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*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, 21 Dec 2010 15:53:05 +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/21/t12929469817ik2wh4yicbv2nw.htm/, Retrieved Tue, 21 Dec 2010 16:56:21 +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/21/t12929469817ik2wh4yicbv2nw.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 «

13.193 15.234 14.718 16.961 13.945 15.876 16.226 18.316 16.748 17.904 17.209 18.950 17.225 18.710 17.236 18.687 17.580 19.568 17.381 19.580 17.260 18.661 15.658 18.674 15.908 17.475 17.725 19.562 16.368 19.555 17.743 19.867 15.703 19.324 18.162 19.074 15.323 19.704 18.375 18.352 13.927 17.795 16.761 18.902 16.239 19.158 18.279 15.698 16.239 18.431 18.414 19.801 14.995 18.706 18.232 19.409 16.263 19.017 20.298 19.891 15.203 17.845 17.502 18.532 15.737 17.770 17.224 17.601 14.940 18.507 17.635 19.392 15.699 17.661 18.243 19.643 15.770 17.344 17.229 17.322 16.152 17.919 16.918 18.114 16.308 17.759 16.021 17.952 15.954 17.762 16.610 17.751 15.458 18.106 15.990 15.349 13.185 15.409 16.007 16.633 14.800 15.974 15.693

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	'Sir Ronald Aylmer Fisher' @ 193.190.124.24

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.29732040102439
beta	0.0796850294543876
gamma	0.343463420691937

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
5	13.945	13.55585625	0.389143749999999
6	15.876	15.5210387136601	0.35496128633987
7	16.226	15.5542177709575	0.671782229042499
8	18.316	18.1233850124274	0.192614987572554
9	16.748	15.2950672855516	1.45293271444844
10	17.904	17.6315980361721	0.272401963827942
11	17.209	17.7780494792103	-0.569049479210292
12	18.95	19.8946057697566	-0.944605769756588
13	17.225	17.0373552695942	0.187644730405818
14	18.71	18.6878131258518	0.0221868741482361
15	17.236	18.5259004693103	-1.28990046931032
16	18.687	20.2895259586921	-1.60252595869208
17	17.58	17.4463690691496	0.133630930850373
18	19.568	18.9759994475667	0.592000552433277
19	17.381	18.6155029721445	-1.23450297214455
20	19.58	20.2701255540762	-0.690125554076211
21	17.26	18.0888496297511	-0.828849629751115
22	18.661	19.3917303036948	-0.730730303694774
23	15.658	18.1145941295205	-2.4565941295205
24	18.674	19.4257451246555	-0.751745124655493
25	15.908	17.079706649638	-1.17170664963801
26	17.475	18.1832450872427	-0.708245087242666
27	17.725	16.375714359147	1.34928564085300
28	19.562	19.1995045241943	0.362495475805666
29	16.368	17.0794127110294	-0.711412711029414
30	19.555	18.4385809179828	1.11641908201720
31	17.743	17.7202851856172	0.022714814382784
32	19.867	19.9302239672163	-0.0632239672162811
33	15.703	17.4330106503664	-1.73001065036636
34	19.324	18.9149695516027	0.40903044839731
35	18.162	17.6901365910526	0.47186340894735
36	19.074	19.9912603765896	-0.917260376589631
37	15.323	16.7960054342244	-1.47300543422443
38	19.704	18.8348635269206	0.86913647307938
39	18.375	17.7371346860348	0.637865313965168
40	18.352	19.7314302295545	-1.3794302295545
41	13.927	16.2327606163285	-2.30576061632854
42	17.795	18.5376816086575	-0.742681608657453
43	16.761	16.8151174998679	-0.0541174998679175
44	18.902	18.0106215314357	0.89137846856428
45	16.239	14.9111574328608	1.32784256713923
46	19.158	18.7073636743971	0.450636325602876
47	18.279	17.5677507266291	0.711249273370886
48	15.698	19.2991095189821	-3.60110951898208
49	16.239	14.9429445739367	1.29605542606332
50	18.431	18.4909068080669	-0.0599068080669376
51	18.414	17.2232168239087	1.19078317609126
52	19.801	18.0285671215959	1.77243287840412
53	14.995	16.5514622356321	-1.55646223563213
54	18.706	18.9559679026148	-0.249967902614841
55	18.232	17.9610228795215	0.270977120478491
56	19.409	18.6388886906002	0.770111309399816
57	16.263	16.0422293680713	0.220770631928687
58	19.017	19.3144318674623	-0.297431867462297
59	20.298	18.4539513743276	1.84404862567244
60	19.891	19.7801052274856	0.110894772514438
61	15.203	16.8993671302384	-1.69636713023842
62	17.845	19.4755794482686	-1.63057944826864
63	17.502	18.7030565532131	-1.20105655321313
64	18.532	18.6009008849753	-0.0689008849753279
65	15.737	15.121622938422	0.615377061577998
66	17.77	18.3469015054496	-0.576901505449644
67	17.224	17.9621453996123	-0.738145399612318
68	17.601	18.2526535090336	-0.651653509033636
69	14.94	14.7332426683359	0.206757331664102
70	18.507	17.5075835492080	0.999416450791983
71	17.635	17.5482331210370	0.0867668789629548
72	19.392	18.1200722082062	1.27192779179382
73	15.699	15.4405210960288	0.258478903971151
74	17.661	18.4835361749248	-0.82253617492475
75	18.243	17.7810454262646	0.461954573735373
76	19.643	18.7781832986364	0.864816701363612
77	15.77	15.7510694054379	0.0189305945621268
78	17.344	18.4743594168996	-1.13035941689960
79	17.229	17.9954535127495	-0.766453512749543
80	17.322	18.7005877194464	-1.37858771944638
81	16.152	14.7251615233627	1.42683847663727
82	17.919	17.5458800663042	0.373119933695754
83	16.918	17.5936403239496	-0.675640323949565
84	18.114	18.1720156218748	-0.0580156218748336
85	16.308	15.2915601256015	1.01643987439847
86	17.759	17.7514890295299	0.00751097047010418
87	16.021	17.4443111756237	-1.42331117562367
88	17.952	17.9386130042716	0.0133869957284318
89	15.954	15.3295566694765	0.624443330523471
90	17.762	17.4110056704459	0.350994329554119
91	16.61	16.8503371815392	-0.240337181539221
92	17.751	18.0608340112344	-0.309834011234361
93	15.458	15.5132281152180	-0.0552281152180303
94	18.106	17.3205740927730	0.785425907226966
95	15.99	16.7506214131922	-0.76062141319222
96	15.349	17.7815930535876	-2.43259305358758
97	13.185	14.6059416273707	-1.42094162737071
98	15.409	16.1194103233798	-0.71041032337981
99	16.007	14.6054350820287	1.4015649179713
100	16.633	15.8008242095102	0.832175790489819
101	14.8	13.8424377072998	0.957562292700159
102	15.974	16.2933444064655	-0.319344406465540
103	15.693	15.4733975528503	0.219602447149711

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
104	16.2199854169822	14.2081481057896	18.2318227281749
105	14.0647609140316	11.9518058970295	16.1777159310337
106	15.9204270096575	13.6965776316747	18.1442763876402
107	15.3307025684690	12.9866004766935	17.6748046602446

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/21/t12929469817ik2wh4yicbv2nw/1xt831292946781.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/21/t12929469817ik2wh4yicbv2nw/1xt831292946781.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/21/t12929469817ik2wh4yicbv2nw/2xt831292946781.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/21/t12929469817ik2wh4yicbv2nw/2xt831292946781.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/21/t12929469817ik2wh4yicbv2nw/3xt831292946781.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/21/t12929469817ik2wh4yicbv2nw/3xt831292946781.ps (open in new window)

Parameters (Session):

par1 = 4 ; par2 = Triple ; par3 = additive ;

Parameters (R input):

par1 = 4 ; 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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