.19130435694922
94	22	21.9408180212416	0.0591819787584171
95	21	21.9436736189835	-0.943673618983464
96	32	21.8981402942793	10.1018597057207
97	21	22.3855665069481	-1.38556650694806
98	21	22.3187113481400	-1.31871134814003
99	18	22.2550820267327	-4.25508202673268
100	18	22.049769482577	-4.04976948257701
101	23	21.854363502037	1.14563649796301
102	19	21.9096417652048	-2.90964176520481
103	20	21.7692482429078	-1.76924824290777
104	21	21.6838800042435	-0.683880004243452
105	20	21.6508820167343	-1.65088201673433
106	17	21.5712250830190	-4.57122508301903
107	18	21.3506582770664	-3.3506582770664
108	19	21.1889852068086	-2.18898520680857
109	22	21.0833641825591	0.916635817440902
110	15	21.1275929026074	-6.12759290260742
111	14	20.8319295803775	-6.8319295803775
112	18	20.5022812128896	-2.50228121288955
113	24	20.3815433002515	3.61845669974848
114	35	20.5561379487859	14.4438620512141
115	29	21.2530707097850	7.74692929021497
116	21	21.6268688532698	-0.626868853269823
117	25	21.5966217185946	3.40337828140542
118	20	21.7608385889838	-1.76083858898382
119	22	21.6758761256798	0.324123874320179
120	13	21.691515471018	-8.69151547101798
121	26	21.272139970684	4.72786002931599
122	17	21.5002645908282	-4.50026459082821
123	25	21.2831217092781	3.71687829072187
124	20	21.4624653114439	-1.46246531144392
125	19	21.3918996979151	-2.39189969791506
126	21	21.2764878188352	-0.276487818835207
127	22	21.2631469673178	0.73685303268222
128	24	21.2987009636587	2.70129903634129
129	21	21.4290417124838	-0.429041712483798
130	26	21.4083399622037	4.59166003779635
131	24	21.6298927779509	2.37010722204915
132	16	21.7442531453006	-5.74425314530065
133	23	21.4670864020683	1.53291359793169
134	18	21.5410512254924	-3.54105122549237
135	16	21.3701914790706	-5.37019147907059
136	26	21.1110736363958	4.88892636360417
137	19	21.3469698902389	-2.34696989023892
138	21	21.2337259254471	-0.233725925447079
139	21	21.2224483838892	-0.222448383889194
140	22	21.2117149965296	0.788285003470364
141	23	21.2497506439192	1.75024935608085
142	29	21.3342021646998	7.66579783530024
143	21	21.7040856232701	-0.70408562327015
144	21	21.6701126916771	-0.670112691677094
145	23	21.6377789926474	1.36222100735265
146	27	21.7035077045652	5.29649229543481
147	25	21.9590694778623	3.04093052213765
148	21	22.1057978319054	-1.10579783190542
149	10	22.0524418296559	-12.0524418296559
150	20	21.4708978124574	-1.47089781245744
151	26	21.3999253211702	4.60007467882977
152	24	21.6218841529114	2.37811584708855
153	29	21.7366309455221	7.26336905447793
154	19	22.0870967580135	-3.08709675801349
155	24	21.9381408306164	2.06185916938357
156	19	22.0376278790675	-3.03762787906751
157	24	21.8910588813064	2.10894111869361
158	22	21.9928176873255	0.00718231267454428
159	17	21.9931642420755	-4.99316424207549

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
160	21.7522383921184	13.3975687204359	30.1069080638009
161	21.7522383921184	13.3878488197791	30.1166279644577
162	21.7522383921184	13.3781402011018	30.126336583135
163	21.7522383921184	13.3684428252098	30.1360339590270
164	21.7522383921184	13.3587566531352	30.1457201311016
165	21.7522383921184	13.3490816461347	30.1553951381022
166	21.7522383921184	13.3394177656875	30.1650590185493
167	21.7522383921184	13.3297649734944	30.1747118107425
168	21.7522383921184	13.3201232314747	30.1843535527621
169	21.7522383921184	13.3104925017658	30.193984282471
170	21.7522383921184	13.3008727467204	30.2036040375164
171	21.7522383921184	13.2912639289054	30.2132128553314

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Nov/30/t1291119290sl31sp861zsq1fz/12msb1291119324.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/30/t1291119290sl31sp861zsq1fz/12msb1291119324.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/30/t1291119290sl31sp861zsq1fz/22msb1291119324.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/30/t1291119290sl31sp861zsq1fz/22msb1291119324.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/30/t1291119290sl31sp861zsq1fz/32msb1291119324.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/30/t1291119290sl31sp861zsq1fz/32msb1291119324.ps (open in new window)

Parameters (Session):

par1 = 12 ; par2 = Single ; par3 = additive ;

Parameters (R input):

par1 = 12 ; par2 = Single ; 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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