Bonjour,
Dans le cadre de mon mémoire en économie du développement, j'ai besoin d'utiliser les ondelettes.
Je me base sur ce papier, et ce qui m'intéresse ici, c'est la page 3 (équation 2). Pour faire simple : ils divisent un taux de croissance en plusieurs périodes.
tdc = D1 + D2 + D3 + D4 + S4
J'ai calculé mes ondelettes via R (module "wavelets"), où j'obtiens au final ceci :
Je comprends le principe des ondelettes, l'utilité de la chose. J'ai une assez bonne idée de comment ça fonctionne (méthode d'Haar), mais il y a quelque chose que je ne comprends définitivement pas : la différence entre V et W. V et W ne correspondent pas à la même bande de fréquence ("frequency band"), d'accord. Mais concrètement, qu'est-ce que cela signifie ?Code:An object of class "modwt" Slot "W": $W1 [,1] [1,] 4.138813e-02 [2,] -2.635967e-18 [3,] -3.029046e-02 [4,] -2.843390e-02 [5,] 3.147502e-02 [6,] -3.797981e-02 [7,] 3.552178e-02 [8,] 1.693556e-02 [9,] -1.608217e-03 [10,] -5.378781e-02 [11,] 2.721954e-02 [12,] 1.626622e-04 [13,] -8.543045e-03 [14,] 4.222788e-02 [15,] -2.863310e-02 [16,] -1.708306e-02 [17,] -2.026039e-02 [18,] -4.778792e-03 [19,] 1.016632e-01 [20,] -8.802020e-02 [21,] 3.909821e-02 [22,] 7.746766e-03 [23,] -5.292898e-03 [24,] -4.533735e-02 [25,] 2.537324e-02 [26,] 2.212471e-02 [27,] -1.944631e-02 [28,] -2.858051e-02 [29,] 6.454173e-02 [30,] -3.776371e-02 [31,] -1.080870e-03 [32,] 1.441985e-03 $W2 [,1] [1,] 0.0215956157 [2,] 0.0421091246 [3,] 0.0055488366 [4,] -0.0445074082 [5,] -0.0278416183 [6,] -0.0017318352 [7,] -0.0044814115 [8,] 0.0249996540 [9,] 0.0338923427 [10,] -0.0200343384 [11,] -0.0409821429 [12,] 0.0004069702 [13,] 0.0095009108 [14,] 0.0126522253 [15,] 0.0236398064 [16,] -0.0160606877 [17,] -0.0415297992 [18,] -0.0311913105 [19,] 0.0359226144 [20,] 0.0552637049 [21,] -0.0176394900 [22,] -0.0010385026 [23,] 0.0246494233 [24,] -0.0240881922 [25,] -0.0352971848 [26,] 0.0137669165 [27,] 0.0250881755 [28,] -0.0226742089 [29,] -0.0060327977 [30,] 0.0313696221 [31,] -0.0060332810 [32,] -0.0192417339 $W3 [,1] [1,] 0.0017481280 [2,] 0.0174976394 [3,] 0.0213533936 [4,] 0.0102345535 [5,] 0.0024258353 [6,] -0.0243187635 [7,] -0.0273079058 [8,] -0.0114857122 [9,] -0.0014560493 [10,] 0.0141165672 [11,] 0.0111605655 [12,] -0.0073310263 [13,] -0.0192855161 [14,] -0.0032840864 [15,] 0.0008297426 [16,] 0.0048253666 [17,] 0.0076253622 [18,] -0.0253302303 [19,] -0.0117485888 [20,] -0.0115898019 [21,] 0.0063379697 [22,] 0.0391487983 [23,] 0.0126465288 [24,] 0.0145492537 [25,] -0.0018189141 [26,] -0.0177239853 [27,] -0.0104283854 [28,] -0.0096142840 [29,] 0.0044231843 [30,] -0.0001059396 [31,] 0.0034946496 [32,] 0.0104116507 $W4 [,1] [1,] 0.0043877912 [2,] -0.0002191125 [3,] 0.0089571537 [4,] 0.0107217854 [5,] 0.0051726378 [6,] 0.0052852879 [7,] 0.0094467655 [8,] 0.0096975228 [9,] 0.0025718747 [10,] -0.0085116601 [11,] -0.0110509262 [12,] -0.0100339486 [13,] -0.0098858897 [14,] 0.0003151423 [15,] -0.0020785161 [16,] -0.0106611992 [17,] -0.0162008597 [18,] -0.0088909179 [19,] 0.0005357309 [20,] -0.0046350475 [21,] 0.0011515890 [22,] -0.0073978743 [23,] -0.0050104532 [24,] -0.0019024918 [25,] 0.0092411938 [26,] 0.0176216905 [27,] 0.0015580417 [28,] 0.0039472107 [29,] 0.0035616629 [30,] 0.0017974441 [31,] -0.0023577962 [32,] 0.0028661682 $W5 [,1] [1,] 0.0033346595 [2,] 0.0130666753 [3,] 0.0063044840 [4,] 0.0069905797 [5,] 0.0067237763 [6,] 0.0007411509 [7,] -0.0001396400 [8,] 0.0067636837 [9,] 0.0102943255 [10,] 0.0043359027 [11,] 0.0042107114 [12,] 0.0076784165 [13,] 0.0020105244 [14,] 0.0063415811 [15,] 0.0072286093 [16,] 0.0058000073 [17,] -0.0033346595 [18,] -0.0130666753 [19,] -0.0063044840 [20,] -0.0069905797 [21,] -0.0067237763 [22,] -0.0007411509 [23,] 0.0001396400 [24,] -0.0067636837 [25,] -0.0102943255 [26,] -0.0043359027 [27,] -0.0042107114 [28,] -0.0076784165 [29,] -0.0020105244 [30,] -0.0063415811 [31,] -0.0072286093 [32,] -0.0058000073 Slot "V": $V1 [,1] [1,] 0.036140777 [2,] 0.077528910 [3,] 0.047238451 [4,] -0.011485907 [5,] -0.008444786 [6,] -0.014949577 [7,] -0.017407609 [8,] 0.035049731 [9,] 0.050377076 [10,] -0.005018946 [11,] -0.031587209 [12,] -0.004205005 [13,] -0.012585388 [14,] 0.021099445 [15,] 0.034694225 [16,] -0.011021930 [17,] -0.048365374 [18,] -0.073404551 [19,] 0.023479855 [20,] 0.037122859 [21,] -0.011799125 [22,] 0.035045853 [23,] 0.037499722 [24,] -0.013130531 [25,] -0.033094648 [26,] 0.014403302 [27,] 0.017081703 [28,] -0.030945116 [29,] 0.005016108 [30,] 0.031794128 [31,] -0.007050454 [32,] -0.006689340 $V2 [,1] [1,] 0.0145451616 [2,] 0.0354197850 [3,] 0.0416896140 [4,] 0.0330215014 [5,] 0.0193968322 [6,] -0.0132177419 [7,] -0.0129261976 [8,] 0.0100500770 [9,] 0.0164847336 [10,] 0.0150153925 [11,] 0.0093949334 [12,] -0.0046119757 [13,] -0.0220862987 [14,] 0.0084472198 [15,] 0.0110544185 [16,] 0.0050387574 [17,] -0.0068355743 [18,] -0.0422132408 [19,] -0.0124427591 [20,] -0.0181408464 [21,] 0.0058403652 [22,] 0.0360843559 [23,] 0.0128502984 [24,] 0.0109576610 [25,] 0.0022025369 [26,] 0.0006363853 [27,] -0.0080064723 [28,] -0.0082709070 [29,] 0.0110489055 [30,] 0.0004245062 [31,] -0.0010171732 [32,] 0.0125523944 $V3 [,1] [1,] 0.0127970336 [2,] 0.0179221456 [3,] 0.0203362204 [4,] 0.0227869479 [5,] 0.0169709969 [6,] 0.0111010216 [7,] 0.0143817082 [8,] 0.0215357892 [9,] 0.0179407829 [10,] 0.0008988253 [11,] -0.0017656321 [12,] 0.0027190506 [13,] -0.0028007825 [14,] 0.0117313062 [15,] 0.0102246760 [16,] 0.0002133909 [17,] -0.0144609365 [18,] -0.0168830105 [19,] -0.0006941703 [20,] -0.0065510445 [21,] -0.0004976046 [22,] -0.0030644424 [23,] 0.0002037696 [24,] -0.0035915927 [25,] 0.0040214511 [26,] 0.0183603706 [27,] 0.0024219130 [28,] 0.0013433770 [29,] 0.0066257212 [30,] 0.0005304457 [31,] -0.0045118227 [32,] 0.0021407437 $V4 [,1] [1,] 0.0084092423 [2,] 0.0181412581 [3,] 0.0113790667 [4,] 0.0120651624 [5,] 0.0117983591 [6,] 0.0058157337 [7,] 0.0049349427 [8,] 0.0118382664 [9,] 0.0153689082 [10,] 0.0094104855 [11,] 0.0092852942 [12,] 0.0127529993 [13,] 0.0070851072 [14,] 0.0114161639 [15,] 0.0123031921 [16,] 0.0108745900 [17,] 0.0017399232 [18,] -0.0079920926 [19,] -0.0012299012 [20,] -0.0019159969 [21,] -0.0016491935 [22,] 0.0043334319 [23,] 0.0052142228 [24,] -0.0016891009 [25,] -0.0052197427 [26,] 0.0007386801 [27,] 0.0008638714 [28,] -0.0026038337 [29,] 0.0030640583 [30,] -0.0012669983 [31,] -0.0021540265 [32,] -0.0007254245 $V5 [,1] [1,] 0.005074583 [2,] 0.005074583 [3,] 0.005074583 [4,] 0.005074583 [5,] 0.005074583 [6,] 0.005074583 [7,] 0.005074583 [8,] 0.005074583 [9,] 0.005074583 [10,] 0.005074583 [11,] 0.005074583 [12,] 0.005074583 [13,] 0.005074583 [14,] 0.005074583 [15,] 0.005074583 [16,] 0.005074583 [17,] 0.005074583 [18,] 0.005074583 [19,] 0.005074583 [20,] 0.005074583 [21,] 0.005074583 [22,] 0.005074583 [23,] 0.005074583 [24,] 0.005074583 [25,] 0.005074583 [26,] 0.005074583 [27,] 0.005074583 [28,] 0.005074583 [29,] 0.005074583 [30,] 0.005074583 [31,] 0.005074583 [32,] 0.005074583 Slot "filter": Filter Class: Daubechies Name: HAAR Length: 2 Level: 1 Wavelet Coefficients: 5.0000e-01 -5.0000e-01 Scaling Coefficients: 5.0000e-01 5.0000e-01 Slot "level": [1] 5 Slot "n.boundary": [1] 2 4 8 16 32 Slot "boundary": [1] "periodic" Slot "series": [,1] [1,] 0.077528910 [2,] 0.077528910 [3,] 0.016947992 [4,] -0.039919805 [5,] 0.023030233 [6,] -0.052929387 [7,] 0.018114169 [8,] 0.051985293 [9,] 0.048768859 [10,] -0.058806751 [11,] -0.004367668 [12,] -0.004042343 [13,] -0.021128432 [14,] 0.063327323 [15,] 0.006061127 [16,] -0.028104988 [17,] -0.068625759 [18,] -0.078183343 [19,] 0.125143054 [20,] -0.050897337 [21,] 0.027299087 [22,] 0.042792620 [23,] 0.032206824 [24,] -0.058467886 [25,] -0.007721410 [26,] 0.036528013 [27,] -0.002364607 [28,] -0.059525625 [29,] 0.069557841 [30,] -0.005969584 [31,] -0.008131324 [32,] -0.005247355 Slot "class.X": [1] "numeric" Slot "attr.X": list() Slot "aligned": [1] FALSE Slot "coe": [1] FALSE
Quand dans le papier il est écrit ceci :
tdc = D1 + D2 + D3 + D4 + S4
Pourquoi 4 fois D et 1 fois S. Pourquoi S commence à 4 ? Si je ne me trompe pas, D correspondrait au W (dans le logiciel R) et S au V.
Il me faudrait donc considérer tdc = W1 + W2 + W3 + W4 + V4. Est-ce bien cela ?
Quand je regarde les fluctuations sur un graphique, V4 correspond à W5, sauf que c'est un peu plus décalé vers le haut. Ce qui ne m'étonne pas. Mais je ne comprends pas ce que concrètement ça induit.
Pourquoi ne pas considérer tdc = W1 + W2 + W3 + W4 + W5, par exemple ?
Je vous remercie pour vos éclaircissements.
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Re : Ondelettes (MODWT)