-
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)
1
-
• 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”:
2
-
plot(decompose(co2))
320
350
obse
rved
320
340
360
tren
d
−3
−1
13
seas
onal
−0.
50.
5
1960 1970 1980 1990
rand
om
Time
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"))
3
-
320
350
data
−3
−1
13
seas
onal
320
340
360
tren
d
−0.
50.
5
1960 1970 1980 1990
rem
aind
er
time
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)
4
-
0 100 200 300 400
320
Original and Trend
0 100 200 300 400
−4
4
Seasonal
0 100 200 300 400
−4
4
Noise
0 100 200 300 400
−4
4
AR component
0 100 200 300 400
−4
4
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
5
-
## [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
6
-
## [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.
29
-
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).
30
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