Manipulate a time series¶

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 $(\vect{x}_0, \dots, \vect{x}_{N-1})$ of the time series are defined by:

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

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

import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt

ot.Log.Show(ot.Log.NONE)

Create the RegularGrid

tMin = 0.0
timeStep = 0.1
N = 100
myTimeGrid = ot.RegularGrid(tMin, timeStep, N)

Case 1: Create a time series from a time grid and values. Be careful that 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

class=TimeSeries name=Unnamed derived from=class=FieldImplementation name=Unnamed mesh=class=Mesh name=Unnamed dimension=1 vertices=class=Sample name=Unnamed implementation=class=SampleImplementation name=Unnamed size=100 dimension=1 description=[t] data=[[0],[0.1],[0.2],[0.3],[0.4],[0.5],[0.6],[0.7],[0.8],[0.9],[1],[1.1],[1.2],[1.3],[1.4],[1.5],[1.6],[1.7],[1.8],[1.9],[2],[2.1],[2.2],[2.3],[2.4],[2.5],[2.6],[2.7],[2.8],[2.9],[3],[3.1],[3.2],[3.3],[3.4],[3.5],[3.6],[3.7],[3.8],[3.9],[4],[4.1],[4.2],[4.3],[4.4],[4.5],[4.6],[4.7],[4.8],[4.9],[5],[5.1],[5.2],[5.3],[5.4],[5.5],[5.6],[5.7],[5.8],[5.9],[6],[6.1],[6.2],[6.3],[6.4],[6.5],[6.6],[6.7],[6.8],[6.9],[7],[7.1],[7.2],[7.3],[7.4],[7.5],[7.6],[7.7],[7.8],[7.9],[8],[8.1],[8.2],[8.3],[8.4],[8.5],[8.6],[8.7],[8.8],[8.9],[9],[9.1],[9.2],[9.3],[9.4],[9.5],[9.6],[9.7],[9.8],[9.9]] simplices=[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24],[24,25],[25,26],[26,27],[27,28],[28,29],[29,30],[30,31],[31,32],[32,33],[33,34],[34,35],[35,36],[36,37],[37,38],[38,39],[39,40],[40,41],[41,42],[42,43],[43,44],[44,45],[45,46],[46,47],[47,48],[48,49],[49,50],[50,51],[51,52],[52,53],[53,54],[54,55],[55,56],[56,57],[57,58],[58,59],[59,60],[60,61],[61,62],[62,63],[63,64],[64,65],[65,66],[66,67],[67,68],[68,69],[69,70],[70,71],[71,72],[72,73],[73,74],[74,75],[75,76],[76,77],[77,78],[78,79],[79,80],[80,81],[81,82],[82,83],[83,84],[84,85],[85,86],[86,87],[87,88],[88,89],[89,90],[90,91],[91,92],[92,93],[93,94],[94,95],[95,96],[96,97],[97,98],[98,99]] values=class=Sample name=Normal implementation=class=SampleImplementation name=Normal size=100 dimension=3 description=[X0,X1,X2] data=[[0.52378,0.12611,0.393794],[-0.248139,-0.16677,0.214858],[0.0247934,0.138394,0.430279],[-1.61823,0.558478,-1.30886],[-0.139764,-0.546122,-0.567461],[0.205057,0.904138,0.547278],[0.788237,1.49744,-1.41748],[0.47696,-1.43288,2.02109],[1.23468,0.157697,0.940324],[0.117709,-1.14723,-1.32423],[0.498823,0.801346,0.900951],[-1.34579,-0.466627,0.246984],[-0.588016,0.250813,-0.859715],[-0.431791,-1.73008,-1.10509],[-1.22603,-0.70203,1.08883],[-0.700387,-1.66383,1.73256],[0.865208,0.270015,-0.27368],[-0.733251,0.521748,0.362264],[-0.123077,0.573721,-0.328373],[-0.10364,-0.52281,-1.68534],[0.195637,-0.0788806,-1.94656],[1.32949,-1.26065,-0.137471],[-0.349241,0.927261,1.02351],[-0.528963,1.2413,0.273859],[0.207583,1.16668,1.04363],[-2.38573,-1.65012,-0.416396],[2.13326,-0.43238,-0.483701],[-0.923412,-0.325859,-1.19977],[0.803572,0.524339,-0.0630568],[-0.835405,0.0402377,-0.452565],[0.0257213,0.368049,-0.841262],[-1.01146,-2.18025,-1.58033],[-0.449535,0.890842,0.0743256],[1.8438,-0.300223,-1.13582],[0.603846,-1.57444,-0.95201],[0.620044,-0.831148,0.179603],[-0.444416,0.959874,0.393199],[0.607432,-0.442113,0.170379],[0.296909,0.291782,0.977481],[-0.190001,0.250078,0.434109],[0.323441,1.75839,-0.023602],[-0.967543,-0.400737,-1.29578],[-0.312378,-0.944492,1.75581],[1.04865,-0.830397,-0.892176],[0.433775,-0.825645,-0.537977],[-0.102559,0.84464,-1.91496],[0.406466,0.321913,-0.305269],[0.488492,0.122734,0.305202],[-0.386326,0.0689214,0.313803],[0.677903,0.495358,-0.490069],[0.315441,-0.379322,0.785857],[-0.35019,0.264677,1.1554],[-0.455589,-0.559667,-0.662289],[-0.84981,-1.48909,-0.438836],[0.545473,-0.369104,0.962339],[0.770285,0.5617,-0.0453834],[-0.548109,-1.08238,0.471951],[0.50024,-1.40894,0.656795],[-1.05172,0.263069,-0.418894],[0.30767,-0.416239,1.35256],[-0.02062,1.30727,0.445701],[-0.428967,-2.15109,1.10333],[1.20183,-0.65679,-0.0034805],[-1.53118,1.56066,0.317831],[0.206523,1.45024,-1.4915],[-2.71262,-1.60122,-1.73553],[0.315804,-0.389312,-0.835347],[0.915355,-1.11271,0.586068],[0.57727,0.0729422,-0.441155],[-0.145206,1.00515,-1.07333],[1.89198,-0.216602,0.214305],[-0.503722,-0.345728,-1.40113],[0.921629,0.229499,0.777413],[-1.17868,0.132514,0.400826],[0.79873,0.175534,0.6878],[1.1929,1.95488,-0.475165],[-0.434939,-0.563952,2.23404],[-0.139733,-0.897931,0.198825],[-1.95284,-0.125675,1.59678],[1.05037,0.207933,-0.607134],[-0.652396,-0.666132,-1.07025],[0.225309,-1.13645,0.0128528],[0.681485,-0.0589669,0.62219],[-0.784616,-0.500505,-0.497468],[-0.59515,-0.597179,1.28317],[1.98506,0.0321232,0.469106],[0.0181719,-1.12972,0.70832],[0.309358,-0.161451,-0.0337435],[0.242744,-0.720621,0.605114],[-0.0771294,0.22276,-1.78651],[-0.52325,0.174664,-0.704638],[-0.988056,-0.361028,-1.76169],[0.313168,0.825269,0.432011],[1.20499,0.884502,0.117486],[-0.0337554,-0.092368,0.437182],[-0.0595837,-0.637973,-0.887564],[-0.337387,-0.0849012,-0.399771],[0.344743,-1.22959,0.656478],[-0.21913,-0.018154,0.365455],[0.793991,-1.21411,-1.73067]] start=0 timeStep=0.1 n=100

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	-1.670492	-1.138823	1.317518
1	0.1	0.004916539	0.207453	0.5854116
2	0.2	0.6318718	0.3769744	1.288444
3	0.3	-1.815059	-1.064866	-0.5376213
4	0.4	1.100027	-1.608281	0.2590714
5	0.5	0.2505065	-1.76158	0.6998717
6	0.6	0.9417183	-0.488464	-0.08899449
7	0.7	1.739529	-0.8246544	-0.0005616102
8	0.8	0.6088398	0.2800388	1.456377
9	0.9	-0.2445882	-0.9281551	-0.8752327
10	1	-0.9090887	0.09793622	0.5108138
11	1.1	-0.2333036	2.977989	-0.9998576
12	1.2	-1.182387	1.048563	1.703911
13	1.3	0.8708819	-1.284252	1.036737
14	1.4	-0.5798028	-1.859145	0.9250127
15	1.5	0.1757063	1.531778	0.5817077
16	1.6	1.024101	0.09670833	0.6195517
17	1.7	1.139809	-0.4087134	-0.2550477
18	1.8	0.5224197	0.3861234	-2.075111
19	1.9	0.6156748	1.977506	0.06924619
20	2	-1.367441	-1.178709	1.05263
21	2.1	2.36118	-1.022332	0.2285195
22	2.2	-0.9436299	0.05570427	0.8479874
23	2.3	1.947942	-1.768569	0.313241
24	2.4	1.393382	-0.5917899	-1.283321
25	2.5	-0.4261665	-2.275057	-1.122716
26	2.6	-0.9587	0.8365301	-0.2694569
27	2.7	0.8057302	1.089693	-0.6405182
28	2.8	-2.792036	-0.2587426	-0.3865208
29	2.9	0.3161664	0.7604701	-1.217716
30	3	-0.1682398	0.5425786	-1.366357
31	3.1	1.074819	0.01435018	0.1129411
32	3.2	-0.755817	-0.2352335	-0.3526476
33	3.3	1.031261	0.1387743	-0.1019698
34	3.4	0.1148558	1.307602	0.2750357
35	3.5	0.8908154	-0.6283213	0.03249425
36	3.6	0.8022597	-0.7750893	1.622367
37	3.7	-0.8908726	0.1450516	1.288845
38	3.8	0.7490533	-1.660132	-0.2574172
39	3.9	-0.6333344	-0.01390061	-0.5187067
40	4	-0.7641469	-0.8802884	-1.04384
41	4.1	1.624155	-0.08771599	-0.2191296
42	4.2	1.651487	0.1178754	-1.152759
43	4.3	-0.6219183	-1.12608	1.060989
44	4.4	1.310402	-1.324693	2.232301
45	4.5	-0.9766673	-0.5351095	-0.4692117
46	4.6	0.8752487	-0.6614909	0.4248837
47	4.7	0.9250794	0.1116855	-1.69762
48	4.8	-1.71699	0.5622419	-0.7709706
49	4.9	-0.2774593	-1.896383	2.738704
50	5	0.2583176	1.411942	0.5341219
51	5.1	0.813716	0.5418584	-0.7792782
52	5.2	-0.2821132	1.738024	-0.3488722
53	5.3	0.5248662	-0.4167478	-0.4363704
54	5.4	1.231982	-1.329614	0.4294014
55	5.5	-0.1988925	-0.5305423	0.1479477
56	5.6	1.087853	-0.1887557	0.8494047
57	5.7	0.9088333	0.7830113	0.6364255
58	5.8	0.3660167	0.2228301	0.7101552
59	5.9	0.2647587	-0.6881035	-1.005743
60	6	0.2375102	-1.817178	-0.7518762
61	6.1	-0.6579323	0.9981225	-0.07624542
62	6.2	-0.7847343	-0.1382298	-0.9096015
63	6.3	-0.6038747	-1.507997	1.923131
64	6.4	-0.8878048	-0.5890475	0.4536556
65	6.5	1.112655	-0.9013951	0.7350295
66	6.6	0.1578329	-1.229718	-0.7125437
67	6.7	0.3872097	-0.221006	-0.6366688
68	6.8	0.1411499	-1.382219	-0.9600347
69	6.9	-0.673238	1.35691	-1.472505
70	7	0.290016	-0.3173095	1.594927
71	7.1	0.427398	0.06673495	0.1377921
72	7.2	0.3715888	-0.409588	0.4125253
73	7.3	-0.7911313	-0.3134569	0.683661
74	7.4	0.2432084	1.35115	-1.572174
75	7.5	0.7591635	-0.5428189	0.894484
76	7.6	1.496315	1.494005	0.07607672
77	7.7	3.032252	-0.0697185	0.9362872
78	7.8	0.09534806	0.5547554	-0.1597999
79	7.9	-0.117537	0.8516132	-2.280891
80	8	0.4156774	0.1686187	-0.4769556
81	8.1	-1.248859	0.5411939	-2.166777
82	8.2	1.047143	0.4078612	-1.554764
83	8.3	-0.04557226	1.203224	0.9711234
84	8.4	-0.9533762	-0.5502569	0.7774226
85	8.5	0.4831444	-0.1721776	1.008082
86	8.6	1.217844	-0.4041445	1.793038
87	8.7	-0.8329919	1.636185	0.6188379
88	8.8	0.9314921	0.3700222	1.658918
89	8.9	-0.7589742	-0.1231373	-1.700695
90	9	-0.8901343	-0.8929079	0.1463011
91	9.1	-0.01782797	0.1991584	-1.521287
92	9.2	1.876469	-1.655881	0.08516664
93	9.3	0.4022144	-1.748718	0.5559068
94	9.4	-1.22224	1.493979	1.512701
95	9.5	-0.07158338	0.5902772	1.96161
96	9.6	0.1669381	-1.754113	1.140951
97	9.7	0.6632621	1.00617	-0.2992668
98	9.8	-0.7329082	-0.5431353	0.4164685
99	9.9	0.7069276	0.3009473	0.534359

Get the number of values of the time series

myTimeSeries.getSize()

Get the dimension of the values observed at each time

myTimeSeries.getMesh().getDimension()

Get the value $X_i$ at index $i$

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

Xi =  [0.607432,-0.442113,0.170379]

Get the time series at index $i$ : $X_i$

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

Xi =  [0.607432,-0.442113,0.170379]

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])

ti =  3.7
Xi1 =  0.6074317802523269

Get all the values $(X_1, \dots, X_n)$ of the time series

myTimeSeries.getValues()

	X0	X1	X2
0	0.5237798	0.12611	0.3937941
1	-0.2481388	-0.1667702	0.2148585
2	0.02479338	0.1383943	0.430279
3	-1.618226	0.5584776	-1.30886
4	-0.139764	-0.5461219	-0.5674607
5	0.2050574	0.9041375	0.5472782
6	0.7882369	1.49744	-1.417477
7	0.4769603	-1.432882	2.021089
8	1.23468	0.1576975	0.9403244
9	0.117709	-1.147233	-1.324232
10	0.4988229	0.801346	0.9009509
11	-1.34579	-0.466627	0.2469844
12	-0.5880163	0.2508129	-0.8597147
13	-0.4317908	-1.730083	-1.105085
14	-1.226028	-0.7020297	1.088834
15	-0.7003868	-1.663834	1.732557
16	0.865208	0.2700146	-0.2736801
17	-0.7332508	0.5217478	0.3622642
18	-0.1230767	0.5737211	-0.3283734
19	-0.1036404	-0.5228103	-1.685336
20	0.1956371	-0.07888063	-1.946558
21	1.329493	-1.260652	-0.1374706
22	-0.3492414	0.9272612	1.023511
23	-0.5289635	1.241303	0.2738588
24	0.2075833	1.166677	1.043628
25	-2.385726	-1.650125	-0.4163961
26	2.133258	-0.4323803	-0.4837005
27	-0.9234119	-0.3258593	-1.199766
28	0.8035719	0.5243393	-0.06305676
29	-0.835405	0.04023769	-0.4525647
30	0.0257213	0.3680492	-0.8412618
31	-1.011461	-2.180247	-1.580328
32	-0.4495348	0.8908425	0.07432561
33	1.843799	-0.3002226	-1.135818
34	0.6038459	-1.574439	-0.9520099
35	0.6200444	-0.8311482	0.1796026
36	-0.4444155	0.9598739	0.3931992
37	0.6074318	-0.4421131	0.1703794
38	0.2969089	0.2917823	0.9774806
39	-0.1900015	0.2500776	0.4341087
40	0.3234411	1.758392	-0.02360201
41	-0.9675434	-0.4007368	-1.295784
42	-0.3123782	-0.9444916	1.755809
43	1.048647	-0.8303966	-0.8921762
44	0.4337751	-0.825645	-0.5379767
45	-0.1025594	0.8446397	-1.914963
46	0.4064664	0.321913	-0.3052688
47	0.4884916	0.1227336	0.3052021
48	-0.3863256	0.06892144	0.3138032
49	0.6779029	0.4953577	-0.490069
50	0.3154414	-0.3793216	0.7858567
51	-0.3501902	0.2646774	1.155399
52	-0.4555892	-0.5596673	-0.6622893
53	-0.8498098	-1.489086	-0.4388359
54	0.5454725	-0.369104	0.9623389
55	0.7702848	0.5617002	-0.04538344
56	-0.5481091	-1.082383	0.4719511
57	0.5002396	-1.408936	0.6567949
58	-1.051715	0.2630691	-0.4188936
59	0.3076702	-0.416239	1.352556
60	-0.02062003	1.307273	0.4457012
61	-0.4289673	-2.151095	1.103334
62	1.201832	-0.65679	-0.003480504
63	-1.531184	1.560664	0.3178313
64	0.2065234	1.45024	-1.491504
65	-2.712616	-1.601219	-1.735534
66	0.3158041	-0.3893124	-0.8353471
67	0.9153545	-1.112714	0.5860682
68	0.5772697	0.07294216	-0.4411553
69	-0.1452064	1.005149	-1.073327
70	1.891982	-0.2166016	0.2143049
71	-0.5037216	-0.345728	-1.401125
72	0.9216287	0.2294988	0.7774133
73	-1.178678	0.1325139	0.4008258
74	0.7987302	0.1755338	0.6877997
75	1.1929	1.954883	-0.4751648
76	-0.434939	-0.5639525	2.234044
77	-0.1397326	-0.8979314	0.1988251
78	-1.952842	-0.1256751	1.596785
79	1.050368	0.2079334	-0.6071344
80	-0.6523957	-0.6661315	-1.07025
81	0.2253092	-1.136449	0.01285282
82	0.681485	-0.05896693	0.62219
83	-0.784616	-0.5005054	-0.4974685
84	-0.5951495	-0.5971793	1.283174
85	1.985056	0.03212315	0.4691064
86	0.01817188	-1.129723	0.7083197
87	0.3093576	-0.1614505	-0.03374345
88	0.2427443	-0.7206208	0.6051135
89	-0.07712943	0.2227604	-1.786512
90	-0.5232505	0.1746638	-0.7046384
91	-0.9880562	-0.3610281	-1.761685
92	0.3131678	0.8252694	0.4320112
93	1.204991	0.8845017	0.1174865
94	-0.03375537	-0.09236796	0.4371824
95	-0.05958368	-0.6379726	-0.8875642
96	-0.3373875	-0.08490124	-0.3997714
97	0.3447429	-1.229588	0.6564776
98	-0.2191304	-0.01815395	0.3654549
99	0.7939912	-1.214108	-1.730674

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

myTimeSeries.getInputMean()

class=Point name=Unnamed dimension=3 values=[0.00695924,-0.100262,-0.0396681]

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()

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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Manipulate a time series¶