Note

Click here to download the full example code

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 $(\underline{x}_0, \dots, \underline{x}_{N-1})$ 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.

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 1.06975 -1.77994 -0.832708 ]
1 : [ 0.1 -0.245372 -0.0205006 -0.170101 ]
2 : [ 0.2 0.529296 -0.725104 -1.16247 ]
3 : [ 0.3 0.199523 0.727148 -0.260688 ]
4 : [ 0.4 -0.136772 0.52023 -0.659133 ]
5 : [ 0.5 -0.180673 -1.04885 0.512371 ]
6 : [ 0.6 0.20648 -0.960832 0.414682 ]
7 : [ 0.7 -1.22871 2.57497 -0.00804901 ]
8 : [ 0.8 -1.8859 0.830757 -0.378346 ]
9 : [ 0.9 0.479046 1.60938 -0.570841 ]
10 : [ 1 0.269096 0.803503 0.583218 ]
11 : [ 1.1 0.449756 -0.693556 1.89666 ]
12 : [ 1.2 0.0270818 -0.258272 -0.37012 ]
13 : [ 1.3 0.0456596 -0.343048 -0.392484 ]
14 : [ 1.4 -2.41093 1.93921 -0.590044 ]
15 : [ 1.5 0.22705 -0.141765 0.855507 ]
16 : [ 1.6 0.286761 0.564812 -0.509701 ]
17 : [ 1.7 1.40334 -1.37852 0.434035 ]
18 : [ 1.8 0.0342518 0.896116 -0.870577 ]
19 : [ 1.9 1.36995 0.272597 0.579223 ]
20 : [ 2 -1.5321 0.957065 0.427663 ]
21 : [ 2.1 -0.36668 0.648699 -0.00464944 ]
22 : [ 2.2 0.171548 -0.0795761 0.455389 ]
23 : [ 2.3 -2.14009 0.933245 0.818686 ]
24 : [ 2.4 -1.54826 0.370246 -0.773089 ]
25 : [ 2.5 -0.0129833 0.187309 -2.13145 ]
26 : [ 2.6 -1.19768 -0.00500185 -0.125673 ]
27 : [ 2.7 -1.89201 3.40565 -0.103576 ]
28 : [ 2.8 0.415448 0.727255 0.978855 ]
29 : [ 2.9 1.15808 0.295275 0.283934 ]
30 : [ 3 1.29426 0.200773 0.342265 ]
31 : [ 3.1 0.164085 -0.608383 0.144346 ]
32 : [ 3.2 0.537733 0.696557 1.18791 ]
33 : [ 3.3 2.18097 -0.194809 0.628316 ]
34 : [ 3.4 0.230866 -0.648071 -0.0280203 ]
35 : [ 3.5 0.871005 1.24473 -0.106358 ]
36 : [ 3.6 -0.234489 -2.0102 0.121701 ]
37 : [ 3.7 -1.33163 -0.825457 -1.21658 ]
38 : [ 3.8 -1.02579 -1.22486 -0.735057 ]
39 : [ 3.9 0.267431 -0.313967 0.328403 ]
40 : [ 4 -1.18542 0.272577 -0.537997 ]
41 : [ 4.1 -0.154628 0.0348939 0.357208 ]
42 : [ 4.2 0.87381 -1.4897 -1.60323 ]
43 : [ 4.3 0.276884 -0.205279 0.313591 ]
44 : [ 4.4 1.52063 2.12789 0.15741 ]
45 : [ 4.5 0.056432 1.05201 -1.06929 ]
46 : [ 4.6 0.0389696 0.108862 1.56022 ]
47 : [ 4.7 0.897858 0.0713179 0.329058 ]
48 : [ 4.8 0.768345 -0.201722 0.148307 ]
49 : [ 4.9 0.498826 -0.540609 0.202215 ]
50 : [ 5 1.52964 -1.19218 0.524954 ]
51 : [ 5.1 -0.127176 1.00122 0.299567 ]
52 : [ 5.2 -0.0732479 -0.592801 0.509773 ]
53 : [ 5.3 1.56808 0.369343 0.687346 ]
54 : [ 5.4 0.26022 1.5601 0.68388 ]
55 : [ 5.5 -0.260408 0.169652 -1.01657 ]
56 : [ 5.6 0.810285 -0.934548 0.440233 ]
57 : [ 5.7 0.102655 0.16255 0.977606 ]
58 : [ 5.8 -0.685128 -0.0411968 -0.161531 ]
59 : [ 5.9 0.00948899 -0.699237 0.835643 ]
60 : [ 6 0.961209 -0.395342 0.250509 ]
61 : [ 6.1 -1.71279 -0.303372 1.71343 ]
62 : [ 6.2 0.287997 -0.346204 -1.24308 ]
63 : [ 6.3 -0.661934 -0.539626 0.78918 ]
64 : [ 6.4 0.525199 0.265505 -0.615353 ]
65 : [ 6.5 0.667728 -0.320656 -0.00603524 ]
66 : [ 6.6 -1.44043 0.0706512 0.400517 ]
67 : [ 6.7 -0.537003 -2.13043 0.186229 ]
68 : [ 6.8 -1.32629 0.242601 -0.897333 ]
69 : [ 6.9 -0.957364 1.58824 -0.238077 ]
70 : [ 7 -0.654398 1.49892 -0.713136 ]
71 : [ 7.1 -1.33516 0.567629 0.640198 ]
72 : [ 7.2 -0.259729 0.192286 -1.40222 ]
73 : [ 7.3 0.560018 -1.35624 1.03452 ]
74 : [ 7.4 -0.378793 -0.125727 -0.587836 ]
75 : [ 7.5 1.07894 -1.66939 1.70834 ]
76 : [ 7.6 -0.845941 -0.178621 -0.195884 ]
77 : [ 7.7 1.81133 0.400036 1.10812 ]
78 : [ 7.8 -0.455236 -0.793417 2.28383 ]
79 : [ 7.9 0.351885 -0.0608221 1.18257 ]
80 : [ 8 2.05724 2.0836 -1.10946 ]
81 : [ 8.1 0.646117 0.314088 -1.25919 ]
82 : [ 8.2 2.51347 1.10677 -1.23708 ]
83 : [ 8.3 -0.405063 1.24478 0.258866 ]
84 : [ 8.4 -0.1138 0.3815 0.155791 ]
85 : [ 8.5 0.402412 1.33272 -0.805619 ]
86 : [ 8.6 0.385421 -1.61086 -0.687429 ]
87 : [ 8.7 -0.021074 -1.40527 -0.602909 ]
88 : [ 8.8 -0.0745371 -0.287633 -0.402623 ]
89 : [ 8.9 -0.489432 -0.580339 1.19649 ]
90 : [ 9 1.00456 0.537257 -0.0877091 ]
91 : [ 9.1 1.42393 0.682015 2.88405 ]
92 : [ 9.2 0.279699 -1.179 -0.143892 ]
93 : [ 9.3 0.681308 0.0143792 0.50997 ]
94 : [ 9.4 -1.06023 0.0448366 0.24992 ]
95 : [ 9.5 1.24773 -0.3856 -0.288073 ]
96 : [ 9.6 -0.589052 0.499575 1.13231 ]
97 : [ 9.7 -0.843781 1.43619 -0.18765 ]
98 : [ 9.8 0.940522 0.715112 -1.43932 ]
99 : [ 9.9 -0.14294 -0.176589 0.905433 ]

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	0.6688361	-0.1848348	-0.2056171
1	0.1	0.8539061	1.082717	0.7860448
2	0.2	-1.839514	-0.4807376	-0.7431111
3	0.3	0.2583894	0.06498678	0.8220976
4	0.4	-0.2202976	-1.267407	0.06548754
5	0.5	-2.506485	0.2182682	-0.3734256
6	0.6	-0.3483342	-1.020392	-0.9373684
7	0.7	0.793814	-0.983334	-0.4151898
8	0.8	0.1049272	-0.4991656	0.3643877
9	0.9	-0.1627931	0.4925782	0.3548167
10	1	-0.8811936	-0.819895	-2.106536
11	1.1	0.1773956	-0.04881701	-0.9867962
12	1.2	-0.8862132	1.219161	0.266691
13	1.3	0.188304	0.8090514	1.619885
14	1.4	-0.5646788	-0.9921044	0.7245245
15	1.5	0.3057475	-0.4119946	2.759856
16	1.6	0.4088039	1.121707	-0.6501654
17	1.7	-1.034288	1.150379	0.5587453
18	1.8	1.332409	-0.3225148	0.4750779
19	1.9	-0.1536095	1.035535	1.381175
20	2	1.225896	-0.1056646	0.3069166
21	2.1	0.4924758	0.4262604	-0.5698308
22	2.2	-0.4156163	-2.609303	-2.173168
23	2.3	-1.324497	-1.45585	0.1801837
24	2.4	1.421198	1.866039	-0.1742316
25	2.5	-1.55547	1.4884	1.303924
26	2.6	-1.061323	-1.305955	-1.629615
27	2.7	-0.2962869	0.8739792	0.1051378
28	2.8	-0.02998592	-1.516032	1.474471
29	2.9	-1.03669	-1.534651	0.8259901
30	3	0.457382	-0.3865615	1.28411
31	3.1	-0.3259461	1.637177	-0.8420178
32	3.2	-0.2924097	0.3615991	0.4570965
33	3.3	0.237978	1.020826	1.699262
34	3.4	-0.5438809	0.4973056	-1.469904
35	3.5	-2.294773	-0.2623551	-1.554523
36	3.6	-2.82731	0.5825531	0.4139608
37	3.7	-0.9302437	0.549059	-0.69065
38	3.8	-0.6021352	-0.7677184	1.285077
39	3.9	-0.22259	1.221741	0.4439343
40	4	-0.7078664	-1.056912	0.5648551
41	4.1	0.2980986	1.342418	1.085837
42	4.2	0.8239627	-0.6283856	-0.8834576
43	4.3	0.8607533	1.456264	0.1421699
44	4.4	-0.3323323	0.8952978	0.1655028
45	4.5	0.02714461	0.1645807	0.2626963
46	4.6	1.638611	0.1818056	-0.1240066
47	4.7	1.56386	-0.5471615	0.4136208
48	4.8	-0.5009097	-1.561814	-2.157897
49	4.9	-0.8845609	-0.03278067	-0.4371368
50	5	0.9263022	0.3640217	1.127778
51	5.1	-0.2958129	0.521623	-0.5048369
52	5.2	-1.126024	-0.1538759	0.9138794
53	5.3	-2.058274	1.093646	0.353957
54	5.4	-0.5708488	1.521397	0.2852253
55	5.5	-1.835236	-0.3044852	0.9165636
56	5.6	0.9140664	0.1075705	0.06927429
57	5.7	-0.6650488	1.951216	0.7997068
58	5.8	-0.8125796	-0.5797791	0.1117721
59	5.9	-0.2133026	-1.116885	-0.872058
60	6	1.629164	3.399959	-0.9405087
61	6.1	0.8080016	-0.5450092	1.626903
62	6.2	-0.06128802	0.308256	-0.9618253
63	6.3	-1.255094	0.4358796	-0.7273887
64	6.4	-0.3513546	-1.318261	-0.47417
65	6.5	-0.1005602	1.643525	-0.4139103
66	6.6	0.8686027	-0.4322521	1.012874
67	6.7	-1.114927	0.469528	0.9161205
68	6.8	-0.356955	1.022334	-2.00257
69	6.9	-1.71516	0.6274581	-1.352094
70	7	-0.03491598	-0.03793251	0.05596954
71	7.1	-0.2810947	0.144073	-2.171863
72	7.2	-0.3389453	0.5843859	-0.8390798
73	7.3	-1.04138	0.3519497	1.069267
74	7.4	-2.866462	1.182504	0.2067203
75	7.5	-0.6907754	-0.7425984	1.164752
76	7.6	-0.09003073	-1.209451	0.7730654
77	7.7	-0.8069562	-1.046643	0.1396704
78	7.8	1.067365	0.1232827	-0.776005
79	7.9	-0.882326	-0.0145659	0.2200673
80	8	0.4727389	-0.3159074	1.723677
81	8.1	0.5338985	0.4875888	-0.5419431
82	8.2	0.7959215	-0.9714537	-0.3666259
83	8.3	0.1363355	1.229809	-0.4606246
84	8.4	0.5330227	-0.9875807	0.2573491
85	8.5	0.415046	-0.7534109	0.07963906
86	8.6	0.5442014	-1.354907	-0.03364811
87	8.7	-0.7464795	-0.6355808	0.7484256
88	8.8	-1.11568	0.1287166	0.8080038
89	8.9	-0.5232872	-0.02984434	0.04724269
90	9	0.3280034	-1.044189	-0.07286712
91	9.1	2.15871	-0.2920541	-1.050486
92	9.2	-0.294708	1.053643	-0.186262
93	9.3	-1.741194	-0.7187186	0.3079888
94	9.4	-0.1860214	-1.403891	0.8369425
95	9.5	-2.217396	-1.196006	0.9390647
96	9.6	0.55349	0.9341016	-1.968257
97	9.7	0.04515048	-0.2381485	0.3987472
98	9.8	-1.37868	-0.6811075	0.339187
99	9.9	0.6905608	-0.2576185	1.481621

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 =  [-1.33163,-0.825457,-1.21658]

Get the time series at index i : Xi

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

Out:

Xi =  [-1.33163,-0.825457,-1.21658]

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 =  -1.3316320019575207

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

myTimeSeries.getValues()

	X0	X1	X2
0	1.069747	-1.779941	-0.8327076
1	-0.2453716	-0.0205006	-0.1701006
2	0.5292955	-0.7251038	-1.162473
3	0.1995235	0.7271477	-0.2606875
4	-0.1367718	0.5202298	-0.6591333
5	-0.1806734	-1.048847	0.5123711
6	0.2064803	-0.960832	0.4146824
7	-1.228714	2.57497	-0.008049008
8	-1.885899	0.830757	-0.3783459
9	0.4790463	1.609382	-0.5708413
10	0.2690964	0.8035033	0.5832181
11	0.4497564	-0.6935559	1.896662
12	0.02708176	-0.258272	-0.37012
13	0.04565963	-0.3430478	-0.3924844
14	-2.410929	1.939206	-0.5900438
15	0.2270499	-0.1417654	0.8555065
16	0.286761	0.5648119	-0.5097008
17	1.403344	-1.378522	0.4340351
18	0.03425181	0.8961165	-0.8705775
19	1.369953	0.2725969	0.5792226
20	-1.532103	0.957065	0.4276634
21	-0.3666802	0.6486989	-0.004649441
22	0.1715484	-0.07957611	0.4553892
23	-2.140093	0.9332446	0.8186856
24	-1.548256	0.370246	-0.773089
25	-0.01298333	0.1873089	-2.131449
26	-1.197682	-0.005001849	-0.1256726
27	-1.892007	3.40565	-0.1035762
28	0.4154477	0.7272545	0.9788553
29	1.158081	0.2952752	0.2839339
30	1.294258	0.2007735	0.342265
31	0.1640854	-0.6083832	0.1443463
32	0.5377329	0.6965567	1.187906
33	2.180975	-0.1948093	0.6283156
34	0.2308662	-0.6480712	-0.02802031
35	0.8710046	1.244731	-0.1063582
36	-0.2344887	-2.010204	0.1217012
37	-1.331632	-0.8254575	-1.216578
38	-1.025789	-1.224865	-0.7350567
39	0.2674311	-0.3139666	0.3284034
40	-1.185418	0.2725766	-0.5379969
41	-0.1546276	0.03489387	0.3572081
42	0.8738098	-1.489697	-1.603233
43	0.2768838	-0.2052791	0.3135911
44	1.520626	2.127892	0.1574096
45	0.05643199	1.05201	-1.069286
46	0.03896958	0.1088619	1.560223
47	0.8978581	0.07131786	0.3290581
48	0.7683447	-0.2017215	0.1483074
49	0.4988259	-0.5406089	0.202215
50	1.52964	-1.192179	0.5249542
51	-0.1271758	1.001217	0.2995675
52	-0.07324792	-0.5928008	0.509773
53	1.568079	0.3693428	0.6873462
54	0.2602205	1.560101	0.6838802
55	-0.2604079	0.1696515	-1.016573
56	0.8102853	-0.9345477	0.4402335
57	0.1026545	0.1625502	0.9776058
58	-0.6851276	-0.04119683	-0.1615313
59	0.009488993	-0.6992373	0.8356431
60	0.9612086	-0.3953424	0.2505092
61	-1.712787	-0.3033722	1.713433
62	0.2879968	-0.3462038	-1.243077
63	-0.6619336	-0.5396257	0.7891796
64	0.525199	0.2655049	-0.6153533
65	0.6677281	-0.3206562	-0.00603524
66	-1.440427	0.07065125	0.4005165
67	-0.5370034	-2.130432	0.1862285
68	-1.326288	0.2426011	-0.8973327
69	-0.9573643	1.588237	-0.2380769
70	-0.6543979	1.498919	-0.7131357
71	-1.335157	0.5676285	0.640198
72	-0.259729	0.1922855	-1.402221
73	0.5600177	-1.356244	1.034522
74	-0.3787931	-0.1257271	-0.5878356
75	1.078941	-1.669386	1.708344
76	-0.8459409	-0.1786205	-0.1958844
77	1.811325	0.4000363	1.108118
78	-0.4552358	-0.7934174	2.283829
79	0.351885	-0.06082214	1.182574
80	2.057236	2.083603	-1.109457
81	0.6461174	0.3140881	-1.259195
82	2.51347	1.106768	-1.237082
83	-0.4050629	1.244775	0.2588656
84	-0.1137998	0.3814998	0.1557911
85	0.4024124	1.332716	-0.8056192
86	0.3854209	-1.61086	-0.6874292
87	-0.02107395	-1.405266	-0.6029087
88	-0.07453712	-0.287633	-0.4026233
89	-0.4894317	-0.5803388	1.196489
90	1.004556	0.5372572	-0.08770909
91	1.423935	0.6820146	2.884055
92	0.2796988	-1.178997	-0.143892
93	0.6813079	0.01437919	0.5099701
94	-1.060234	0.04483657	0.2499197
95	1.24773	-0.3856004	-0.2880728
96	-0.5890517	0.4995753	1.132313
97	-0.8437811	1.43619	-0.1876503
98	0.940522	0.7151117	-1.439318
99	-0.1429401	-0.1765888	0.9054335

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

myTimeSeries.getInputMean()

[0.0424435,0.0709075,0.0473796]

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 0.120 seconds)

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Manipulate a time series¶