In a causal world the direction of the time arrow dictates how past causal events in a variable X produce future effects in Y. X is said to cause an effect in Y, if the predictability (uncertainty) about the future states of Y increases (decreases) as its own past and the past of X are taken into consideration. Causality is thus intrinsic dependent on the observation of the past events of both variables involved, to the prediction (or uncertainty reduction) of future event of the other variable. We will show that this temporal notion of causality leads to another natural spatiotemporal definition for it, and that can be exploited to detect the arrow of influence from X to Y, either by considering shorter time-series of X and longer time-series of Y (an approach that explores the time nature of causality) or lower precision measured time-series in X and higher precision measured time-series in Y (an approach that explores the spatial nature of causality). Causality has thus space and time signatures, causing a break of symmetry in the topology of the probabilistic space, or causing a break of symmetry in the length of the measured time-series, a consequence of the fact that information flows from X to Y.

1.
2.
I.
Guyon
,
D.
Janzing
, and
B.
Schölkopf
, ``Causality: Objectives and assessment,'' in JMLR Workshop and Conference Proceedings (
2010
), Vol.
6
, pp.
1–38
.
3.
Y.
Kano
and
S.
Shimizu
,
``Causal inference using nonnormality,''
in
Proceedings of the International Symposium on the Science of Modeling, the 30th Anniversary of the Information Criterion
(
2013
), pp.
261
270
.
4.
C. W. J.
Granger
,
``Investigating Causal Relations by Econometric Models and Cross-spectral Methods,''
Econometrica
37
,
424
(
1969
).
6.
Y.
Hirata
and
K.
Aihara
,
Phys. Rev. E
81
,
016203
(
2010
).
7.
S.
Haufe
,
V. V.
Nikulin
,
K.-R.
Müller
, and
G.
Nolte
,
NeuroImage
64
,
120
(
2013
).
8.
Y.
Chen
,
G.
Rangarajan
,
J.
Feng
, and
M.
Ding
,
Phys. Lett. A
324
,
26
(
2004
).
9.
N.
Ancona
,
D.
Marinazzo
, and
S.
Stramaglia
,
Phys. Rev. E
70
,
056221
(
2004
).
10.
E.
Bollt
,
P.
Gora
,
A.
Ostruszka
, and
K.
Zyczkowski
,
SIAM J. Appl. Dyn. Syst.
7
,
341
(
2008
).
11.
C. W.
Granger
,
J. Econ. Dyn. Control
2
,
329
(
1980
).
13.
P.-O.
Amblard
, and
O. J.
Michel
,
J. Comput. Neurosci.
30
,
7
(
2011
).
14.
M.
Takigawa
,
G.
Wang
,
H.
Kawasaki
, and
H.
Fukuzako
,
Int. J. Psychophysiol.
21
,
65
(
1996
).
15.
L. A.
Baccalá
and
K.
Sameshima
,
Biol. Cybern.
84
,
463
(
2001
).
16.
M.
Kaminski
and
K. J.
Blinowska
,
Biol. Cybern.
65
,
203
(
1991
).
17.
H.
Marko
,
IEEE Trans. Commun.
21
,
1345
(
1973
).
18.
V. A.
Vakorin
,
O. A.
Krakovska
, and
A. R.
McIntosh
,
J. Neurosci. Methods
184
,
152
(
2009
).
19.
I.
Vlachos
and
D.
Kugiumtzis
,
Phys. Rev. E
82
,
016207
(
2010
).
20.
P.
Myrberg
,
J. Math. Pures Appl. (9)
41
,
339
(
1962
).
21.
G.
Bouma
,
``Normalized (Pointwise) mutual information in collocation extraction
,''
Proc. Ger. Soc. Comput. Linguist
, (
2009
), pp.
31
40
.
22.
N. J.
Newton
, arXiv:1604.01969 (
2016
).
23.
Y.
Liu
and
S.
Aviyente
,
``The relationship between transfer entropy and directed information
,'' in
2012 IEEE Statistical Signal Processing Workshop (SSP)
, (
2012
), pp.
73
76
.
24.
R.
Szmoski
,
F.
Ferrari
,
S. d. S.
Pinto
,
M.
Baptista
, and
R.
Viana
,
Phys. Lett. A
377
,
760
(
2013
).
25.
26.
S.
Mangiarotti
,
Chaos Solitons Fractals
81
,
184
(
2015
).
27.
F. S.
Borges
et al.,
Phys. Rev. E
97
,
022303
(
2018
).
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