Descomposición temporal con RFrancisco Parra

5 de julio de 2017

Componentes de una serie temporal

Tradicionalmente, en los métodos de descomposición de series temporales, se parte de la idea de que la serietemporal se puede descomponer en todos o algunos de los siguientes componentes:

• Tendencia (T), que representa la evolución de la serie en el largo plazo

• Fluctuación cíclica (C), que refleja las fluctuaciones de carácter periódico, pero no necesariamenteregular, a medio plazo en torno a la tendencia. Este componente es frecuente hallarlo en las serieseconómicas, y se debe a los cambios en la actividad económica.

Para la obtención de la tendencia es necesario disponer de una serie larga y de un número de ciclos completo,para que ésta no se vea influida por la fase del ciclo en que finaliza la serie, por lo que, a veces, resultadifícil separar ambos componentes. En estos casos resulta útil englobar ambos componentes en uno solo,denominado ciclo-tendencia o tendencia generalizada.

• Variación Estacional (S): recoge aquellos comportamientos de tipo regular y repetitivo que se dan alo largo de un período de tiempo, generalmente igual o inferior a un año, y que son producidos porfactores tales como las variaciones climatológicas, las vacaciones, las fiestas, etc.

• Movimientos Irregulares (I), que pueden ser aleatorios, la cual recoge los pequeños efectos accidentales,o erráticos, como resultado de hechos no previsibles, pero identificables a posteriori (huelgas, catástrofes,etc.)

En este punto, cabe señalar que en una serie concreta no tienen por qué darse los cuatro componentes. Así,por ejemplo, una serie con periodicidad anual carece de estacionalidad.

La asociación de estos cuatro componentes en una serie temporal, Y , puede responder a distintos esquemas;así, puede ser de tipo aditivo:

Yt = Tt + Ct + St + etTambién puede tener una forma multiplicativa:

Yt = TtCtStetO bien ser una combinación de ambos, por ejemplo:

Yt = TtCtSt + etEn R “ts” es la función genérica de R para que los datos tengan forma de serie temporal. Su sintasis es lasiguiente:

ts(data = NA, start = 1, end = numeric(), frequency = 1, deltat = 1, ts.eps = getOption(“ts.eps”), class = ,names = )

De esta sintasis hay que tener presentes los siguiente argumentos:

• data: Vector, “data frame” o matriz de datos

• start: Referencia de la primera observacion, es un vector con dos valores numericos, el primero relativoal año y el segundo relativo al trimestre y mes de inicio (1 para el primer trimestre y 1 para enero enseries de datos mensuales)

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• end: Referencia de la ultima observación frequency Número de observaciones por año (4 en seriestrimestrales, 12 en series anuales)

Un ejemplo de elaboración de un objeto “ts” es el siguiente:ts(1:10, frequency = 4, start = c(1959, 2)) # 2nd Quarter of 1959

## Qtr1 Qtr2 Qtr3 Qtr4## 1959 1 2 3## 1960 4 5 6 7## 1961 8 9 10

Herramientas R

En este post se introducen algunas de las herramientas de R para la implementación de la descomposiciónYt = Tt + St + et, y para la estimación de la tendencia Tt. Hay que hacer énfasis en que algunos de estosmétodos no asumen un modelo global para la tendencia, sino local, es decir, no son modelos con parámetrosfijos, de la forma, por ejemplo, Tt = β0 + β1t, sino que β0 y β1 cambian en el tiempo para permitir mayorflexibilidad.

1. Las funciones decompose() y stl() realizan una descomposición de la serie Yt en las tres componentes.

2. La librería “timsac” incluye la función decomp() que realiza una descomposición temporal incluyendouna componente autoregresiva TAt y otra para fechas de intervenciones,Rt, Yt = Tt +St +Rt +TAt +et.

3. La librería “descomponer” incluye la función descomponer() que realiza una descomposición temporalde Yt transformado las series de tiempo en series de frecuencia.

4. Hay librerías con varios tipos de filtros para suavizamiento y extracción de componentes de tendencia yestacionales como “mFilter”, que tiene implementado el filtro Hodrick-Prescott, o la libreria “stats”,que incluye la función filter(), con la que se pueden implementar varios tipos de medias móviles y filtroslineales,por ejemplo,fitros tipo Henderson y filtros recursivos.

La función R “decompose”, obtiene las series de tendencia, estacionalidad e irregular de una serie temporala través de medias móviles, además permite obtener los componentes en base a un esquema aditivo ómultiplicativo.

Es una función generica de R, lo que significa que no requiere de la instalación de ninguna librería, su uso esel siguiente:

decompose(x, type = c(“additive”, “multiplicative”), filter = NULL)

El modelo aditivo que usa la función es:

Yt = Tt + St + etY el multiplicativo:

Yt = TtStetLa función calcula el componente de tendencia utilizando medias móviles, (si filter = NULL, se utilizanmedias móviles simétricas), los índices de estacionalidad son promedios de los indices de estacionalidad que seobtienen al desestacionalizar la serie por el modelo elegido, por último, el componente irregular se obtieneeliminando la tendencia y estacionalidad de la serie temporal.

La función requiere que los datos tengan forma de serie temporal.

La base de datos “co2” de R, contiene 468 observaciones mensuales desde 1959 a 1979 de las concentracionesatmosféricas de CO2 en partes por millón (ppm) en Mauna Loa (Hawai).

A continuación se realiza la descomposición temporal mediante la función “descomponse”:

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plot(decompose(co2))

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Decomposition of additive time series

STL es un método para estimar las componentes Tt y St con base en Regresión Loess, desarrollado porCleveland et al. [1990]. STL consiste de una secuencia de dos aplicaciones iteradas de Regresión Loess.Para aplicar este método se debe especificar una frecuencia de muestreo relacionada con el periódo dela componente estacional. La forma de especificar esta frecuencia es declarando la variable en donde seencuentran los datos como un objeto “ts”con frecuencia (52, 12, 4, 1), es decir, semanal,mensual, trimestral oanual, respectivamente.

El ejercicio anterior realizado con la función “stl”.plot(stl(co2,"per"))

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La función decomp() de la libreria timsac, al utilizar tecnica ARIMA (ver apartado siguiente) precisa qeu sele indique al estructura del modelo, su síntaxis es:

decomp(y, trend.order=2, ar.order=2, frequency=12,seasonal.order=1, log=FALSE, trade=FALSE,diff=1,year=1980, month=1, miss=0, omax=99999.9, plot=TRUE)

La opción trade, realiza el ajuste de los efectos calendario.

Esta función utiliza el modelo aditivo: yt = Tt +ARt + St + TDt +Wtdonde Tt es el componente de tendencia, ARt es un proceso Autoregresivo, St es la estacionalidad, TDt elefecto calendario y W − t el componente irregular.library(timsac)decomp(co2, trade=TRUE,plot=TRUE)

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Trading Day Effect

## $trend## [1] 315.5460 315.6103 315.6746 315.7389 315.8033 315.8677 315.9322## [8] 315.9966 316.0610 316.1251 316.1891 316.2529 316.3166 316.3801## [15] 316.4435 316.5068 316.5700 316.6330 316.6957 316.7582 316.8204## [22] 316.8824 316.9442 317.0058 317.0673 317.1286 317.1897 317.2507## [29] 317.3116 317.3722 317.4327 317.4929 317.5528 317.6125 317.6718## [36] 317.7309 317.7897 317.8483 317.9068 317.9650 318.0231 318.0811## [43] 318.1388 318.1964 318.2538 318.3109 318.3679 318.4248 318.4816## [50] 318.5384 318.5952 318.6521 318.7090 318.7659 318.8229 318.8800## [57] 318.9372 318.9947 319.0522 319.1101 319.1682 319.2265 319.2851## [64] 319.3440 319.4034 319.4632 319.5235 319.5843 319.6454 319.7071## [71] 319.7694 319.8321 319.8956 319.9597 320.0245 320.0901 320.1564## [78] 320.2236 320.2916 320.3603 320.4297 320.4997 320.5704 320.6417## [85] 320.7137 320.7864 320.8597 320.9337 321.0082 321.0833 321.1590## [92] 321.2353 321.3120 321.3893 321.4672 321.5457 321.6246 321.7040## [99] 321.7839 321.8645 321.9456 322.0273 322.1096 322.1926 322.2762## [106] 322.3602 322.4448 322.5299 322.6155 322.7016 322.7882 322.8754## [113] 322.9632 323.0515 323.1402 323.2294 323.3190 323.4089 323.4993## [120] 323.5900 323.6809 323.7722 323.8637 323.9554 324.0473 324.1395## [127] 324.2317 324.3241 324.4165 324.5089 324.6013 324.6939 324.7865## [134] 324.8793 324.9721 325.0650 325.1580 325.2511 325.3444 325.4378## [141] 325.5314 325.6250 325.7187 325.8126 325.9067 326.0010 326.0956## [148] 326.1905 326.2859 326.3816 326.4777 326.5741 326.6708 326.7678## [155] 326.8651 326.9628 327.0608 327.1591 327.2578 327.3568 327.4562## [162] 327.5559 327.6559 327.7562 327.8566 327.9572 328.0578 328.1584## [169] 328.2589 328.3594 328.4597 328.5599 328.6600 328.7598 328.8593

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## [176] 328.9585 329.0573 329.1557 329.2538 329.3516 329.4493 329.5470## [183] 329.6445 329.7419 329.8393 329.9368 330.0343 330.1319 330.2296## [190] 330.3274 330.4254 330.5237 330.6222 330.7212 330.8205 330.9202## [197] 331.0204 331.1211 331.2222 331.3239 331.4261 331.5289 331.6323## [204] 331.7362 331.8409 331.9462 332.0522 332.1589 332.2663 332.3745## [211] 332.4834 332.5932 332.7038 332.8152 332.9274 333.0404 333.1542## [218] 333.2688 333.3841 333.5000 333.6166 333.7338 333.8515 333.9697## [225] 334.0885 334.2076 334.3272 334.4472 334.5676 334.6883 334.8094## [232] 334.9307 335.0523 335.1743 335.2964 335.4188 335.5413 335.6640## [239] 335.7869 335.9100 336.0333 336.1567 336.2803 336.4040 336.5278## [246] 336.6517 336.7757 336.8997 337.0237 337.1477 337.2718 337.3958## [253] 337.5198 337.6437 337.7675 337.8911 338.0146 338.1379 338.2610## [260] 338.3840 338.5068 338.6294 338.7518 338.8741 338.9963 339.1184## [267] 339.2402 339.3620 339.4836 339.6051 339.7266 339.8481 339.9697## [274] 340.0914 340.2132 340.3352 340.4572 340.5793 340.7015 340.8238## [281] 340.9462 341.0689 341.1918 341.3150 341.4386 341.5625 341.6868## [288] 341.8115 341.9366 342.0622 342.1881 342.3144 342.4410 342.5679## [295] 342.6951 342.8225 342.9501 343.0779 343.2061 343.3346 343.4632## [302] 343.5921 343.7213 343.8508 343.9804 344.1103 344.2404 344.3708## [309] 344.5016 344.6327 344.7641 344.8959 345.0280 345.1605 345.2933## [316] 345.4264 345.5597 345.6935 345.8276 345.9621 346.0970 346.2324## [323] 346.3683 346.5046 346.6413 346.7786 346.9164 347.0547 347.1934## [330] 347.3326 347.4722 347.6123 347.7528 347.8936 348.0349 348.1765## [337] 348.3185 348.4607 348.6032 348.7460 348.8889 349.0319 349.1750## [344] 349.3180 349.4610 349.6038 349.7464 349.8886 350.0305 350.1720## [351] 350.3128 350.4530 350.5925 350.7313 350.8692 351.0064 351.1426## [358] 351.2780 351.4124 351.5458 351.6782 351.8095 351.9397 352.0690## [365] 352.1972 352.3244 352.4505 352.5756 352.6997 352.8228 352.9449## [372] 353.0661 353.1862 353.3054 353.4235 353.5407 353.6571 353.7725## [379] 353.8871 354.0009 354.1139 354.2262 354.3378 354.4486 354.5586## [386] 354.6680 354.7767 354.8847 354.9920 355.0987 355.2047 355.3104## [393] 355.4158 355.5210 355.6261 355.7310 355.8357 355.9405 356.0453## [400] 356.1501 356.2551 356.3602 356.4655 356.5713 356.6775 356.7844## [407] 356.8918 357.0000 357.1089 357.2187 357.3293 357.4408 357.5532## [414] 357.6665 357.7808 357.8961 358.0126 358.1303 358.2491 358.3689## [421] 358.4899 358.6117 358.7346 358.8583 358.9830 359.1086 359.2351## [428] 359.3624 359.4907 359.6198 359.7497 359.8803 360.0114 360.1432## [435] 360.2755 360.4083 360.5415 360.6751 360.8091 360.9434 361.0780## [442] 361.2130 361.3483 361.4838 361.6196 361.7554 361.8913 362.0273## [449] 362.1634 362.2996 362.4359 362.5722 362.7087 362.8453 362.9821## [456] 363.1190 363.2560 363.3932 363.5305 363.6680 363.8056 363.9434## [463] 364.0813 364.2195 364.3579 364.4966 364.6354 364.7744#### $seasonal## [1] -0.04509067 0.62853011 1.37016228 2.50599037 2.98546092## [6] 2.33026948 0.81189092 -1.25254315 -3.06943987 -3.25147520## [11] -2.06243298 -0.95131404 -0.04507447 0.62847432 1.37019365## [16] 2.50598812 2.98549318 2.33020883 0.81192146 -1.25249530## [21] -3.06952468 -3.25141656 -2.06248598 -0.95127370 -0.04505984## [26] 0.62840386 1.37025279 2.50598586 2.98551378 2.33014922## [31] 0.81193562 -1.25242316 -3.06962730 -3.25134602 -2.06253619## [36] -0.95125622 -0.04501747 0.62833025 1.37027131 2.50605960## [41] 2.98546047 2.33013072 0.81194339 -1.25236613 -3.06968886## [46] -3.25133162 -2.06256591 -0.95122610 -0.04499214 0.62829263

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## [51] 1.37024431 2.50614196 2.98542701 2.33012019 0.81191057## [56] -1.25224270 -3.06983875 -3.25125760 -2.06261571 -0.95117470## [61] -0.04500051 0.62826612 1.37024204 2.50621976 2.98537047## [66] 2.33010221 0.81190390 -1.25211787 -3.06999903 -3.25120173## [71] -2.06263665 -0.95112294 -0.04502108 0.62824188 1.37021211## [76] 2.50633776 2.98530978 2.33007067 0.81191263 -1.25202319## [81] -3.07011379 -3.25119333 -2.06263629 -0.95107989 -0.04500672## [86] 0.62817192 1.37021662 2.50640275 2.98530167 2.33006419## [91] 0.81183170 -1.25182873 -3.07031594 -3.25111838 -2.06270272## [96] -0.95098463 -0.04499414 0.62807936 1.37024177 2.50643429## [101] 2.98533464 2.33004123 0.81174517 -1.25162362 -3.07055895## [106] -3.25097723 -2.06281139 -0.95085341 -0.04503472 0.62800015## [111] 1.37030242 2.50644854 2.98534903 2.33001953 0.81169877## [116] -1.25146040 -3.07077472 -3.25085214 -2.06290779 -0.95074976## [121] -0.04505330 0.62792177 1.37035119 2.50646930 2.98537492## [126] 2.32999759 0.81163052 -1.25128399 -3.07099121 -3.25074246## [131] -2.06297030 -0.95065521 -0.04511448 0.62790542 1.37035445## [136] 2.50651217 2.98537782 2.33002690 0.81149974 -1.25104597## [141] -3.07126617 -3.25061762 -2.06300783 -0.95058611 -0.04515498## [146] 0.62788303 1.37036382 2.50650739 2.98543137 2.33004782## [151] 0.81136999 -1.25081052 -3.07154160 -3.25051126 -2.06303076## [156] -0.95049800 -0.04521867 0.62786016 1.37034535 2.50657123## [161] 2.98545028 2.33007407 0.81123633 -1.25060461 -3.07176683## [166] -3.25043013 -2.06304797 -0.95042604 -0.04524601 0.62776388## [171] 1.37043276 2.50656644 2.98544977 2.33015546 0.81107903## [176] -1.25039692 -3.07198026 -3.25036544 -2.06307347 -0.95034409## [181] -0.04525479 0.62763841 1.37052441 2.50658547 2.98543068## [186] 2.33023304 0.81095481 -1.25022851 -3.07217908 -3.25029879## [191] -2.06313022 -0.95024659 -0.04520531 0.62745607 1.37062097## [196] 2.50661487 2.98538076 2.33036485 0.81080814 -1.25008044## [201] -3.07235616 -3.25024229 -2.06318657 -0.95014991 -0.04513526## [206] 0.62723928 1.37074059 2.50664164 2.98533706 2.33047087## [211] 0.81068080 -1.24991679 -3.07256479 -3.25017640 -2.06323522## [216] -0.95005591 -0.04506729 0.62703494 1.37084230 2.50666088## [221] 2.98531586 2.33057775 0.81053171 -1.24972519 -3.07280728## [226] -3.25009122 -2.06327259 -0.94996756 -0.04503440 0.62688210## [231] 1.37092230 2.50667211 2.98530501 2.33068480 0.81035668## [236] -1.24947152 -3.07312103 -3.24997331 -2.06330027 -0.94988391## [241] -0.04503648 0.62679393 1.37095166 2.50667596 2.98534309## [246] 2.33076002 0.81019196 -1.24922929 -3.07343870 -3.24983154## [251] -2.06335677 -0.94976343 -0.04508827 0.62676793 1.37091652## [256] 2.50671437 2.98539310 2.33080444 0.81006531 -1.24903939## [261] -3.07372547 -3.24968485 -2.06342017 -0.94962299 -0.04520940## [266] 0.62685524 1.37077133 2.50680421 2.98545840 2.33080912## [271] 0.80996236 -1.24886002 -3.07398490 -3.24957902 -2.06346494## [276] -0.94949161 -0.04529380 0.62690972 1.37062865 2.50689478## [281] 2.98552850 2.33080535 0.80986136 -1.24867579 -3.07423898## [286] -3.24947100 -2.06352835 -0.94934497 -0.04538592 0.62698792## [291] 1.37043898 2.50702092 2.98558145 2.33082463 0.80973101## [296] -1.24847571 -3.07449540 -3.24935856 -2.06358932 -0.94922399## [301] -0.04543524 0.62701439 1.37030122 2.50712575 2.98562530## [306] 2.33084796 0.80963497 -1.24835749 -3.07468115 -3.24928188## [311] -2.06359295 -0.94917407 -0.04545324 0.62702474 1.37021450## [316] 2.50716997 2.98569492 2.33087313 0.80954167 -1.24828473

7

## [321] -3.07480061 -3.24922052 -2.06361324 -0.94914540 -0.04543581## [326] 0.62706058 1.37005868 2.50726112 2.98574550 2.33089503## [331] 0.80946315 -1.24820433 -3.07495530 -3.24912169 -2.06364440## [336] -0.94913080 -0.04541733 0.62710080 1.36993270 2.50732134## [341] 2.98579757 2.33090723 0.80939141 -1.24807130 -3.07520095## [346] -3.24896523 -2.06368403 -0.94911302 -0.04542917 0.62716874## [351] 1.36981208 2.50738037 2.98583763 2.33088993 0.80938603## [356] -1.24797542 -3.07545155 -3.24879715 -2.06374040 -0.94907057## [361] -0.04542544 0.62717349 1.36974063 2.50742076 2.98589166## [366] 2.33086211 0.80938132 -1.24788383 -3.07568286 -3.24865276## [371] -2.06379150 -0.94899194 -0.04548228 0.62719740 1.36971302## [376] 2.50740543 2.98596787 2.33085219 0.80934089 -1.24777959## [381] -3.07589588 -3.24852320 -2.06384621 -0.94890601 -0.04552483## [386] 0.62719586 1.36966996 2.50744961 2.98598718 2.33086631## [391] 0.80931156 -1.24771559 -3.07606553 -3.24839781 -2.06390547## [396] -0.94886933 -0.04551013 0.62719525 1.36961216 2.50750511## [401] 2.98597649 2.33087554 0.80931649 -1.24767262 -3.07621427## [406] -3.24828080 -2.06396763 -0.94884111 -0.04549277 0.62720820## [411] 1.36955022 2.50755042 2.98599300 2.33082597 0.80936934## [416] -1.24765000 -3.07632226 -3.24822352 -2.06400082 -0.94878824## [421] -0.04553240 0.62728168 1.36943948 2.50763518 2.98599539## [426] 2.33074034 0.80946042 -1.24763892 -3.07640739 -3.24820105## [431] -2.06401109 -0.94872040 -0.04562131 0.62739336 1.36932638## [436] 2.50769991 2.98600587 2.33066638 0.80953599 -1.24762211## [441] -3.07648796 -3.24817201 -2.06402790 -0.94867245 -0.04567043## [446] 0.62744859 1.36928889 2.50769536 2.98605292 2.33059012## [451] 0.80957838 -1.24755528 -3.07661554 -3.24809023 -2.06409892## [456] -0.94859114 -0.04570630 0.62747915 1.36921701 2.50774907## [461] 2.98607848 2.33052816 0.80959452 -1.24749003 -3.07671249## [466] -3.24803777 -2.06413818 -0.94853776#### $ar## [1] -0.0744194632 -0.1400708483 -0.4113924400 -0.5763758789 -0.5456206909## [6] -0.3488335108 -0.2388912437 0.0192639114 0.3772047353 0.4079522581## [11] 0.3795899569 0.1947108409 -0.0112415121 -0.1832020023 -0.2508505737## [16] -0.0711953124 0.2093096732 0.3956443913 0.4259718143 0.3124550556## [21] 0.2064402853 0.0733524399 -0.0200097645 -0.1045980380 -0.2174650444## [26] -0.2234130481 -0.2149361623 -0.2344739211 -0.0488036412 0.0142063080## [31] 0.1587626518 0.3183396837 0.4345457981 0.5292588906 0.3593964112## [36] 0.1634611266 0.0624619581 0.0227249233 0.0688569836 -0.0160790986## [41] -0.0492603753 0.1076268815 0.3422163688 0.4676035931 0.5802889994## [46] 0.3697494471 0.2158711680 0.0901788275 0.0114956820 -0.1236820472## [51] -0.1315802016 0.0532110111 0.2089361438 0.1640411704 0.0379192992## [56] 0.0355454964 0.0896521266 0.0772047133 0.0184452777 0.0868260926## [61] 0.1787330002 0.1437787495 -0.0256039808 -0.2517263117 -0.2523749063## [66] -0.1243051921 -0.0024617493 0.0920666105 0.0735038074 0.0612833744## [71] -0.1324587184 -0.3212906549 -0.4389277887 -0.4495350849 -0.5878955850## [76] -0.7270949734 -0.8800446557 -0.6961304394 -0.3417991033 -0.1978201693## [81] 0.0415982885 0.0183592280 -0.0174152105 -0.1931646253 -0.1749892209## [86] -0.0614207299 -0.0055628478 0.0380846055 0.0271223097 0.1431094249## [91] 0.2273170133 0.2184007580 0.1446235881 0.0309969847 0.1647359947## [96] 0.2759014789 0.3208139207 0.0940650615 -0.1241715453 -0.1617784510## [101] -0.2183058522 -0.3517601478 -0.3910856962 -0.2186402075 -0.0741128064## [106] 0.0877983862 0.1444915198 0.0838930217 -0.1307142770 -0.3036981662

8

## [111] -0.4138092242 -0.4745386999 -0.4078397304 -0.2200272882 -0.0583223455## [116] -0.0418002116 -0.0683633120 -0.1059103482 -0.1142126622 0.0183313613## [121] 0.0966485847 0.0369392704 0.1033108621 0.1004837625 0.1423092173## [126] 0.2322210487 0.4638163100 0.5446909264 0.6131753919 0.4321293907## [131] 0.2442513373 0.2008295265 0.2006210768 0.2843140663 0.3261049949## [136] 0.2340375764 -0.0133023321 -0.0398964127 0.0962251001 0.2871226969## [141] 0.4134343034 0.4022786861 0.2530802534 0.1343826715 0.0517918111## [146] -0.1734063024 -0.4902337009 -0.7248614980 -0.5645017372 -0.3295802332## [151] -0.1540152673 -0.1300797792 -0.2250316614 -0.1843368808 -0.1788404791## [156] -0.2372753720 -0.3693691114 -0.4717802649 -0.6487468612 -0.5389980050## [161] -0.5970660171 -0.7238121599 -0.5889843734 -0.3503316225 -0.0678733859## [166] 0.1883605776 0.2771129912 0.2447019081 0.2459653267 0.3116889674## [171] 0.3139812909 0.3779754351 0.5926326269 0.8210526146 1.0558780203## [176] 1.2912315207 1.2680991558 1.0549053368 0.6799544229 0.2220476340## [181] 0.0310458645 0.1988718522 0.2565940713 0.1970834893 0.0969478309## [186] 0.0015586513 0.1267685568 0.2102777803 0.1508497912 0.0713858681## [191] -0.0484059034 -0.1638167945 -0.2248554401 -0.2171227689 -0.2723939397## [196] -0.2516786375 -0.1868380492 -0.1377493053 -0.1709348269 -0.1353744113## [201] -0.0626272975 -0.1204039317 -0.1934736923 -0.2088431846 -0.1906594148## [206] -0.1654589944 -0.1620237590 -0.2839970975 -0.4445130654 -0.4878716266## [211] -0.4868234788 -0.5400637225 -0.5719254941 -0.6701024532 -0.6634530705## [216] -0.5508471730 -0.4731170457 -0.4669559566 -0.2926368979 -0.1349741364## [221] -0.0464349606 0.0256738045 0.0618539922 0.0630138987 0.1732832223## [226] 0.0897746057 0.0637712172 0.1228127782 0.1450235864 0.0909266080## [231] 0.1688972932 0.1009070146 0.0225526935 0.1409313559 0.2228767705## [236] 0.2357302612 0.1391396841 0.0189717660 -0.0191899109 -0.0650637212## [241] -0.0349650670 -0.0816819074 -0.0288388476 -0.1123815334 -0.0887379587## [246] 0.0210884336 0.0537492844 0.0665000455 -0.0896069141 -0.1410983136## [251] -0.0511900362 0.0948626691 0.2289061760 0.2228264543 0.4173879624## [256] 0.3405680473 0.3354348441 0.3950382737 0.3396632421 0.3152855941## [261] 0.3190335040 0.3565381594 0.2530730904 0.1849804413 0.2290030611## [266] 0.4264755096 0.4979406127 0.4334357665 0.2803124236 0.0856756411## [271] -0.1322719528 -0.2805302554 -0.2890558422 -0.1717967701 -0.0240798723## [276] 0.0633801704 0.1672563128 0.2544950900 0.2974099081 0.1360442943## [281] -0.0039745457 -0.1400386392 -0.2319434406 -0.3949569285 -0.5101819296## [286] -0.5617105542 -0.5559502585 -0.5473701211 -0.5710330613 -0.4722126516## [291] -0.3893343201 -0.1256475918 0.1078563507 0.2385547965 0.3146127017## [296] 0.3191909676 0.0526422889 -0.0278143856 0.0127273867 0.2019783305## [301] 0.1680736228 0.1296581401 0.1557100707 0.3150433409 0.2798470220## [306] 0.2036523333 0.0995116694 -0.0669554648 -0.2727790408 -0.1745490774## [311] -0.0205925941 0.0006955346 -0.0402447542 0.1148491232 0.3350162473## [316] 0.2770612738 0.1838695762 0.0164205110 -0.1433083501 -0.1928141502## [321] -0.1973639964 -0.2709176353 -0.2561440191 -0.2876911023 -0.4197132346## [326] -0.5281324008 -0.4852117897 -0.2958132770 -0.2400558188 -0.3221432480## [331] -0.4497056732 -0.4498389343 -0.3205868023 -0.4487505074 -0.4907449780## [336] -0.4984653939 -0.5455868253 -0.6644465241 -0.6284399645 -0.4666368363## [341] -0.3345247624 -0.3497196306 -0.4244430908 -0.2689177105 -0.1624603675## [346] -0.1286344070 -0.0767266351 0.0193493727 0.2613370539 0.4846572094## [351] 0.4462249252 0.4506882488 0.4657075282 0.5194127533 0.5366761252## [356] 0.5299872253 0.5347247045 0.5930662915 0.6001328904 0.6347238599## [361] 0.6975604841 0.5396311889 0.4321614201 0.4818504776 0.4078922796## [366] 0.3598468518 0.3400506357 0.2089673011 0.1353643597 0.1892309858## [371] 0.2489113282 0.3066783329 0.3997187751 0.4693839579 0.3590291123## [376] 0.1863262732 0.1698728307 0.0505210819 -0.0177709158 -0.0320306383

9

## [381] -0.0639718357 0.1045679105 0.3113325916 0.3739726999 0.2861900389## [386] 0.4608715063 0.7842991281 1.0043381522 0.9900273627 0.6342288763## [391] 0.1756963168 -0.0901389944 -0.1970245687 -0.1130118316 0.0105018422## [396] 0.0814423044 0.0907384739 0.1324858562 0.2559996420 0.3565608750## [401] 0.3493288431 0.2267567437 -0.1356754458 -0.3810698664 -0.5097989325## [406] -0.5041936082 -0.6240826831 -0.6438317278 -0.5597082334 -0.5716299912## [411] -0.4647807149 -0.4479817974 -0.4272235585 -0.5833749489 -0.9248644442## [416] -1.1161787766 -1.1319215262 -0.9731861884 -0.8104990317 -0.5532156549## [421] -0.3034731536 -0.2559704810 -0.1950635107 -0.1886491801 -0.3087667798## [426] -0.4327123143 -0.4971985384 -0.5456432853 -0.5031253917 -0.3443827014## [431] -0.1287856161 0.0191473827 0.0756576426 0.1779992476 0.1969780334## [436] 0.3532482116 0.3233684536 0.2845632429 0.1884919069 0.0185591239## [441] 0.0078625084 0.0186933985 0.1891836592 0.3000551687 0.5141638934## [446] 0.7148658327 0.6545995289 0.4064525741 0.3514770285 0.3576015868## [451] 0.3450725416 0.1829656306 0.0115572246 -0.0124319923 -0.0052779142## [456] 0.0769406146 0.0596077668 -0.0047860080 -0.0761904024 0.0262566242## [461] -0.0953217865 -0.3634938302 -0.4446836902 -0.5390971567 -0.6820167564## [466] -0.4068361525 -0.0500716056 0.3012091373#### $trad## [1] 0.0213755473 0.0042542801 -0.0256298274 0.0002571471 0.0164366011## [6] -0.0166937482 0.0213755473 0.0314231224 -0.0082281677 -0.0189406746## [11] 0.0271688423 -0.0525415227 0.0164366011 0.0000000000 0.0278767539## [16] -0.0231949548 0.0314231224 -0.0082281677 -0.0189406746 -0.0256298274## [21] 0.0002571471 0.0164366011 -0.0166937482 0.0213755473 0.0314231224## [26] 0.0000000000 -0.0525415227 0.0253726804 -0.0256298274 0.0002571471## [31] 0.0164366011 0.0278767539 -0.0231949548 0.0314231224 -0.0082281677## [36] -0.0189406746 -0.0256298274 0.0000000000 0.0213755473 -0.0046817992## [41] 0.0278767539 -0.0231949548 0.0314231224 -0.0525415227 0.0253726804## [46] -0.0256298274 0.0002571471 0.0164366011 0.0278767539 -0.0443133550## [51] 0.0164366011 -0.0166937482 0.0213755473 -0.0046817992 0.0278767539## [56] -0.0189406746 0.0271688423 -0.0525415227 0.0253726804 -0.0256298274## [61] 0.0213755473 0.0000000000 0.0314231224 -0.0082281677 -0.0189406746## [66] 0.0271688423 -0.0525415227 0.0164366011 -0.0166937482 0.0213755473## [71] -0.0046817992 0.0278767539 -0.0189406746 0.0000000000 -0.0256298274## [76] 0.0002571471 0.0164366011 -0.0166937482 0.0213755473 0.0314231224## [81] -0.0082281677 -0.0189406746 0.0271688423 -0.0525415227 0.0164366011## [86] 0.0000000000 0.0278767539 -0.0231949548 0.0314231224 -0.0082281677## [91] -0.0189406746 -0.0256298274 0.0002571471 0.0164366011 -0.0166937482## [96] 0.0213755473 0.0314231224 -0.0527986698 0.0213755473 -0.0046817992## [101] 0.0278767539 -0.0231949548 0.0314231224 -0.0525415227 0.0253726804## [106] -0.0256298274 0.0002571471 0.0164366011 0.0278767539 0.0000000000## [111] -0.0189406746 0.0271688423 -0.0525415227 0.0253726804 -0.0256298274## [116] 0.0213755473 -0.0046817992 0.0278767539 -0.0231949548 0.0314231224## [121] -0.0525415227 0.0000000000 0.0164366011 -0.0166937482 0.0213755473## [126] -0.0046817992 0.0278767539 -0.0189406746 0.0271688423 -0.0525415227## [131] 0.0253726804 -0.0256298274 0.0213755473 0.0000000000 0.0314231224## [136] -0.0082281677 -0.0189406746 0.0271688423 -0.0525415227 0.0164366011## [141] -0.0166937482 0.0213755473 -0.0046817992 0.0278767539 -0.0189406746## [146] -0.0089360793 0.0278767539 -0.0231949548 0.0314231224 -0.0082281677## [151] -0.0189406746 -0.0256298274 0.0002571471 0.0164366011 -0.0166937482## [156] 0.0213755473 0.0314231224 0.0000000000 -0.0525415227 0.0253726804## [161] -0.0256298274 0.0002571471 0.0164366011 0.0278767539 -0.0231949548## [166] 0.0314231224 -0.0082281677 -0.0189406746 -0.0256298274 0.0000000000

10

## [171] 0.0213755473 -0.0046817992 0.0278767539 -0.0231949548 0.0314231224## [176] -0.0525415227 0.0253726804 -0.0256298274 0.0002571471 0.0164366011## [181] 0.0278767539 0.0000000000 -0.0189406746 0.0271688423 -0.0525415227## [186] 0.0253726804 -0.0256298274 0.0213755473 -0.0046817992 0.0278767539## [191] -0.0231949548 0.0314231224 -0.0525415227 0.0211184003 0.0314231224## [196] -0.0082281677 -0.0189406746 0.0271688423 -0.0525415227 0.0164366011## [201] -0.0166937482 0.0213755473 -0.0046817992 0.0278767539 -0.0189406746## [206] 0.0000000000 -0.0256298274 0.0002571471 0.0164366011 -0.0166937482## [211] 0.0213755473 0.0314231224 -0.0082281677 -0.0189406746 0.0271688423## [216] -0.0525415227 0.0164366011 0.0000000000 0.0278767539 -0.0231949548## [221] 0.0314231224 -0.0082281677 -0.0189406746 -0.0256298274 0.0002571471## [226] 0.0164366011 -0.0166937482 0.0213755473 0.0314231224 0.0000000000## [231] -0.0525415227 0.0253726804 -0.0256298274 0.0002571471 0.0164366011## [236] 0.0278767539 -0.0231949548 0.0314231224 -0.0082281677 -0.0189406746## [241] -0.0256298274 0.0445705021 -0.0189406746 0.0271688423 -0.0525415227## [246] 0.0253726804 -0.0256298274 0.0213755473 -0.0046817992 0.0278767539## [251] -0.0231949548 0.0314231224 -0.0525415227 0.0000000000 0.0164366011## [256] -0.0166937482 0.0213755473 -0.0046817992 0.0278767539 -0.0189406746## [261] 0.0271688423 -0.0525415227 0.0253726804 -0.0256298274 0.0213755473## [266] 0.0000000000 0.0314231224 -0.0082281677 -0.0189406746 0.0271688423## [271] -0.0525415227 0.0164366011 -0.0166937482 0.0213755473 -0.0046817992## [276] 0.0278767539 -0.0189406746 0.0000000000 -0.0256298274 0.0002571471## [281] 0.0164366011 -0.0166937482 0.0213755473 0.0314231224 -0.0082281677## [286] -0.0189406746 0.0271688423 -0.0525415227 0.0164366011 0.0361049216## [291] -0.0525415227 0.0253726804 -0.0256298274 0.0002571471 0.0164366011## [296] 0.0278767539 -0.0231949548 0.0314231224 -0.0082281677 -0.0189406746## [301] -0.0256298274 0.0000000000 0.0213755473 -0.0046817992 0.0278767539## [306] -0.0231949548 0.0314231224 -0.0525415227 0.0253726804 -0.0256298274## [311] 0.0002571471 0.0164366011 0.0278767539 0.0000000000 -0.0189406746## [316] 0.0271688423 -0.0525415227 0.0253726804 -0.0256298274 0.0213755473## [321] -0.0046817992 0.0278767539 -0.0231949548 0.0314231224 -0.0525415227## [326] 0.0000000000 0.0164366011 -0.0166937482 0.0213755473 -0.0046817992## [331] 0.0278767539 -0.0189406746 0.0271688423 -0.0525415227 0.0253726804## [336] -0.0256298274 0.0213755473 0.0042542801 -0.0256298274 0.0002571471## [341] 0.0164366011 -0.0166937482 0.0213755473 0.0314231224 -0.0082281677## [346] -0.0189406746 0.0271688423 -0.0525415227 0.0164366011 0.0000000000## [351] 0.0278767539 -0.0231949548 0.0314231224 -0.0082281677 -0.0189406746## [356] -0.0256298274 0.0002571471 0.0164366011 -0.0166937482 0.0213755473## [361] 0.0314231224 0.0000000000 -0.0525415227 0.0253726804 -0.0256298274## [366] 0.0002571471 0.0164366011 0.0278767539 -0.0231949548 0.0314231224## [371] -0.0082281677 -0.0189406746 -0.0256298274 0.0000000000 0.0213755473## [376] -0.0046817992 0.0278767539 -0.0231949548 0.0314231224 -0.0525415227## [381] 0.0253726804 -0.0256298274 0.0002571471 0.0164366011 0.0278767539## [386] -0.0443133550 0.0164366011 -0.0166937482 0.0213755473 -0.0046817992## [391] 0.0278767539 -0.0189406746 0.0271688423 -0.0525415227 0.0253726804## [396] -0.0256298274 0.0213755473 0.0000000000 0.0314231224 -0.0082281677## [401] -0.0189406746 0.0271688423 -0.0525415227 0.0164366011 -0.0166937482## [406] 0.0213755473 -0.0046817992 0.0278767539 -0.0189406746 0.0000000000## [411] -0.0256298274 0.0002571471 0.0164366011 -0.0166937482 0.0213755473## [416] 0.0314231224 -0.0082281677 -0.0189406746 0.0271688423 -0.0525415227## [421] 0.0164366011 0.0000000000 0.0278767539 -0.0231949548 0.0314231224## [426] -0.0082281677 -0.0189406746 -0.0256298274 0.0002571471 0.0164366011## [431] -0.0166937482 0.0213755473 0.0314231224 -0.0527986698 0.0213755473## [436] -0.0046817992 0.0278767539 -0.0231949548 0.0314231224 -0.0525415227

11

## [441] 0.0253726804 -0.0256298274 0.0002571471 0.0164366011 0.0278767539## [446] 0.0000000000 -0.0189406746 0.0271688423 -0.0525415227 0.0253726804## [451] -0.0256298274 0.0213755473 -0.0046817992 0.0278767539 -0.0231949548## [456] 0.0314231224 -0.0525415227 0.0000000000 0.0164366011 -0.0166937482## [461] 0.0213755473 -0.0046817992 0.0278767539 -0.0189406746 0.0271688423## [466] -0.0525415227 0.0253726804 -0.0256298274#### $noise## [1] -2.791231e-02 2.069586e-01 -1.077216e-01 -1.087578e-01 -1.295490e-01## [6] 1.675347e-01 -1.365404e-01 -1.447424e-01 3.195003e-01 -8.267179e-02## [11] 1.265418e-01 -1.378955e-02 -6.697445e-03 -1.536207e-02 -1.707197e-01## [16] -4.842503e-02 7.375692e-02 7.938114e-02 9.533367e-02 -5.249799e-02## [21] 4.243754e-02 -4.076106e-02 -5.003686e-03 5.868201e-02 -1.061583e-01## [26] 6.444272e-03 8.750840e-02 -2.375899e-01 1.973595e-01 -1.068186e-01## [31] 2.114219e-04 4.333866e-02 -6.453709e-02 2.381904e-01 -2.043571e-02## [36] -7.412732e-02 -1.517150e-03 -9.939516e-02 1.627031e-01 -3.033439e-02## [41] -1.371899e-01 -4.561959e-02 1.255778e-01 -1.091080e-01 3.202451e-01## [46] -1.336963e-01 8.547447e-03 -5.015759e-02 1.040326e-01 -7.866655e-02## [51] -1.502836e-01 2.527945e-02 1.552887e-01 5.462470e-02 -1.205841e-01## [56] -3.435075e-02 6.577056e-02 6.193816e-02 -1.234497e-01 -2.010131e-02## [61] 8.673403e-02 7.147705e-02 7.886462e-02 -1.902769e-01 -5.744489e-02## [66] 3.380815e-02 -1.041158e-02 9.936358e-02 -9.225902e-02 1.713960e-01## [71] -3.957511e-02 -3.760729e-02 -1.226952e-01 1.415730e-01 -5.122941e-02## [76] 1.003936e-01 -2.781451e-01 -1.308755e-01 2.668933e-01 -2.318856e-01## [81] 2.670365e-01 -1.079590e-01 1.824680e-01 -1.949453e-01 -5.018613e-02## [86] 7.681556e-02 -2.227947e-02 8.502716e-02 -1.420530e-01 4.171397e-02## [91] 6.075421e-02 2.378898e-02 9.340341e-02 -2.456503e-01 7.743241e-02## [96] -2.194486e-02 2.381731e-01 -3.333452e-02 -1.713828e-01 4.554133e-02## [101] 8.948228e-02 -5.240648e-02 -1.817316e-01 9.019040e-02 -5.685651e-02## [106] 6.856092e-02 3.321936e-02 1.205994e-01 -6.761504e-02 -3.588329e-02## [111] 4.221343e-03 -7.450440e-02 -8.815366e-02 1.315525e-02 1.120112e-01## [116] -7.528289e-03 4.837088e-03 9.949277e-03 -1.389473e-01 5.104113e-02## [121] 1.500038e-01 -1.770429e-01 1.162093e-01 -4.567454e-02 1.359564e-02## [126] -1.569903e-01 1.849557e-01 -9.852310e-02 2.341938e-01 -1.770329e-02## [131] -1.179755e-01 3.156786e-02 -7.342911e-02 2.847567e-02 6.998317e-02## [136] 1.726577e-01 -2.011151e-01 -6.840403e-02 -1.958081e-02 3.965425e-02## [141] 7.315098e-02 1.019568e-01 -5.413351e-02 -6.431586e-02 1.155628e-01## [146] 6.341369e-02 6.378888e-03 -3.289833e-01 2.175870e-02 2.613350e-02## [151] 8.388394e-02 1.024414e-01 -1.744432e-01 5.062091e-02 2.342159e-02## [156] 5.358752e-02 -7.762969e-02 1.547981e-01 -3.468426e-01 2.102085e-01## [161] 8.103525e-02 -2.423992e-01 -1.457949e-02 -2.311312e-02 -1.381335e-02## [166] 1.134144e-01 7.633274e-02 -4.373575e-02 -6.402227e-02 1.011489e-01## [171] -2.552972e-02 -1.098059e-01 4.404931e-02 1.218136e-02 -5.771540e-02## [176] 2.031735e-01 7.118068e-02 8.535538e-02 1.190604e-01 -1.597535e-01## [181] -2.829753e-01 1.765390e-01 6.733569e-02 7.226783e-03 5.081556e-02## [186] -2.139402e-01 6.361054e-02 1.166856e-01 -3.355537e-02 3.364892e-02## [191] -6.668737e-04 -3.101907e-02 -6.962957e-02 9.738595e-02 -8.011743e-02## [196] -2.691371e-02 -3.467751e-06 8.914880e-02 -7.954578e-02 -5.487591e-02## [201] 1.255535e-01 -9.624354e-03 -5.091336e-02 -1.513167e-02 -6.158265e-03## [206] -1.798672e-02 9.471499e-02 2.822636e-02 -1.135340e-01 -3.036519e-02## [211] 6.132845e-02 -6.464741e-02 8.893909e-02 -9.593587e-02 -8.784538e-02## [216] 3.304698e-02 9.754243e-02 -1.888328e-01 3.986594e-02 5.148270e-02## [221] -1.692218e-02 1.818076e-02 5.503780e-02 -1.674076e-01 2.307858e-01## [226] -8.375165e-02 -7.102372e-02 3.855489e-02 1.009889e-01 -1.861185e-01

12

## [231] 1.733530e-01 2.634167e-02 -1.945566e-01 7.387359e-02 2.391770e-02## [236] 7.709352e-02 1.587170e-02 -8.444373e-02 5.377875e-02 -9.614769e-02## [241] 1.223175e-01 -1.564164e-01 1.865181e-01 -1.154506e-01 -7.185726e-02## [246] 9.107342e-02 -5.397935e-02 1.816875e-01 -1.059343e-01 -8.464223e-02## [251] -1.402004e-02 -1.234100e-02 1.889063e-01 -3.032981e-01 3.377522e-01## [256] -1.217110e-01 -6.679761e-02 1.409263e-01 -4.864476e-02 -1.302731e-03## [261] -5.926453e-02 1.562763e-01 -3.687074e-02 -4.385849e-02 -1.414657e-01## [266] 1.283189e-01 6.962638e-02 3.602445e-02 9.611082e-03 3.127402e-02## [271] -3.170631e-02 -7.512685e-02 -6.995991e-02 -1.141863e-02 6.898345e-02## [276] -3.691683e-02 9.819801e-03 -2.066675e-02 1.861264e-01 -7.695674e-02## [281] 1.578545e-02 -6.296949e-02 8.888020e-02 -5.282442e-02 -3.593006e-02## [286] -4.237586e-02 -4.501829e-03 5.773483e-02 -1.366465e-01 9.696159e-02## [291] -1.866387e-01 4.886500e-02 7.116717e-02 2.432440e-03 -2.586663e-02## [296] 2.889372e-01 -2.150052e-01 -1.217880e-02 -1.670079e-01 2.516292e-01## [301] -4.023719e-02 -1.881773e-02 -1.587088e-01 2.117469e-01 -2.376329e-02## [306] -1.602568e-03 3.899565e-02 1.070092e-01 -2.794779e-01 -3.216949e-03## [311] 1.197854e-01 7.612166e-02 -1.802000e-01 -8.236462e-02 2.704150e-01## [316] -6.776383e-02 6.323148e-02 3.874625e-03 -8.815410e-02 -3.234494e-02## [321] 9.982487e-02 -1.201443e-01 3.469237e-02 8.084305e-02 -1.364825e-02## [326] -9.753047e-02 -1.376753e-01 1.205503e-01 6.949286e-02 3.331850e-02## [331] -9.984993e-02 -1.652963e-01 2.955858e-01 -1.532333e-01 -2.588135e-02## [336] 1.671327e-02 9.117671e-02 -1.375921e-01 -8.907866e-02 1.306209e-02## [341] 1.033567e-01 7.356301e-02 -2.513076e-01 8.751813e-02 5.487244e-02## [346] -2.727159e-02 6.865207e-03 -1.263384e-01 -1.289249e-02 2.562039e-01## [351] -1.066980e-01 2.214936e-02 -3.545642e-02 4.665820e-02 2.363010e-02## [356] 7.232026e-03 -5.217509e-02 8.130152e-02 -2.207372e-02 -7.280102e-02## [361] 2.382865e-01 -5.626844e-02 -1.591013e-01 1.763509e-01 -4.536038e-02## [366] -4.533646e-02 1.336275e-01 -4.455403e-02 -9.617154e-02 3.519542e-02## [371] 1.817628e-02 -3.481620e-02 -1.483207e-02 1.480301e-01 5.633724e-02## [376] -1.897949e-01 1.592195e-01 -6.066889e-02 -4.008211e-02 9.145874e-02## [381] -1.794247e-01 -1.664129e-02 1.044648e-01 1.799098e-01 -2.371712e-01## [386] -8.176847e-02 8.285964e-02 1.001599e-01 2.305686e-01 6.092994e-02## [391] -1.576140e-01 -3.362805e-02 -1.199122e-01 2.911133e-03 4.196018e-02## [396] 5.210213e-02 -2.234314e-02 -7.017565e-02 1.768405e-02 6.403380e-02## [401] 8.561279e-03 2.250154e-01 -1.466255e-01 -3.896314e-02 -1.348093e-01## [406] 1.767447e-01 -1.090779e-01 -1.051928e-01 1.451941e-01 -1.742756e-01## [411] 1.115476e-01 -9.061780e-02 1.016116e-01 1.527515e-01 -1.566510e-01## [416] -8.373343e-02 -1.261752e-01 6.004209e-02 -1.017499e-01 -3.440425e-02## [421] 1.827142e-01 -9.303228e-02 1.319343e-02 9.589162e-02 -2.163336e-02## [426] -5.835864e-02 2.162738e-02 -5.352451e-02 -7.142507e-02 -4.365986e-02## [431] 4.980316e-02 7.794318e-02 -9.290199e-02 1.341854e-01 -2.031966e-01## [436] 2.154164e-01 -5.877130e-02 3.284774e-02 1.014733e-01 -1.617708e-01## [441] 7.521131e-02 -1.579201e-01 1.362634e-01 -1.116644e-01 -2.594070e-02## [446] 1.922568e-01 1.637147e-01 -2.086152e-01 1.626809e-03 -3.142897e-03## [451] 1.351165e-01 1.099619e-02 -1.289453e-01 3.733526e-02 -8.950941e-02## [456] 1.012238e-01 1.259665e-02 4.409586e-02 -2.299854e-01 2.146759e-01## [461] 1.222524e-01 -2.257129e-01 4.588352e-02 1.560236e-01 -3.863293e-01## [466] 4.085872e-02 -5.657069e-02 2.385952e-01#### $aic## [1] 208.8332#### $lkhd## [1] -77.41658

13

#### $sigma2## [1] 0.02687679#### $tau1## [1] 0.0004375694#### $tau2## [1] 1.0001#### $tau3## [1] 0.0001000002#### $arcoef## [1] 1.167818 -0.297576#### $tdf## [1] 0.036104922 -0.052798670 0.044570502 -0.044313355 0.021118400## [6] 0.004254280 -0.008936079

La función descomponer() de la libreria descomponer descompone la tendencia siguiendo un modelo multi-plicativo Yt = TDtSt + IRt, tiene la siguiente sintesis:

descomponer(y,frequency,type)

En type se elige un modelo lineal (1) ó cuadrático (2) para la tendencia.

La representación gráfica se realiza con gdescomponer(y,freq,type,year,q).library(descomponer)

## Loading required package: taRifx

gdescomponer(co2,12,1,1959,1)

14

Time

TD

ST

1960 1970 1980 1990

320

Time

TD

1960 1970 1980 1990

320

Time

ST

1960 1970 1980 1990

−3

2

Las series de tendencia y estacionalidad son por lo general muy similares en todos los métodos.# Representación gráfica de la tendenciaplot(co2)lines (decompose(co2)$trend,col=2)lines(stl(co2,"per")$time.series[,2],col=3)lines(decomp(co2, trade=TRUE,plot=FALSE)$seasonal,col=4)lines(ts(descomponer(co2,12,1)$datos$TD,frequency = 12, start = c(1959, 1)),col=5)legend("bottomleft", c("Original", "decompose",

"stl","decomp","descomponer"),lwd=c(1,2,2,2), col=c("black",2,3,4,5))

grid()

15

Time

co2

1960 1970 1980 1990

320

330

340

350

360

Originaldecomposestldecompdescomponer

# Respresentación gráfica de la estacionalidad

plot (decompose(co2)$seasonal,col=2)lines(stl(co2,"per")$time.series[,1],col=3)lines(decomp(co2, trade=TRUE,plot=FALSE)$seasonal,col=4)lines(ts(descomponer(co2,12,1)$datos$ST,frequency = 12, start = c(1959, 1)),col=5)legend("bottomleft", c("decompose",

"stl","decomp","descomponer"),lwd=c(2,2,2), col=c(2,3,4,5))

grid()

16

Time

deco

mpo

se(c

o2)$

seas

onal

1960 1970 1980 1990

−3

−2

−1

01

23

decomposestldecompdescomponer

Evaluación de los métodos

Para evaluar el grado de ajuste, proponemos realizar varios test sobre al serie irregular resultante de ladescomposición de la serie original.

Un primer test será sobre la normalidad de los residuos, en concreto:

• Test de Jarque-Bera (jarque.bera.test)

• Test de Kolmogorov-Smirnov (lillie.test)

• Test de Cramer-von Mises (cvm.test)

• Test de Anderson-Darling (ad.test)

• Test de Shapiro-Francia (sf.test)

El test de Durbin (1969) permite evaluar si la serie temporal no esta autocorrelacionda (cov(yj , ys) = 0).

El test de Durbin esta basado en el siguiente estadístico:

sj =∑j

r=1pr∑m

r=1pr

(11)

donde m = 12n para n par y12 (n− 1) para n impar.

El estadístico sj ha en encontrarse entre unos límites inferior y superior de valores críticos que han sidotabulados por Durbin (1969). Si bien hay que tener presente que el valor po no se considera en el cálculo delestadístico esto es, po = v̂1 = 0.

17

Resultados equivalentes al test de Durbin se obtienen con la función “cpgram”:Plots a cumulative periodogram.

El test de durbin se realiza con la función gtd de la libreria descomponer, que utiliza la mismas bandascríticas que la función “cpgram”

Evaluación de la serie irregular de la función descompose.

Histograma de la serie y test de normalidad:

Distribución de los errores

decompose(co2)$random

Den

sity

−1.0 −0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

#### Lilliefors (Kolmogorov-Smirnov) normality test#### data: decompose(co2)$random## D = 0.039482, p-value = 0.08579

#### Cramer-von Mises normality test#### data: decompose(co2)$random## W = 0.10601, p-value = 0.0927

#### Anderson-Darling normality test#### data: decompose(co2)$random## A = 0.67575, p-value = 0.07715

##

18

## Shapiro-Francia normality test#### data: decompose(co2)$random## W = 0.99465, p-value = 0.1042

Test de Durbin y cpgram:

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

frequency

Series: decompose(co2)$random

Evaluación de la serie irregular de la función stl.

Histograma de la serie y test de normalidad:

19

Distribución de los errores

stl(co2, "per")$time.series[, 3]

Den

sity

−1.0 −0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

#### Jarque Bera Test#### data: stl(co2, "per")$time.series[, 3]## X-squared = 5.468, df = 2, p-value = 0.06496

#### Lilliefors (Kolmogorov-Smirnov) normality test#### data: stl(co2, "per")$time.series[, 3]## D = 0.036161, p-value = 0.1442

#### Cramer-von Mises normality test#### data: stl(co2, "per")$time.series[, 3]## W = 0.084161, p-value = 0.1826

#### Anderson-Darling normality test#### data: stl(co2, "per")$time.series[, 3]## A = 0.60695, p-value = 0.1142

#### Shapiro-Francia normality test#### data: stl(co2, "per")$time.series[, 3]

20

## W = 0.99448, p-value = 0.08428

Test de Durbin y cpgram:

21

Test Periodograma

frecuencia

dens

idad

acu

mul

ada

0 50 100 150 200

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

frequency

Series: stl(co2, "per")$time.series[, 3]

22

Evaluación de la serie irregular de la función decomp.

Histograma de la serie y test de normalidad:

Distribución de los errores

decomp(co2, trade = TRUE, plot = FALSE)$noise

Den

sity

−0.4 −0.2 0.0 0.2 0.4

0.0

0.5

1.0

1.5

2.0

2.5

3.0

#### Jarque Bera Test#### data: decomp(co2, trade = TRUE, plot = FALSE)$noise## X-squared = 0.43576, df = 2, p-value = 0.8042

#### Lilliefors (Kolmogorov-Smirnov) normality test#### data: decomp(co2, trade = TRUE, plot = FALSE)$noise## D = 0.029146, p-value = 0.4338

#### Cramer-von Mises normality test#### data: decomp(co2, trade = TRUE, plot = FALSE)$noise## W = 0.035025, p-value = 0.7702

#### Anderson-Darling normality test#### data: decomp(co2, trade = TRUE, plot = FALSE)$noise## A = 0.24021, p-value = 0.7748

23

#### Shapiro-Francia normality test#### data: decomp(co2, trade = TRUE, plot = FALSE)$noise## W = 0.99813, p-value = 0.839

Test de Durbin y cpgram:

24

Test Periodograma

frecuencia

dens

idad

acu

mul

ada

0 50 100 150 200

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

frequency

Series: decomp(co2, trade = TRUE, plot = FALSE)$noise

25

Evaluación de la serie irregular de la función descomponer.

Histograma de la serie y test de normalidad:

Distribución de los errores

descomponer(co2, 12, 1)$datos$IR

Den

sity

−1.0 −0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

#### Jarque Bera Test#### data: descomponer(co2, 12, 1)$datos$IR## X-squared = 136.64, df = 2, p-value < 2.2e-16

#### Lilliefors (Kolmogorov-Smirnov) normality test#### data: descomponer(co2, 12, 1)$datos$IR## D = 0.038772, p-value = 0.08929

#### Cramer-von Mises normality test#### data: descomponer(co2, 12, 1)$datos$IR## W = 0.11079, p-value = 0.07974

#### Anderson-Darling normality test#### data: descomponer(co2, 12, 1)$datos$IR## A = 0.896, p-value = 0.02204

26

#### Shapiro-Francia normality test#### data: descomponer(co2, 12, 1)$datos$IR## W = 0.97342, p-value = 7.476e-07

Test de Durbin y cpgram:

27

Test Periodograma

frecuencia

dens

idad

acu

mul

ada

0 50 100 150 200

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

frequency

Series: descomponer(co2, 12, 1)$datos$IR

28

Conclusiones

La descomposición temporal de las concentraciones atmosféricas de CO2 en partes por millón (ppm) enMauna Loa (Hawai), utilizando las funciones de R: decompose, stl, decomp y descomponer, presentan en losanálisis gráficos una gran simulitud, tanto en lo relativo a la serie de tenencia (Tt), como a la estacionalidad(St).

El estudio de normalidad de la serie irregular o residual, acepta la hipótesis de normalidad en todos lasfunciones en los test de KS y CVM. Los mejores resultados se obtienen para la función decomp tanto si serealiza el ajuste estacional como si se prescinde de él, los peores resultados se dan para la serie irregularde la función descomponer que rechaza la hipótesis de normalidad a un α = 0, 05 en los test JB, AD y SF,mejorando los resultados del test cuando se elige en descomponer un tipo de tendencia cuadrática.

Función JB KS CVM AD SFdecompose .. 0.039482(0.08579)*0.10601(0.0927)0.67575(0.07715)0.99465(0.10042)stl 5.468(0.06496)0.036161(0.1442)0.084161(0.1826)0.60695(0.1142)0.99548(0.08465)decomp** 0.43576(0.8042)0.029146(0.4338)0.035025(0.7702)0.24021(0.7748)0.99813(0.839)decomp*** 0.80442(0.6687)0.01935(0.9403)0.032249(0.8147)0.24395(0.7629)0.99758(0.6633)descomponer 136,64(0.0000)0.038772(0.08429)0.11079(0.07974)0.896(0.02204)0.97342(0.0000)descomponer**** 32,5849(0.0000)0.029092(0.4368)0.074419(0.2444)0.58337(0.1283)0.98769(0.0000)

(*)p-valor

(**) con ajuste calendario

(***) sin ajuste calendario

(****) tendencia cuadrática

Ninguna de las funciones deja una serie irregular incorrelacionada, tal y como se aprecia en los test gráficos“gtd” y “cpgram”.

Bibliografía

R. B. Cleveland, W. S. Cleveland, J.E. McRae, and I. Terpenning (1990) STL: A Seasonal-Trend DecompositionProcedure Based on Loess. Journal of Official Statistics, 6, 3–73.

Durbib, J., “Tests for Serial Correlation in Regression Analysis based on the Periodogram ofLeast-SquaresResiduals,” Biometrika, 56, (No. 1, 1969), 1-15.

Harvey, A.C. (1978), Linear Regression in the Frequency Domain, International Economic Review, 19, 507-512.

Keeling, C. D. and Whorf, T. P., Scripps Institution of Oceanography (SIO), University of California, LaJolla, California USA 92093-0220.

G.Kitagawa (1981) A Nonstationary Time Series Model and Its Fitting by a Recursive Filter Journal of TimeSeries Analysis, Vol.2, 103-116.

W.Gersch and G.Kitagawa (1983) The prediction of time series with Trends and Seasonalities Journal ofBusiness and Economic Statistics, Vol.1, 253-264.

M. Kendall and A. Stuart (1983) The Advanced Theory of Statistics, Vol.3, Griffin. pp. 410–414.

G.Kitagawa (1984) A smoothness priors-state space modeling of Time Series with Trend and SeasonalityJournal of American Statistical Association, VOL.79, NO.386, 378-389.

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Parra, F. (2014), Amplitude time-frequency regression, (http://econometria.wordpress.com/2013/08/21/estimation-of-time-varying-regression-coefficients/)

Venables and Ripley, “Modern Applied Statistics with S” (4th edition, 2002).

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http://econometria.wordpress.com/2013/08/21/estimation-of-time-varying-regression-coefficients/http://econometria.wordpress.com/2013/08/21/estimation-of-time-varying-regression-coefficients/

Componentes de una serie temporalHerramientas REvaluación de los métodosEvaluación de la serie irregular de la función descompose.Evaluación de la serie irregular de la función stl.Evaluación de la serie irregular de la función decomp.Evaluación de la serie irregular de la función descomponer.

ConclusionesBibliografía