Toolbox Reference Manual - TOCSY - Toolbox for Complex Systems ...

General Information
Lmax and divergence
entropy
L max = max ({l i ; i = 1 . . . N l }) respective DIV = 1
L max
,
N
∑
ENTR = − p(l) ln p(l) with p(l) =
l=l min
laminarity (Marwan et al., 2002)
LAM = ∑N v=v min
vP ε (v)
∑ N v=1 vPε (v) ,
P ε (l)
∑ N l=l min
P ε (l) ,
(where P ε (v) = {v i ; i = 1 . . . N v } denotes the frequency distribution of
the lengths l of vertical structures)
trapping time
TT = ∑N v=v min
vP ε (v)
∑ N v=v min
P ε (v) ,
recurrence times of first type (Gao and Cai, 2000)
T 1 j = ∣ ∣ { i, j : ⃗x i ,⃗x j ∈ R i
}∣ ∣ ,
recurrence times of second type
T 2 j
= ∣ ∣ { i, j : ⃗x i ,⃗x j ∈ R i ; ⃗x j−1 ̸∈ R i
}∣ ∣
(where R i are the recurrence points which belong to the state ⃗x i ).
Further quantifiers are based on complex network theory, as clustering
coefficient (Marwan et al., 2009)
with RR i = ∑ N j=1 Rm, ε
i,j
transitivity
C =
N
∑
i=1
∑ N j,k=1 Rm, ε
i,j
R m, ε
j,k Rm, ε
k,i
RR i
the local recurrence rate, or
C = ∑N i,j,k=1 Rm, ε
i,j
R m, ε
∑i,j,k=1 N Rm, ε
i,j
R m, ε
k,i
j,k Rm, ε
k,i
Above definitions are for the entire recurrence plot (or for squared windows
in it, reveiling some time dependencies). But most of these measures
can be quantified for each diagonal line (parallel to the main diagonal)
as well, which is even interesting for cross recurrence plots, for
example
.
4 Reference Manual