Time series signatures



In [1]:

    
import os, sys
sys.path.append(os.path.abspath('../../main/python'))



In [2]:

    
import datetime as dt

import numpy as np
import pandas as pd

import thalesians.tsa.signatures as signatures

import importlib
importlib.reload(signatures)









    Out[2]:





<module 'thalesians.tsa.signatures' from 'S:\\dev\\tsa\\src\\main\\python\\thalesians\\tsa\\signatures.py'>

Basic example

This is the data from the example in [CK16], Section 2.1.1, equations (2.1)-(2.3):



In [3]:

    
df = pd.DataFrame(
    np.array(((1.,1.),(3.,4.),(5.,2.),(8.,6.))),
    columns=('A', 'B'))



In [4]:

    
df

This reproduces equation (2.11) in [CK16]:



In [5]:

    
signatures.signature(df)









    Out[5]:





{(1,): 7.0, (1, 1): 24.5, (1, 2): 19.0, (2,): 5.0, (2, 1): 16.0, (2, 2): 12.5}

(An even simpler) variation on the basic example



In [6]:

    
df = pd.DataFrame(
    np.array([[1.], [3.], [5.], [9.], [10.]]),
    columns=['A'])



In [7]:

    
signatures.signature(df, max_word_length=5)









    Out[7]:





{(1,): 9.0,
 (1, 1): 40.5,
 (1, 1, 1): 121.49999999999999,
 (1, 1, 1, 1): 273.375,
 (1, 1, 1, 1, 1): 492.07499999999999}



In [8]:

    
df = pd.DataFrame(
    np.array([[1.], [3.], [2.], [-9.], [10.]]),
    columns=['A'])



In [9]:

    
signatures.signature(df, max_word_length=5)









    Out[9]:





{(1,): 9.0,
 (1, 1): 40.5,
 (1, 1, 1): 121.49999999999987,
 (1, 1, 1, 1): 273.37500000000074,
 (1, 1, 1, 1, 1): 492.07500000000215}

Slightly more involved



In [10]:

    
df = pd.DataFrame([
    [63.62, 41.48, 177.14, 102.22, 105.35, 759.44, 33.31, 13.97],
    [63.73, 41.68, 174.09, 102.73, 102.71, 761.53, 32.43, 13.72],
    [62.81, 41.05, 169.84, 102.97, 100.70, 759.33, 31.27, 13.11],
    [60.27, 40.27, 164.62,  97.92,  96.45, 741.00, 29.99, 12.70],
    [58.92, 39.70, 163.94,  97.33,  96.96, 730.91, 29.53, 12.54],
    [58.83, 39.86, 165.73,  97.51,  98.53, 733.07, 30.25, 12.77],
    [58.96, 40.62, 165.71,  99.37,  99.96, 745.34, 30.30, 12.85],
    [57.34, 39.01, 158.99,  95.44,  97.39, 719.57, 30.49, 12.20],
    [58.20, 39.93, 161.39,  98.37,  99.52, 731.39, 30.30, 12.19],
    [57.04, 39.05, 155.61,  94.97,  97.13, 710.49, 29.57, 11.97],
    [57.01, 39.13, 156.82,  95.26,  96.66, 719.08, 29.40, 11.95],
    [55.51, 38.87, 153.75,  94.35,  96.79, 718.56, 29.42, 11.90],
    [55.25, 38.52, 151.65,  94.16,  96.30, 726.67, 29.55, 12.01],
    [56.95, 39.37, 156.86,  97.94, 101.42, 745.46, 29.28, 12.14],
    [55.66, 38.29, 151.12,  97.01,  99.44, 733.62, 29.23, 11.98],
    [57.08, 39.02, 154.45,  97.34,  99.99, 733.79, 29.65, 12.26],
    [57.04, 39.18, 153.72,  94.45,  93.42, 717.58, 29.21, 11.85],
    [57.28, 39.22, 157.06, 109.11,  94.09, 748.30, 29.02, 11.71],
    [59.50, 40.06, 161.56, 112.21,  97.34, 761.35, 29.64, 11.94],
    [58.86, 39.80, 159.65, 115.09,  96.43, 770.77, 30.11, 12.07],
    [57.03, 38.75, 151.70, 114.61,  94.48, 780.91, 29.65, 11.51],
    [57.41, 39.28, 152.68, 112.69,  96.35, 749.38, 28.92, 11.46],
    [58.40, 40.11, 156.49, 110.49,  96.60, 730.03, 28.64, 11.53],
    [57.75, 40.09, 156.47, 104.07,  94.02, 703.76, 28.54, 11.45],
    [56.54, 39.53, 149.25,  99.75,  95.01, 704.16, 28.68, 11.59]],
    index=[
        dt.datetime(2016, 1, 4),
        dt.datetime(2016, 1, 5),
        dt.datetime(2016, 1, 6),
        dt.datetime(2016, 1, 7),
        dt.datetime(2016, 1, 8),
        dt.datetime(2016, 1, 11),
        dt.datetime(2016, 1, 12),
        dt.datetime(2016, 1, 13),
        dt.datetime(2016, 1, 14),
        dt.datetime(2016, 1, 15),
        dt.datetime(2016, 1, 19),
        dt.datetime(2016, 1, 20),
        dt.datetime(2016, 1, 21),
        dt.datetime(2016, 1, 22),
        dt.datetime(2016, 1, 25),
        dt.datetime(2016, 1, 26),
        dt.datetime(2016, 1, 27),
        dt.datetime(2016, 1, 28),
        dt.datetime(2016, 1, 29),
        dt.datetime(2016, 2, 1),
        dt.datetime(2016, 2, 2),
        dt.datetime(2016, 2, 3),
        dt.datetime(2016, 2, 4),
        dt.datetime(2016, 2, 5),
        dt.datetime(2016, 2, 8)],
    columns=[
        'JPM UN Equity',
        'USB UN Equity',
        'GS UN Equity',
        'FB UW Equity',
        'AAPL UW Equity',
        'GOOGL UW Equity',
        'GM UN Equity',
        'F UN Equity'])



In [11]:

    
df









    Out[11]:







  
    
      
      JPM UN Equity
      USB UN Equity
      GS UN Equity
      FB UW Equity
      AAPL UW Equity
      GOOGL UW Equity
      GM UN Equity
      F UN Equity
    
  
  
    
      2016-01-04
      63.62
      41.48
      177.14
      102.22
      105.35
      759.44
      33.31
      13.97
    
    
      2016-01-05
      63.73
      41.68
      174.09
      102.73
      102.71
      761.53
      32.43
      13.72
    
    
      2016-01-06
      62.81
      41.05
      169.84
      102.97
      100.70
      759.33
      31.27
      13.11
    
    
      2016-01-07
      60.27
      40.27
      164.62
      97.92
      96.45
      741.00
      29.99
      12.70
    
    
      2016-01-08
      58.92
      39.70
      163.94
      97.33
      96.96
      730.91
      29.53
      12.54
    
    
      2016-01-11
      58.83
      39.86
      165.73
      97.51
      98.53
      733.07
      30.25
      12.77
    
    
      2016-01-12
      58.96
      40.62
      165.71
      99.37
      99.96
      745.34
      30.30
      12.85
    
    
      2016-01-13
      57.34
      39.01
      158.99
      95.44
      97.39
      719.57
      30.49
      12.20
    
    
      2016-01-14
      58.20
      39.93
      161.39
      98.37
      99.52
      731.39
      30.30
      12.19
    
    
      2016-01-15
      57.04
      39.05
      155.61
      94.97
      97.13
      710.49
      29.57
      11.97
    
    
      2016-01-19
      57.01
      39.13
      156.82
      95.26
      96.66
      719.08
      29.40
      11.95
    
    
      2016-01-20
      55.51
      38.87
      153.75
      94.35
      96.79
      718.56
      29.42
      11.90
    
    
      2016-01-21
      55.25
      38.52
      151.65
      94.16
      96.30
      726.67
      29.55
      12.01
    
    
      2016-01-22
      56.95
      39.37
      156.86
      97.94
      101.42
      745.46
      29.28
      12.14
    
    
      2016-01-25
      55.66
      38.29
      151.12
      97.01
      99.44
      733.62
      29.23
      11.98
    
    
      2016-01-26
      57.08
      39.02
      154.45
      97.34
      99.99
      733.79
      29.65
      12.26
    
    
      2016-01-27
      57.04
      39.18
      153.72
      94.45
      93.42
      717.58
      29.21
      11.85
    
    
      2016-01-28
      57.28
      39.22
      157.06
      109.11
      94.09
      748.30
      29.02
      11.71
    
    
      2016-01-29
      59.50
      40.06
      161.56
      112.21
      97.34
      761.35
      29.64
      11.94
    
    
      2016-02-01
      58.86
      39.80
      159.65
      115.09
      96.43
      770.77
      30.11
      12.07
    
    
      2016-02-02
      57.03
      38.75
      151.70
      114.61
      94.48
      780.91
      29.65
      11.51
    
    
      2016-02-03
      57.41
      39.28
      152.68
      112.69
      96.35
      749.38
      28.92
      11.46
    
    
      2016-02-04
      58.40
      40.11
      156.49
      110.49
      96.60
      730.03
      28.64
      11.53
    
    
      2016-02-05
      57.75
      40.09
      156.47
      104.07
      94.02
      703.76
      28.54
      11.45
    
    
      2016-02-08
      56.54
      39.53
      149.25
      99.75
      95.01
      704.16
      28.68
      11.59



In [12]:

    
signatures.signature(df, max_word_length=2)









    Out[12]:





{(1,): -7.0799999999999983,
 (1, 1): 25.063199999999984,
 (1, 2): 6.0017499999999737,
 (1, 3): 106.58764999999991,
 (1, 4): -3.3171999999999926,
 (1, 5): 18.630049999999962,
 (1, 6): 228.75960000000023,
 (1, 7): 12.240049999999982,
 (1, 8): 7.3664499999999871,
 (2,): -1.9499999999999957,
 (2, 1): 7.8042499999999917,
 (2, 2): 1.9012499999999917,
 (2, 3): 32.697049999999983,
 (2, 4): -7.0254000000000296,
 (2, 5): 6.5714999999999826,
 (2, 6): 69.278749999999874,
 (2, 7): 4.4688999999999863,
 (2, 8): 2.6193999999999957,
 (3,): -27.889999999999986,
 (3, 1): 90.873549999999923,
 (3, 2): 21.688449999999875,
 (3, 3): 388.92604999999969,
 (3, 4): 39.679849999999973,
 (3, 5): 107.85864999999977,
 (3, 6): 1052.2296999999999,
 (3, 7): 56.102949999999929,
 (3, 8): 28.720899999999958,
 (4,): -2.4699999999999989,
 (4, 1): 20.804799999999979,
 (4, 2): 11.841900000000017,
 (4, 3): 29.208449999999942,
 (4, 4): 3.0504499999999974,
 (4, 5): 29.961249999999836,
 (4, 6): -223.68689999999989,
 (4, 7): -0.11485000000001663,
 (4, 8): 0.83894999999997721,
 (5,): -10.339999999999989,
 (5, 1): 54.577149999999932,
 (5, 2): 13.591499999999954,
 (5, 3): 180.52394999999979,
 (5, 4): -4.421449999999882,
 (5, 5): 53.457799999999892,
 (5, 6): 380.99625000000077,
 (5, 7): 27.010449999999974,
 (5, 8): 14.981799999999987,
 (6,): -55.280000000000086,
 (6, 1): 162.62280000000035,
 (6, 2): 38.517250000000068,
 (6, 3): 489.52950000000146,
 (6, 4): 360.22850000000022,
 (6, 5): 190.59894999999949,
 (6, 6): 1527.9392000000046,
 (6, 7): 38.192600000000233,
 (6, 8): 19.059950000000054,
 (7,): -4.6300000000000026,
 (7, 1): 20.540350000000025,
 (7, 2): 4.5595999999999988,
 (7, 3): 73.027750000000097,
 (7, 4): 11.550950000000023,
 (7, 5): 20.86375,
 (7, 6): 217.75380000000027,
 (7, 7): 10.718450000000011,
 (7, 8): 5.5547000000000066,
 (8,): -2.3800000000000008,
 (8, 1): 9.4839500000000125,
 (8, 2): 2.0215999999999958,
 (8, 3): 37.657300000000035,
 (8, 4): 5.0396500000000195,
 (8, 5): 9.6273999999999962,
 (8, 6): 112.5064500000002,
 (8, 7): 5.4647000000000023,
 (8, 8): 2.8322000000000016}



In [13]:

    
signatures.signature(df[['JPM UN Equity','USB UN Equity']], max_word_length=2)









    Out[13]:





{(1,): -7.0799999999999983,
 (1, 1): 25.063199999999984,
 (1, 2): 6.0017499999999737,
 (2,): -1.9499999999999957,
 (2, 1): 7.8042499999999917,
 (2, 2): 1.9012499999999917}



In [14]:

    
signatures.signature(df[['JPM UN Equity']])









    Out[14]:





{(1,): -7.0799999999999983}

Bibliography

[CK16] Ilya Chevyrev and Andrey Kormilitzin. A Primer on the Signature Method in Machine Learning. arXiv:1603.03788v1 [stat.ML] 11 Mar 2016, https://arxiv.org/pdf/1603.03788.pdf

	JPM UN Equity	USB UN Equity	GS UN Equity	FB UW Equity	AAPL UW Equity	GOOGL UW Equity	GM UN Equity	F UN Equity
2016-01-04	63.62	41.48	177.14	102.22	105.35	759.44	33.31	13.97
2016-01-05	63.73	41.68	174.09	102.73	102.71	761.53	32.43	13.72
2016-01-06	62.81	41.05	169.84	102.97	100.70	759.33	31.27	13.11
2016-01-07	60.27	40.27	164.62	97.92	96.45	741.00	29.99	12.70
2016-01-08	58.92	39.70	163.94	97.33	96.96	730.91	29.53	12.54
2016-01-11	58.83	39.86	165.73	97.51	98.53	733.07	30.25	12.77
2016-01-12	58.96	40.62	165.71	99.37	99.96	745.34	30.30	12.85
2016-01-13	57.34	39.01	158.99	95.44	97.39	719.57	30.49	12.20
2016-01-14	58.20	39.93	161.39	98.37	99.52	731.39	30.30	12.19
2016-01-15	57.04	39.05	155.61	94.97	97.13	710.49	29.57	11.97
2016-01-19	57.01	39.13	156.82	95.26	96.66	719.08	29.40	11.95
2016-01-20	55.51	38.87	153.75	94.35	96.79	718.56	29.42	11.90
2016-01-21	55.25	38.52	151.65	94.16	96.30	726.67	29.55	12.01
2016-01-22	56.95	39.37	156.86	97.94	101.42	745.46	29.28	12.14
2016-01-25	55.66	38.29	151.12	97.01	99.44	733.62	29.23	11.98
2016-01-26	57.08	39.02	154.45	97.34	99.99	733.79	29.65	12.26
2016-01-27	57.04	39.18	153.72	94.45	93.42	717.58	29.21	11.85
2016-01-28	57.28	39.22	157.06	109.11	94.09	748.30	29.02	11.71
2016-01-29	59.50	40.06	161.56	112.21	97.34	761.35	29.64	11.94
2016-02-01	58.86	39.80	159.65	115.09	96.43	770.77	30.11	12.07
2016-02-02	57.03	38.75	151.70	114.61	94.48	780.91	29.65	11.51
2016-02-03	57.41	39.28	152.68	112.69	96.35	749.38	28.92	11.46
2016-02-04	58.40	40.11	156.49	110.49	96.60	730.03	28.64	11.53
2016-02-05	57.75	40.09	156.47	104.07	94.02	703.76	28.54	11.45
2016-02-08	56.54	39.53	149.25	99.75	95.01	704.16	28.68	11.59