<s>
Convergent	B-General_Concept
cross	I-General_Concept
mapping	I-General_Concept
(	O
CCM	O
)	O
is	O
a	O
statistical	O
test	O
for	O
a	O
cause-and-effect	B-Application
relationship	I-Application
between	O
two	O
variables	O
that	O
,	O
like	O
the	O
Granger	B-General_Concept
causality	I-General_Concept
test	O
,	O
seeks	O
to	O
resolve	O
the	O
problem	O
that	O
correlation	O
does	O
not	O
imply	O
causation	O
.	O
</s>
<s>
While	O
Granger	B-General_Concept
causality	I-General_Concept
is	O
best	O
suited	O
for	O
purely	O
stochastic	O
systems	O
where	O
the	O
influences	O
of	O
the	O
causal	O
variables	O
are	O
separable	O
(	O
independent	O
of	O
each	O
other	O
)	O
,	O
CCM	O
is	O
based	O
on	O
the	O
theory	O
of	O
dynamical	O
systems	O
and	O
can	O
be	O
applied	O
to	O
systems	O
where	O
causal	O
variables	O
have	O
synergistic	O
effects	O
.	O
</s>
<s>
Convergent	B-General_Concept
Cross	I-General_Concept
Mapping	I-General_Concept
(	O
CCM	O
)	O
leverages	O
a	O
corollary	O
to	O
the	O
Generalized	O
Takens	O
Theorem	O
that	O
it	O
should	O
be	O
possible	O
to	O
cross	O
predict	O
or	O
cross	O
map	O
between	O
variables	O
observed	O
from	O
the	O
same	O
system	O
.	O
</s>
<s>
CCM	O
leverages	O
this	O
property	O
to	O
infer	O
causality	B-Application
by	O
predicting	O
using	O
the	O
library	O
of	O
points	O
(	O
or	O
vice-versa	O
for	O
the	O
other	O
direction	O
of	O
causality	B-Application
)	O
,	O
while	O
assessing	O
improvements	O
in	O
cross	O
map	O
predictability	O
as	O
larger	O
and	O
larger	O
random	O
samplings	O
of	O
are	O
used	O
.	O
</s>
<s>
The	O
basic	O
steps	O
of	O
convergent	B-General_Concept
cross	I-General_Concept
mapping	I-General_Concept
for	O
a	O
variable	O
of	O
length	O
against	O
variable	O
are	O
:	O
</s>
