Les séries chronologiques en mathématiques

Résumé du document

Cours de mathématiques sur les séries chronologiques présentant de nombreux schémas, tableaux, calculs et explications qui permettent une parfaite compréhension du cours.

Extraits

[...] Séries chronologiques I. Définitions Définition : Une série chronologique (ou chronique) est constituée d'observations relevées au cours du temps. Si t t n sont les n instants successifs d'observations, et y t1 la valeur mesuré à l'instant t on note la série chronologique : y t tεT T = t t n But : Analyser Prévoir La série chronologique peut s'écrire comme une statistique bidimensionnelle : ti, t i = n en abscisse : t i en ordonnée : y t i Les observations sont rangées par ordre chronologique. [...]

[...] Cycle : noté t t Regroupe les variations autour de la tendance. Variation saisonnière : notée t t Elles sont de nature périodique (trimestre, semestre, année). p : période s t+p = s t On a donc besoin de connaître seulement s s s p Fluctuation irrégulière : notée (å t t De faible intensité et de faible durée. [...]

[...] On doit avoir : s t = 1 Si ce n'est pas le cas, on calcule un coefficient correcteur : â = s t / p On prend s' t = s t â au lieu de s t y t = â t + b + s' t e t = y t y t y iTrimestreAn 1An 2An 31567223437894678 Ex : y = chômage picscalex1000100090000039c1b000009001610000000001610000026060f002220574d 46430100000000000100f3660000000002000000002000005c0200005c22000001000000 6c0000001100000012000000fa010000ac00000000000000000000003a350000d8130000 20454d46000001005c220000f30100000500000000000000000000000000000000050000 c003000052010000fe0000000000000000000000000000005028050030e0030046000000 2c00000020000000454d462b014001001c000000100000000110c0db0100000060000000 60000000460000005c00000050000000454d462b224004000c000000000000001e400100 0c00000000000000244001000c00000000000000214007000c0000000000000030400200 10000000040000000000803f044000000c00000000000000180000000c00000000000000 190000000c000000ffffff00140000000c0000000d000000120000000c00000002000000 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0102ffffff0005000000090200000000040000002c010000070000001604900082010000 000007000000fc020000000000000000040000002d01080004000000f001070004000000 2d01050004000000f0010200040000002d010100040000002701ffff030000001e000500 00000102ffffff0005000000090200000000040000002701ffff030000001e0005000000 0102ffffff0005000000090200000000040000002701ffff030000000000 y t = a t + b + s t + å t = f t + s t + å t â = cov / t b = picscalex75010009000003a600000000008500000000000400000003010800050000000 b0200000000050000000c0212000d00030000001e0085000000410b2000cc0011000c000 000000011000c0000000000280000000c000000110000000100040000000000000000000 000000000000000000000000000000000000000ffffff00fefefe004444440099999900c ccccc00666666005555550077777700aaaaaa0011111100eeeeee00bbbbbb00222222003 3333300dddddd0022866ef222220f0222a825a522220a082283228a222202022222fcba4 222080a2228ec7a72220202222d922702220202222a42290b22070a22237222a422020d2 2260222372202022cb50c22602202092542372290b20203227263222a420202225675222 89202022222222222220b052222222222220202233333333334020222222222222207020 40000002701ffff030000000000 - â t iy it i i y if ty t f ty t = f t + s tå t = y t y ty i i - y(y i .230825- 1122444.6154 2.8462709 0.23084911104100407 .077 0.230816 0.2308642478726505120482050 t = 78 / 12 = 6.5 y = 72 / 12 = 6 t = 650 / 12 ( 6.5 = 11.91 cov = 512 /12 ( 6.5 * = 3.66 â = 3.66 / 11.91 = 0.308 b = y ( 0.308 * 6.5 ) = 4 Et donc : f t = 0.308 t + 4 s t = y t f t y t f tTrimestreAn 1An 2An 3s t (moyenne) .44152- 2.6154 - 2.8462 - 3.077 - s t : écart saisonnier si s t différent de on l'ajoute à f t Qualité de l'ajustement linéaire : á = y ( y = 1/n i y = 50 / 12 = 4.17 R = ( cov / á y á t ) = (cov / (á á = 3.66 / ( 4.17 * 11.91 ) = 0.27 L'ajustement linéaire est de mauvaise qualité. Prévoir le taux de chômage au 2nd trimestre de la 5ième année : par l'ajustement linéaire par le modèle additif et calculer les erreurs commises. [...]

[...] Variation accidentelle : Valeur isolée, anormale, élevée ou faible. On se limitera à : f t , s t , å t On étudiera deux modèles : - Modèle additif - Modèle multiplicatif Modèle additif : y t = f t + s t + å t 1 t n s t = 0 å t = 0 Modèle multiplicatif : y t = f t + s + å t s t = 0 å t = 0 II. [...]

[...] Dans le nuage de points, on relie les points par des segments de droite. Ex : y t tiy ti190076Y 1191092Y 21950151Y 92 76 t t Convention : On supposera dans la suite que les instants d'observations sont équidistants, on appellera t = n la série chronologique. Représenter les données Examiner les données : - Tendance - Cycle - Saison (phénomène périodique) Tendance : notée t t Evolution à long terme, décrit la forme générale de la série. [...]