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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: Wed, 02 Jun 2010 09:05:52 +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/Jun/02/t1275469605l3azjhaq10zf4wa.htm/, Retrieved Wed, 02 Jun 2010 11:06:47 +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/Jun/02/t1275469605l3azjhaq10zf4wa.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:

KDGP2W62

Dataseries X:

» Textbox « » Textfile « » CSV «

112 118 129 99 116 168 118 129 205 147 150 267 126 129 124 97 102 127 222 214 118 141 154 226 89 77 82 97 127 121 117 117 106 112 134 169 75 108 115 85 101 108 109 124 105 95 135 164 88 85 112 87 91 87 87 142 95 108 139 159 61 82 124 93 108 75 87 103 90 108 123 129 57 65 67 71 76 67 110 118 99 85 107 141 58 65 70 86 93 74 87 73 101 100 96 157 63 115 70 66 67 83 79 77 102 116 100 135 71 60 89 74 73 91 86 74 87 87 109 137 43 69 73 77 69 76 78 70 83 65 110 132 54 55 66 65 60 65 96 55 71 63 74 106 34 47 56 53 53 55 67 52 46 51 58 91 33 40 46 45 41 55 57 54 46 52 48 77 30 35 42 48 44 45 46

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	4 seconds
R Server	184.73.214.54 @ 184.73.214.54

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.152823208325162
beta	0.147474162937433
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
3	129	124	5
4	99	130.876803415252	-31.8768034152516
5	116	131.399552768518	-15.3995527685179
6	168	134.093341398697	33.9066586013035
7	118	145.086433902012	-27.0864339020116
8	129	146.147906503835	-17.1479065038352
9	205	148.341746239256	56.6582537607441
10	147	163.091814142575	-16.0918141425755
11	150	166.361314408516	-16.3613144085161
12	267	169.220886069219	97.7791139307813
13	126	191.727458497205	-65.7274584972051
14	129	187.765101009866	-58.765101009866
15	124	183.542336352517	-59.5423363525171
16	97	177.858858195858	-80.8588581958583
17	102	167.095366307872	-65.0953663078718
18	127	157.273816651726	-30.273816651726
19	222	152.091512555264	69.9084874447363
20	214	163.79495035709	50.2050496429099
21	118	173.618740616409	-55.6187406164089
22	141	166.016693770282	-25.0166937702818
23	154	162.527536800942	-8.52753680094168
24	226	161.366116557114	64.6338834428858
25	89	172.842143795057	-83.8421437950573
26	77	159.737997994439	-82.7379979944389
27	82	144.937885758919	-62.9378857589193
28	97	131.745229190883	-34.745229190883
29	127	122.077995126074	4.92200487392613
30	121	118.583764596877	2.41623540312274
31	117	114.761050182157	2.23894981784284
32	117	110.961702690435	6.03829730956465
33	106	107.879071644938	-1.87907164493795
34	112	103.544133344615	8.45586665538512
35	134	100.979187354215	33.0208126457851
36	169	102.912540952811	66.087459047189
37	75	111.38868998095	-36.3886899809497
38	108	103.383995957073	4.61600404292662
39	115	101.749803904766	13.2501960952339
40	85	101.733742742788	-16.7337427427879
41	101	96.7583035452347	4.24169645476528
42	108	95.0839953885221	12.9160046114779
43	109	95.0264169618007	13.9735830381993
44	124	95.4453903378498	28.5546096621502
45	105	98.7362317767934	6.2637682232066
46	95	98.761684828785	-3.76168482878495
47	135	97.1702371037051	37.8297628962949
48	164	102.787515183994	61.2124848160055
49	88	113.357790676834	-25.3577906768345
50	85	110.1266183568	-25.1266183568003
51	112	106.364484006244	5.63551599375556
52	87	107.430528021343	-20.4305280213427
53	91	104.052623052806	-13.052623052806
54	87	101.508060030129	-14.5080600301295
55	87	98.414097422823	-11.4140974228231
56	142	95.5357191754259	46.4642808245741
57	95	102.54968793916	-7.54968793916002
58	108	101.138917800934	6.86108219906647
59	139	102.085079256316	36.914920743684
60	159	108.456133833816	50.5438661661844
61	61	118.049138683103	-57.0491386831028
62	82	109.913691816814	-27.9136918168143
63	124	105.601713292812	18.3982867071882
64	93	108.781930832383	-15.781930832383
65	108	106.382932999056	1.61706700094427
66	75	106.679350442923	-31.6793504429228
67	87	101.173329987367	-14.1733299873671
68	103	98.0232046764737	4.97679532352632
69	90	97.9118273542362	-7.91182735423615
70	108	95.6524567544646	12.3475432455354
71	123	96.7674706122127	26.2325293877873
72	129	100.595647567825	28.4043524321754
73	57	105.395891864019	-48.3958918640192
74	65	97.368555237026	-32.368555237026
75	67	91.0610621187327	-24.0610621187327
76	71	85.4808711725545	-14.4808711725545
77	76	81.0383934750891	-5.03839347508912
78	67	77.925392848289	-10.9253928482889
79	110	73.6664913246873	36.3335086753127
80	118	77.4487122872918	40.5512877127082
81	99	82.7894313962958	16.2105686037042
82	85	84.7756689983068	0.224331001693187
83	107	84.3238943352109	22.6761056647891
84	141	87.8143340670082	53.1856659329918
85	58	97.1660132852881	-39.1660132852881
86	65	91.5215095567116	-26.5215095567116
87	70	87.211651604308	-17.211651604308
88	86	83.9366488502318	2.06335114976824
89	93	83.6538165818968	9.34618341810318
90	74	84.6946094790651	-10.6946094790651
91	87	82.4316746174781	4.56832538252191
92	73	82.6042289477417	-9.60422894774166
93	101	80.3944329879273	20.6055670120727
94	100	83.2657924152725	16.7342075847275
95	96	85.9226650548375	10.0773349451625
96	157	87.7893307418358	69.2106692581642
97	63	100.252776006629	-37.2527760066291
98	115	95.6065524990148	19.3934475009852
99	70	100.054265940226	-30.0542659402255
100	66	96.267873909996	-30.267873909996
101	67	91.7666761812713	-24.7666761812713
102	83	87.5480108006196	-4.54801080061955
103	79	86.3167260530232	-7.31672605302317
104	77	84.4974168291247	-7.49741682912469
105	102	82.4815210188172	19.5184789811828
106	116	85.0341783087312	30.9658216912688
107	100	90.0341466621093	9.96585333789074
108	135	92.0494376447146	42.9505623552854
109	71	100.073554898544	-29.0735548985441
110	60	96.435470968159	-36.435470968159
111	89	90.8511518985503	-1.85115189855031
112	74	90.5103991377743	-16.5103991377743
113	73	87.5572704789984	-14.5572704789984
114	91	84.5745410926405	6.42545890735946
115	86	84.9432733482468	1.05672665175322
116	74	84.5153546661178	-10.5153546661178
117	87	82.0819638499783	4.91803615002172
118	87	82.1179934491259	4.88200655087414
119	109	82.2585449885855	26.7414550114145
120	137	86.3424124404373	50.6575875595627
121	43	95.2229140982885	-52.2229140982885
122	69	87.20391481212	-18.2039148121201
123	73	83.973537870898	-10.973537870898
124	77	81.6008144988228	-4.60081449882281
125	69	80.0983004177545	-11.0983004177545
126	76	77.35269202735	-1.35269202734997
127	78	76.0659525158704	1.93404748412962
128	70	75.325091627755	-5.32509162775503
129	83	73.354851692913	9.64514830708693
130	65	73.8897891405026	-8.88978914050257
131	110	71.3918045831791	38.6081954168209
132	132	77.0227456432757	54.9772543567243
133	54	86.3943072883557	-32.3943072883557
134	55	81.6833806847044	-26.6833806847044
135	66	77.243840189369	-11.243840189369
136	65	74.9104120435154	-9.91041204351536
137	60	72.5574070040354	-12.5574070040354
138	65	69.5168674593012	-4.5168674593012
139	96	67.6033101782577	28.3966898217423
140	55	71.3596979974331	-16.3596979974331
141	71	67.9075647569084	3.09243524309156
142	63	67.4978646080997	-4.49786460809969
143	74	65.8268199739596	8.17318002604037
144	106	66.2764078719013	39.7235921280987
145	34	72.4429004248403	-38.4429004248403
146	47	65.7973329056275	-18.7973329056275
147	56	61.7304196288082	-5.73041962880821
148	53	59.5312847743102	-6.53128477431024
149	53	57.062560473317	-4.06256047331697
150	55	54.8795546864849	0.120445313515056
151	67	53.3385237976484	13.6614762023516
152	52	54.174771867893	-2.17477186789298
153	46	52.541859834204	-6.54185983420397
154	51	50.0941184057962	0.905881594203848
155	58	48.8049810007872	9.1950189992128
156	91	48.989848676552	42.010151323448
157	33	55.1364328793738	-22.1364328793738
158	40	50.9810309846959	-10.9810309846959
159	46	48.2829486894455	-2.28294868944553
160	45	46.8626833384443	-1.86268333844433
161	41	45.4646641080699	-4.46466410806987
162	55	43.5683795739889	11.4316204260111
163	57	44.3590560992747	12.6409439007253
164	54	45.6194402715341	8.38055972846588
165	46	46.4176155189542	-0.417615518954165
166	52	45.8618133984532	6.13818660154681
167	48	46.4462292165428	1.55377078345722
168	77	46.3650579707762	30.6349420292238
169	30	51.4185988483316	-21.4185988483316
170	35	48.0344294741013	-13.0344294741013
171	42	45.637792637953	-3.63779263795298
172	48	44.595193330938	3.40480666906205
173	44	44.7056023892176	-0.705602389217582
174	45	44.1719430516691	0.828056948330897
175	46	43.8913247671066	2.10867523289338

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
176	43.8539388921958	-15.2163826921205	102.924260476512
177	43.4942985028783	-16.4773925973345	103.465989603091
178	43.1346581135609	-17.9657320511246	104.235048278246
179	42.7750177242435	-19.6974516049384	105.247487053425
180	42.4153773349261	-21.6845732756973	106.515327945549
181	42.0557369456086	-23.9350586876601	108.046532578877
182	41.6960965562912	-26.4529932513262	109.845186363909
183	41.3364561669738	-29.2389529045126	111.91186523846
184	40.9768157776564	-32.2905001125367	114.244131667849
185	40.6171753883389	-35.602748355276	116.837099131954
186	40.2575349990215	-39.1689382717573	119.6840082698
187	39.8978946097041	-42.9809803672515	122.77676958666

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Jun/02/t1275469605l3azjhaq10zf4wa/1y9uk1275469548.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Jun/02/t1275469605l3azjhaq10zf4wa/1y9uk1275469548.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Jun/02/t1275469605l3azjhaq10zf4wa/2y9uk1275469548.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Jun/02/t1275469605l3azjhaq10zf4wa/2y9uk1275469548.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Jun/02/t1275469605l3azjhaq10zf4wa/3y9uk1275469548.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Jun/02/t1275469605l3azjhaq10zf4wa/3y9uk1275469548.ps (open in new window)

Parameters (Session):

par1 = 12 ; par2 = Double ; par3 = multiplicative ;

Parameters (R input):

par1 = 12 ; par2 = Double ; par3 = multiplicative ;

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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