This function computes the signature space of a matrix obtained from a time series' state space by calculating the successive differences of each embedded vector. The signature space reveals changes between successive points in the time series, capturing dynamics that are crucial for understanding complex system behaviors.
Value
Matrix where each row represents the vector of differences between successive elements of the corresponding row in matrix M. The orientation of the matrix may vary depending on the embedding dimension.
Examples
data(climate_indices)
ts <- climate_indices$AO
E <- 3
tau <- 1
stateSpaceMatrix <- stateSpace(ts, E, tau)
signatureMatrix <- signatureSpace(stateSpaceMatrix, E)
print(signatureMatrix)
#> [,1] [,2]
#> [1,] 1.5361 -0.1174
#> [2,] -0.1174 -0.3427
#> [3,] -0.3427 0.9067
#> [4,] 0.9067 1.1833
#> [5,] 1.1833 -0.8947
#> [6,] -0.8947 -0.7226
#> [7,] -0.7226 0.6382
#> [8,] 0.6382 -1.1975
#> [9,] -1.1975 1.7185
#> [10,] 1.7185 0.8197
#> [11,] 0.8197 -3.3605
#> [12,] -3.3605 1.1320
#> [13,] 1.1320 -0.4996
#> [14,] -0.4996 1.0142
#> [15,] 1.0142 -0.7357
#> [16,] -0.7357 1.8763
#> [17,] 1.8763 -1.3437
#> [18,] -1.3437 0.4370
#> [19,] 0.4370 0.4978
#> [20,] 0.4978 -0.8338
#> [21,] -0.8338 -0.8398
#> [22,] -0.8398 1.3037
#> [23,] 1.3037 -0.0590
#> [24,] -0.0590 -0.2153
#> [25,] -0.2153 -1.3131
#> [26,] -1.3131 2.0751
#> [27,] 2.0751 -0.2508
#> [28,] -0.2508 -0.6175
#> [29,] -0.6175 0.9984
#> [30,] 0.9984 -0.8046
#> [31,] -0.8046 -0.7960
#> [32,] -0.7960 -0.1274
#> [33,] -0.1274 0.9798
#> [34,] 0.9798 -1.0280
#> [35,] -1.0280 0.3323
#> [36,] 0.3323 1.8573
#> [37,] 1.8573 0.1002
#> [38,] 0.1002 0.3797
#> [39,] 0.3797 -1.6625
#> [40,] -1.6625 -0.9714
#> [41,] -0.9714 1.1849
#> [42,] 1.1849 0.3574
#> [43,] 0.3574 0.1955
#> [44,] 0.1955 -0.7687
#> [45,] -0.7687 0.8719
#> [46,] 0.8719 0.3063
#> [47,] 0.3063 0.3919
#> [48,] 0.3919 -3.1650
#> [49,] -3.1650 1.2388
#> [50,] 1.2388 -0.1707
#> [51,] -0.1707 0.2969
#> [52,] 0.2969 0.7534
#> [53,] 0.7534 -0.1820
#> [54,] -0.1820 0.9673
#> [55,] 0.9673 -0.9309
#> [56,] -0.9309 1.2020
#> [57,] 1.2020 -2.0568
#> [58,] -2.0568 0.8741
#> [59,] 0.8741 0.7188
#> [60,] 0.7188 -1.2077
#> [61,] -1.2077 -2.0833
#> [62,] -2.0833 2.1024
#> [63,] 2.1024 0.7628
#> [64,] 0.7628 -0.4719
#> [65,] -0.4719 0.0116
#> [66,] 0.0116 0.4468
#> [67,] 0.4468 -0.8785
#> [68,] -0.8785 0.1425
#> [69,] 0.1425 -0.6956
#> [70,] -0.6956 1.4119
#> [71,] 1.4119 -3.2517
#> [72,] -3.2517 1.3659
#> [73,] 1.3659 1.9912
#> [74,] 1.9912 0.1010
#> [75,] 0.1010 -1.0846
#> [76,] -1.0846 0.0856
#> [77,] 0.0856 -0.0430
#> [78,] -0.0430 0.3882
#> [79,] 0.3882 0.1158
#> [80,] 0.1158 0.9207
#> [81,] 0.9207 -2.2526
#> [82,] -2.2526 -0.7301
#> [83,] -0.7301 1.3800
#> [84,] 1.3800 -2.3365
#> [85,] -2.3365 4.8349
#> [86,] 4.8349 -1.8278
#> [87,] -1.8278 0.2639
#> [88,] 0.2639 0.1677
#> [89,] 0.1677 -0.5429
#> [90,] -0.5429 -0.8180
#> [91,] -0.8180 0.8029
#> [92,] 0.8029 1.4480
#> [93,] 1.4480 -0.4989
#> [94,] -0.4989 -0.8659
#> [95,] -0.8659 -1.2074
#> [96,] -1.2074 -0.3256
#> [97,] -0.3256 -0.2733
#> [98,] -0.2733 2.1335
#> [99,] 2.1335 -0.0618
#> [100,] -0.0618 -1.0355
#> [101,] -1.0355 0.2440
#> [102,] 0.2440 -0.3694
#> [103,] -0.3694 1.1222
#> [104,] 1.1222 -0.3665
#> [105,] -0.3665 -0.4558
#> [106,] -0.4558 0.0019
#> [107,] 0.0019 0.7986
#> [108,] 0.7986 -1.3309
#> [109,] -1.3309 0.8691
#> [110,] 0.8691 -0.3636
#> [111,] -0.3636 -0.2854
#> [112,] -0.2854 0.9066
#> [113,] 0.9066 -0.2039
#> [114,] -0.2039 0.3980
#> [115,] 0.3980 0.7847
#> [116,] 0.7847 -1.0069
#> [117,] -1.0069 -0.0671
#> [118,] -0.0671 1.7135
#> [119,] 1.7135 1.4272
#> [120,] 1.4272 0.1733
#> [121,] 0.1733 -1.7490
#> [122,] -1.7490 -1.7805
#> [123,] -1.7805 1.1390
#> [124,] 1.1390 -0.5438
#> [125,] -0.5438 0.5206
#> [126,] 0.5206 -0.3147
#> [127,] -0.3147 0.1522
#> [128,] 0.1522 0.2876
#> [129,] 0.2876 -0.9569
#> [130,] -0.9569 -0.6775
#> [131,] -0.6775 1.6444
#> [132,] 1.6444 2.4009
#> [133,] 2.4009 -0.4116
#> [134,] -0.4116 -1.1112
#> [135,] -1.1112 -0.9360
#> [136,] -0.9360 -0.6385
#> [137,] -0.6385 -0.6001
#> [138,] -0.6001 0.1156
#> [139,] 0.1156 -0.0302
#> [140,] -0.0302 0.8707
#> [141,] 0.8707 -0.1397
#> [142,] -0.1397 0.7561
#> [143,] 0.7561 -0.5535
#> [144,] -0.5535 -1.5992
#> [145,] -1.5992 0.3492
#> [146,] 0.3492 1.0570
#> [147,] 1.0570 -0.0437
#> [148,] -0.0437 -0.6019
#> [149,] -0.6019 -0.0726
#> [150,] -0.0726 0.9849
#> [151,] 0.9849 -0.9091
#> [152,] -0.9091 -0.1397
#> [153,] -0.1397 0.5366
#> [154,] 0.5366 1.3285
#> [155,] 1.3285 -1.0632
#> [156,] -1.0632 0.5717
#> [157,] 0.5717 -0.1375
#> [158,] -0.1375 -1.5047
#> [159,] -1.5047 1.8619
#> [160,] 1.8619 -1.6434
#> [161,] -1.6434 0.4931
#> [162,] 0.4931 0.3442
#> [163,] 0.3442 -1.1756
#> [164,] -1.1756 0.2744
#> [165,] 0.2744 1.0829
#> [166,] 1.0829 0.9097
#> [167,] 0.9097 1.8686
#> [168,] 1.8686 -3.3108
#> [169,] -3.3108 0.5798
#> [170,] 0.5798 -1.1997
#> [171,] -1.1997 -1.1721
#> [172,] -1.1721 1.0880
#> [173,] 1.0880 0.0088
#> [174,] 0.0088 0.1177
#> [175,] 0.1177 0.0324
#> [176,] 0.0324 -0.2044
#> [177,] -0.2044 1.5668
#> [178,] 1.5668 -1.1059
#> [179,] -1.1059 -0.1838
#> [180,] -0.1838 -0.5736
#> [181,] -0.5736 2.7425
#> [182,] 2.7425 -1.6563
#> [183,] -1.6563 -0.3401
#> [184,] -0.3401 1.7217
#> [185,] 1.7217 -1.2556
#> [186,] -1.2556 0.4768
#> [187,] 0.4768 -0.9116
#> [188,] -0.9116 0.2581
#> [189,] 0.2581 1.6054
#> [190,] 1.6054 -0.8856
#> [191,] -0.8856 -1.0476
#> [192,] -1.0476 1.5827
#> [193,] 1.5827 -1.0357
#> [194,] -1.0357 -1.3563
#> [195,] -1.3563 0.0719
#> [196,] 0.0719 0.7794
#> [197,] 0.7794 -0.1053
#> [198,] -0.1053 0.7607
#> [199,] 0.7607 -1.0926
#> [200,] -1.0926 0.6240
#> [201,] 0.6240 -0.7983
#> [202,] -0.7983 -1.4038
#> [203,] -1.4038 0.9267
#> [204,] 0.9267 1.3636
#> [205,] 1.3636 -1.6464
#> [206,] -1.6464 -0.0419
#> [207,] -0.0419 1.2987
#> [208,] 1.2987 0.7231
#> [209,] 0.7231 0.2179
#> [210,] 0.2179 -0.5899
#> [211,] -0.5899 -1.2647
#> [212,] -1.2647 1.3225
#> [213,] 1.3225 -0.0461
#> [214,] -0.0461 -1.8572
#> [215,] -1.8572 1.2640
#> [216,] 1.2640 2.3455
#> [217,] 2.3455 -0.7979
#> [218,] -0.7979 -0.7672
#> [219,] -0.7672 -1.2847
#> [220,] -1.2847 0.1461
#> [221,] 0.1461 0.3844
#> [222,] 0.3844 0.5512
#> [223,] 0.5512 0.0739
#> [224,] 0.0739 -0.8942
#> [225,] -0.8942 0.0386
#> [226,] 0.0386 0.5900
#> [227,] 0.5900 -2.0095
#> [228,] -2.0095 1.8974
#> [229,] 1.8974 -0.0712
#> [230,] -0.0712 0.2165
#> [231,] 0.2165 0.4665
#> [232,] 0.4665 -1.1393
#> [233,] -1.1393 0.4990
#> [234,] 0.4990 0.8620
#> [235,] 0.8620 -1.7002
#> [236,] -1.7002 1.3442
#> [237,] 1.3442 -1.7437
#> [238,] -1.7437 2.8028
#> [239,] 2.8028 -1.2431
#> [240,] -1.2431 0.3718
#> [241,] 0.3718 -1.9737
#> [242,] -1.9737 1.7760
#> [243,] 1.7760 -0.0585
#> [244,] -0.0585 0.4811
#> [245,] 0.4811 -0.7090
#> [246,] -0.7090 -0.6701
#> [247,] -0.6701 0.7312
#> [248,] 0.7312 -0.0649
#> [249,] -0.0649 0.6167
#> [250,] 0.6167 0.4322
#> [251,] 0.4322 0.2271
#> [252,] 0.2271 -0.1944
#> [253,] -0.1944 -1.5272
#> [254,] -1.5272 0.1729
#> [255,] 0.1729 1.2476
#> [256,] 1.2476 -0.3830
#> [257,] -0.3830 -1.2355
#> [258,] -1.2355 0.7933
#> [259,] 0.7933 0.2510
#> [260,] 0.2510 -0.0781
#> [261,] -0.0781 -1.8983
#> [262,] -1.8983 -0.7729
#> [263,] -0.7729 1.3956
#> [264,] 1.3956 0.3364
#> [265,] 0.3364 -1.0641
#> [266,] -1.0641 2.5925
#> [267,] 2.5925 -0.4540
#> [268,] -0.4540 -0.4673
#> [269,] -0.4673 -0.0157
#> [270,] -0.0157 0.5515
#> [271,] 0.5515 -1.2271
#> [272,] -1.2271 1.4141
#> [273,] 1.4141 0.1111
#> [274,] 0.1111 -2.1410
#> [275,] -2.1410 2.7037
#> [276,] 2.7037 -0.0778
#> [277,] -0.0778 -0.4015
#> [278,] -0.4015 -0.1536
#> [279,] -0.1536 -0.3470
#> [280,] -0.3470 0.1713
#> [281,] 0.1713 -0.2451
#> [282,] -0.2451 -0.5561
#> [283,] -0.5561 0.1858
#> [284,] 0.1858 -1.4458
#> [285,] -1.4458 0.0634
#> [286,] 0.0634 -0.1670
#> [287,] -0.1670 1.1204
#> [288,] 1.1204 0.5995
#> [289,] 0.5995 0.8052
#> [290,] 0.8052 -1.1111
#> [291,] -1.1111 1.1948
#> [292,] 1.1948 -1.1188
#> [293,] -1.1188 0.1774
#> [294,] 0.1774 -0.3557
#> [295,] -0.3557 0.7470
#> [296,] 0.7470 -1.1364
#> [297,] -1.1364 1.3122
#> [298,] 1.3122 -0.3772
#> [299,] -0.3772 -1.9510
#> [300,] -1.9510 0.1573
#> [301,] 0.1573 1.8466
#> [302,] 1.8466 -0.7275
#> [303,] -0.7275 0.3151
#> [304,] 0.3151 -0.1416
#> [305,] -0.1416 0.0354
#> [306,] 0.0354 -0.5197
#> [307,] -0.5197 1.5752
#> [308,] 1.5752 -1.3704
#> [309,] -1.3704 1.1937
#> [310,] 1.1937 0.5518
#> [311,] 0.5518 -0.8739
#> [312,] -0.8739 -1.6268
#> [313,] -1.6268 -0.0773
#> [314,] -0.0773 1.3017
#> [315,] 1.3017 -0.7172
#> [316,] -0.7172 0.3802
#> [317,] 0.3802 0.3530
#> [318,] 0.3530 0.0563
#> [319,] 0.0563 0.7763
#> [320,] 0.7763 -0.7726
#> [321,] -0.7726 0.1979
#> [322,] 0.1979 -2.3316
#> [323,] -2.3316 1.9334
#> [324,] 1.9334 0.0147
#> [325,] 0.0147 -1.4480
#> [326,] -1.4480 1.7421
#> [327,] 1.7421 0.0175
#> [328,] 0.0175 0.9150
#> [329,] 0.9150 -0.9681
#> [330,] -0.9681 -0.3679
#> [331,] -0.3679 0.8717
#> [332,] 0.8717 -1.6356
#> [333,] -1.6356 1.5504
#> [334,] 1.5504 1.7604
#> [335,] 1.7604 -0.2479
#> [336,] -0.2479 -3.3407
#> [337,] -3.3407 2.4890
#> [338,] 2.4890 -0.6378
#> [339,] -0.6378 0.3494
#> [340,] 0.3494 -1.4487
#> [341,] -1.4487 0.1585
#> [342,] 0.1585 0.3628
#> [343,] 0.3628 0.2126
#> [344,] 0.2126 0.2046
#> [345,] 0.2046 -0.9022
#> [346,] -0.9022 1.3398
#> [347,] 1.3398 -0.0021
#> [348,] -0.0021 0.1191
#> [349,] 0.1191 -0.3525
#> [350,] -0.3525 -1.0406
#> [351,] -1.0406 -0.7497
#> [352,] -0.7497 1.1149
#> [353,] 1.1149 -0.3901
#> [354,] -0.3901 0.3994
#> [355,] 0.3994 -0.2461
#> [356,] -0.2461 2.0024
#> [357,] 2.0024 -1.5836
#> [358,] -1.5836 0.5556
#> [359,] 0.5556 0.1519
#> [360,] 0.1519 -1.4720
#> [361,] -1.4720 0.7936
#> [362,] 0.7936 0.8512
#> [363,] 0.8512 0.2212
#> [364,] 0.2212 -2.5444
#> [365,] -2.5444 -0.0052
#> [366,] -0.0052 1.3022
#> [367,] 1.3022 0.9282
#> [368,] 0.9282 -2.4144
#> [369,] -2.4144 1.9988
#> [370,] 1.9988 -3.8717
#> [371,] -3.8717 0.8260
#> [372,] 0.8260 -1.6789
#> [373,] -1.6789 3.8336
#> [374,] 3.8336 0.1576
#> [375,] 0.1576 -0.6441
#> [376,] -0.6441 0.9056
#> [377,] 0.9056 0.4483
#> [378,] 0.4483 -0.5519
#> [379,] -0.5519 -0.7480
#> [380,] -0.7480 0.3976
#> [381,] 0.3976 0.0913
#> [382,] 0.0913 -2.2553
#> [383,] -2.2553 0.9479
#> [384,] 0.9479 3.2585
#> [385,] 3.2585 -0.1513
#> [386,] -0.1513 0.8507
#> [387,] 0.8507 -2.3099
#> [388,] -2.3099 -0.8227
#> [389,] -0.8227 0.3862
#> [390,] 0.3862 -0.5910
#> [391,] -0.5910 1.7273
#> [392,] 1.7273 0.1351
#> [393,] 0.1351 0.6594
#> [394,] 0.6594 0.7616
#> [395,] 0.7616 -2.4405
#> [396,] -2.4405 0.1834
#> [397,] 0.1834 1.0734
#> [398,] 1.0734 -1.0717
#> [399,] -1.0717 0.2030
#> [400,] 0.2030 -0.8408
#> [401,] -0.8408 0.8402
#> [402,] 0.8402 -0.1538
#> [403,] -0.1538 0.7582
#> [404,] 0.7582 -2.2862
#> [405,] -2.2862 1.4034
#> [406,] 1.4034 -1.6380
#> [407,] -1.6380 1.1391
#> [408,] 1.1391 -0.3979
#> [409,] -0.3979 -2.1780
#> [410,] -2.1780 3.5076
#> [411,] 3.5076 0.1718
#> [412,] 0.1718 0.0547
#> [413,] 0.0547 -0.5598
#> [414,] -0.5598 0.1653
#> [415,] 0.1653 -0.6151
#> [416,] -0.6151 0.7237
#> [417,] 0.7237 1.7662
#> [418,] 1.7662 -0.5541
#> [419,] -0.5541 -2.4437
#> [420,] -2.4437 1.0126
#> [421,] 1.0126 1.1620
#> [422,] 1.1620 -0.2338
#> [423,] -0.2338 -0.5078
#> [424,] -0.5078 -0.9716
#> [425,] -0.9716 0.0185
#> [426,] 0.0185 0.1174
#> [427,] 0.1174 0.4734
#> [428,] 0.4734 -1.2363
#> [429,] -1.2363 0.6041
#> [430,] 0.6041 0.9432
#> [431,] 0.9432 0.6787
#> [432,] 0.6787 -0.0490
#> [433,] -0.0490 0.7948
#> [434,] 0.7948 -0.6217
#> [435,] -0.6217 -0.4529
#> [436,] -0.4529 -0.3358
#> [437,] -0.3358 -1.5349
#> [438,] -1.5349 0.4189
#> [439,] 0.4189 0.5245
#> [440,] 0.5245 -0.0856
#> [441,] -0.0856 2.1951
#> [442,] 2.1951 -0.5009
#> [443,] -0.5009 -2.8928
#> [444,] -2.8928 1.4252
#> [445,] 1.4252 0.3037
#> [446,] 0.3037 -1.3313
#> [447,] -1.3313 1.0154
#> [448,] 1.0154 0.3486
#> [449,] 0.3486 -0.2281
#> [450,] -0.2281 0.3876
#> [451,] 0.3876 0.3086
#> [452,] 0.3086 -2.6983
#> [453,] -2.6983 1.3064
#> [454,] 1.3064 2.3973
#> [455,] 2.3973 -0.8444
#> [456,] -0.8444 -0.6021
#> [457,] -0.6021 1.0255
#> [458,] 1.0255 -1.4541
#> [459,] -1.4541 -0.6414
#> [460,] -0.6414 1.1318
#> [461,] 1.1318 0.2325
#> [462,] 0.2325 -0.4842
#> [463,] -0.4842 -0.6424
#> [464,] -0.6424 1.1827
#> [465,] 1.1827 -0.7679
#> [466,] -0.7679 0.0186
#> [467,] 0.0186 -0.2218
#> [468,] -0.2218 0.3935
#> [469,] 0.3935 -1.0538
#> [470,] -1.0538 1.4850
#> [471,] 1.4850 0.6357
#> [472,] 0.6357 -0.7997
#> [473,] -0.7997 0.2319
#> [474,] 0.2319 0.2243
#> [475,] 0.2243 -0.2516
#> [476,] -0.2516 -0.1717
#> [477,] -0.1717 -1.5290
#> [478,] -1.5290 1.2259
#> [479,] 1.2259 -0.8229
#> [480,] -0.8229 1.8627
#> [481,] 1.8627 0.9666
#> [482,] 0.9666 -2.3714
#> [483,] -2.3714 -0.9760
#> [484,] -0.9760 0.6300
#> [485,] 0.6300 -0.2884
#> [486,] -0.2884 0.1679
#> [487,] 0.1679 1.0280
#> [488,] 1.0280 -0.3884
#> [489,] -0.3884 -1.1112
#> [490,] -1.1112 1.6055
#> [491,] 1.6055 2.0069
#> [492,] 2.0069 0.9982
#> [493,] 0.9982 -0.7758
#> [494,] -0.7758 -1.7133
#> [495,] -1.7133 -0.9552
#> [496,] -0.9552 -0.0947
#> [497,] -0.0947 -0.2900
#> [498,] -0.2900 0.0306
#> [499,] 0.0306 1.0126
#> [500,] 1.0126 -0.7031
#> [501,] -0.7031 2.1581
#> [502,] 2.1581 -3.8224
#> [503,] -3.8224 -0.7476
#> [504,] -0.7476 1.2929
#> [505,] 1.2929 3.2999
#> [506,] 3.2999 -2.3136
#> [507,] -2.3136 0.0438
#> [508,] 0.0438 1.0052
#> [509,] 1.0052 -0.2144
#> [510,] -0.2144 -0.8395
#> [511,] -0.8395 -0.0423
#> [512,] -0.0423 0.1058
#> [513,] 0.1058 0.2388
#> [514,] 0.2388 0.1051
#> [515,] 0.1051 0.6502
#> [516,] 0.6502 0.6961
#> [517,] 0.6961 -1.2392
#> [518,] -1.2392 -0.9078
#> [519,] -0.9078 1.8261
#> [520,] 1.8261 -1.2977
#> [521,] -1.2977 0.0991
#> [522,] 0.0991 -0.1949
#> [523,] -0.1949 -0.4852
#> [524,] -0.4852 2.0009
#> [525,] 2.0009 -1.0068
#> [526,] -1.0068 -3.0581
#> [527,] -3.0581 2.0449
#> [528,] 2.0449 2.2747
#> [529,] 2.2747 -1.3201
#> [530,] -1.3201 -1.2534
#> [531,] -1.2534 2.1074
#> [532,] 2.1074 -1.4205
#> [533,] -1.4205 0.1317