In chaos theory, the correlation sum is the estimator of the correlation integral, which reflects the mean probability that the states at two different times are close:
C
(
ε
)
=
1
N
2
∑
i
≠
j
i
,
j
=
1
N
Θ
(
ε
−
|
|
x
→
(
i
)
−
x
→
(
j
)
|
|
)
,
x
→
(
i
)
∈
R
m
,
where
N
is the number of considered states
x
→
(
i
)
,
ε
is a threshold distance,
|
|
⋅
|
|
a norm (e.g. Euclidean norm) and
Θ
(
⋅
)
the Heaviside step function. If only a time series is available, the phase space can be reconstructed by using a time delay embedding (see Takens' theorem):
x
→
(
i
)
=
(
u
(
i
)
,
u
(
i
+
τ
)
,
…
,
u
(
i
+
τ
(
m
−
1
)
)
,
where
u
(
i
)
is the time series,
m
the embedding dimension and
τ
the time delay.
The correlation sum is used to estimate the correlation dimension.
