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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: Wed, 29 Dec 2010 18:25:49 +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/t1293647065wovfxqzhm09h5u2.htm/, Retrieved Wed, 29 Dec 2010 19:24:28 +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/t1293647065wovfxqzhm09h5u2.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 «

16 17 23 24 27 31 40 47 43 60 64 65 65 55 57 57 57 65 69 70 71 71 73 68 65 57 41 21 21 17 9 11 6 -2 0 5 3 7 4 8 9 14 12 12 7 15 14 19 39 12 11 17 16 25 24 28 25 31 24 24

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	2 seconds
R Server	'George Udny Yule' @ 72.249.76.132

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.783753045323201
beta	0.247471046463912
gamma	0.506225351407864

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	65	48.3731303418804	16.6268696581196
14	55	54.7597647424338	0.240235257566248
15	57	60.3915863099934	-3.39158630999342
16	57	61.0191375177296	-4.01913751772965
17	57	61.1669736368374	-4.1669736368374
18	65	68.7240657657245	-3.72406576572452
19	69	67.9893160272941	1.01068397270592
20	70	74.9531358907643	-4.95313589076432
21	71	65.9071024269052	5.09289757309479
22	71	86.9308106075684	-15.9308106075684
23	73	75.5539108742788	-2.55391087427877
24	68	71.1241835639579	-3.12418356395786
25	65	66.50335351124	-1.50335351123994
26	57	49.7056244922747	7.29437550772531
27	41	54.6558637201746	-13.6558637201746
28	21	39.3665264520586	-18.3665264520586
29	21	17.6670724252086	3.33292757479144
30	17	22.0190753905381	-5.01907539053807
31	9	11.404849810512	-2.40484981051201
32	11	4.99359355438016	6.00640644561984
33	6	-2.28274161783329	8.28274161783329
34	-2	11.6386351374146	-13.6386351374146
35	0	-3.33374098224702	3.33374098224702
36	5	-8.92582526160245	13.9258252616024
37	3	-2.41366698844914	5.41366698844914
38	7	-13.8929163140608	20.8929163140608
39	4	0.993484728273314	3.00651527172669
40	8	3.05111468163762	4.94888531836238
41	9	11.3262382368390	-2.32623823683904
42	14	18.5565514492181	-4.55655144921807
43	12	16.9087081285208	-4.90870812852085
44	12	17.2878833466315	-5.28788334663147
45	7	7.05026692294998	-0.0502669229499846
46	15	16.0661210302261	-1.06612103022612
47	14	19.2691932847397	-5.26919328473967
48	19	12.8891943350033	6.11080566499674
49	39	15.6238615415665	23.3761384584335
50	12	26.6805727037811	-14.6805727037811
51	11	11.5917364809972	-0.591736480997204
52	17	10.2075649999494	6.79243500005063
53	16	18.654447259605	-2.65444725960500
54	25	24.8429923352278	0.157007664772184
55	24	27.2247071730139	-3.22470717301387
56	28	29.5826798088764	-1.58267980887636
57	25	24.2415000729461	0.758499927053904
58	31	35.3559996428529	-4.35599964285288
59	24	36.4583954033569	-12.4583954033569
60	24	25.2330970453288	-1.23309704532877

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
61	22.2210880965942	5.52524183831081	38.9169343548777
62	4.37577581047097	-18.9740402380642	27.7255918590061
63	-1.23227575846366	-31.6934580800694	29.2289065631421
64	-4.79701750418589	-42.8461059199842	33.2520709116124
65	-7.47799742877338	-53.5836923290109	38.6276974714641
66	-3.15652895130559	-57.772114982362	51.4590570797508
67	-5.55379405615687	-69.1154745352238	58.0078864229101
68	-4.14897265059504	-77.0762155428065	68.7782702416164
69	-11.3467410302785	-94.0433039535915	71.3498218930346
70	-4.88702351195378	-97.7421591046425	87.968112080735
71	-3.91311618317705	-107.302765113256	99.4765327469018
72	-4.38444606583316	-118.672346166318	109.903454034652

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/29/t1293647065wovfxqzhm09h5u2/1ytcl1293647147.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/29/t1293647065wovfxqzhm09h5u2/1ytcl1293647147.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/29/t1293647065wovfxqzhm09h5u2/2rkb61293647147.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/29/t1293647065wovfxqzhm09h5u2/2rkb61293647147.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/29/t1293647065wovfxqzhm09h5u2/3rkb61293647147.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/29/t1293647065wovfxqzhm09h5u2/3rkb61293647147.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=0, beta=0)

if (par2 == 'Double') fit <- HoltWinters(x, gamma=0)

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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