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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 09:10:04 +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/t1275469839i0slf4lhsipwwow.htm/, Retrieved Wed, 02 Jun 2010 11:10:43 +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/t1275469839i0slf4lhsipwwow.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	1 seconds
R Server	'Gwilym Jenkins' @ 72.249.127.135

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.108312065002691
beta	0.0447363123542684
gamma	0.508451758294713

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	126	130.144329985792	-4.14432998579227
14	129	126.500495253810	2.49950474619028
15	124	122.354081329985	1.64591867001492
16	97	98.6532008959596	-1.65320089595963
17	102	103.969303655171	-1.96930365517079
18	127	130.957447008567	-3.95744700856656
19	222	124.120396173949	97.8796038260514
20	214	147.196630483076	66.8033695169239
21	118	247.025283584059	-129.025283584059
22	141	168.447717966043	-27.4477179660433
23	154	170.508706787079	-16.5087067870791
24	226	306.535574045117	-80.5355740451167
25	89	136.537427208121	-47.5374272081215
26	77	131.007607161597	-54.0076071615966
27	82	120.031537133320	-38.0315371333203
28	97	91.577857547966	5.42214245203392
29	127	96.7465563567895	30.2534436432105
30	121	125.192476020700	-4.19247602069954
31	117	159.372314553024	-42.3723145530239
32	117	150.984352890117	-33.9843528901169
33	106	145.979206824267	-39.9792068242669
34	112	125.173964372512	-13.1739643725124
35	134	130.490232858689	3.50976714131144
36	169	216.313367529353	-47.313367529353
37	75	91.408556262659	-16.4085562626590
38	108	85.2572380806106	22.7427619193894
39	115	88.5708575729344	26.4291424270656
40	85	86.9470388632552	-1.94703886325516
41	101	100.594290301228	0.405709698771844
42	108	108.651244557116	-0.651244557116428
43	109	122.915443023605	-13.9154430236054
44	124	120.686466528317	3.31353347168286
45	105	116.608811846146	-11.6088118461458
46	95	111.142687541814	-16.1426875418138
47	135	122.530013312232	12.4699866877679
48	164	181.603317031655	-17.6033170316554
49	88	79.372836041403	8.62716395859707
50	85	93.2271108848331	-8.2271108848331
51	112	94.3116887318487	17.6883112681513
52	87	79.9955798602076	7.00442013979236
53	91	94.7471269494024	-3.7471269494024
54	87	101.318032018014	-14.3180320180141
55	87	107.204925963829	-20.2049259638287
56	142	111.267506769873	30.7324932301269
57	95	104.005922320230	-9.0059223202303
58	108	97.0067359818533	10.9932640181467
59	139	123.361895894440	15.6381041055598
60	159	167.705575370262	-8.7055753702617
61	61	80.9401698420455	-19.9401698420455
62	82	83.6285428740918	-1.62854287409179
63	124	96.2000056045196	27.7999943954804
64	93	79.1437619306942	13.8562380693058
65	108	89.4608523965795	18.5391476034205
66	75	93.7103883869437	-18.7103883869437
67	87	96.3479479513994	-9.34794795139945
68	103	124.399034851706	-21.3990348517055
69	90	94.8592569025689	-4.85925690256887
70	108	97.1815709055674	10.8184290944326
71	123	124.260709860798	-1.26070986079783
72	129	153.690896006709	-24.6908960067094
73	57	66.3727638663265	-9.37276386632645
74	65	77.6218205080587	-12.6218205080587
75	67	99.7972165841832	-32.7972165841832
76	71	73.0566072532419	-2.05660725324189
77	76	81.3348544944935	-5.33485449449346
78	67	68.2282282047864	-1.22822820478639
79	110	74.8272017538354	35.1727982461646
80	118	98.4709340938534	19.5290659061466
81	99	82.7695044049151	16.2304955950849
82	85	93.5620660782976	-8.56206607829762
83	107	110.784022872340	-3.78402287234039
84	141	126.895693388692	14.1043066113076
85	58	56.9815450210338	1.01845497896620
86	65	67.117718661446	-2.11771866144595
87	70	80.1277986229931	-10.1277986229931
88	86	70.2263055166748	15.7736944833252
89	93	79.2560646358973	13.7439353641027
90	74	70.1300321267206	3.86996787327944
91	87	94.5733573880377	-7.57335738803773
92	73	106.505458314919	-33.505458314919
93	101	84.5823799120424	16.4176200879576
94	100	84.2536434892917	15.7463565107083
95	96	105.793131473077	-9.79313147307718
96	157	128.562123007903	28.4378769920971
97	63	56.1922708231173	6.80772917688268
98	115	65.5936916768001	49.4063083231999
99	70	81.723808433297	-11.7238084332969
100	66	83.9727498076714	-17.9727498076714
101	67	88.9025328194664	-21.9025328194664
102	83	71.6372986993604	11.3627013006396
103	79	92.0380082626993	-13.0380082626993
104	77	91.2764619592525	-14.2764619592525
105	102	94.3007181888119	7.6992818111881
106	116	92.7705581452625	23.2294418547375
107	100	103.971587758270	-3.97158775826976
108	135	146.170874039924	-11.1708740399241
109	71	59.5021981548414	11.4978018451586
110	60	87.5171761947385	-27.5171761947385
111	89	68.8851857487601	20.1148142512399
112	74	71.4389346389515	2.56106536104855
113	73	76.3970394594134	-3.39703945941342
114	91	76.2095992984226	14.7904007015774
115	86	85.9084813097835	0.091518690216489
116	74	86.0628792474626	-12.0628792474626
117	87	99.7270875519514	-12.7270875519514
118	87	102.838239976045	-15.8382399760448
119	109	97.5259660590252	11.4740339409748
120	137	136.905885828239	0.0941141717607934
121	43	63.2396644545201	-20.2396644545201
122	69	68.8687087401573	0.131291259842712
123	73	74.2291547650164	-1.22915476501640
124	77	66.9170447748802	10.0829552251198
125	69	69.7169310064325	-0.716931006432461
126	76	77.2536129219603	-1.25361292196033
127	78	78.2061454112974	-0.206145411297371
128	70	73.0196354980326	-3.01963549803261
129	83	85.8653897252149	-2.86538972521487
130	65	88.12084853166	-23.1208485316599
131	110	93.3263300791336	16.6736699208664
132	132	125.029808490760	6.97019150924022
133	54	49.2546571025369	4.74534289746305
134	55	66.1521491231288	-11.1521491231288
135	66	69.3534333911984	-3.35343339119842
136	65	66.9423302065172	-1.9423302065172
137	60	63.6858791137508	-3.68587911375084
138	65	69.8710429102855	-4.87104291028547
139	96	70.5905635284038	25.4094364715962
140	55	67.2565228295534	-12.2565228295534
141	71	78.0677197600627	-7.06771976006274
142	63	70.83685969607	-7.83685969606996
143	74	93.7691243556734	-19.7691243556734
144	106	114.023879247739	-8.0238792477388
145	34	44.9249058521102	-10.9249058521102
146	47	51.1003199004015	-4.10031990040145
147	56	57.0629918616688	-1.06299186166878
148	53	55.4626702062255	-2.46267020622553
149	53	51.6868352715647	1.31316472843534
150	55	56.61167105174	-1.61167105174005
151	67	68.3218912568018	-1.32189125680181
152	52	49.129003200485	2.87099679951496
153	46	60.922317768643	-14.9223177686430
154	51	53.4414431683052	-2.44144316830518
155	58	67.2538939852502	-9.25389398525019
156	91	87.8602272553183	3.13977274468171
157	33	31.8866936673493	1.11330633265069
158	40	40.4317564609978	-0.431756460997846
159	46	46.6239868350619	-0.623986835061935
160	45	44.6069382832251	0.39306171677493
161	41	42.983478674085	-1.98347867408499
162	55	45.3697282045682	9.63027179543177
163	57	56.2348181087636	0.765181891236367
164	54	41.8845674455068	12.1154325544932
165	46	45.862980259851	0.137019740148986
166	52	45.6786180656055	6.32138193439446
167	48	56.1374797060579	-8.13747970605792
168	77	79.5606153668407	-2.56061536684074
169	30	28.6641418770312	1.33585812296876
170	35	35.6703079828056	-0.670307982805568
171	42	41.0649225714864	0.935077428513608
172	48	39.8698997016363	8.13010029836374
173	44	38.3273921188424	5.67260788115765
174	45	46.3172147008443	-1.31721470084428
175	46	51.5423765375909	-5.54237653759093

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
176	42.4875893851717	18.3906768986515	66.5845018716918
177	40.045151863656	15.5057695576652	64.5845341696468
178	42.2454244940831	17.0578184924456	67.4330304957205
179	44.8817649762370	18.8858100540598	70.8777198984143
180	68.1783163837826	38.4890526344703	97.867580133095
181	25.5258205745446	0.405737900418821	50.6459032486705
182	30.6618291190903	4.27149977168498	57.0521584664957
183	36.0128494642620	7.87293400122939	64.1527649272945
184	37.608126838853	8.2280037222346	66.9882499554713
185	34.4994959754093	5.08686697715785	63.9121249736607
186	37.8673896381498	6.40300500653447	69.3317742697651
187	40.5791551960481	-0.292870962640883	81.451181354737

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Jun/02/t1275469839i0slf4lhsipwwow/1wm9t1275469801.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Jun/02/t1275469839i0slf4lhsipwwow/1wm9t1275469801.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Jun/02/t1275469839i0slf4lhsipwwow/27wre1275469801.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Jun/02/t1275469839i0slf4lhsipwwow/27wre1275469801.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Jun/02/t1275469839i0slf4lhsipwwow/3l6on1275469801.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Jun/02/t1275469839i0slf4lhsipwwow/3l6on1275469801.ps (open in new window)

Parameters (Session):

par1 = 12 ; par2 = Triple ; par3 = multiplicative ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; 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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