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Exponential Smoothing

*Unverified author*

R Software Module: /rwasp_exponentialsmoothing.wasp (opens new window with default values)

Title produced by software: Exponential Smoothing

Date of computation: Mon, 16 Aug 2010 13:46:38 +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/Aug/16/t1281966373o4r9nju7is3c5pd.htm/, Retrieved Mon, 16 Aug 2010 15:46:13 +0200

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/Aug/16/t1281966373o4r9nju7is3c5pd.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 «

244 243 242 240 260 259 244 234 235 235 236 238 240 233 233 233 247 251 233 226 233 233 227 225 223 212 206 202 223 221 212 205 204 200 195 193 196 180 174 164 192 189 181 167 166 164 167 164 161 141 134 111 147 144 142 140 143 137 140 130 129 112 101 74 104 103 100 98 99 91 92 94 93 76 64 32 62 69 69 68 68 59 66 73 70 57 48 22 64 74 67 61 61 52 54 69 69 53 50 22 69 78 74 63 67 59 60 80 77 58 54 32 78 86 84 78 72 64 62 72

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	0.644806046017667
beta	8.40256683676266e-19
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
3	242	242	0
4	240	241	-1
5	260	239.355193953982	20.6448060460177
6	259	251.667089711317	7.3329102886834
7	244	255.395394600365	-11.3953946003648
8	234	247.047575265292	-13.0475752652925
9	235	237.634419848361	-2.63441984836135
10	235	234.935730002389	0.0642699976110066
11	236	233.977171685426	2.0228283145739
12	238	234.281503612719	3.71849638728094
13	240	235.679212565333	4.32078743466732
14	233	237.465282426763	-4.46528242676334
15	233	233.58604132081	-0.586041320809898
16	233	232.208158333936	0.7918416660645
17	247	231.718742627703	15.2812573722974
18	251	240.572189772112	10.4278102278880
19	233	246.296104853779	-13.2961048537790
20	226	236.722696055577	-10.7226960555775
21	233	228.808636809331	4.19136319066868
22	233	230.511253135730	2.48874686426961
23	227	231.116012160819	-4.11601216081894
24	225	227.461982634041	-2.46198263404065
25	223	224.874481346421	-1.87448134642074
26	212	222.665804441101	-10.6658044411013
27	206	214.788429251837	-8.7884292518371
28	202	208.121596935254	-6.12159693525402
29	223	203.174354220119	19.825645779881
30	221	214.958050485191	6.04194951480909
31	212	217.853936062073	-5.85393606207333
32	205	213.079282696248	-8.07928269624759
33	204	206.869712366221	-2.86971236622122
34	200	204.01930448215	-4.01930448215012
35	195	200.427632651274	-5.42763265127383
36	193	195.927862302170	-2.92786230216956
37	196	193.039958987823	2.96004101217659
38	180	193.948611328935	-13.9486113289351
39	174	183.954462410487	-9.95446241048722
40	164	176.535764863349	-12.5357648633495
41	192	167.452627888006	24.5473721119941
42	189	182.280921839665	6.71907816033485
43	181	185.613424061114	-4.61342406111433
44	167	181.638660333664	-14.6386603336644
45	166	171.199563644919	-5.19956364491861
46	164	166.846853570021	-2.84685357002144
47	167	164.011185175945	2.98881482405537
48	164	164.938391044923	-0.938391044922753
49	161	163.333310825628	-2.33331082562773
50	141	160.828777898024	-19.8287778980245
51	134	147.043062024237	-13.0430620242368
52	111	137.632816772425	-26.6328167724255
53	147	119.459815495085	27.5401845049152
54	144	136.217892972296	7.7821070277038
55	142	140.235842634516	1.76415736548381
56	140	140.373381969907	-0.373381969906745
57	143	139.132623018237	3.8673769817631
58	137	140.626331078307	-3.62633107830729
59	140	137.288050874153	2.71194912584701
60	130	138.036732066991	-8.03673206699148
61	129	131.854598639971	-2.8545986399713
62	112	129.013936177964	-17.013936177964
63	101	117.043247263854	-16.0432472638541
64	74	105.698464430365	-31.6984644303646
65	104	84.2591029161895	19.7408970838105
66	103	95.988152709643	7.01184729035695
67	100	99.5094342362178	0.4905657637822
68	98	98.8257540066738	-0.825754006673833
69	99	97.2933028306472	1.70669716935276
70	91	97.3937914841671	-6.39379148416714
71	92	92.2710360781999	-0.271036078199899
72	94	91.0962703762877	2.90372962371231
73	93	91.968612793658	1.03138720634200
74	76	91.6336575000926	-15.6336575000926
75	64	80.5529806226635	-16.5529806226635
76	32	68.8795186375568	-36.8795186375568
77	62	44.0993820458390	17.9006179541610
78	69	54.6418087301344	14.3581912698656
79	69	62.9000572708218	6.09994272917818
80	68	65.8333372229574	2.16666277704259
81	68	66.2304144812759	1.7695855187241
82	59	66.3714539226945	-7.37145392269451
83	66	60.6182958654004	5.38170413459955
84	73	63.0884512292685	9.91154877073149
85	70	68.4794778020351	1.52052219796485
86	57	68.459919708387	-11.4599197083870
87	48	60.070494193542	-12.0704941935420
88	22	51.287366559125	-29.287366559125
89	64	31.4026955298656	32.5973044701344
90	74	51.421634536087	22.5783654639130
91	67	64.9803010964146	2.01969890358544
92	61	65.2826151605817	-4.2826151605817
93	61	61.5211590122717	-0.521159012271696
94	52	60.1851125302223	-8.18511253022231
95	54	53.9073024834	0.0926975165999977
96	69	52.9670744025545	16.0329255974455
97	69	62.3052017631388	6.69479823686123
98	53	65.6220481431353	-12.6220481431353
99	50	56.4832751873156	-6.4832751873156
100	22	51.3028201485382	-29.3028201485382
101	69	31.4081845513925	37.5918154486075
102	78	54.6476144334349	23.3523855665651
103	74	68.7053738356918	5.29462616430823
104	63	71.119380797841	-8.11938079784105
105	67	64.8839549694734	2.11604503052661
106	59	65.2483935988026	-6.24839359880259
107	60	60.2193916283966	-0.219391628396593
108	80	59.0779265799608	20.9220734200392
109	77	71.5686060164276	5.43139398357241
110	58	74.070801695339	-16.0708016953391
111	54	62.7082515978335	-8.70825159783347
112	32	56.0931183173074	-24.0931183173074
113	78	39.5577299588886	38.4422700411114
114	86	63.345538104041	22.6544618959589
115	84	76.9532721038322	7.04672789616777
116	78	80.4970448559226	-2.49704485592257
117	72	77.8869352356464	-5.88693523564638
118	64	73.0910038031871	-9.09100380318715
119	62	66.2290695865225	-4.22906958652247
120	72	62.5021399481033	9.49786005189665

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
121	67.626417533796	41.7023153221299	93.550519745462
122	66.626417533796	35.7802779369105	97.4725571306814
123	65.626417533796	30.54209652597	100.710738541622
124	64.626417533796	25.7633909773141	103.489444090278
125	63.626417533796	21.3208613303911	105.931973737201
126	62.626417533796	17.1381186182608	108.114716449331
127	61.626417533796	13.1639517403727	110.088883327219
128	60.626417533796	9.362045467727	111.890789599865
129	59.626417533796	5.70553942635055	113.547295641241
130	58.626417533796	2.1739034318747	115.078931635717
131	57.626417533796	-1.24897291906858	116.501807986661
132	56.626417533796	-4.57600769206326	117.828842759655

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Aug/16/t1281966373o4r9nju7is3c5pd/17fve1281966393.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Aug/16/t1281966373o4r9nju7is3c5pd/17fve1281966393.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Aug/16/t1281966373o4r9nju7is3c5pd/20ocz1281966393.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Aug/16/t1281966373o4r9nju7is3c5pd/20ocz1281966393.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Aug/16/t1281966373o4r9nju7is3c5pd/30ocz1281966393.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Aug/16/t1281966373o4r9nju7is3c5pd/30ocz1281966393.ps (open in new window)

Parameters (Session):

par1 = 12 ; par2 = Double ; par3 = additive ;

Parameters (R input):

par1 = 12 ; par2 = Double ; 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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