Box Jenkins (1)

8/2/2019 Box Jenkins (1)

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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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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


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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(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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(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


421

41

1603132

1620350

38625.634

196.534-281.143

564.285

566.023

Number of ResidualsNumber of Parameters

Residual df


Squares

Residual Sum of Squares

Residual Variance

Model Std. ErrorLog-Likelihood


Criterion (AIC)

Schwarz's Bayesian

Criterion (BIC)

Parameter Estimates

.634

.125

5.066

.000

Estimates

Std Error

t

Approx Sig

MA1

Non-Seasonal Lags



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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


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


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.


42

2

39

1268226.611

1336414.106

25544.245

159.826

-276.531

559.062

564.275

Number of Residuals


Residual df

Adjusted ResidualSum of Squares


Residual Variance

Model Std. Error

Log-Likelihood


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


42

14

27

949178.0

1041062

28699.741

169.410

-270.689

571.379

597.444

Number of Residuals


Residual dfAdjusted Residual Sum of

Squares

Residual Sum o f Squares

Residual Variance

Mode l Std. Error

Log-Likelihood


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



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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




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


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


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


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


Melard's algorithm was used for estim ation.

M dl R 2 P 1


64/109

64

Modle AR : p = 2, P = 1 avec cste2 12

1 2(1 )(1 ) t tB B B w a

Demande SPSS

R l


65/109

65

Rsultats2 12

1 2(1 )(1 ) t tB B B w a


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


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



M dl AR 2 P 1


66/109

66

Modle AR : p = 2, P = 1 sans cste2 12

1 2(1 )(1 ) t tB B B w a

Demande SPSS

R lt t


67/109

67

Rsultats2 12

1 2(1 )(1 ) t tB B B w a


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



Rsultats


68/109

68

Rsultats2 12

1 2(1 )(1 ) t tB B B w a

Rsultats avec Time Series Modeler


69/109

69


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.



70/109

70

Rsultats avec Time Series Modeler2 12

1 2(1 )(1 ) t tB B B w a


71/109

71

7.3 tude de la voie AR/MA

2 12

1 2(1 ) (1 )t tB B w B a Modle avecconstante


72/109

72

Rsultats

2 121 2(1 ) (1 )t tB B w B a


42

3

38

1112464

1256550

19325.966

139.018

-274.630

557.261

564.211

Number of Residuals


Residual df


Squares


Residual Variance

Model Std. Error

Log-Likelihood


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


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.


73/109

73

7.3 tude de la voie AR/MA

2 12

1 2(1 ) (1 )t tB B w B a Modle sansconstante


74/109

74

Rsultats

2 121 2(1 ) (1 )t tB B w B a


42

339

1190270

1287295

24282.930

155.830

-275.152

556.304

561.517

Number of Residuals

Number of ParametersResidual df


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




75/109

75

Rsultats

2 12

1 2(1 ) (1 )t tB B w B a


76/109

76

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)


77/109

77

Rsultats (avec Time Series Modeler)2 12

1 2(1 ) (1 )t tB B w B a

8 Le modle multiplicatif usuel


78/109

78

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


79/109

79

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


80/109

80

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


81/109

81

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


82/109

82

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


83/109

83

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%


84/109

84

Intervalle de prvision 95%dezt+hralis linstant t

2 2

.975 1 1 ( ) ( ) 1 ...t hz h t N r


85/109

85

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


86/109

86

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


87/109

87

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


88/109

88

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


89/109

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


90/109

90

Visualisation de la saisonnalit

Anne

198519801975197019651960

IPI

160

140

120

100

80

60

Trimestre

4

3

2

1

Visualisation de la tendance


91/109

91

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 )


92/109

92

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


93/109

93

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


94/109

94

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



tape 2 : Modlisation de la srie Noise


95/109

95

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


96/109

96

4

68.2(1 )(1 )( )tNoise B B z I


75

8

66

493.364

494.199

7.294

2.701

-177.255

372.509

393.367

Number of Residuals


Residual df


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



Noise ~ ARIMA(8,1,0)*(0,1,0)4

Modlisation de la srie Noise


97/109

97

4

68.2(1 )(1 )( )tNoise B B z I


75

2

73

549.094

550.078

7.344

2.710

-181.075

366.150

370.785

Number of Residuals


Residual df


Squares


Residua l Variance

Model Std. Error

Log-Likelihood


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


Noise ~ ARIMA(0,1,0)*(2,1,0)4sans constante

tape 3 : estimation du modle complet


98/109

98

tape 3 : estimation du modle complet

4 8 4

1 2 68.2(1 ) (1 )(1 )( )t tB B B B z I a


75

2

71

547.971

551.748

7.533

2.745

-181.015

370.031

379.301

Number of Residuals


Residual df


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 Lags

i22

Regression

Coefficients

Constant


tape 3 : estimation du modle complett t


99/109

99

sans constante

4 8 4

1 2 68.2

(1 ) (1 )(1 )( )t t

B B B B z I a


75

2

72

550.462

554.558

7.464

2.732

-181.173

368.347

375.299

Number of Residuals


Residual df


Squares


Residual Variance

Model Std. Error

Log-Likelihood


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 Lags

i22

Regression

Coefficients


Utilisation de Time Series Modeler


100/109

100

4 8 4

1 2 68.2

(1 ) (1 )(1 )( )t t

B B B B z I a

Fentre 1



101/109

101

4 8 4

1 2 68.2

(1 ) (1 )(1 )( )t t

B B B B z I a

Fentre 2



102/109

102

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


103/109

103

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


104/109

104

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


105/109

105

p



106/109

106

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 :



107/109

107

p



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


109/109

pour All models

Rponse :

Model Description

ARIMA(0,1,0)(0,1,1)Model_1IPIModel ID

Model Type

Documents

Box Jenkins (1)