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La Mthodologie de Box-Jenkins
Michel Tenenhaus
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1. Les donnes
Une srie chronologique assez longue
(n 50).
Exemple : Ventes danti-inflammatoires en
France de janvier 1978 juillet 1982. Objectif : Prvoir les ventes daot
dcembre 1982.
1( ,..., ,..., )t nz z z
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date ventes date ventes date ventes
JAN 1978 3 741 JAN 1980 4 687 JAN 1982 4 764
FEB 1978 3 608 FEB 1980 4 704 FEB 1982 4 726
MAR 1978 3 735 MAR 1980 4 579 MAR 1982 5 080
APR 1978 3 695 APR 1980 4 800 APR 1982 4 952
MAY 1978 3 810 MAY 1980 4 485 MAY 1982 4 633
JUN 1978 3 819 JUN 1980 4 617 JUN 1982 4 830
JUL 1978 3 291 JUL 1980 4 491 JUL 1982 4 460
AUG 1978 3 053 AUG 1980 3 832
SEP 1978 3 908 SEP 1980 4 669
OCT 1978 4 035 OCT 1980 5 193
NOV 1978 3 933 NOV 1980 4 544DEC 1978 4 004 DEC 1980 4 676
JAN 1979 3 961 JAN 1981 4 709
FEB 1979 4 025 FEB 1981 4 705
MAR 1979 4 336 MAR 1981 4 677
APR 1979 4 335 APR 1981 4 627
MAY 1979 4 412 MAY 1981 4 555
JUN 1979 4 268 JUN 1981 4 570
JUL 1979 3 968 JUL 1981 4 457
AUG 1979 3 505 AUG 1981 3 589
SEP 1979 4 434 SEP 1981 4 636
OCT 1979 4 854 OCT 1981 5 077
NOV 1979 4 592 NOV 1981 4 623
DEC 1979 4 264 DEC 1981 4 591
March totaldes anti-inflammatoires
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March total des anti-inflammatoires
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2. Stabiliser la srie
Il faut TRANSFORMER la srie observe
de manire - enlever la tendance,
- enlever la saisonnalit,
- stabiliser la variance.
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Pour enlever la tendance
Faire des diffrences rgulires dordre d:
1 (1 )t t tz z B z 1o t tBz z
d= 2 21(1 ) (1 ) (1 )t t tB z B z B z
d= 1
Diffrence rgulire dordre d:
(1 )dt tw B z
Dans la pratiqued= 0,1, rarement 2
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March total des anti-inflammatoires :Diffrence rgulire dordre d = 1
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Dans la pratiqueD = 0,1,trs trs rarement 2
Pour enlever la saisonnalit
Faire des diffrences saisonnires dordreD :
(1 )s
t t s t z z B z
D = 2 2(1 ) (1 ) (1 )s s st t s t B z B z B z
D = 1
Diffrence saisonnire dordreD :
(1 )s Dt tw B z
Ordre de la saisonnalit : s = 12 (mois) ou 4 (trimestre)
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March total des anti-inflammatoires :Diffrence saisonnire (s = 12) dordre D = 1
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Pour enlever tendance et saisonnalit
Formule gnrale :
(1 ) (1 )d s Dt tw B B z
On peut choisir detD minimisantlcart-type de wt.
Application March total : s = 12, d= 1,D = 112
12 1 13(1 )(1 ) ( ) ( )t t t t t t w B B z z z z z
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March total des anti-inflammatoires :Diffrence rgulire/saisonnire (s = 12, d = 1, D = 1)
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Calcul des sries diffrenciesDonnes (20 premiers mois)
JAN 1978 3741 . . .
FEB 1978 3608 -133 . .
MAR 1978 3735 127 . .
APR 1978 3695 -40 . .
MAY 1978 3810 115 . .
JUN 1978 3819 9 . .
JUL 1978 3291 -528 . .
AUG 1978 3053 -238 . .
SEP 1978 3908 855 . .
OCT 1978 4035 127 . .
NOV 1978 3933 -102 . .
DEC 1978 4004 71 . .
JAN 1979 3961 -43 220 .
FEB 1979 4025 64 417 197MAR 1979 4336 311 601 184
APR 1979 4335 -1 640 39
MAY 1979 4412 77 602 -38
JUN 1979 4268 -144 449 -153
JUL 1979 3968 -300 677 228
AUG 1979 3505 -463 452 -225
1
2
3
4
5
6
7
8
9
10
11
12
13
1415
16
17
18
19
20
DAT E ventes DIFF(ventes,1) SDIFF(ventes,1,12) DIFF(ventes_2,1)
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Calcul des carts-typesDescriptive Statistics
55 3053 4347.71 478.613
54 -868 13.31 382.030
43 -243 263.47 279.368
42 -436 -5.17 242.719
ventes
DIFF(ventes,1)
SDIFF(ventes,1,12)DIFF(SDIFF(ventes,1,12),1)
N Minimum Mean Std. Deviation
s = 12, d= 1,D = 1
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Dveloppement de zt
12 1 13( ) ( )t t t t t w z z z z De
On dduit
12 1 13( )t t t t t z z z z w
valeur
1 an avant
valuationde la tendance
1 an avant
terme
alatoire
On va modliser la srie stationnaire wt.
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Pour stabiliser la variance
On utilise souvent les transformations ( ) out tLog z z
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3. Le modle statistique
On suppose que la srie stabilise (w1,,wN)provient dun processus stationnaire (wt) :
2( )( )
( , )
t
t w
k t t k
E w
Var w
Cor w w
Indpendantde la priode t
Dans des conditions assez gnrales tout processusstationnaire peut tre approch par des modlesAR(p), MA(q) ou ARMA(p,q).
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AR(p) : Auto-rgressif dordre p
2
( ) 0
( )
( , ) 0 pour tout 1,2,...
t
t
t t k
E a
Var a
Cor a a k
1 1 ...t t p t p t w w w a
o atest un bruit blanc :
Remarque : 1(1 ... )p
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MA(q) : Moyenne Mobile dordre q
1 1 ...t t t q t qw a a a
Remarque :
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ARMA(p,q)
1 1 1 1... ...t t p t p t t q t qw w w a a a
Remarque : 1(1 ... )p
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Question
Comment choisir le modlecorrespondant le mieux aux donnes
tudies ?Rponse
On utilise les autocorrlations ket les autocorrlations partielles kk.
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4. Autocorrlation
1
2
1
( , )
( )( )
= estimation de
( )
k t t k
N
t t k
t kk kN
tt
Cor w w
w w w w
r
w w
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Exemple : March TotalDiffrence rgulire/saisonnire : d= 1,D = 1
Autocorrlationscalcules
Autocorrelations
Series: ventes
-.515 .154 11.937 1 .001
.016 .191 11.948 2 .003
.189 .191 13.635 3 .003-.200 .195 15.581 4 .004
.062 .200 15.770 5 .008
.174 .201 17.326 6 .008
-.243 .204 20.449 7 .005
.076 .211 20.759 8 .008
.081 .212 21.127 9 .012
-.210 .212 23.686 10 .008
.344 .217 30.755 11 .001
-.312 .230 36.747 12 .000
.114 .240 37.574 13 .000
-.139 .241 38.842 14 .000
.140 .243 40.184 15 .000
-.072 .245 40.549 16 .001
Lag
1
2
34
5
6
7
8
9
10
11
12
13
14
15
16
Autocorrel
ation Std. Errora
Value df Sig.b
Box-Ljung Statistic
The underlying process assumed is MA with the order equal to
the lag number minus one. The Bartlett approximation is used.
a.
Based on the asymptotic chi-square approximation.b.
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Exemple : March TotalDiffrence rgulire/saisonnire : d= 1,D = 1
Corrlogrammeobserv
Formulede Bartlett
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Variance des autocorrlations rk
Formule de Bartlett(Hypothse : h= 0 pour hk)
2 2 2
1 1
1
( ) (1 2 ... 2 ) estimation de ( )k k ks r r r Var r N
Formule de Box-Jenkins pour un bruit blanc
(Hypothse : h
= 0 pourh
1)2 1( ) estimation de ( )
2k k
N ks r Var r
N N
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Test : H0 : k= 0
On rejette H0 : k= 0 au risque = 0.05 si
2 ( )k kr s r
Application March total :
1
= 0, k
= 0 pour k > 1
Corrlogrammethorique
0
k
1 k
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5. Autocorrlation partielle
Rgression de wtsur wt-1,,wt-k:
0 1 1 ...t k k t kk t k t w w w
Autocorrlation partielle dordre k: kk
Cest une corrlation partielle:
1 1( , | ,..., )kk t t k t t k Cor w w w w
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Calcul pratique de estimation de kk
1 2 1
1 3 2
1 2 1
1 2 1
1 3 2
1 2 1
1
1
1
1
1
k
k
k k k
kkk k
k k
k k
Soit :1
11 11
12
1 2 2 122 2
1 1
1
1
1 1
1
Etc
On obtient les estimations des kken remplaant les kpar rk.kk
kk
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Partial Autocorrelations
Series: ventes
-.515 .154
-.339 .154
.039 .154
-.073 .154
-.073 .154
.186 .154
-.012 .154
-.097 .154
.001 .154
-.139 .154
.238 .154
-.116 .154
.029 .154
-.343 .154
.022 .154
-.053 .154
Lag
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Partial
Autocorrelation Std. Error
Exemple : March TotalDiffrence rgulire/saisonnire : d= 1,D = 1
Autocorrlations partielles calcules
Rejet de
H0 : kk=0si:
2 /kk N
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Corrlogramme partiel observ
Corrlogrammepartiel thorique
0
kk
1 k
142
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6. Autocorrlations et autocorrlations partielles desmodles AR(p) et MA(q)
Corrlogramme Corrlogramme partiel
(a) (a)
(b) (b)
10.5t t tw w a
(a) :
10.5t t tw w a (b) :
AR(1)
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Corrlogramme Corrlogramme partiel
(a) (a)
(b) (b)
AR(2)
1 2.8 .15t t t tw w w a
(a) :
(b) :
1 2.5t t t tw w w a
Le dernier pic significatif du corrlogramme partiel donne
lordrep du modle AR(p).
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Corrlogramme Corrlogramme partiel
(a) (a)
(b) (b)
MA(1)
1.7t t tw a a
(a) :
(b) :
1.7t t tw a a
C l d diff t MA( )
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MA(q)
1 2.5 .3t t t t w a a a (a) : q = 2
(b) : q = 5
5.7t t tw a a
Corrlogramme de diffrents processus MA(q)
(a)
(b)
(c)
(c) : q = 6
1 6.3 .6t t t t w a a a
Le dernier pic significatif ducorrlogramme donne lordre q
du modle MA(q).
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7. tude de la srie March Total
Les autocorrlations suggrent un modle MA(1).
Les autocorrlations partielles suggrent unmodle AR(14).
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7.1 tude de la voie moyenne mobile
On suppose que wtsuit un modle MA(1) :
1
2( )
t t t
t
w a a
Var a
et on a = E(wt) = .
On choisit les paramtres , et 2 laidede la mthode du maximum de vraisemblance.
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Maximum de vraisemblance
On suppose que le vecteur alatoirew = (w1,,wN) suit une loi multinormale.
Densit de probabilit de w :2
1
2 1 '
/ 2 2
( ,..., | , , )
1 1exp ( ) ( , ) ( )
2(2 ) ( , )
N
N
p w w
w - w -
On recherche maximisantla vraisemblance
2
, et
2
1
( ,..., | , , )Np w w
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Qualit de lajustement dans ARIMA
2 ( ) 2
2 ( ) ( )
AIC Log r
SBC Log rLog N
On recherche le modle minimisant SBC.
o r est le nombre de paramtres (hors 2).
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Modle MA(1) avec constante
1t t tw a a
Residual Diagnostics
42
1
40
1585179
1591466
39100.764
197.739
-280.918
565.835
569.311
Number of Residuals
Number of Parameters
Residua l df
Adjusted Residual Sum of
Squares
Residua l Sum o f Squares
Residual Variance
Model Std. Error
Log-Likelihood
Akaike's Information
Criterion (AIC)
Schwarz's Bayesian
Criterion (BIC)
Parameter Estimates
.657 -7.772
.123 10.990
5.326 -.707
.000 .484
Estimates
Std Error
t
Approx Sig
MA1
Non-
Seasonal
Lags
Constant
Melard's algorithm was used for estimation.
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Modle MA(1) sans constante
1t t tw a a
Residual Diagnostics
421
41
1603132
1620350
38625.634
196.534-281.143
564.285
566.023
Number of ResidualsNumber of Parameters
Residual df
Adjusted Residual Sum of
Squares
Residual Sum of Squares
Residual Variance
Model Std. ErrorLog-Likelihood
Akaike's Information
Criterion (AIC)
Schwarz's Bayesian
Criterion (BIC)
Parameter Estimates
.634
.125
5.066
.000
Estimates
Std Error
t
Approx Sig
MA1
Non-Seasonal Lags
Melard's algorithm was used for estimation.
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Modlisation de zt
12 1 13 1( ) ( )t t t t t t t w z z z z a a De
On dduit
12 1 13 1( )t t t t t t z z z z a a
march1 an avant
valuation
de la tendance1 an avant
chocalatoire
en tchocalatoireen t-1
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Calcul des prvisions et des erreurs
Modle : 12 1 13 1t t t t t t z z z z a a
Prvision de zt ralise en t-1 :
12 1 13 1
t t t t t z z z z a
Erreur de prvision lhorizon 1 :
t t ta z z
Calcul pratique des prvisions et des erreurs sur lhistorique:
12 1 13 1
ett t t t t t t t z z z z a a z z
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MA(1) sans constante
DATE. ventes Fit Error
1 JAN 1978 3741 . .
2 FEB 1978 3608 . .
3 MAR 1978 3735 . .
4 APR 1978 3695 . .
5 MAY 1978 3810 . .
6 JUN 1978 3819 . .
7 JUL 1978 3291 . .
8 AUG 1978 3053 . .
9 SEP 1978 3908 . .
10 OCT 1978 4035 . .
11 NOV 1978 3933 . .
12 DEC 1978 4004 . .
13 JAN 1979 3961 . .
14 FEB 1979 4025 3828.00 197.00
15 MAR 1979 4336 4062.93 273.07
16 APR 1979 4335 4140.81 194.19
17 MAY 1979 4412 4331.83 80.17
18 JUN 1979 4268 4370.99 -102.99
19 JUL 1979 3968 3804.86 163.14
20 AUG 1979 3505 3626.87 -121.87
21 SEP 1979 4434 4437.16 -3.16
22 OCT 1979 4854 4563.00 291.00
23 NOV 1979 4592 4567.61 24.39
24 DEC 1979 4264 4647.55 -383.55
Rsultats
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Rsultats (suite)MA(1) sans constante
DATE. ventes Fit Error
25 JAN 1980 4687 4464.06 222.9426 FEB 1980 4704 4609.72 94.28
27 MAR 1980 4579 4955.25 -376.25
28 APR 1980 4800 4816.44 -16.44
29 MAY 1980 4485 4887.42 -402.42
30 JUN 1980 4617 4596.02 20.98
31 JUL 1980 4491 4303.71 187.29
32 AUG 1980 3832 3909.31 -77.31
33 SEP 1980 4669 4809.99 -140.9934 OCT 1980 5193 5178.35 14.65
35 NOV 1980 4544 4921.72 -377.72
36 DEC 1980 4676 4455.37 220.63
37 JAN 1981 4709 4959.18 -250.18
38 FEB 1981 4705 4884.55 -179.55
39 MAR 1981 4677 4693.78 -16.78
40 APR 1981 4627 4908.64 -281.64
41 MAY 1981 4555 4490.48 64.5242 JUN 1981 4570 4646.11 -76.11
43 JUL 1981 4457 4492.23 -35.23
44 AUG 1981 3589 3820.33 -231.33
45 SEP 1981 4636 4572.60 63.40
46 OCT 1981 5077 5119.82 -42.82
47 NOV 1981 4623 4455.14 167.86
48 DEC 1981 4591 4648.62 -57.62
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Rsultats (fin)
MA(1) sans constante
DATE. ventes Fit Error
49 JAN 1982 4764 4660.52 103.48
50 FEB 1982 4726 4694.42 31.58
51 MAR 1982 5080 4677.99 402.01
52 APR 1982 4952 4775.23 176.77
53 MAY 1982 4633 4767.98 -134.98
54 JUN 1982 4830 4733.54 96.46
55 JUL 1982 4460 4655.87 -195.87
Vrifier les calculs pour 55 55 etz a
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Graphique des ventes observes et prdites
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Graphique des rsidus
Limite 2
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Qualit de lajustement dansTime Series Modeler
2 ( )
Normalized BIC 2 ( )t
a Log N
Log rN r N
2 2
2 2
Stationary R-Squared 1 1t t t t
t t
t t
t t
w w z z
w w w w
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Validation du modletude des ( )k tr a
Autocorrelations
Series: Error for ventes from ARIMA, MOD_2, NOCON
-.087 .149 .342 1 .558
.072 .147 .581 2 .748
.188 .145 2.253 3 .522-.079 .143 2.556 4 .635
.128 .141 3.379 5 .642
.164 .140 4.768 6 .574
-.168 .138 6.265 7 .509
.031 .136 6.316 8 .612
.063 .134 6.535 9 .685
-.115 .132 7.304 10 .696
.208 .130 9.894 11 .540-.281 .127 14.747 12 .256
-.076 .125 15.119 13 .300
-.157 .123 16.750 14 .270
.062 .121 17.017 15 .318
-.054 .119 17.222 16 .371
Lag
1
2
34
5
6
7
8
9
10
1112
13
14
15
16
Autocorrel
ation Std. Errora
Value df Sig.b
Box-Ljung Statistic
The underlying process assumed is independence (white
noise).
a.
Based on the asymptotic chi-square approximation.b.
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Validation du modleCorrlogramme des ( )k tr a
Formule deBox-Jenkins
Corrlogrammethorique des erreurs bt
0
k(bt)
12 k
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Validation du modle : Utilisation dela statistique de Ljung-Box
La statistique de Ljung-Box
22
1
( )( 2)
mk t
m
k
r aN N
N k
suit une loi du khi-deux m-rddl lorsque les rsidusforment un bruit blanc.On accepte le modle tudi si les niveaux designification
2 2Prob( ( ) )mm r
sont > .05 pour diffrentes valeurs de m.
Utili ti d dl ti i i
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Utilisation du modle estim en prvision
Modle : 12 1 13 1t t t t t t z z z z a a
Prvision de z55+h ralise en t= 55 :
56 44 55 43 55 z z z z a h = 1
57 45 56 44 z z z z h = 2
Et ainsi de suite
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Application
AUG 1982 3716.13 3319.22 4113.04 196.53
SEP 1982 4763.13 4340.43 5185.82 209.30
OCT 1982 5204.13 4757.13 5651.12 221.34
NOV 1982 4750.13 4280.09 5220.17 232.75
DEC 1982 4718.13 4226.12 5210.14 243.62
1
2
3
4
5
DATE. Fit for ventes 95% LCL 95% UCL SE of Fit
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Intervalle de prvision 95% de z55+h
Chaque modle a sa propre formule de constructionde lintervalle de prvision.
2
55 .975
( ) 1 ( 1)(1 )hz t N r h
Modle MA(1) :
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Amlioration du modle MA(1)
On suppose maintenant le modle est significatif.12
( ) .281tr a
1
12 , o bruit blanct t t
t t t t
w b b
b a a a
De 12(1 ) et (1 )t t t t w B b b B a
on dduit :12(1 )(1 )
t tw B B a
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Demande SPSS
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Rsultats
Parameter Estimates
.715 .765 -11.468
.107 .399 5.219
6.693 1.918 -2.197.000 .062 .034
Estimates
Std Error
t
Approx Sig
MA1
Non-Seasonal
Lags
Seasonal MA1
Seasonal
Lags
Constant
Melard's algori thm was used for estimation.
Residual Diagnostics
42
2
39
1268226.611
1336414.106
25544.245
159.826
-276.531
559.062
564.275
Number of Residuals
Number of Parameters
Residual df
Adjusted ResidualSum of Squares
Residual Sum of Squares
Residual Variance
Model Std. Error
Log-Likelihood
Akaike's Information
Criterion (AIC)
Schwarz's Bayesian
Criterion (BIC)
12(1 )(1 )t t
w B B a
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7.2 tude de la voie autorgressive
On suppose que wtsuit un modle AR(14) :
1 1 14 14
2
...
( )
t t t t
t
w w w a
Var a
et on a = (1 - 1 --14).
On choisit les paramtres , 1,,14 et 2 laidede la mthode du maximum de vraisemblance.
est appelConstant dansSPSS
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Rsultats
1 1 14 14...t t t t w w w a
Residual Diagnostics
42
14
27
949178.0
1041062
28699.741
169.410
-270.689
571.379
597.444
Number of Residuals
Number of Parameters
Residual dfAdjusted Residual Sum of
Squares
Residual Sum o f Squares
Residual Variance
Mode l Std. Error
Log-Likelihood
Akaike's Information
Criterion (AIC)Schwarz's Bayesian
Criterion (BIC)
Parameter Estimates
-.680 .156 -4.367 .000
-.441 .169 -2.614 .014
.059 .188 .311 .758
.034 .184 .185 .855
.107 .191 .560 .580
.138 .214 .644 .525
-.051 .254 -.200 .843
-.016 .240 -.067 .947
-.006 .232 -.026 .980
-.054 .237 -.228 .821
.185 .234 .791 .436
-.307 .227 -1.355 .187
-.428 .208 -2.059 .049
-.572 .156 -3.668 .001
-10.788 9.983 -1.081 .289
AR1
AR2
AR3AR4
AR5
AR6
AR7
AR8
AR9
AR10
AR11
AR12
AR13
AR14
Non-Seasonal
Lags
Constant
Estimates Std Error t Approx Sig
Melard's algorithm was used for estimation.
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Modle AR : p = (1,2,12,13,14) avec cste
1 1 2 2 12 12 13 13 14 14t t t t t t tw w w w w w a
Demande SPSS
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Rsultats
1 1 2 2 12 12 13 13 14 14t t t t t t tw w w w w w a
Parameter Es timates
-.775 .127 -6.083 .000-.490 .122 -4.006 .000
-.512 .138 -3.711 .001
-.594 .159 -3.733 .001
-.526 .145 -3.619 .001
-12.797 7.487 -1.709 .096
AR1AR2
AR12
AR13
AR14
Non-SeasonalLags
Constant
Estimates Std Error t Approx Sig
Melard's algorithm was used for estimation.
Residual Diagnostics
42
5
36
1093774.600
1192109.813
25711.840
160.349
-273.114
558.228
568.654
Number of Residuals
Number of
Parameters
Residual dfAdjusted Residual
Sum of Squares
Residual Sum of
Squares
Residual Variance
Mode l Std. Error
Log-Likelihood
Akaike's Information
Criterion (AIC)
Schwarz's Bayesian
Criterion (BIC)
l ( )
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Modle AR : p = (1,2,12,13,14) sans cste
1 1 2 2 12 12 13 13 14 14t t t t t t t w w w w w w a
Demande SPSS
l
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Rsultats
1 1 2 2 12 12 13 13 14 14t t t t t t t w w w w w w a
Residual Diagnostics
42
5
37
1172013
1233379
27877.941
166.967-274.563
559.127
567.815
Number of
Residuals
Number of
Parameters
Residual df
Adjusted Residual
Sum of Squares
Residual Sum ofSquares
Residual Variance
Model Std. ErrorLog-Likelihood
Akaike's Information
Criterion (AIC)
Schwarz's Bayesian
Criterion (BIC)
Parameter Es timates
-.747 .134 -5.591 .000
-.460 .129 -3.568 .001
-.454 .148 -3.066 .004
-.508 .171 -2.975 .005
-.467 .154 -3.041 .004
AR1
AR2
AR12
AR13
AR14
Non-Seasonal
Lags
Estimates Std Error t Approx Sig
Melard's algorithm was used for estim ation.
M dl R 2 P 1
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Modle AR : p = 2, P = 1 avec cste2 12
1 2(1 )(1 ) t tB B B w a
Demande SPSS
R l
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Rsultats2 12
1 2(1 )(1 ) t tB B B w a
Residual Diagnostics
42
3
38
1196121
1286077
27725.190
166.509
-274.998
557.997
564.948
Number of
Residuals
Number of
ParametersResidual df
Adjusted Residual
Sum of Squares
Residual Sum of
Squares
Residual Variance
Model Std. Error
Log-Likelihood
Akaike's Information
Criterion (AIC)
Schwarz's Bayesian
Criterion (BIC)
Parameter Estimates
-.759 .139 -5.445 .000
-.523 .132 -3.970 .000
-.557 .146 -3.812 .000
-12.289 8.308 -1.479 .147
AR1
AR2
Non-Seasonal
Lags
Seasonal AR1Seasonal Lags
Constant
Estimates Std Error t Approx Sig
Melard's algorithm was used for estimation.
M dl AR 2 P 1
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Modle AR : p = 2, P = 1 sans cste2 12
1 2(1 )(1 ) t tB B B w a
Demande SPSS
R lt t
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Rsultats2 12
1 2(1 )(1 ) t tB B B w a
Residual Diagnostics
42
3
39
1256636
1315334
29246.908
171.017
-276.033
558.066
563.279
Number of
Residuals
Number of
Parameters
Residual df
Adjusted Residual
Sum of Squares
Residual Sum of
Squares
Residual Variance
Model Std. Error
Log-LikelihoodAkaike's Information
Criterion (AIC)
Schwarz's Bayesian
Criterion (BIC)
Parameter Estimates
-.731 .143 -5.101 .000-.481 .135 -3.562 .001
-.489 .154 -3.186 .003
AR1
AR2
Non-Seasonal
Lags
Seasonal AR1Seasonal Lags
Estimates Std Error t Approx Sig
Melard's algorithm was used for estimation.
Rsultats
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Rsultats2 12
1 2(1 )(1 ) t tB B B w a
Rsultats avec Time Series Modeler
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Rsultats avec Time Series Modeler
2 12
1 2(1 )(1 ) t tB B B w a
Forecast
3818 4792 5192 4688 4742
4163 5150 5567 5123 5197
3472 4434 4817 4253 4288
Forecast
UCL
LCL
Modelventes-Model_1
Aug 1982 Sep 1982 Oct 1982 Nov 1982 Dec 1982
For each model , forecasts start after the last non-missing in the range o f the requested
estimation period, and end at the last period for which non-missing values of all the predict
are available or at the end date of the requested forecast period, whichever is earlier.
Rsultats avec Time Series Modeler
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Rsultats avec Time Series Modeler2 12
1 2(1 )(1 ) t tB B B w a
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7.3 tude de la voie AR/MA
2 12
1 2(1 ) (1 )t tB B w B a Modle avecconstante
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Rsultats
2 121 2(1 ) (1 )t tB B w B a
Residual Diagnostics
42
3
38
1112464
1256550
19325.966
139.018
-274.630
557.261
564.211
Number of Residuals
Number of Parameters
Residual df
Adjusted Residual Sum of
Squares
Residual Sum of Squares
Residual Variance
Model Std. Error
Log-Likelihood
Akaike's Information
Criterion (AIC)
Schwarz's Bayesian
Criterion (BIC)
Parameter Estimates
-.765 .123 -6.228 .000
-.558 .114 -4.911 .000
.965 2.964 .326 .747-11.009 6.504 -1.693 .099
AR1
AR2
Non-Seasonal
Lags
Seasonal MA1Seasonal LagsConstant
Estimates Std Error t Approx Sig
Melard's algorithm was used for estimati on.
Warnings
Our tests have determined that the estimated model lies close to the boundary of t
invertibility region. Although the moving average parameters are probably correctl
estimated, their standard errors and covariances should be considered suspect.
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7.3 tude de la voie AR/MA
2 12
1 2(1 ) (1 )t tB B w B a Modle sansconstante
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Rsultats
2 121 2(1 ) (1 )t tB B w B a
Residual Diagnostics
42
339
1190270
1287295
24282.930
155.830
-275.152
556.304
561.517
Number of Residuals
Number of ParametersResidual df
Adjusted Residual Sum of
Squares
Residual Sum of Squares
Residual Variance
Model Std. Error
Log-Likelihood
Akaike's InformationCriterion (AIC)
Schwarz's Bayesian
Criterion (BIC)
Parameter Estimates
-.736 .134 -5.488 .000
-.506 .124 -4.074 .000
.745 .360 2.071 .045
AR1
AR2
Non-Seasonal
Lags
Seasonal MA1Seasonal Lags
Estimates Std Error t Approx Sig
Melard's algorithm was used for estimation.
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Rsultats
2 12
1 2(1 ) (1 )t tB B w B a
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Rsultats (avec Time Series Modeler)
2 12
1 2(1 ) (1 )t tB B w B a
Forecast
3861 4854 5206 4810 4798
4184 5187 5553 5215 5220
3539 4521 4858 4405 4375
Forecast
UCL
LCL
Model
ventes-Model_1
Aug 1982 Sep 1982 Oct 1982 Nov 1982 Dec 1982
For each model, forecasts start after the last non-missing in the range of the requested
estimation period, and end at the last period for which non-missing values of all the predict
are available or at the end date of the requested forecast period, whichever is earlier.
Rsultats (avec Time Series Modeler)
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Rsultats (avec Time Series Modeler)2 12
1 2(1 ) (1 )t tB B w B a
8 Le modle multiplicatif usuel
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8. Le modle multiplicatif usuelARIMA(p,d,q)*(P,D,Q)s
( ) (1 ) (1 ) ( )s d s D st w tB B B B z B B a
1
1
1
1
( ) 1 ...
( ) 1 ...
( ) 1 ...
( ) 1 ...
p
p
s s sP
P
q
qs s sQ
Q
B B B
B B B
B B B
B B B
o :
Tous ces polynmes doivent tre inversibles.
wtbruitblanc
9 Prvision
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9. Prvision
(1 ) (1 )d s D t tB B B z B a
Le modle gnral
peut scrire :
1 1 1 1... ...t t p t p t t q t qz z z a a a
Prvision lhorizon h
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Prvision l horizon h
Modle
1 1 1 1... ...t h t h p t h p t h t h q t h qz z z a a a
Prvision
1 1 1 1 ( ) ... ...t t h p t h p t h q t h qz h z z a a
avec :si 0
( ) si 0
t h j
t h j
t
z h jz
z h j h j
1 (1) si 0
0 si 0
t h j t h j
t h j
z z h ja
h j
10 C l l d li t ll d i i
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10. Calcul de lintervalle de prvision
( )(1 ) (1 ) ( )s d s D st tB B B B z B B a
De
on dduit (formellement) :
1 1
1 1 2 2
' ( )(1 ) (1 ) ( )
' ...
s d s D s
t t
t t t
z B B B B B B a
a a a
Prvision de zt h linstant t
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Prvision dezt+h l instant t
On a
1 1 2 2 1 1
1 1
' ...
...
t h t h t h t h h t
h t h t
z a a a a
a a
Futur
Pass
1 1 ( ) ' ...
t h t h t z h a a
Do la prvision dezt+h linstant t
E d i i lh i h
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Erreur de prvision lhorizon h
1 1 2 2 1 1
( ) ( )
...
t t h t
t h t h t h h t
e h z z h
a a a a
Do :
2 2 2
1 1[ ( )] 1 ...t hVar e h
[ ( )] 0tE e h
I t ll d i i 95%
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Intervalle de prvision 95%dezt+hralis linstant t
2 2
.975 1 1 ( ) ( ) 1 ...t hz h t N r
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Exemple March Total
12(1 )(1 ) (1 )t tB B z B a Modle :
On dduit :
1 12 1
2 12 24
2 11
1 2 11
(1 ) (1 ) (1 )
(1 ...)(1 ...)(1 )
(1 (1 ) (1 ) ... (1 ) ...)
t t
t
t
z B B B a
B B B B B a
B B B a
Remarque : (1 ) pour 11h h
M h T t l I t ll d i i
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March Total : Intervalle de prvision lhorizon h 12
2 2
.975 1 1
2
.975
( ) ( ) 1 ...
( ) ( ) 1 ( 1)(1 )
t h
t
z h t N r
z h t N r h
11. Le modle gnral de TS Modeler
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g a SLe modle fonction de transfert
1
srie dpendante
,..., sries prdicteurs
( )
, ou
t
t kt
t t
i
Y
X X
Z f Y
f f Log
, (1 ) (1 )
( ) ( )
( ) ( )
d s D
i
s
i i i
s
i i i
B B
Num B B
Den B B
1( ) ( ) ( ) ( ) ( )
k
s sit i i it t
i i
NumB B Z f X B B aDen
Nt= Noise
Application la srie IPI
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Application la srie IPI
Anne Trimestre 1 Trimestre 2 Trimestre 3 Trimestre 4
63
64
65
66
67
68
69
70
.
.
.
82
68
77
76
81
84
89
95
100
137
74
79
79
84
85
77
99
104
136
64
65
67
71
72
78
82
87
111
78
79
83
87
90
99
103
110
140
Indice de la Production Industrielle de la France (1963 - 1982)
Visualisation de la srie IPI
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89
Visualisation de la srie IPI
Date
Q11982
Q11981
Q11980
Q11979
Q11978
Q11977
Q11976
Q11975
Q11974
Q11973
Q11972
Q11971
Q11970
Q11969
Q11968
Q11967
Q11966
Q11965
Q11964
Q11963
IPI
160
140
120
100
80
60
40
Cette srieprsente unetendance etune saisonnalit
Visualisation de la saisonnalit
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Visualisation de la saisonnalit
Anne
198519801975197019651960
IPI
160
140
120
100
80
60
Trimestre
4
3
2
1
Visualisation de la tendance
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Date
Q11982
Q11981
Q11980
Q11979
Q11978
Q11977
Q11976
Q11975
Q11974
Q11973
Q11972
Q11971
Q11970
Q11969
Q11968
Q11967
Q11966
Q11965
Q11964
Q11963
160
140
120
100
80
60
40
IPI
MA(IPI,4,4)
Visualisation de la tendance
Moyenne mobile centredordre 4 :
4
X5.0XXXX5.0Z
2t1tt1t2t
t
Tendance Zt
(a) Indice de la production industrielle ( 23.85 )
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7773696561575349454137332925211713951
Trimestre
150
125
100
75
ipi
(b) Diffrence saisonnire de IPI ( 5.49 )
(c) Diffrence rgulire/saisonnire de IPI ( 4.76 )
Modle avec intervention
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Modle avec intervention
4
68.2( ) ( ) (1 )(1 )( ) ( ) ( )s s
t tB B B B z I B B a
Effetmai 68
Nt= Noise = Srie corrige stationnarise
tapes1. Construction de la srie Noise 2. Modlisation de la srie Noise 3. Estimation du modle complet
Etape 1 : Construction de la srie Noise
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Etape 1 : Construction de la srie Noise
4
68.2(1 )(1 )( )
t tNoise B B z I a
Parameter Estimates
-15.250 1.626 -9.380 .000
-.160 .375 -.426 .671
i22Regression Coeffi cients
Constant
Estimates Std Error t Approx Sig
Melard's algorithm was used for estimation.
tape 2 : Modlisation de la srie Noise
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tape : od sat o de a s e Noise
4
68.2(1 )(1 )( )
tNoise B B z I
Noise suit un AR(8)
Modlisation de la srie Noise
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4
68.2(1 )(1 )( )tNoise B B z I
Residual Diagnostics
75
8
66
493.364
494.199
7.294
2.701
-177.255
372.509
393.367
Number of Residuals
Number of Parameters
Residual df
Adjusted Residual Sum of
Squares
Residual Sum o f Squares
Residual Variance
Model Std. Error
Log-Likelihood
Akaike's Informati on
Criterion (AIC)
Schwarz's Bayesian
Criterion (BIC)
Parameter Estimates
.095 .118 .803 .425
.016 .121 .135 .893
-.215 .119 -1.800 .076
-.520 .125 -4.175 .000
-.081 .121 -.668 .506-.085 .119 -.714 .478
-.116 .124 -.934 .354
-.259 .127 -2.042 .045
.066 .150 .437 .663
AR1
AR2
AR3
AR4
AR5AR6
AR7
AR8
Non-Seasonal
Lags
Constant
Estimates Std Error t Approx Sig
Melard's algorithm was used for estimation.
Noise ~ ARIMA(8,1,0)*(0,1,0)4
Modlisation de la srie Noise
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4
68.2(1 )(1 )( )tNoise B B z I
Residual Diagnostics
75
2
73
549.094
550.078
7.344
2.710
-181.075
366.150
370.785
Number of Residuals
Number of Parameters
Residual df
Adjusted Residual Sum of
Squares
Residual Sum of Squares
Residua l Variance
Model Std. Error
Log-Likelihood
Akaike's Information
Criterion (AIC)
Schwarz's Bayesian
Criterion (BIC)
Parameter Estimates
-.628 -.292
.115 .118
-5.476 -2.474
.000 .016
Estimates
Std Error
t
Approx Sig
Seasonal AR1 Seasonal AR2
Seasonal Lags
Melard's algorithm was used for estimation.
Noise ~ ARIMA(0,1,0)*(2,1,0)4sans constante
tape 3 : estimation du modle complet
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tape 3 : estimation du modle complet
4 8 4
1 2 68.2(1 ) (1 )(1 )( )t tB B B B z I a
Residual Diagnostics
75
2
71
547.971
551.748
7.533
2.745
-181.015
370.031
379.301
Number of Residuals
Number of Parameters
Residual df
Adjusted Residual Sum of
Squares
Residual Sum o f Squares
Residual Variance
Mode l Std. Error
Log-LikelihoodAkai ke's Information
Criterion (AIC)
Schwarz's Bayesian
Criterion (BIC)
Parameter Estimates
-.632 -.295 -15.089 -.097
.116 .118 1.679 .170
-5.440 -2.509 -8.987 -.569
.000 .014 .000 .571
Estimates
Std Error
t
Approx Sig
Seasonal AR1 Seasonal AR2
Seasonal Lags
i22
Regression
Coefficients
Constant
Melard's algorithm was used for estimation.
tape 3 : estimation du modle complett t
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sans constante
4 8 4
1 2 68.2
(1 ) (1 )(1 )( )t t
B B B B z I a
Residual Diagnostics
75
2
72
550.462
554.558
7.464
2.732
-181.173
368.347
375.299
Number of Residuals
Number of Parameters
Residual df
Adjusted Residual Sum of
Squares
Residual Sum of Squares
Residual Variance
Model Std. Error
Log-Likelihood
Akaike's Information
Criterion (AIC)Schwarz's Bayesian
Criterion (BIC)
Parameter Estimates
-.631 -.292 -15.095
.116 .117 1.671
-5.459 -2.498 -9.033.000 .015 .000
Estimates
Std Error
tApprox Sig
Seasonal AR1 Seasonal AR2
Seasonal Lags
i22
Regression
Coefficients
Melard's algorithm was used for estimation.
Utilisation de Time Series Modeler
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4 8 4
1 2 68.2
(1 ) (1 )(1 )( )t t
B B B B z I a
Fentre 1
Utilisation de Time Series Modeler
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4 8 4
1 2 68.2
(1 ) (1 )(1 )( )t t
B B B B z I a
Fentre 2
Utilisation de Time Series Modeler
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4 8 4
1 2 68.2(1 ) (1 )(1 )( )t tB B B B z I a
Fentre 3
Utilisation de Time Series Modeler pour la prvision
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p p
4 8 4
1 2 68.2(1 ) (1 )(1 )( )t tB B B B z I a
Forecast LCL UCL
Q1 1983 136.1 130.7 141.6
Q2 1983 133.4 125.7 141.1
Q3 1983 110.3 100.9 119.8
Q4 1983 138.1 127.2 149.0
Model Statistics
1 .678 18.846 16 .277 0
Model
IPI-Model_1
Number of
Predictors
Stationary
R-squared
Model Fit
statistics
Statistics DF Sig.
Ljung-Box Q(18)
Number of
Outliers
Utilisation de Time Series Modeler pour la prvisionLa syntaxe SPSS
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La syntaxe SPSSPREDICT THRU END.
* Time Series Modeler.
TSMODEL
/MODELSUMMARY PRINT=[ MODELFIT]
/MODELSTATISTICS DISPLAY=YES MODELFIT=[ SRSQUARE]
/MODELDETAILS PRINT=[ PARAMETERS FORECASTS]
/SERIESPLOT OBSERVED FORECAST FIT FORECASTCI
/OUTPUTFILTER DISPLAY=ALLMODELS
/SAVE NRESIDUAL(NResidual)
/AUXILIARY CILEVEL=95 MAXACFLAGS=24
/MISSING USERMISSING=EXCLUDE
/MODEL DEPENDENT=ipi INDEPENDENT=i22
PREFIX='Model'
/ARIMA AR=[0] DIFF=1 MA=[0] ARSEASONAL=[1,2]
DIFFSEASONAL=1MASEASONAL=[0]
TRANSFORM=NONE CONSTANT=NO
/TRANSFERFUNCTION VARIABLES=i22
DIFF=1
DIFFSEASONAL=1
/AUTOOUTLIER DETECT=OFF.
Utilisation de Expert Modeler
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p
Utilisation de Expert Modeler
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p
Model Description
ARIMA(0,1,0)(0,1,1)Model_1IPIModel IDModel Type
Model Statistics
1 .660 27.437 17 .052 0
Model
IPI-Model_1
Number of
Predictors
Stationary
R-squared
Model Fit
statistics
Statistics DF Sig.
Ljung-Box Q(18)
Number of
Outliers
ARIMA Model Parameters
1
1
.507 .109 4.657 .000
-15.315 1.728 -8.863 .000
1
1
Difference
Seasonal Difference
Lag 1MA, Seasonal
No TransformationIPI
Lag 0Numerator
Difference
Seasonal Difference
No Transformationi2 2
IPI-Model_1
Estimate SE t Sig.
4 4
68.2 1(1 )(1 )( ) (1 )t tB B z I B a
Rponse :
Utilisation de Expert Modeler
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107
p
Utilisation de Expert Modeler
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108/109
108
p
Forecast
136 134 111 139141 142 121 150
130 126 101 128
ForecastUCL
LCL
Model
IPI-Model_1
Q1 1983 Q2 1983 Q3 1983 Q4 1983
For each model, forecasts start after the last non-missing in the range of the
requested estimation period, a nd end at the l ast period for which
non-missing values of all the predictors are available or at the end date of t
requested forecast period, whichever is earlier.
Utilisation de Expert Modelerpour All models
8/2/2019 Box Jenkins (1)
109/109
pour All models
Rponse :
Model Description
ARIMA(0,1,0)(0,1,1)Model_1IPIModel ID
Model Type
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