Note

Click here to download the full example code

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.

from __future__ import print_function
import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt
import math as m
ot.Log.Show(ot.Log.NONE)

Create the RegularGrid

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

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

[ t X0 X1 X2 ]
0 : [ 0 -0.182484 -0.440356 1.13247 ]
1 : [ 0.1 0.674547 1.34247 1.39874 ]
2 : [ 0.2 0.736782 -0.252381 1.3757 ]
3 : [ 0.3 -0.858775 0.00109033 0.579508 ]
4 : [ 0.4 0.383663 0.494226 0.707441 ]
5 : [ 0.5 -2.79738 3.04752 -0.0763745 ]
6 : [ 0.6 -1.14715 2.57154 1.52295 ]
7 : [ 0.7 -0.412344 0.197543 -1.32056 ]
8 : [ 0.8 0.0784782 1.15086 0.357457 ]
9 : [ 0.9 0.80829 -0.846094 0.197997 ]
10 : [ 1 -0.309455 -0.798818 -1.49124 ]
11 : [ 1.1 0.908445 -0.572821 1.15664 ]
12 : [ 1.2 0.50848 -1.56675 -0.0885204 ]
13 : [ 1.3 -0.439673 -0.216443 0.992476 ]
14 : [ 1.4 1.24972 -0.0733184 1.40725 ]
15 : [ 1.5 -0.461717 1.11996 -1.24464 ]
16 : [ 1.6 0.773224 -0.531869 -0.52065 ]
17 : [ 1.7 0.383878 0.590867 0.971656 ]
18 : [ 1.8 0.42521 -0.679881 -0.224992 ]
19 : [ 1.9 0.911013 -0.0652468 -1.73899 ]
20 : [ 2 0.0937754 0.11441 0.44022 ]
21 : [ 2.1 -0.01304 -1.13674 0.638542 ]
22 : [ 2.2 0.637729 -0.495112 -0.512163 ]
23 : [ 2.3 -0.975287 -0.810021 0.330267 ]
24 : [ 2.4 -1.91617 0.0245578 0.862084 ]
25 : [ 2.5 0.863564 0.257896 0.263776 ]
26 : [ 2.6 -0.168931 -0.522402 0.468233 ]
27 : [ 2.7 -0.140062 1.35497 -0.597185 ]
28 : [ 2.8 -1.11299 -1.42881 2.20836 ]
29 : [ 2.9 -1.23704 -0.445937 0.762529 ]
30 : [ 3 -0.756441 0.394736 -0.89202 ]
31 : [ 3.1 -0.635901 -0.737606 -0.191263 ]
32 : [ 3.2 -0.427053 -0.299966 -1.87562 ]
33 : [ 3.3 -1.15349 0.631904 -0.255809 ]
34 : [ 3.4 0.527049 -0.525721 1.36335 ]
35 : [ 3.5 -0.549281 0.734446 0.517316 ]
36 : [ 3.6 0.0623652 -1.39255 -2.64948 ]
37 : [ 3.7 -0.330858 -1.40182 0.339849 ]
38 : [ 3.8 0.892018 0.769703 0.328253 ]
39 : [ 3.9 -0.772016 -0.333568 2.05861 ]
40 : [ 4 0.813765 0.70262 0.130501 ]
41 : [ 4.1 -0.859126 1.60644 -0.351036 ]
42 : [ 4.2 0.0271934 -1.44884 1.15448 ]
43 : [ 4.3 -1.12584 1.55685 -0.19887 ]
44 : [ 4.4 -0.529622 1.17651 0.293214 ]
45 : [ 4.5 -0.407357 0.346712 -1.03687 ]
46 : [ 4.6 0.015817 -0.676278 0.424196 ]
47 : [ 4.7 -1.28905 1.08282 -0.953553 ]
48 : [ 4.8 -1.63306 1.36541 0.164474 ]
49 : [ 4.9 -1.16748 -0.534041 -0.564179 ]
50 : [ 5 0.300872 1.68764 -0.119298 ]
51 : [ 5.1 -0.543738 -2.44822 0.252724 ]
52 : [ 5.2 0.0290843 -1.11237 -0.737089 ]
53 : [ 5.3 1.6428 -2.0151 1.18185 ]
54 : [ 5.4 -0.0548356 -0.134918 0.569179 ]
55 : [ 5.5 0.7776 0.266954 -1.28617 ]
56 : [ 5.6 0.192363 -0.632593 -0.197716 ]
57 : [ 5.7 -1.00735 0.503821 -0.888276 ]
58 : [ 5.8 1.88178 -0.789546 0.426544 ]
59 : [ 5.9 -0.30418 0.433553 1.7036 ]
60 : [ 6 -0.478431 -0.898794 -0.104475 ]
61 : [ 6.1 -0.744107 1.10736 -0.421768 ]
62 : [ 6.2 1.84934 0.907834 -0.400522 ]
63 : [ 6.3 0.287392 0.543246 -0.428477 ]
64 : [ 6.4 -1.34214 0.109366 -0.629774 ]
65 : [ 6.5 -1.29804 -1.05483 -0.790768 ]
66 : [ 6.6 2.26413 0.00432486 -0.208023 ]
67 : [ 6.7 -0.965814 -1.04571 0.800641 ]
68 : [ 6.8 -2.11142 -0.475061 0.664259 ]
69 : [ 6.9 0.752243 1.42432 1.53569 ]
70 : [ 7 -0.534013 -0.085785 0.128866 ]
71 : [ 7.1 0.24489 -0.797195 0.988479 ]
72 : [ 7.2 -0.0338217 0.695057 2.77684 ]
73 : [ 7.3 0.557462 0.441871 1.11801 ]
74 : [ 7.4 0.110938 -0.262389 0.341779 ]
75 : [ 7.5 -0.766122 -0.570663 0.718851 ]
76 : [ 7.6 -0.782492 0.0530888 0.469527 ]
77 : [ 7.7 -0.824322 0.0659858 1.3081 ]
78 : [ 7.8 0.0867897 0.226602 -1.13576 ]
79 : [ 7.9 -0.11632 0.506479 0.543716 ]
80 : [ 8 0.527622 -0.879927 -0.853346 ]
81 : [ 8.1 -0.812964 -0.388733 0.646043 ]
82 : [ 8.2 -0.968639 1.69371 0.772567 ]
83 : [ 8.3 -1.66345 1.85585 -1.34847 ]
84 : [ 8.4 1.4947 0.208781 1.15428 ]
85 : [ 8.5 -0.508857 0.471101 -1.04333 ]
86 : [ 8.6 0.225843 -0.159677 0.269437 ]
87 : [ 8.7 1.96162 -1.1784 -0.872668 ]
88 : [ 8.8 0.0748948 0.935867 1.68141 ]
89 : [ 8.9 -0.508926 1.74994 1.02931 ]
90 : [ 9 0.527057 -1.10613 -0.332096 ]
91 : [ 9.1 -0.320962 -1.78858 -0.662092 ]
92 : [ 9.2 0.593153 1.02311 -0.762216 ]
93 : [ 9.3 -1.06471 -1.74861 0.805991 ]
94 : [ 9.4 -0.225593 1.92683 -1.94564 ]
95 : [ 9.5 -0.379446 0.401309 1.17893 ]
96 : [ 9.6 -0.553239 -1.15422 -0.0443501 ]
97 : [ 9.7 0.407933 -1.3633 0.392107 ]
98 : [ 9.8 1.03737 0.624896 0.0248253 ]
99 : [ 9.9 2.54438 -0.59357 -0.849044 ]

Case 2: Get a time series from a Process

myProcess = ot.WhiteNoise(ot.Normal(3), myTimeGrid)
myTimeSeries2 = myProcess.getRealization()
myTimeSeries2

	t	X0	X1	X2
0	0	-0.8828219	0.4980987	-1.355246
1	0.1	-0.5372505	0.06518352	0.5361502
2	0.2	-0.1752852	0.1363254	3.087135
3	0.3	0.2017424	-0.9372854	0.8240501
4	0.4	-0.5542514	-2.939255	-0.2608888
5	0.5	-1.337974	-0.6713168	0.1290662
6	0.6	1.200179	-1.153398	0.1501093
7	0.7	2.021389	0.4048002	-1.588321
8	0.8	-0.2116392	-0.3148359	0.5790985
9	0.9	-0.023225	1.259079	1.173906
10	1	-1.95625	-1.3848	-0.3217505
11	1.1	1.241243	-1.147135	-0.4986612
12	1.2	0.4974517	-0.227134	-0.9424234
13	1.3	0.01796889	-0.2453247	-0.2136053
14	1.4	0.3282077	1.111701	1.47936
15	1.5	-1.324153	0.5101362	1.170227
16	1.6	1.654765	-0.6048047	1.260798
17	1.7	-0.4303438	0.7634444	1.257361
18	1.8	2.206263	0.2056162	-0.9989515
19	1.9	-0.6315028	-0.9982424	-1.252102
20	2	-2.133005	-0.6429958	-0.5764426
21	2.1	-0.5161815	1.439152	-0.5802348
22	2.2	0.6908353	0.2409871	1.614409
23	2.3	-0.8676221	-0.3794585	0.1487301
24	2.4	0.3659664	-0.5051051	0.7310519
25	2.5	1.890257	0.2846552	0.9271769
26	2.6	0.4909163	2.140718	0.1315835
27	2.7	-1.539999	-0.02612867	-1.663261
28	2.8	0.09603311	0.7470867	1.736009
29	2.9	0.1859428	0.7696002	1.110721
30	3	0.2810756	2.080717	-0.8239839
31	3.1	2.051948	-0.7929625	-0.1298837
32	3.2	2.10165	0.3877531	1.186128
33	3.3	-1.032992	0.1772762	0.7248854
34	3.4	-2.422873	0.3085703	-0.8755152
35	3.5	-0.3806235	-0.4749865	1.639512
36	3.6	-2.301016	0.662737	-0.154703
37	3.7	1.869129	0.06195638	0.7639706
38	3.8	-1.05445	-0.4374065	0.3126223
39	3.9	0.6484891	-1.034977	-0.461004
40	4	-0.9253823	1.205154	-0.4920414
41	4.1	0.5250973	-2.057055	-0.5560234
42	4.2	0.8234611	0.5511538	-1.348432
43	4.3	1.129766	-0.4567406	-0.2960206
44	4.4	-0.04858309	0.4198652	1.019731
45	4.5	0.2619137	2.059704	0.1056127
46	4.6	0.1887279	0.917922	0.4719192
47	4.7	1.440201	2.463454	-1.305822
48	4.8	-0.320028	-0.5290385	-0.03588321
49	4.9	-0.3387673	0.269285	-1.67791
50	5	-1.893451	1.045713	0.1414545
51	5.1	1.840738	0.02085951	0.7341872
52	5.2	0.3694536	0.01631195	-0.5654601
53	5.3	0.03670462	0.5372229	-0.4234632
54	5.4	-0.4065987	-0.2864694	1.122679
55	5.5	0.5949969	-0.12097	-1.716656
56	5.6	-0.970214	0.6843236	0.3490545
57	5.7	0.4498215	-0.4501522	-0.02359276
58	5.8	-0.6908011	0.7983409	-2.055452
59	5.9	1.19464	1.021232	-0.008772303
60	6	-0.2725593	-0.429363	1.270387
61	6.1	-1.071065	0.2382513	0.03229052
62	6.2	0.6391206	-0.4283499	-0.9460564
63	6.3	-0.02129602	-0.6210193	1.669518
64	6.4	0.8789429	-0.8569577	0.7040885
65	6.5	-0.8044128	-0.5791309	1.158013
66	6.6	1.474823	1.203803	1.656491
67	6.7	-0.2605618	-2.72704	0.5875819
68	6.8	-2.815473	0.2202595	1.707816
69	6.9	0.9120917	1.192637	-0.4587521
70	7	1.524778	-0.246746	0.7623076
71	7.1	-1.748163	-0.5658546	-2.084908
72	7.2	0.6480018	1.28735	0.2404443
73	7.3	-0.6897505	1.711613	0.4393317
74	7.4	-0.8528524	-0.8352737	1.109671
75	7.5	0.3668992	0.2840582	2.098185
76	7.6	-0.5463	0.5214562	-0.7110372
77	7.7	-1.294471	0.1793523	1.753766
78	7.8	0.4673149	1.326889	-1.088676
79	7.9	-0.7117908	-1.291751	1.001505
80	8	0.6587044	-1.465987	-2.289392
81	8.1	0.6074961	-0.4320503	-0.207302
82	8.2	0.9224311	0.4791631	0.2511686
83	8.3	-1.860983	1.481042	0.04964255
84	8.4	-0.1871141	1.187763	-1.256839
85	8.5	-0.3439743	0.183169	-0.5022868
86	8.6	-0.4769816	-1.563596	0.4836341
87	8.7	0.6520573	1.268271	-0.09222983
88	8.8	1.368659	1.201644	-1.463297
89	8.9	-0.08670679	0.6765859	2.165009
90	9	1.49675	-0.1054785	-2.160915
91	9.1	0.14953	-0.5258041	-0.3039998
92	9.2	-0.05036142	0.8297279	-0.2839957
93	9.3	0.2642588	-0.08572813	0.2829766
94	9.4	-0.4570203	-0.8522508	-0.1272129
95	9.5	-1.938921	0.5054613	-0.2439029
96	9.6	-0.09140759	-2.067526	-1.783602
97	9.7	1.277923	-0.04433687	0.1895748
98	9.8	-0.9785647	-0.6177446	1.358516
99	9.9	0.6591985	-0.6847979	-0.0930263

Get the number of values of the time series

myTimeSeries.getSize()

Out:

Get the dimension of the values observed at each time

myTimeSeries.getMesh().getDimension()

Out:

Get the value Xi at index i

i = 37
print('Xi = ', myTimeSeries.getValueAtIndex(i))

Out:

Xi =  [-0.330858,-1.40182,0.339849]

Get the time series at index i : Xi

i = 37
print('Xi = ', myTimeSeries[i])

Out:

Xi =  [-0.330858,-1.40182,0.339849]

Get a the marginal value at index i of the time series

i = 37
# get the time stamp:
print('ti = ', myTimeSeries.getTimeGrid().getValue(i))
# get the first component of the corresponding value :
print('Xi1 = ', myTimeSeries[i, 0])

Out:

ti =  3.7
Xi1 =  -0.3308581965168566

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

myTimeSeries.getValues()

	X0	X1	X2
0	-0.1824838	-0.4403559	1.132474
1	0.6745465	1.34247	1.398739
2	0.7367821	-0.252381	1.3757
3	-0.8587751	0.001090329	0.579508
4	0.3836626	0.4942256	0.707441
5	-2.797382	3.047523	-0.07637454
6	-1.147148	2.571537	1.522949
7	-0.4123436	0.1975434	-1.320555
8	0.07847825	1.150862	0.3574573
9	0.8082896	-0.8460937	0.1979967
10	-0.3094551	-0.7988185	-1.491241
11	0.9084451	-0.5728211	1.156641
12	0.5084799	-1.566754	-0.08852041
13	-0.4396728	-0.2164431	0.9924761
14	1.24972	-0.07331844	1.407254
15	-0.4617167	1.11996	-1.244644
16	0.7732243	-0.5318689	-0.52065
17	0.3838784	0.5908666	0.9716562
18	0.4252095	-0.6798812	-0.2249922
19	0.9110131	-0.06524682	-1.738988
20	0.09377542	0.1144101	0.4402198
21	-0.01304001	-1.136739	0.6385415
22	0.6377289	-0.4951121	-0.5121626
23	-0.9752874	-0.8100208	0.3302668
24	-1.916166	0.02455777	0.8620839
25	0.8635642	0.2578959	0.2637762
26	-0.1689309	-0.522402	0.4682332
27	-0.1400617	1.354969	-0.5971853
28	-1.112985	-1.428811	2.208364
29	-1.237043	-0.4459371	0.7625289
30	-0.7564409	0.3947358	-0.8920199
31	-0.6359006	-0.7376064	-0.1912633
32	-0.4270532	-0.2999664	-1.875619
33	-1.153489	0.6319036	-0.2558087
34	0.527049	-0.5257207	1.363352
35	-0.5492806	0.7344462	0.5173155
36	0.06236516	-1.392551	-2.649476
37	-0.3308582	-1.401821	0.3398492
38	0.8920179	0.7697032	0.3282534
39	-0.7720165	-0.3335678	2.058606
40	0.8137651	0.7026201	0.1305007
41	-0.8591256	1.606435	-0.3510362
42	0.02719343	-1.448844	1.154483
43	-1.12584	1.556847	-0.1988698
44	-0.529622	1.17651	0.2932139
45	-0.4073566	0.3467122	-1.036869
46	0.01581702	-0.6762781	0.4241955
47	-1.289054	1.082821	-0.9535527
48	-1.633061	1.365407	0.1644739
49	-1.167478	-0.5340412	-0.5641792
50	0.3008715	1.687641	-0.119298
51	-0.543738	-2.448224	0.2527243
52	0.02908425	-1.11237	-0.7370888
53	1.642798	-2.0151	1.181847
54	-0.05483563	-0.1349175	0.5691794
55	0.7776005	0.2669536	-1.286173
56	0.1923625	-0.6325928	-0.1977164
57	-1.007348	0.5038206	-0.8882761
58	1.881779	-0.789546	0.4265443
59	-0.3041803	0.4335531	1.703603
60	-0.478431	-0.8987942	-0.104475
61	-0.7441071	1.10736	-0.4217685
62	1.849343	0.9078336	-0.4005217
63	0.287392	0.5432461	-0.4284768
64	-1.342138	0.1093661	-0.6297741
65	-1.29804	-1.054826	-0.7907677
66	2.264132	0.004324857	-0.2080225
67	-0.9658142	-1.045708	0.8006405
68	-2.111424	-0.4750612	0.6642585
69	0.7522429	1.424315	1.535688
70	-0.5340135	-0.08578498	0.1288659
71	0.24489	-0.797195	0.9884785
72	-0.03382174	0.6950574	2.776844
73	0.5574622	0.4418714	1.118012
74	0.1109384	-0.2623889	0.3417789
75	-0.7661217	-0.5706632	0.7188506
76	-0.7824923	0.05308881	0.4695268
77	-0.8243218	0.06598585	1.308104
78	0.08678973	0.2266022	-1.135762
79	-0.1163204	0.5064792	0.5437157
80	0.5276222	-0.8799274	-0.8533455
81	-0.8129643	-0.3887334	0.6460432
82	-0.9686388	1.693706	0.7725672
83	-1.663448	1.855849	-1.348466
84	1.494702	0.2087813	1.154285
85	-0.5088572	0.4711006	-1.043329
86	0.2258432	-0.1596775	0.2694374
87	1.961616	-1.178402	-0.8726679
88	0.07489477	0.9358671	1.681411
89	-0.5089258	1.749942	1.029307
90	0.5270572	-1.106128	-0.332096
91	-0.3209623	-1.788576	-0.6620917
92	0.5931534	1.02311	-0.7622161
93	-1.064708	-1.748614	0.8059907
94	-0.225593	1.926831	-1.94564
95	-0.3794463	0.4013089	1.178926
96	-0.5532393	-1.154224	-0.0443501
97	0.4079327	-1.3633	0.392107
98	1.037367	0.6248955	0.02482525
99	2.544377	-0.59357	-0.8490439

Compute the temporal Mean It corresponds to the mean of the values of the time series

myTimeSeries.getInputMean()

[-0.0791696,0.0141695,0.101313]

Draw the marginal i of the time series using linear interpolation

graph = myTimeSeries.drawMarginal(0)
view = viewer.View(graph)

with no interpolation

graph = myTimeSeries.drawMarginal(0, False)
view = viewer.View(graph)
plt.show()

Total running time of the script: ( 0 minutes 0.136 seconds)

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