Time series manipulation¶

The objective here is to create and manipulate a time series. A time series is a particular field where the mesh $\mathcal{M}$ 1-d and regular, eg a time grid $(t_0, \dots, t_{N-1})$ .

It is possible to draw a time series, using interpolation between the values: see the use case on the Field.

A time series can be obtained as a realization of a multivariate stochastic process $X: \Omega \times [0,T] \rightarrow \mathbb{R}^d$ of dimension $d$ where $[0,T]$ is discretized according to the regular grid $(t_0, \dots, t_{N-1})$ . The values $(\underline{x}_0, \dots, \underline{x}_{N-1})$ of the time series are defined by:

$\forall i \in [0, N-1],\quad \underline{x}_i= X(\omega)(t_i)$

A time series is stored in the TimeSeries object that stores the regular time grid and the associated values.

[1]:

from __future__ import print_function
import openturns as ot
import math as m

[2]:

# Create the RegularGrid
tMin = 0.
timeStep = 0.1
N = 100
myTimeGrid = ot.RegularGrid(tMin, timeStep, N)

[15]:

# Case 1: Create a time series from a time grid and values
# Care! The number of steps of the time grid must correspond to the size of the values
myValues = ot.Normal(3).getSample(myTimeGrid.getVertices().getSize())
myTimeSeries = ot.TimeSeries(myTimeGrid, myValues)
myTimeSeries

[15]:

[ t X0 X1 X2 ]
0 : [ 0 2.31733 -0.469879 -0.801703 ]
1 : [ 0.1 -0.0108486 -0.0553915 -1.7548 ]
2 : [ 0.2 -0.894538 -0.360249 -0.770978 ]
3 : [ 0.3 0.55883 1.62925 1.34049 ]
4 : [ 0.4 0.494815 -0.0180664 -0.0365295 ]
5 : [ 0.5 -1.01369 1.03588 -0.12276 ]
6 : [ 0.6 -0.177026 1.4058 0.0981434 ]
7 : [ 0.7 1.01776 0.833219 2.12309 ]
8 : [ 0.8 -1.65891 1.91107 -0.499354 ]
9 : [ 0.9 -0.707992 -0.836693 -0.08448 ]
10 : [ 1 -0.0772068 -0.754153 1.35719 ]
11 : [ 1.1 -0.262538 0.43282 -1.88328 ]
12 : [ 1.2 1.1694 -0.0711754 -1.57923 ]
13 : [ 1.3 1.07961 0.818312 -0.769237 ]
14 : [ 1.4 -1.63959 -2.49198 0.392813 ]
15 : [ 1.5 1.54502 -0.592087 2.36899 ]
16 : [ 1.6 -0.81979 -0.103824 -0.98551 ]
17 : [ 1.7 0.987314 -0.346144 -1.34504 ]
18 : [ 1.8 1.07011 0.927288 -0.155148 ]
19 : [ 1.9 -0.461614 1.68461 0.712465 ]
20 : [ 2 0.929992 -0.483778 -0.174459 ]
21 : [ 2.1 0.662006 -0.698763 -0.0269453 ]
22 : [ 2.2 -1.28184 0.386923 0.152988 ]
23 : [ 2.3 -0.467685 -0.514194 0.384462 ]
24 : [ 2.4 0.261137 -0.578211 -1.33245 ]
25 : [ 2.5 0.864973 0.941687 0.329529 ]
26 : [ 2.6 0.0412185 0.141166 -0.0393573 ]
27 : [ 2.7 0.27118 -0.383442 0.760954 ]
28 : [ 2.8 0.0351336 -0.295452 0.13985 ]
29 : [ 2.9 -0.863809 1.23659 1.40946 ]
30 : [ 3 0.681355 -0.220565 1.25437 ]
31 : [ 3.1 -0.583736 -1.26391 -0.235933 ]
32 : [ 3.2 -0.633479 2.00508 -1.06159 ]
33 : [ 3.3 -0.973466 0.182045 0.466031 ]
34 : [ 3.4 -2.404 -0.675804 -0.43258 ]
35 : [ 3.5 0.615584 0.946263 0.254775 ]
36 : [ 3.6 -0.173574 0.741881 -0.0834884 ]
37 : [ 3.7 -1.23036 -1.46178 -0.519229 ]
38 : [ 3.8 1.90887 -1.15193 1.02327 ]
39 : [ 3.9 0.169987 -0.0977635 -0.423342 ]
40 : [ 4 -0.638673 -0.260154 -1.43577 ]
41 : [ 4.1 1.15734 -1.18448 -0.242118 ]
42 : [ 4.2 -1.75983 0.224354 -0.497826 ]
43 : [ 4.3 0.894821 0.595231 -0.248979 ]
44 : [ 4.4 0.0926772 -0.824599 0.824924 ]
45 : [ 4.5 -1.64514 -0.313514 1.07257 ]
46 : [ 4.6 -0.281387 1.01541 -0.343512 ]
47 : [ 4.7 1.73504 -1.54906 -0.570394 ]
48 : [ 4.8 -0.634918 0.397189 -0.45656 ]
49 : [ 4.9 -0.853559 0.0403166 -0.299548 ]
50 : [ 5 0.450826 0.0339842 -0.428601 ]
51 : [ 5.1 0.877138 0.262852 0.269669 ]
52 : [ 5.2 -0.86813 -0.323261 -0.719086 ]
53 : [ 5.3 -1.67268 -0.0197131 -1.12035 ]
54 : [ 5.4 -0.597611 -0.330729 -0.0366483 ]
55 : [ 5.5 -0.0950169 0.912443 1.13421 ]
56 : [ 5.6 0.159206 -0.617572 1.47189 ]
57 : [ 5.7 1.41735 -0.150357 -0.463105 ]
58 : [ 5.8 -1.24986 -1.85139 1.21562 ]
59 : [ 5.9 -0.383809 -0.135983 -0.966214 ]
60 : [ 6 3.0788 -0.710827 -0.216468 ]
61 : [ 6.1 -0.226418 0.0258637 0.921209 ]
62 : [ 6.2 0.0419902 0.103587 0.249475 ]
63 : [ 6.3 -0.667934 0.12365 -0.14698 ]
64 : [ 6.4 -0.271669 0.485002 0.201355 ]
65 : [ 6.5 -0.259562 0.00503527 -1.99632 ]
66 : [ 6.6 -0.735897 0.244483 -1.93819 ]
67 : [ 6.7 0.553995 -0.879113 1.10496 ]
68 : [ 6.8 0.533898 0.043558 -0.593256 ]
69 : [ 6.9 0.955203 1.44755 -0.420213 ]
70 : [ 7 -1.02309 -0.567824 -0.632136 ]
71 : [ 7.1 -0.671702 -0.562391 0.0522118 ]
72 : [ 7.2 0.567886 -0.804104 -1.0274 ]
73 : [ 7.3 0.0207672 1.36971 0.726692 ]
74 : [ 7.4 1.2459 1.99545 -0.41103 ]
75 : [ 7.5 0.266026 1.93123 0.98723 ]
76 : [ 7.6 0.407197 0.223757 -0.806872 ]
77 : [ 7.7 -0.880356 -1.38558 1.47653 ]
78 : [ 7.8 0.487294 -0.616603 -0.270488 ]
79 : [ 7.9 2.22406 0.934046 1.08349 ]
80 : [ 8 0.412462 -1.05112 -0.246497 ]
81 : [ 8.1 0.236328 0.880083 -0.623114 ]
82 : [ 8.2 -1.05763 -0.325425 0.72565 ]
83 : [ 8.3 -1.21886 0.220351 2.01841 ]
84 : [ 8.4 0.759168 2.49498 0.803174 ]
85 : [ 8.5 -1.08808 -0.321161 2.63821 ]
86 : [ 8.6 -0.66262 -0.645395 -0.440328 ]
87 : [ 8.7 1.31516 0.381884 0.226424 ]
88 : [ 8.8 0.346054 0.743799 -1.39891 ]
89 : [ 8.9 0.124891 0.507658 -0.102776 ]
90 : [ 9 0.191792 -0.310295 0.0640703 ]
91 : [ 9.1 -1.09636 -0.302375 0.387491 ]
92 : [ 9.2 -1.44132 -0.573967 -0.098096 ]
93 : [ 9.3 0.230417 -1.3587 -0.984013 ]
94 : [ 9.4 1.16441 -0.634025 0.0382606 ]
95 : [ 9.5 0.67274 -0.22672 0.504295 ]
96 : [ 9.6 -1.35244 -0.0812011 0.472061 ]
97 : [ 9.7 1.42725 0.112269 -0.870752 ]
98 : [ 9.8 -0.247597 0.139603 1.04401 ]
99 : [ 9.9 0.0305908 -0.845472 -0.273887 ]

[4]:

# Case 2: Get a time series from a Process
myProcess = ot.WhiteNoise(ot.Normal(3), myTimeGrid)
myTimeSeries2 = myProcess.getRealization()
myTimeSeries2

[4]:

[ t X0 X1 X2 ]
0 : [ 0 1.73945 0.935939 1.27511 ]
1 : [ 0.1 -0.595004 -0.0230083 1.85337 ]
2 : [ 0.2 0.356188 -1.31765 -1.195 ]
3 : [ 0.3 -0.443684 -0.826204 -0.748816 ]
4 : [ 0.4 0.434034 -0.644208 -0.559902 ]
5 : [ 0.5 0.0546682 -0.0564792 0.757535 ]
6 : [ 0.6 -0.663158 0.683646 0.591043 ]
7 : [ 0.7 -2.20872 -0.779031 -0.703086 ]
8 : [ 0.8 -0.0566955 0.588101 0.738839 ]
9 : [ 0.9 0.727128 -1.1831 0.853199 ]
10 : [ 1 1.03198 0.104467 0.51551 ]
11 : [ 1.1 -1.73251 0.369231 0.667104 ]
12 : [ 1.2 -0.51281 -0.477754 1.26264 ]
13 : [ 1.3 1.57846 1.89005 0.390297 ]
14 : [ 1.4 0.235759 0.244948 0.57169 ]
15 : [ 1.5 -0.420542 0.553255 -2.19382 ]
16 : [ 1.6 0.421279 0.753758 -0.512419 ]
17 : [ 1.7 -0.306225 -1.21103 0.0174802 ]
18 : [ 1.8 -1.16501 0.0643894 0.717609 ]
19 : [ 1.9 -0.304268 -0.391375 0.258215 ]
20 : [ 2 -1.49078 1.06182 0.374969 ]
21 : [ 2.1 -0.165467 0.357871 0.885118 ]
22 : [ 2.2 1.7379 -0.713196 1.70642 ]
23 : [ 2.3 0.428636 -0.277595 0.411295 ]
24 : [ 2.4 0.370445 0.187854 1.43266 ]
25 : [ 2.5 -0.416128 -0.122868 0.950488 ]
26 : [ 2.6 -0.411128 -0.98049 0.564196 ]
27 : [ 2.7 -1.55666 0.624888 1.05797 ]
28 : [ 2.8 -0.769529 1.54237 0.374802 ]
29 : [ 2.9 0.506105 0.419768 1.58128 ]
30 : [ 3 0.00956375 -0.38302 0.163699 ]
31 : [ 3.1 1.2561 0.00620397 0.272004 ]
32 : [ 3.2 -0.153784 -0.404166 2.09224 ]
33 : [ 3.3 0.675043 -0.383208 -0.355239 ]
34 : [ 3.4 -1.3053 -1.51577 0.172158 ]
35 : [ 3.5 0.57768 -0.173049 0.662172 ]
36 : [ 3.6 0.697838 0.789578 0.479307 ]
37 : [ 3.7 -0.449433 1.7657 0.281658 ]
38 : [ 3.8 0.127981 -0.747966 0.512891 ]
39 : [ 3.9 1.08613 -0.551952 1.57891 ]
40 : [ 4 -0.222394 0.205503 1.33851 ]
41 : [ 4.1 0.045253 2.15649 1.00985 ]
42 : [ 4.2 1.18898 1.28765 -0.279847 ]
43 : [ 4.3 -0.567342 0.0789431 -0.374098 ]
44 : [ 4.4 0.36891 -0.216249 -0.964493 ]
45 : [ 4.5 1.14255 0.271953 -0.62676 ]
46 : [ 4.6 -1.24554 -0.793869 0.808652 ]
47 : [ 4.7 0.551898 1.26214 -0.398731 ]
48 : [ 4.8 -2.13505 0.420672 -0.164956 ]
49 : [ 4.9 -1.27789 -0.748335 0.447628 ]
50 : [ 5 1.6065 -1.23832 -0.857113 ]
51 : [ 5.1 -0.796012 2.55533 -1.40966 ]
52 : [ 5.2 0.161757 1.07153 -0.0756152 ]
53 : [ 5.3 -0.38581 -0.695899 -0.240171 ]
54 : [ 5.4 0.0140339 -1.32783 -0.421805 ]
55 : [ 5.5 -1.31457 -0.314242 0.870097 ]
56 : [ 5.6 -1.05029 -0.661749 -0.398957 ]
57 : [ 5.7 -0.306869 2.24097 -1.84225 ]
58 : [ 5.8 -1.12629 -0.343091 1.35707 ]
59 : [ 5.9 -1.14144 0.719911 0.691288 ]
60 : [ 6 0.0157415 0.897854 -0.0268917 ]
61 : [ 6.1 -1.17253 -1.22323 1.33729 ]
62 : [ 6.2 -0.412087 1.40508 0.839941 ]
63 : [ 6.3 -0.171826 -0.612136 0.43319 ]
64 : [ 6.4 1.25989 -2.15135 0.638385 ]
65 : [ 6.5 1.1986 0.576051 -2.72106 ]
66 : [ 6.6 -0.456634 0.834452 0.189723 ]
67 : [ 6.7 -1.5599 0.141706 -0.312877 ]
68 : [ 6.8 -1.09669 1.30628 -0.603543 ]
69 : [ 6.9 0.0844715 -1.01157 -0.45413 ]
70 : [ 7 0.184323 1.11103 0.0332655 ]
71 : [ 7.1 -0.402727 0.812135 0.137958 ]
72 : [ 7.2 -0.552396 0.712298 0.657798 ]
73 : [ 7.3 -0.2971 0.288142 -0.843501 ]
74 : [ 7.4 1.4814 0.904859 0.90054 ]
75 : [ 7.5 0.211378 0.408217 -0.290709 ]
76 : [ 7.6 -1.03768 0.403674 -0.0361247 ]
77 : [ 7.7 -0.0439957 1.57857 1.44503 ]
78 : [ 7.8 -0.908343 -1.329 -0.476214 ]
79 : [ 7.9 1.02238 -1.19788 2.59603 ]
80 : [ 8 0.149855 -0.390626 -0.311655 ]
81 : [ 8.1 -0.451449 0.237096 0.624386 ]
82 : [ 8.2 -0.5553 0.765645 0.509255 ]
83 : [ 8.3 0.416132 -1.42826 -0.155242 ]
84 : [ 8.4 1.19865 -0.189811 -1.09981 ]
85 : [ 8.5 0.727195 0.566469 -1.437 ]
86 : [ 8.6 -0.253593 0.898235 0.732674 ]
87 : [ 8.7 -0.109548 -0.485969 -0.35672 ]
88 : [ 8.8 -1.25939 0.0349764 -0.409638 ]
89 : [ 8.9 0.795517 -2.10814 -0.914884 ]
90 : [ 9 1.09502 -0.0632852 2.00363 ]
91 : [ 9.1 -0.124881 0.0604086 -0.449676 ]
92 : [ 9.2 0.15909 -2.04317 1.06144 ]
93 : [ 9.3 -1.44415 0.751101 0.844058 ]
94 : [ 9.4 -1.23908 0.743606 1.43464 ]
95 : [ 9.5 0.0770002 -2.18739 0.237613 ]
96 : [ 9.6 0.805032 0.784197 1.02133 ]
97 : [ 9.7 -0.35903 -1.16653 1.01787 ]
98 : [ 9.8 0.275839 0.907344 0.359928 ]
99 : [ 9.9 1.17828 1.57853 2.0749 ]

[5]:

# Get the number of values of the time series
myTimeSeries.getSize()

[5]:

[6]:

# Get the dimension of the values observed at each time
myTimeSeries.getMesh().getDimension()

[6]:

[7]:

# Get the value Xi at index i
i = 37
print('Xi = ', myTimeSeries.getValueAtIndex(i))

Xi =  [-0.488205,-0.465482,0.332084]

[8]:

# Get the time series at index i : (ti, Xi)
i = 37
print('(ti, Xi) = ', myTimeSeries[i])

(ti, Xi) =  [-0.488205,-0.465482,0.332084]

[9]:

# Get a the marginal value at index i of the time series
i = 37
# get the time stamp:
print('ti = ', myTimeSeries[i, 0])
# get the first component of the corresponding value :
print('Xi1 = ', myTimeSeries[i, 1])

ti =  -0.4882047479037244
Xi1 =  -0.46548206392049335

[10]:

# Get all the values (X1, .., Xn) of the time series
myTimeSeries.getValues()

[10]:

	X0	X1	X2
0	0.6082016512187646	-1.2661731022166567	-0.43826561996041397
1	1.2054782008285756	-2.1813852346165143	0.3500420865302907
2	-0.3550070491856397	1.437249310140903	0.8106679824694837
3	0.79315601145977	-0.4705255986325704	0.26101793529769673
4	-2.2900619818700854	-1.2828852904549808	-1.311781115463341
5	-0.09078382658049489	0.9957932259165571	-0.13945281896393122
6	-0.5602056000378475	0.4454896972990519	0.32292503034661274
7	0.44578529818450985	-1.0380765948630941	-0.8567122780208447
8	0.4736169171884015	-0.12549774541256004	0.35141776801611424
9	1.7823586399387168	0.070207359297043	-0.7813664602347197
10	-0.7215334320191932	-0.24122348956683715	-1.7879641991225605
11	0.4013597437727011	1.367825523115209	1.0043431266393292
12	0.7415483840772669	-0.043612337965035816	0.5393446780678338
13	0.2999504132732002	0.40771715787264406	-0.48511195560000947
14	-0.38299201615804984	-0.7528165991274193	0.25792642458131404
15	1.9687596027732095	-0.6712905330814658	1.8557922404430598
16	0.05215932560232187	0.790445824639787	0.7163525542248972
17	-0.7436220201526069	0.18435600073165376	-1.530734483444384
18	0.6550274609707977	0.5380714996799529	1.738212597715617
19	-0.9587222803460276	0.3779221311281753	-0.1810041948595859
20	1.6729651981107247	-1.0389584433401373	-0.3535524450885604
21	1.2138135523904243	-0.7770331122619051	-1.3685314491601195
22	0.1034744243555785	-0.8918195408099401	0.905601680864366
23	0.33479445398050983	-0.4836415989409058	0.6779582665237674
24	1.7093788628314481	1.07062031445028	-0.506924729621585
25	-1.6608639532729972	2.2462287040616475	0.7596015459405085
26	-0.510763820034195	-0.6330662015316597	-0.9570721500415863
27	0.5440465956139983	0.8145606872090909	-0.734708377960548
28	-0.11146080893135524	0.9944818782078243	-0.16062532311409344
29	-0.9387705743377833	-1.968691542444217	-0.6576034692788746
30	0.3387510861377688	1.0155577081272245	0.6371672460165964
31	-0.08990711915988879	-0.8558864367182618	1.2712829469472742
32	-0.2382525840159452	1.3262988324629563	2.119675592985771
33	-0.9015813901152602	-1.5169648184616176	-1.2993795893316618
34	0.23037243903915736	-3.097374220265686	0.013229996975342417
35	-1.2574295730740137	1.0277604824132867	-0.7664306747065759
36	0.21751211240280865	1.0453331498743215	0.3315688008777966
37	-0.4882047479037244	-0.46548206392049335	0.33208385645183086
38	-0.16772582180258397	3.012626743702593	0.9420404910001581
39	0.6118901404192247	0.6117151564272073	-1.5374973590145564
40	-2.406701632898191	0.6629359832717704	-0.6561602005666336
41	-0.7516114592509675	0.4381769975654701	-0.4553345919601327
42	1.8603777479857533	0.21972123726271536	1.725462994560952
43	-0.5434054870737179	-0.7367488371221942	-0.5082064350443634
44	-2.258672040864401	-0.5963998333071584	-0.3146800263075891
45	-1.782738221104927	-0.6847337811346507	0.061115707883683305
46	0.8737197289205034	-1.4629534274978957	-0.3187856285938157
47	1.263141994155593	-0.42672559757717077	-1.8923429557066818
48	-0.5143905498504552	0.6472293778560264	0.0037024864831827547
49	0.7296877765012115	-0.2472337723951604	0.4791910385893783
50	-0.03360982636241379	-0.03672706471415314	0.11025616355141858
51	-0.3768703589700057	-0.09558941438605024	0.10912241770686365
52	-0.198754145656359	0.4736195376455026	0.16163728851497228
53	0.3844829079369488	0.11646763738621115	-0.10008045244292718
54	1.491564303161293	1.2230054393315812	0.526646331161638
55	-0.6569234219822457	-0.13122822803845577	-1.453470810242808
56	1.1741448211259624	0.9293947008749075	-0.3371134579838932
57	0.5786882960372883	-0.5824589478380001	-1.388861072987769
58	-0.4997483415305477	-1.5551578034867246	0.48308301365700856
59	0.20500419543073403	-0.09725248024468533	0.5925630820144326
60	-0.6020444666415793	-1.2100861975899475	-0.8866978667261813
61	-0.14111369030042462	0.4419834280530504	0.5191620132847282
62	-1.5145511107546767	-0.6769173970437052	0.6676776291109286
63	-1.4058454178947792	-0.02953347109931314	-0.6318286705576991
64	-0.34215693175874956	2.0533857387192125	1.1587027449019125
65	-1.457170091897887	-0.8443667458052184	-0.28861027376722903
66	0.4192711129639974	-0.8360643654354077	0.8582685706406981
67	-0.9065659143816405	-0.9168098650471502	1.1632207448691083
68	0.30191831259808755	0.49033126865016113	0.4754245747137407
69	-0.7887042563466362	-0.6694491869900636	-0.1379279554775434
70	-0.9715312223420239	-1.1878377947381524	1.428203280692695
71	-0.5892298619089735	-1.7321759670632966	0.8249934440800486
72	3.0279901522056356	1.6947972773244453	-1.6482672001700636
73	-0.9964693298966726	0.7731214398905927	-0.5194757998273593
74	-0.03519734280661111	-0.43986558279364807	-0.259332190949003
75	-0.8754189647636796	-2.5398626160475524	-0.056670934740152897
76	-0.021727906394664178	0.5992195203104552	0.1468678550666206
77	-0.7453603509332029	-0.5215960035243518	0.5920202454509629
78	-0.4700387195138181	-2.172109456702745	-0.43261732336113257
79	0.2677501726668674	-0.367989726099373	1.1484168621665602
80	-0.034328296438353126	0.46108184356376175	-0.6224243706780185
81	-1.6250556819326551	-0.543099206860823	-0.2695348828774449
82	0.02088179929670331	0.6238537218570768	0.7671373238839353
83	0.8887983653765579	1.4803058800425652	0.6610018495637071
84	1.408952186863387	0.5761247344316582	1.8932649144636948
85	0.8586112321424536	-0.9073478946471832	-0.5375026761534027
86	-0.6384336188895413	1.348563813060152	-2.2660834614598864
87	0.4232323917550925	-0.9961412477351472	-1.087506356723854
88	0.11107999528804083	0.677663085654946	-1.0550178569789195
89	-0.004096591671766315	0.56283328125277	-0.029615966971302244
90	0.07020645731636742	-0.235269945857301	-1.2903078142197937
91	-1.0186404579648396	-1.7113094070040313	0.9433259509800255
92	-0.5423194949602862	-0.9991112989354737	-1.4045695396487468
93	1.9460618018795466	0.7795719081491099	1.1384752363617938
94	0.711147617175394	-0.4533857740632992	0.6183193642974254
95	0.7220435333040667	0.6600207498477129	0.4659188068304761
96	-0.40772976610396877	1.4591902726984554	-0.4115654109111582
97	0.5494386852470321	1.4501932963738728	-0.327249469535079
98	-1.3979567876733754	1.301150630693703	-0.48525934352978206
99	-0.2724066372034578	-0.33882294820034387	-0.7907566133559019

[11]:

# Compute the temporal Mean
# It corresponds to the mean of the values of the time series
myTimeSeries.getInputMean()

[11]:

[-0.025392,-0.027327,-0.0146978]

[12]:

# Draw the marginal i of the time series using linear interpolation
myTimeSeries.drawMarginal(0)

[12]:

../../_images/examples_probabilistic_modeling_timeseries_manipulation_13_0.png

[13]:

# with no interpolation
myTimeSeries.drawMarginal(0, False)

[13]:

../../_images/examples_probabilistic_modeling_timeseries_manipulation_14_0.png

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Time series manipulation¶