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.

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 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 0.192 seconds)

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Manipulate a time series¶