Page 661 - Mathematical Techniques of Fractional Order Systems

632 Mathematical Techniques of Fractional Order Systems

TABLE 21.1 Some Measures of Recurrence Quantification
Measures Symbols Definitions
N
The Recurrence Rate RR P
RR 5 1
N 2 R ij
i; j51
The Determinism Rate DET P N lP lðÞ
l5l min
DET 5 P N
lPðlÞ
l51
The Lamirarity Rate LAM P N vP vðÞ
v5v min
LAM 5 P N
vP vðÞ
v51
The Trapping Time TT P N
v5v min vP v ðÞ
TT 5 P N
Pv ðÞ
v5v min
The Ratio rENT P N lP lðÞ
2 l5l min
r ENT 5 N P N 2

lP lðÞ
l51
The Shannon Entropy ENT P
N
ENT 52 plðÞlog plðÞ
l5l min
The Longest Diagonal Lines maxL max L 5 max l i ; i 5 1; .. .; N l gÞ
f
ð
The Trend T P M ð i 2 M=2Þ ð RR i 2 RR i iÞ
h
T 5 i51 P M
ð i2M=2Þ 2
i51
The ratio: Is the relation between the percentage of recurrence and the
percentage of determinism, which gives the ratio of the recurrence points,
forming the structure of the diagonal pieces (off main diagonal), and rela-
tive to the total number of recurrence points. It is a deterministic measure
of the dynamic system.
The Shannon entropy: Is the frequency of the distribution whose probabil-
ities are derived from the percentage of appearance of oblique segments
of variable lengths. It was correlated with the inverse of the largest
Lyapunov exponent; which is contrary to the usual sense of Shannon
entropy which habitually measures the degree of complexity of a dynam-
ical behavior.
The longest diagonal lines: It gives a measure of the sensitivity to the ini-
tial conditions; it is inversely proportional to the Lyapunov exponent.
The trend: It quantifies the filling of the recurring points observed when
moving away from the principal diagonal and gives a measure of the non-
stationarity described by the recurrence diagram.
With these measures, it will be able to study the spatial and temporal var-
iability of the abstract mechanism, behind an endogenous structural change
and to assess the stability, over time, of its spatial distribution. That is, to