Module R’Stat1 : Les séries temporelles

Novembre 2019 ; IRD-Montpellier-France

CC BY-NC-ND 3.0

Définitions

Série temporelle

Mesures répétées à un pas de temps donné. Par exemple la température chaque jour, la valeur d’une action chaque semaine, le nombre d’insectes chaque mois, …

Une série temporelle est régulière si le pas de temps est toujours le même et irrégulière dans le cas contraire. La plupart des analyses statistiques ne s’appliquent qu’aux séries temporelles régulières. La quasi totalité des séries temporelles en biologie sont irrégulières !

!! On ne va que survoler la thématique des séries temporelles qui fait l’objet de nombreux livres et de nombreux packages R !!! Vue d’ensemble ici

Décomposition

Une série temporelle peut être décomposée en :

  • une tendance ou orientation (en général avec un modèle linéaire)
  • une composante saisonnière (quelque chose qui revient avec périodicité)
  • une composante aléatoire (bruit de fond)
  • une composante cyclique (des évènements qui reviennent avec des variations, par exemple El Niño)

Pour quoi faire ?

Pour décrire, expliquer, faire des hypothèses, modéliser, prévoir, …

ts

##              Jan         Feb         Mar         Apr         May
## 2016             -0.66960810  1.13983300 -1.85598599  0.81628493
## 2017 -1.12656474 -0.14393231  1.06762113 -2.30920531  0.47295250
## 2018  0.20322220 -0.20668717  0.15708461 -0.72950695  1.09791403
## 2019  0.96931212  0.15772934 -1.21621748 -0.94901891  3.12868932
##              Jun         Jul         Aug         Sep         Oct
## 2016 -0.17076981  1.08981231  0.74192109 -0.72764400 -0.70373971
## 2017  0.30791470  1.58896050 -0.17352878 -0.87229436 -0.18136641
## 2018 -0.43094538 -0.84162924  0.38198947  1.09339724 -0.71494964
## 2019 -0.16476848  0.38624069  0.82786204 -1.49346438  0.74670559
##              Nov         Dec
## 2016  0.00253112 -1.66704299
## 2017  0.97877421 -0.12194153
## 2018 -0.40475779  0.79466085
## 2019

##  Time-Series [1:45] from 2016 to 2020: -0.67 1.14 -1.856 0.816 -0.171 ...

## Time Series:
## Start = c(2019, 32) 
## End = c(2019, 61) 
## Frequency = 365 
##  [1]  0.1892583  2.4191292 -0.9180494 -0.7277307 -1.1643339  0.1161126
##  [7]  0.5771136 -1.8024935  0.6341600  0.6972268  1.5070810 -0.9685960
## [13]  1.7469175  0.7793346  1.5068857 -1.9686809  0.2755536  0.8376550
## [19]  0.5363980  0.4532773 -2.3447569 -0.9409205  1.4031314 -1.1792448
## [25] -3.0580679 -0.9762208  1.5576742 -0.5317512  2.0538006  0.6494285

Mais concrètement, avec des données réelles ?

Exemple : données météo

Dans la pratique nos séries temporelles viennent d’un fichier. Et le fichier comportera souvent (si ce n’est tout le temps) des données manquantes !

Charger les données

Dates au format POSIXct

## 'data.frame':    188184 obs. of  19 variables:
##  $ raspid      : Factor w/ 8 levels "EG","EM","IC",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ timestamp   : POSIXct, format: "2019-09-01 00:11:16" "2019-09-01 00:12:19" ...
##  $ temperature : num  22.4 22.4 24.1 22.3 22.3 ...
##  $ gas         : int  5797414 5797414 6552984 5797414 5797414 6593137 5797414 11127198 6593137 5797414 ...
##  $ humidity    : num  59.2 59.2 49.4 59.1 59 ...
##  $ pressure    : num  1009 1009 1010 1010 1010 ...
##  $ lightvisible: int  261 260 299 260 260 301 260 276 301 260 ...
##  $ lightir     : int  254 253 648 253 254 673 255 392 670 253 ...
##  $ lightuv     : int  2 2 23 2 2 24 2 10 24 2 ...
##  $ ndvi_0      : int  307200 307200 237139 307200 307200 230841 307200 224441 241853 307200 ...
##  $ ndvi_1      : int  0 0 0 0 0 0 0 954 0 0 ...
##  $ ndvi_2      : int  0 0 0 0 0 0 0 15 0 0 ...
##  $ ndvi_3      : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ ndvi_4      : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ ndvi_5      : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ ndvi_6      : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ ndvi_7      : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ ndvi_8      : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ ndvi_9      : int  0 0 0 0 0 0 0 0 0 0 ...

Aperçu des données

Séparer les différentes mesures

Mettre chaque mesure dans une colonne d’un data.frame

Identifier le nombre de données manquantes

## [1]   0  51 286 563 271 325 174 718
## [1] "0%"    "7.1%"  "39.7%" "78.2%" "37.6%" "45.1%" "24.2%" "99.7%"

##  [1]   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17
## [18]  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34
## [35] 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127

Que faire ?

##                    date EM
## 1   2019-09-01 00:00:00 NA
## 2   2019-09-01 01:00:00 NA
## 3   2019-09-01 02:00:00 NA
## 4   2019-09-01 03:00:00 NA
## 5   2019-09-01 04:00:00 NA
## 6   2019-09-01 05:00:00 NA
## 7   2019-09-01 06:00:00 NA
## 8   2019-09-01 07:00:00 NA
## 9   2019-09-01 08:00:00 NA
## 10  2019-09-01 09:00:00 NA
## 11  2019-09-01 10:00:00 NA
## 12  2019-09-01 11:00:00 NA
## 13  2019-09-01 12:00:00 NA
## 14  2019-09-01 13:00:00 NA
## 15  2019-09-01 14:00:00 NA
## 16  2019-09-01 15:00:00 NA
## 17  2019-09-01 16:00:00 NA
## 18  2019-09-01 17:00:00 NA
## 19  2019-09-01 18:00:00 NA
## 20  2019-09-01 19:00:00 NA
## 21  2019-09-01 20:00:00 NA
## 22  2019-09-01 21:00:00 NA
## 23  2019-09-01 22:00:00 NA
## 24  2019-09-01 23:00:00 NA
## 25  2019-09-02 00:00:00 NA
## 26  2019-09-02 01:00:00 NA
## 27  2019-09-02 02:00:00 NA
## 28  2019-09-02 03:00:00 NA
## 29  2019-09-02 04:00:00 NA
## 30  2019-09-02 05:00:00 NA
## 31  2019-09-02 06:00:00 NA
## 32  2019-09-02 07:00:00 NA
## 33  2019-09-02 08:00:00 NA
## 34  2019-09-02 09:00:00 NA
## 111 2019-09-05 14:00:00 NA
## 112 2019-09-05 15:00:00 NA
## 113 2019-09-05 16:00:00 NA
## 114 2019-09-05 17:00:00 NA
## 115 2019-09-05 18:00:00 NA
## 116 2019-09-05 19:00:00 NA
## 117 2019-09-05 20:00:00 NA
## 118 2019-09-05 21:00:00 NA
## 119 2019-09-05 22:00:00 NA
## 120 2019-09-05 23:00:00 NA
## 121 2019-09-06 00:00:00 NA
## 122 2019-09-06 01:00:00 NA
## 123 2019-09-06 02:00:00 NA
## 124 2019-09-06 03:00:00 NA
## 125 2019-09-06 04:00:00 NA
## 126 2019-09-06 05:00:00 NA
## 127 2019-09-06 06:00:00 NA

Remplacer les valeurs manquantes par la moyenne de la température à la même heure ?

## [1] 18.86718
## [1] 1.354671

Autres idées ?

  • Remplacer par la médiane ?
  • Prendre une valeur au hasard ?

Le package imputeTS

## [1] "Length of time series:"
## [1] 720
## [1] "-------------------------"
## [1] "Number of Missing Values:"
## [1] 51
## [1] "-------------------------"
## [1] "Percentage of Missing Values:"
## [1] "7.08%"
## [1] "-------------------------"
## [1] "Stats for Bins"
## [1] "  Bin 1 (180 values from 1 to 180) :      51 NAs (28.3%)"
## [1] "  Bin 2 (180 values from 181 to 360) :      0 NAs (0%)"
## [1] "  Bin 3 (180 values from 361 to 540) :      0 NAs (0%)"
## [1] "  Bin 4 (180 values from 541 to 720) :      0 NAs (0%)"
## [1] "-------------------------"
## [1] "Longest NA gap (series of consecutive NAs)"
## [1] "34 in a row"
## [1] "-------------------------"
## [1] "Most frequent gap size (series of consecutive NA series)"
## [1] "34 NA in a row (occuring 1 times)"
## [1] "-------------------------"
## [1] "Gap size accounting for most NAs"
## [1] "34 NA in a row (occuring 1 times, making up for overall 34 NAs)"
## [1] "-------------------------"
## [1] "Overview NA series"
## [1] "  17 NA in a row: 1 times"
## [1] "  34 NA in a row: 1 times"

?na.kalman

Mais concrètement, avec des données réelles ? (2)

Exemple : mesures humaines

Quand les relevés sont faits par des humains (sondage, piégeage, mesure physique, …), par exemple tous les 10 jours, et bien il y a toujours un jour qui tombe un samedi ou un dimanche, ou alors il y avait (encore) une réunion… Bref : la série n’est pas régulière !

Option 1 (tristement la plus courante) : fermer les yeux très fort, des fois que…

Option 2 : régler le problème en faisant des hypothèses sur la distribution des données

Données théoriques

Cas 1 : mesures accumulée (qt/unité de temps)

transformation avec l’écart à la moyenne

## [1] 10
##            x     y
## 1 2017-01-11  91.0
## 2 2017-01-21 107.0
## 3 2017-01-31  96.0
## 4 2017-02-10  98.0
## 5 2017-02-20  73.5
## 6 2017-03-02 146.4

## 
##  Welch Two Sample t-test
## 
## data:  df$y and newDf$y
## t = -0.055922, df = 185.83, p-value = 0.9555
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -6.783969  6.409969
## sample estimates:
## mean of x mean of y 
##   102.480   102.667

Cas 2 : mesures ponctuelles (qt)

## Time differences in days
##   [1] 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 2 2 0 0 0 0 0 0
##  [36] 0 0 0 3 0 0 0 0 0 0 0 3 0 2 0 0 0 1 0 1 0 0 3 0 0 0 0 0 0 0 3 0 2 0 0
##  [71] 0 1 0 0 0 2 0 0 0 0 3 0 3 0 0 3 0 0 0 0 3 0 0 2 0 0 0 2 0 0

Ici nous ne voyons que quelques options simples…

Quand enfin tout va bien :

On peut travailler avec les séries temporelles !

Remarque : il existe des cas où il est possible de travailler avec des séries temporelles irrégulières…

Données météo

ts

Décomposition de la série temporelle

## $x
## Time Series:
## Start = c(1, 1) 
## End = c(30, 24) 
## Frequency = 24 
##   [1] 22.273168 21.037693 19.545947 19.601642 20.272462 19.633283 18.663302
##   [8] 18.842599 18.952720 18.888261 20.229870 22.692596 26.987959 23.698750
##  [15] 23.572174 23.595759 23.792630 23.868487 23.479944 23.118936 21.901572
##  [22] 20.851323 19.750172 18.828811 18.170570 17.156384 15.625388 14.836387
##  [29] 14.332974 13.505771 12.691493 12.327269 15.104249 25.970046 29.875693
##  [36] 31.029839 27.884496 25.673849 23.735647 23.160990 22.987806 22.951892
##  [43] 22.776070 22.457206 21.160521 18.764015 17.192274 16.225878 15.475508
##  [50] 15.767551 14.993237 13.637285 12.847993 12.366448 11.997856 12.059026
##  [57] 14.833673 25.359393 30.982937 32.263853 30.973705 28.392066 26.315561
##  [64] 25.570692 25.317558 25.191575 24.909684 24.427783 23.152553 20.810441
##  [71] 19.339344 18.785671 18.043124 16.968638 15.960633 15.516684 15.009115
##  [78] 14.515707 13.960835 13.786954 16.558752 26.637185 33.429596 37.187980
##  [85] 33.200888 30.754551 28.439324 27.676275 27.208279 25.131789 23.463481
##  [92] 22.950316 21.746190 20.591493 19.283965 17.896718 16.889231 16.462669
##  [99] 15.721064 14.967312 14.241619 13.534458 13.819779 14.011788 15.381909
## [106] 19.527094 26.169399 24.817092 23.658671 22.817428 21.866071 21.749189
## [113] 20.583212 20.847342 20.336534 19.729609 18.616887 17.571560 16.730744
## [120] 15.650727 14.611713 13.677903 12.774007 11.928756 11.488338 10.445090
## [127]  9.785215  9.490543 13.013782 20.982035 27.348493 28.989091 26.327581
## [134] 23.868914 22.052043 21.799528 22.681662 22.358594 22.283560 21.180504
## [141] 19.953914 18.815304 18.076015 18.066657 17.275898 17.534194 17.084725
## [148] 15.900326 15.405079 14.775662 14.756472 14.937939 15.718014 16.437155
## [155] 17.456090 21.553071 23.632371 21.516606 20.941787 21.158051 21.089910
## [162] 21.098212 20.427968 19.392754 18.201889 17.301275 16.193213 15.772353
## [169] 15.596163 14.977783 14.250391 14.380874 14.429904 13.987645 13.492822
## [176] 13.369217 14.638076 15.930267 17.670892 18.596478 20.549309 20.507721
## [183] 20.023010 20.174056 20.183313 19.163933 18.868809 18.226108 17.235332
## [190] 16.098130 15.205946 14.338718 13.565615 12.401697 11.490968 10.755788
## [197] 10.181052  9.970885  9.496437  9.730891 11.043747 11.730658 15.724685
## [204] 21.403540 22.820831 23.154222 22.456789 22.384927 22.244221 21.767020
## [211] 21.332995 20.552358 19.413248 18.479709 17.312134 16.061849 15.759386
## [218] 14.232512 14.525676 14.260465 13.735099 13.652934 12.979094 13.942733
## [225] 14.934362 25.855404 35.307196 35.844549 30.030939 27.114479 24.508851
## [232] 24.011384 23.726512 23.410510 23.051560 22.321516 20.672962 18.437857
## [239] 17.077550 15.827745 15.256909 15.807926 15.382368 14.255089 13.247725
## [246] 13.323123 13.602413 13.696499 14.597156 22.578972 26.643313 32.021619
## [253] 33.649198 34.049110 32.032218 30.802225 30.629678 30.409218 25.537535
## [260] 24.915091 22.173637 20.645121 20.266355 19.781422 19.562891 18.977543
## [267] 18.257786 18.179340 18.670128 17.865170 17.318850 16.735506 19.912244
## [274] 27.660883 26.958622 29.562826 28.855475 29.654877 29.533642 28.800028
## [281] 28.756264 28.910441 27.891218 27.041027 25.155037 23.284454 22.198048
## [288] 21.253095 21.069306 20.808934 20.252855 19.571523 18.842130 18.350169
## [295] 18.347208 18.474997 18.915331 20.456100 21.689673 24.027937 27.911338
## [302] 27.226457 26.016061 25.676371 25.287717 25.235596 24.564019 24.214486
## [309] 23.056758 22.040766 21.250645 20.176885 19.415807 18.903006 18.394833
## [316] 18.434643 17.986421 17.658704 17.035880 16.931720 18.949915 24.694450
## [323] 30.059023 32.387052 30.438951 28.105093 27.575785 27.297345 27.277255
## [330] 27.131669 26.833475 26.167188 24.012244 22.719762 21.999014 21.050847
## [337] 19.997681 19.483331 19.009420 18.152190 17.648819 17.233622 16.650121
## [344] 15.892233 18.922222 28.137668 33.434139 35.349500 33.294833 30.573072
## [351] 28.849624 28.548474 29.208639 28.995399 28.883166 27.659218 25.513217
## [358] 23.795690 22.207162 20.906830 19.913498 18.677992 18.135342 17.483910
## [365] 16.679949 16.228280 15.725731 14.858238 18.637734 28.473319 33.761778
## [372] 36.090283 32.935859 31.006329 29.443196 29.572441 29.272053 28.937840
## [379] 28.014056 26.379846 24.037833 23.123233 22.911334 22.152245 21.424401
## [386] 20.441123 19.776926 19.496262 18.916870 18.975991 18.672859 18.268573
## [393] 19.039169 20.194892 21.289437 22.818148 23.032515 23.172309 24.151985
## [400] 24.073651 23.928194 23.679304 23.086658 22.111833 20.871231 19.558094
## [407] 18.753966 17.825820 17.063395 16.226577 15.605881 15.049799 14.518382
## [414] 14.027992 13.473836 12.928122 14.942325 22.238381 25.402256 27.373390
## [421] 24.544779 22.037196 20.939779 20.953883 21.185071 21.161466 20.790119
## [428] 19.965345 19.150871 18.019724 17.128739 16.204239 15.520638 14.936637
## [435] 14.424953 13.914276 13.427478 13.255047 12.844323 12.535071 14.800741
## [442] 21.819015 25.400480 26.562190 24.313820 21.956278 21.178739 20.931148
## [449] 21.129568 20.991602 20.765040 19.906613 18.668144 17.545121 16.785112
## [456] 15.830768 15.172085 14.239577 13.558684 13.060923 12.436020 11.778427
## [463] 11.706237 11.179291 14.334541 22.097345 24.278825 26.767397 24.263607
## [470] 22.138234 21.832285 22.543867 23.277355 23.707731 23.606258 22.714415
## [477] 20.987463 19.014916 18.756278 18.478427 17.763930 17.286915 16.455929
## [484] 15.767813 15.450823 14.949740 14.640766 14.448510 16.495409 24.514944
## [491] 29.046622 31.443405 30.457745 28.236048 27.811476 28.440177 29.282042
## [498] 29.690836 29.462963 27.087847 24.860397 23.575124 22.854359 23.088607
## [505] 22.083166 21.610661 23.048832 23.022102 22.155055 21.178182 19.683091
## [512] 20.907097 22.439173 23.734245 24.488457 24.732701 24.669364 22.198222
## [519] 22.041499 21.014358 20.533218 20.883797 20.282452 19.485184 19.328040
## [526] 19.040160 18.787566 18.679883 18.306158 17.716366 16.985405 15.715023
## [533] 15.681799 16.591989 16.988440 17.512741 18.710900 25.833324 27.874963
## [540] 28.811880 29.209444 26.411353 25.109691 24.935417 24.858303 24.408488
## [547] 23.052416 21.647369 22.098294 21.448676 20.705877 19.895755 19.243340
## [554] 18.801728 18.896194 18.690894 18.792377 18.809365 18.613196 18.476035
## [561] 18.960849 18.669911 18.991928 20.447427 19.671160 19.333512 19.894936
## [568] 21.006347 22.020844 22.635527 22.092908 21.764413 20.727869 20.099892
## [575] 18.713823 18.790602 18.791798 18.775196 18.644714 18.272876 17.995686
## [582] 18.061843 17.749637 17.812439 17.951964 18.207031 18.629527 19.063772
## [589] 20.199925 21.167332 22.204942 22.312129 22.925220 22.272804 22.567911
## [596] 21.604462 20.284283 19.629150 19.886898 20.072369 19.916455 19.824117
## [603] 19.871968 19.941244 19.834506 19.718843 19.871108 19.922359 19.938340
## [610] 20.249781 20.440735 20.748568 22.387607 23.541684 23.264886 23.475665
## [617] 23.783680 23.597770 23.463625 22.952468 22.353129 22.159903 21.777211
## [624] 21.577343 21.323003 20.868141 20.667298 19.892130 19.503286 19.202190
## [631] 18.326262 18.268799 18.387367 18.938714 21.820070 25.260246 27.904239
## [638] 24.334307 23.397945 23.611952 23.677009 22.857083 22.303764 21.538957
## [645] 20.384773 19.636562 19.141064 18.839228 18.251159 18.128187 18.327976
## [652] 18.186528 17.579058 17.843713 17.591160 17.772105 18.179013 20.842664
## [659] 21.481960 21.459506 26.079887 22.867390 23.031381 22.934364 23.296074
## [666] 22.656706 22.234687 21.622160 20.422534 18.609749 17.815129 18.122608
## [673] 17.504113 16.811428 16.751899 16.369796 15.937425 16.145097 16.352012
## [680] 16.084533 17.036401 23.457710 21.040759 19.258029 19.850014 20.758968
## [687] 21.278523 21.248565 21.344463 21.562048 21.324814 20.751625 20.082450
## [694] 19.012439 18.165759 18.306857 18.060205 17.706881 17.472676 17.041263
## [701] 16.663433 16.087747 15.605027 15.148431 16.393796 23.893106 28.146876
## [708] 27.256473 28.150391 22.714666 22.365711 22.434982 22.258832 21.895940
## [715] 21.318422 19.391462 18.007200 18.262268 18.920834 19.224666
## 
## $seasonal
## Time Series:
## Start = c(1, 1) 
## End = c(30, 24) 
## Frequency = 24 
##   [1] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
##   [7] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
##  [13]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
##  [19]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
##  [25] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
##  [31] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
##  [37]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
##  [43]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
##  [49] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
##  [55] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
##  [61]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
##  [67]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
##  [73] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
##  [79] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
##  [85]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
##  [91]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
##  [97] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [103] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [109]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [115]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [121] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [127] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [133]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [139]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [145] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [151] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [157]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [163]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [169] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [175] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [181]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [187]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [193] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [199] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [205]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [211]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [217] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [223] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [229]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [235]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [241] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [247] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [253]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [259]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [265] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [271] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [277]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [283]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [289] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [295] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [301]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [307]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [313] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [319] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [325]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [331]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [337] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [343] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [349]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [355]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [361] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [367] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [373]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [379]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [385] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [391] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [397]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [403]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [409] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [415] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [421]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [427]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [433] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [439] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [445]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [451]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [457] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [463] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [469]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [475]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [481] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [487] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [493]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [499]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [505] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [511] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [517]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [523]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [529] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [535] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [541]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [547]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [553] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [559] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [565]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [571]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [577] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [583] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [589]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [595]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [601] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [607] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [613]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [619]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [625] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [631] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [637]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [643]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [649] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [655] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [661]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [667]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [673] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [679] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [685]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [691]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## [697] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
## [703] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [709]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [715]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## 
## $trend
## Time Series:
## Start = c(1, 1) 
## End = c(30, 24) 
## Frequency = 24 
##   [1]       NA       NA       NA       NA       NA       NA       NA
##   [8]       NA       NA       NA       NA       NA 21.33453 21.16820
##  [15] 21.00566 20.82471 20.60169 20.35030 20.09823 19.83808 19.62217
##  [22] 19.68953 20.03802 20.41266 20.60504 20.66486 20.70941 20.70376
##  [29] 20.67794 20.64207 20.60832 20.57987 20.55064 20.49172 20.39494
##  [36] 20.28742 20.17705 20.09197 20.04986 20.01171 19.95579 19.90112
##  [43] 19.86293 19.84290 19.83167 19.81331 19.82366 19.87243 19.96250
##  [50] 20.08349 20.19387 20.29782 20.39656 20.49175 20.58286 20.66837
##  [57] 20.75092 20.83505 20.92242 21.02048 21.12730 21.20581 21.25099
##  [64] 21.31030 21.39448 21.48428 21.56995 21.64684 21.71878 21.78134
##  [71] 21.85893 22.01249 22.16148 22.25709 22.35056 22.43867 22.52193
##  [78] 22.56007 22.52870 22.46779 22.40771 22.37384 22.36813 22.34846
##  [85] 22.30590 22.27132 22.25578 22.23935 22.21191 22.17548 22.15210
##  [92] 22.15385 22.13401 21.96137 21.66199 21.25301 20.79648 20.43233
##  [99] 20.13003 19.86961 19.60810 19.38082 19.22642 19.09417 18.96188
## [106] 18.83377 18.71767 18.61768 18.52344 18.41798 18.29857 18.17387
## [113] 18.05320 17.93148 17.78307 17.60482 17.46129 17.44227 17.49714
## [120] 17.60862 17.75114 17.82865 17.85443 17.85935 17.90412 17.97932
## [127] 18.05137 18.12216 18.18024 18.23401 18.28794 18.36630 18.47214
## [134] 18.60798 18.77813 18.95068 19.11502 19.28683 19.48062 19.69768
## [141] 19.86750 19.82916 19.52838 19.16737 18.95630 18.85115 18.77901
## [148] 18.74252 18.69599 18.63657 18.57165 18.49575 18.42201 18.35396
## [155] 18.28320 18.19617 18.11338 18.02513 17.91282 17.82212 17.77015
## [162] 17.73341 17.69067 17.63166 17.57648 17.54342 17.53734 17.48022
## [169] 17.35439 17.26914 17.22898 17.18934 17.14995 17.09077 17.01799
## [176] 16.96120 16.91676 16.87156 16.82592 16.77549 16.70332 16.60735
## [183] 16.49619 16.36318 16.19914 16.02694 15.86000 15.70094 15.55026
## [190] 15.38789 15.25985 15.27778 15.38359 15.48605 15.59189 15.68865
## [197] 15.77764 15.87481 15.98038 16.08018 16.17402 16.26901 16.36250
## [204] 16.44228 16.52388 16.60773 16.70909 16.84533 16.99239 17.14314
## [211] 17.29240 17.45270 17.62151 17.99683 18.69906 19.40788 19.85895
## [218] 20.09166 20.21692 20.29356 20.35832 20.42344 20.49349 20.56615
## [225] 20.62925 20.65462 20.64886 20.63910 20.62375 20.64610 20.69677
## [232] 20.71451 20.70424 20.68722 20.69333 20.70119 20.68903 20.61375
## [239] 20.36499 20.10485 20.10059 20.32044 20.62165 20.91986 21.20515
## [246] 21.49477 21.69237 21.79820 21.88349 21.96074 22.07316 22.22196
## [253] 22.39404 22.54978 22.67572 22.81738 23.01210 23.21969 23.39174
## [260] 23.53248 23.70653 23.92313 24.03557 23.99092 23.83982 23.64841
## [267] 23.50481 23.41104 23.33030 23.26004 23.27786 23.37118 23.47758
## [274] 23.59468 23.68991 23.76082 23.82286 23.89240 23.97211 24.04268
## [281] 24.07527 24.08896 24.12048 24.17815 24.19362 24.02275 23.76288
## [288] 23.53780 23.40282 23.33256 23.20868 23.07033 22.93299 22.78417
## [295] 22.63829 22.51009 22.40749 22.33786 22.29222 22.25006 22.19319
## [302] 22.11903 22.04062 21.97822 21.93671 21.90448 21.86275 21.80328
## [309] 21.77185 21.86087 22.12353 22.47204 22.69885 22.76981 22.82061
## [316] 22.88688 22.96209 23.04304 23.12983 23.21779 23.27838 23.31243
## [323] 23.34216 23.37596 23.40629 23.43050 23.45540 23.46232 23.44940
## [330] 23.43351 23.41662 23.38693 23.36469 23.43585 23.57790 23.70993
## [337] 23.83115 23.94206 24.02001 24.07262 24.13892 24.21799 24.29952
## [344] 24.37330 24.43566 24.48934 24.51609 24.51743 24.51267 24.49414
## [351] 24.45915 24.42702 24.39291 24.35179 24.31158 24.27078 24.24331
## [358] 24.24438 24.25820 24.28046 24.28841 24.28996 24.31135 24.34505
## [365] 24.36770 24.36783 24.34852 24.30376 24.24637 24.20162 24.20228
## [372] 24.24290 24.30032 24.36853 24.43946 24.51559 24.60411 24.70796
## [379] 24.82660 24.95905 25.03846 24.87436 24.44205 23.90571 23.42289
## [386] 23.05336 22.77991 22.55512 22.32923 22.10835 21.89614 21.70457
## [393] 21.54969 21.40944 21.24855 21.07181 20.89082 20.71216 20.53746
## [400] 20.35793 20.17366 19.97894 19.76755 19.54798 19.35137 19.30859
## [407] 19.43684 19.61743 19.74383 19.75169 19.66112 19.52921 19.40706
## [414] 19.29746 19.19716 19.10460 19.02404 18.95615 18.89024 18.82260
## [421] 18.75667 18.69766 18.64618 18.59792 18.55154 18.51271 18.48349
## [428] 18.46219 18.45105 18.43936 18.43059 18.41365 18.39194 18.38544
## [435] 18.38874 18.39324 18.39161 18.38692 18.38285 18.38111 18.36983
## [442] 18.34988 18.33284 18.31790 18.30286 18.28107 18.24850 18.21268
## [449] 18.17424 18.12282 18.06835 18.01640 17.97844 17.97452 17.95696
## [456] 17.93786 17.94109 17.94384 17.96124 18.00846 18.08680 18.18813
## [463] 18.30391 18.42160 18.52841 18.60735 18.67904 18.77526 18.88442
## [470] 19.00190 19.12575 19.24250 19.36170 19.49058 19.61779 19.74703
## [477] 19.86016 19.95554 20.10524 20.30198 20.52845 20.78453 21.03613
## [484] 21.28354 21.53148 21.78122 22.02788 22.24101 22.41281 22.58850
## [491] 22.76888 22.95031 23.13634 23.31640 23.54383 23.83231 24.12311
## [498] 24.39254 24.62735 24.86695 25.12534 25.23290 25.12167 24.88691
## [505] 24.62651 24.38013 24.13413 23.85922 23.52225 23.15650 22.78176
## [512] 22.43211 22.15847 21.94873 21.76953 21.59295 21.42242 21.26260
## [519] 21.05515 20.77659 20.48950 20.25910 20.10741 19.98056 19.83217
## [526] 19.79823 19.91251 20.06805 20.24762 20.42997 20.58167 20.72728
## [533] 20.89907 21.06261 21.19375 21.29650 21.39926 21.50715 21.59729
## [540] 21.66259 21.70744 21.74958 21.81200 21.91381 22.04061 22.15161
## [547] 22.23165 22.28557 22.31084 22.16681 21.83251 21.47319 21.10022
## [554] 20.75405 20.49795 20.30745 20.16649 20.07043 20.01351 19.99596
## [561] 19.96985 19.91319 19.84359 19.77907 19.74664 19.73668 19.73089
## [568] 19.71694 19.69163 19.65946 19.62590 19.59408 19.55924 19.52858
## [575] 19.51138 19.47501 19.45720 19.50642 19.59275 19.66807 19.71412
## [582] 19.72540 19.72774 19.73431 19.72173 19.70268 19.71732 19.76846
## [589] 19.81859 19.86388 19.91130 19.97162 20.04469 20.11752 20.19624
## [596] 20.28439 20.36973 20.45367 20.53396 20.60679 20.68747 20.78251
## [603] 20.85406 20.90038 20.94251 20.98800 21.03426 21.08100 21.15219
## [610] 21.24801 21.34012 21.41085 21.47151 21.52256 21.56088 21.57643
## [617] 21.56851 21.55084 21.50790 21.44126 21.37450 21.31487 21.31630
## [624] 21.43903 21.64795 21.77939 21.79868 21.80429 21.80491 21.78725
## [631] 21.74766 21.69405 21.62359 21.53001 21.42252 21.31056 21.18952
## [638] 21.06844 20.96262 20.87835 20.80273 20.73434 20.69073 20.66506
## [645] 20.65037 20.68570 20.71832 20.63210 20.51491 20.44634 20.40814
## [652] 20.38639 20.36433 20.35222 20.34661 20.34690 20.34942 20.32882
## [659] 20.27980 20.23725 20.20676 20.16376 20.10349 20.03281 19.96076
## [666] 19.89117 19.82997 19.76900 19.71003 19.74071 19.78600 19.73094
## [673] 19.55529 19.38157 19.30113 19.22949 19.15371 19.09025 19.04849
## [680] 19.01140 18.98618 18.98748 19.00317 19.01432 19.02974 19.05998
## [687] 19.09365 19.12266 19.15177 19.16570 19.14895 19.11388 19.08099
## [694] 19.07667 19.23379 19.54847 19.88803 20.10169 20.16509 20.21245
## [701] 20.25622 20.28223 20.28905 20.26058 20.18901 20.13015 20.13025
## [708] 20.16510       NA       NA       NA       NA       NA       NA
## [715]       NA       NA       NA       NA       NA       NA
## 
## $random
## Time Series:
## Start = c(1, 1) 
## End = c(30, 24) 
## Frequency = 24 
##   [1]           NA           NA           NA           NA           NA
##   [6]           NA           NA           NA           NA           NA
##  [11]           NA           NA -0.221730503 -1.851056787 -1.033924346
##  [16] -0.693383092 -0.347876085  0.180204528  0.653710821  1.368726026
##  [21]  1.661519109  1.771309526  1.172809702  0.498758532  0.254392575
##  [26] -0.274127161 -1.407679506 -1.652503882 -1.682270457 -2.145139856
##  [31] -2.574777389 -2.836557388 -1.665534988  3.863130527  4.769209661
##  [36]  4.365616094  1.832288854  1.200275080  0.085346218 -0.315158064
##  [41] -0.506803621 -0.287215961  0.185128074  0.702178256  0.710963558
##  [46] -0.439784006 -1.170727692 -1.563942452 -1.798134752 -1.081587324
##  [51] -1.524282042 -2.445659833 -2.885869178 -3.134139467 -3.242961133
##  [56] -3.193301160 -2.136390660  2.909138879  5.348975166  4.866575038
##  [61]  3.971246844  2.804644476  1.464132722  0.795957943  0.384266268
##  [66]  0.369311397  0.611728605  0.868808885  0.815886204 -0.361386381
##  [71] -1.058932986 -1.144207224 -1.429494884 -2.054107139 -2.713577346
##  [76] -2.707113007 -2.850117169 -3.053198640 -3.225814716 -3.264793047
##  [81] -2.068097182  2.648141384  6.349923846  8.462725186  5.019833424
##  [86]  4.101628158  2.583102433  1.972492443  1.457551016 -0.381679168
##  [91] -1.416626027 -1.115661097 -1.005708895 -0.760361952 -0.917366274
##  [96] -1.273674716 -1.218393382 -0.735311081 -0.732618894 -0.687422768
## [101] -0.703790339 -0.855199309 -0.064593510  0.333652035  0.200882324
## [106] -0.921878354  2.740191034 -0.177389680 -0.739930107  0.017842979
## [111] -0.032931223  0.110888439 -1.008804726 -0.422124496 -0.174537529
## [116]  0.212658542  0.537708908  0.738806749  0.694257454  0.124716974
## [121] -0.450571223 -0.916397239 -1.404075900 -1.715725419 -1.753088813
## [126] -2.543067333 -2.924108997 -3.215577277 -1.385602411  1.132829196
## [131]  4.349006097  4.245988338  1.980284061  0.879325576 -0.326522141
## [136] -0.615582939  0.027831263 -0.266227017  0.074930376 -0.429306248
## [141] -0.531477205 -0.404340369  0.008290222  0.981898229  1.008452143
## [146]  1.917396287  1.982061841  1.372681731  1.371781330  1.130254962
## [151]  1.526861734  1.858225860  1.076863679 -3.532008383 -5.538648958
## [156] -3.019901663 -0.356167660 -0.890128777 -0.571470242 -0.128500774
## [161] -0.219050782  0.026813048  0.009291879 -0.151039786  0.007520294
## [166]  0.367365151  0.116530928  0.374748399  0.930630281  0.942991125
## [171]  0.697756723  1.406404708  1.942643688  1.888040092  1.816878613
## [176]  1.824054985  1.502173455 -2.556488963 -3.866573295 -4.555808902
## [181] -2.029166931 -0.481231931 -0.073616548  0.346442570  0.445359930
## [186] -0.200992152  0.280804051  0.613034405  1.067183248  1.319754428
## [191]  1.406751508  1.143546705  0.870885198  0.149999657 -0.424570312
## [196] -0.717989635 -0.933899785 -0.912761907 -1.141897362 -0.933250848
## [201] -1.349413486 -6.153547809 -5.349358338 -1.415537959  0.421791421
## [206]  2.164888796  2.147259967  2.075163445  1.713020078  1.285895776
## [211]  1.312585326  1.187521709  1.173854585  1.092394988  0.073729077
## [216] -1.263421831 -1.410704942 -2.624803336 -2.014898254 -1.818220009
## [221] -1.960530830 -1.779345182 -2.172346566 -1.207375147 -1.914029869
## [226]  3.585584866  9.946793095  8.828653103  3.532029553  2.086768533
## [231]  0.211641420 -0.167560165 -0.516546430 -0.614695515 -0.369780634
## [236] -0.291805323 -0.633959782 -1.566381147 -1.826788269 -2.194493936
## [241] -2.154821433 -1.278163692 -1.562932876 -2.449899748 -3.294733812
## [246] -3.180487746 -2.747914222 -2.685658930 -3.505481162 -0.996968707
## [251] -0.141389388  3.422857980  5.380001839  7.117724227  5.756063616
## [256]  4.520412821  4.078763531  3.851538783 -0.582215973 -0.529523111
## [261] -2.150776463 -2.668496884 -2.308561892 -2.126881984 -1.588073348
## [266] -1.436513855 -1.570673374 -1.016828618  0.002522830 -0.403710449
## [271] -0.616959851 -1.219635400  0.215516832  2.451002406 -1.442831736
## [276] -0.574787808 -0.842541487  1.380874062  1.961091657  1.292911550
## [281]  1.142180635  1.483498238  1.042726853  0.950746549  0.343531155
## [286] -0.128783370 -0.104177568 -0.202093270  0.355342650  0.710724001
## [291]  0.720517979  0.716069382  0.571835547  0.557165861  1.050962350
## [296]  1.380947932  0.288699890 -3.496962353 -5.314084360 -4.598918639
## [301] -0.157008193  0.725817237  0.375006469  0.233712640 -0.187809875
## [306] -0.006869962 -0.026742259  0.499071276  0.667018600  0.789406860
## [311]  0.587768472 -0.212543734 -0.594183281 -0.632455752 -0.749430230
## [316] -0.237360845 -0.312980100 -0.393176630 -0.751901940 -0.870029578
## [321] -0.547604123 -0.233176620  2.005316751  2.634290266  1.157500453
## [326]  0.292981135  0.519949746  0.370591997  0.289040013  0.360170828
## [331]  0.688849587  0.868130290  0.029663248 -0.106576360 -0.118229348
## [336] -0.576471699 -1.144607785 -1.224380033 -1.334247239 -1.705556262
## [341] -1.827408928 -1.993199519 -2.307349352 -3.065029567 -1.732576346
## [346]  2.033128042  4.206505073  4.455273186  2.907001621  1.697322717
## [351]  0.790032529  0.657017059  1.276909443  1.305626892  1.843576651
## [356]  1.476303378  0.652015078  0.160821817 -0.590381599 -1.291015449
## [361] -1.686056487 -2.377617955 -2.499662638 -2.646269270 -3.025062671
## [366] -3.148382408 -3.280745087 -4.029484047 -1.827779880  2.656496608
## [371]  4.847951678  5.470583630  2.760377871  2.256190275  1.403294548
## [376]  1.592418285  1.129123442  0.891892203  0.459445870 -0.491336702
## [381] -1.618516796 -1.141613919 -0.070061338  0.329149860  0.690373599
## [386]  0.622115398  0.673359276  1.156011357  1.250329215  1.858804055
## [391]  2.118760226  1.980038171  1.270340686 -2.829747899 -4.670660319
## [396] -4.630460670 -3.733463852 -1.921461555  0.014083929  0.251283311
## [401]  0.215716695  0.362373536  0.591102919  0.651725140  0.901978957
## [406]  0.859019194  0.777778115  0.291003997  0.008418272 -0.290765282
## [411] -0.378894737 -0.264536284 -0.225987589 -0.278303479 -0.381279258
## [416] -0.760436141 -0.300856011  1.667034791  1.800474364  2.173993892
## [421] -0.087052791 -1.042069889 -1.306840131 -1.108475170 -0.905282981
## [426] -0.689230381 -0.421379732 -0.408975070  0.081934015  0.189872647
## [431]  0.158804967 -0.126800704 -0.182444619 -0.214456596 -0.287435939
## [436] -0.264093254 -0.301439468 -0.140704301 -0.196486171 -0.429998304
## [441]  0.211769430  1.853932022  2.356101408  1.867493093  0.135806873
## [446] -0.706399991 -0.670199953 -0.745963700 -0.583489824 -0.469209385
## [451] -0.031319106 -0.021915190  0.071818442  0.180108847  0.288812528
## [456] -0.024482592 -0.080149225 -0.469909544 -0.726211798 -0.732661388
## [461] -0.988086806 -1.418540612 -1.255627841 -1.826268574 -0.413015488
## [466]  1.874793263  0.888243231  1.615333222 -0.495971905 -1.245276810
## [471] -0.893901004 -0.163070175  0.376835396  0.879162385  1.260463668
## [476]  1.055251876  0.509417166 -0.331114807  0.111694470  0.259054726
## [481] -0.075659026 -0.263264278 -0.903857010 -1.300854235 -1.417960409
## [486] -1.840317920 -2.045073685 -2.376464113 -2.136546598  0.311242150
## [491]  1.566196170  2.116299915  1.446251653  0.538044018  0.667211878
## [496]  1.143431314  1.620114179  1.960305917  2.107604826  0.308761102
## [501] -0.882826644 -1.048263906 -0.806659071  0.284314074  0.145515836
## [506]  0.464881518  2.591046683  3.377753141  3.295499257  3.012844146
## [511]  2.243375032  3.891024258  4.061564089  0.170317107 -1.992611320
## [516] -3.237051020 -2.628210408 -3.445982003 -2.614082575 -3.226669985
## [521] -3.495100052 -2.713287635 -2.552968752 -2.407506487 -1.122018181
## [526] -0.148557629  0.335708790  0.694448431  0.747400455  0.520741417
## [531]  0.080083848 -0.797383251 -0.554580240  0.520544211  1.136736708
## [536]  1.632278037  1.092496644  2.710973298  1.566127442  0.772492464
## [541]  1.626842126  0.280165710 -0.302746287 -0.442823271 -0.721118492
## [546] -1.081104328 -1.907240839 -2.550330769 -0.830436981 -0.108625483
## [551]  0.334020106  0.505177855  0.831981121  1.282030425  2.074589989
## [556]  2.598310709  3.288584377  3.730094518  3.941732904  3.896116945
## [561]  2.771860470 -2.858482813 -5.563208287 -5.708440530 -5.950635992
## [566] -4.784772147 -3.436386422 -2.175025884 -1.209602173 -0.361920187
## [571] -0.260995494  0.258200275  0.550744174  1.180828767  0.663096615
## [576]  1.398208214  2.023459545  2.503129317  2.728314851  2.819673087
## [581]  2.944258904  3.327602605  3.363939170  3.494170565  2.011087116
## [586] -3.110853395 -5.799331684 -7.081486413 -5.493825690 -3.078150447
## [591] -1.306791374 -1.123927732 -0.658283102 -1.182700242 -0.356331741
## [596] -0.592059760 -0.703333357 -0.215007334  0.813592566  1.548187686
## [601]  1.917842126  2.275953349  2.694254760  3.255732438  3.554690849
## [606]  3.722011080  4.178892936  4.257392931  2.567007689 -2.613431568
## [611] -5.610926096 -7.039085172 -4.959061989 -2.362486859 -1.896435918
## [616] -1.565200246 -1.323641988 -1.291059892 -0.772277669 -0.400925895
## [621]  0.360741067  1.454540531  1.921569050  2.220928955  2.363911031
## [626]  2.323097837  2.544967555  2.302712724  2.361073150  2.406101683
## [631]  1.920648888  1.990791256  0.544633010 -4.206498812 -4.313996181
## [636] -2.427112779  0.839562057 -1.115739662 -1.165114080 -0.730835202
## [641] -0.664537566 -1.215245004 -1.114968232 -1.038237921 -0.883488708
## [646] -0.439625643 -0.116601544  0.289744945  0.425110847  0.916199333
## [651]  1.596183328  2.015012335  1.877416437  2.482653350  2.586595261
## [656]  2.841239457  1.610445259 -1.101353830 -3.509385667 -5.154543515
## [661] -0.002028251 -1.677977557 -0.672549848 -0.562881047 -0.203502171
## [666] -0.572452964 -0.323288725 -0.058967702  0.094612498 -0.521448246
## [671] -0.510213706  0.474278735  0.637681211  0.664203211  1.127115039
## [676]  1.355174967  1.446405187  2.046012147  2.645569031  2.489175580
## [681]  1.831082071  2.855031915 -2.673956549 -6.133086820 -5.054884059
## [686] -2.682619732 -1.415566208 -1.338527537 -1.346123187 -0.941640409
## [691] -0.552137965 -0.274387814  0.383571540  0.545276765  0.392625545
## [696]  0.841002112  0.861036329  0.839536225  0.983935055  1.043679964
## [701]  1.069904657  0.796684883  0.658023058  0.303890343 -0.014355606
## [706]  2.147761885  3.305085994  0.714574994           NA           NA
## [711]           NA           NA           NA           NA           NA
## [716]           NA           NA           NA           NA           NA
## 
## $figure
##  [1] -2.6888577 -3.2343496 -3.6763473 -4.2148717 -4.6626931 -4.9911641
##  [7] -5.3420451 -5.4160386 -3.7808564  1.6151991  4.7115420  6.3767990
## [13]  5.8751578  4.3816065  3.6004367  3.4644348  3.5388148  3.3379867
## [19]  2.7280069  1.9121318  0.6178873 -0.6095121 -1.4606554 -2.0826124
## 
## $type
## [1] "additive"
## 
## attr(,"class")
## [1] "decomposed.ts"

## 
##  Shapiro-Wilk normality test
## 
## data:  decompEG$random
## W = 0.97655, p-value = 4.045e-09

ARMA : “auto-regression” et “moving average”

=> seuleument pour les séries temporelles stationnaires :

  • pas de tendance
  • variance stable

prévisions avec ARIMA : auto-regressive integrated moving average

AR : auto-regressive (p = délais dans l’autocorrélation)

I : integrated (d = différenciation ; rendre stationnaire une ts)

MA : moving average (q = délais dans l’accumulation des erreurs)

Prévisions avec ARIMA

Prévisions avec Holt-Winters Exponential Smoothing

Pour approfondir

Forecasts using Exponential Smoothing

La cross-corrélation

Corrélation entre deux variables avec délais ?

D’après E. E. Holmes, M. D. Scheuerell, and E. J. Ward, 2019. Applied Time Series Analysis for Fisheries and Environmental Sciences

lynx : Annual numbers of lynx trappings for 1821–1934 in Canada. Taken from Brockwell & Davis (1991), this appears to be the series considered by Campbell & Walker (1977).

sunspot.year : Yearly numbers of sunspots from 1700 to 1988 (rounded to one digit).

Pour approfondir

Applied Time Series Analysis for Fisheries and Environmental Sciences

TD

Tester la cross-correlation et les auto-corrélations pour les données de bdd.csv pour les valeurs agrégées par jour des mois de juillet et août 2019, puis par heure.

Le modèle linéaire généralisé