IEEE Computational Intelligence Magazine - August 2021 - 57

where H(X, Y) represents the joint information of X and Y
and can be calculated using the following expression:
n
m
HXYPxy Px y
j
(, )( ,) (, ).
i =1 =1
=-/ /
For
Pxl x+
the time series
Ix xll x+
ij log
ij
" () ; 12 f= ,, ,,
() are the probabilities that xl
appear in " xi iN () ; 12 f= ,, ,
.
xi iN Pxl
and x ,l x+
as follows:
() (, )( )( )( ,).
IIxx Hx Hx Hx x
x== +ll
ll ll++ +xx x
(16)
, () and
respectively,
, Based on these definitions,
the MI entropy (, ) for a time delay x can be specified
(17)
According to the MI algorithm, the value of x when
I ()x reaches a local minimum for the first time is taken as
the final solution.
C. False Nearest Neighbors Algorithm
The FNN algorithm, which was proposed by Kennel in
1992 [48], is used to calculate the embedding dimensionality
for a chaotic time series. This algorithm is based on the
premise that a chaotic time series, such as a series of wind
speed data, can be regarded as a set of continuously varying
particles in a high-dimensional space mapped to a onedimensional
space. If the number of embedding dimensions
is too small, the particles will be compressed and folded
onto one another due to the insufficient extent of their spatial
orbits, meaning that two adjacent points in the onedimensional
space may correspond to two particles
separated by a large distance in the high-dimensional space.
Two such adjacent particles are defined as false nearest
neighbor points. In this case, the embedding dimensionality
should be gradually increased to fully expand the spatial
orbits. With an increase in the number of embedding
dimensions, the particles are expected to gradually separate,
and the number of false nearest neighbor points is expected
to gradually decrease. Once the embedding dimensionality
is set to a sufficient value such that all false nearest neighbor
points are eliminated, the corresponding solution is considered
to be the optimal solution.
Suppose that () (( ), (),, (( )))
zm yi yi ym 1
zmj
between () and () can be calculated using the following
equation:
is the nearest neighbor point of ;zi
zmi
zmj
m =max
Rm zm zmii j
=()
() () .
i
(18)
This distance changes as the number of dimensions
increases. Thus, the updated distance between ()
j ()
zm 1
+ can be expressed as follows:
() () () () .
Rm Rm zi mz jm
ii1 ij
22
zm 1+ and
+= xx++ -+ (19)
where t0
and tM
i =+ -+ixxf
is a vector in an m-dimensional phase space and that
()
the distance
According to Wolf's method, the maximum Lyapunov
exponent can be calculated as
tt L
L
M - 0 i =0
1 / l
i
M
ln
i
time, respectively, and MN () .
=- m 1 xrepresent
the initial time and the final
In addition,
the interval for time series prediction can be calculated
as follows [52], [53]:
Tt = 1
mmax
.
(23)
AUGUST 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 57
,
(22)
If () is considerably larger than (),Rmi
Rm 1
i
+
the two
points are considered to be false nearest neighbor points.
By modifying this equation, a more convenient function
for judging false nearest neighbor points can be obtained
as follows:
R =
x
set to [10, 50]. If Rx
zi mz jm
i
ij
() ()+- +
Rm
xx
()
.
(20)
Rx is compared against a given threshold i , whose range is
is larger than this threshold, the points are
determined to be false nearest neighbor points. For a realworld
chaotic time series, such as a wind speed series, the initial
number of embedding dimensions is usually set to 2. Once
the proportion of false nearest neighbor points is less than 5%,
the corresponding number of embedding dimensions is selected
as the final solution. In certain extreme cases, the proportion
of false nearest neighbor points cannot decrease to 5%; in
such a case, the critical dimensionality at which the number of
false nearest neighbor points stops decreasing is considered the
final solution.
D. Maximum Lyapunov Exponent
Before a machine learning technique is employed to forecast
the future wind speed, the chaotic characteristics of the historical
wind speed series must be confirmed. Lyapunov exponents
are commonly used for this purpose. In 1983, it was
proven that if at least one of the Lyapunov exponents in a
dynamical system is positive, the corresponding time series can
be considered chaotic [50]. Therefore, the maximum Lyapunov
exponent of the time series needs to be calculated. If this
exponent exceeds zero, the time series is considered to have
chaotic characteristics. Among the various approaches for calculating
Lyapunov exponents, Wolf's method [51], which is
based on the phase space reconstruction theory of Takens, is
considered to be the most effective.
For a series () (( ), (),, (( ))
Xt xt xt xm 1
Xti
expressed as follows:
LX () .tX tin i
=()
(21)
=+
-+ re -xxf
t
constructed using Takens' theory, as described above, the
distance Li
between () and the closest point () can be
Xtn
IEEE Computational Intelligence Magazine - August 2021

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