IEEE Computational Intelligence Magazine - August 2021 - 56

wind speed time series data are processed through a normalization
operation. Finally, the DNR model can be employed
for wind speed prediction.
A. Phase Space Reconstruction
It is difficult to forecast the future trend of a chaotic time
series such as a wind speed series because of its irregularity.
A chaotic time series can be considered to represent a
type of random motion in a definite dynamical system.
Thus, we need to restore the original dynamical system
of the chaotic time series. The most common method
to accomplish this is the delayed coordinate approach
proposed by Takens
" xi iN (); 12 f= ,, ,
,
delay x and vector embedding dimensionality m have
Algorithm 1 Pseudocode for the SMS algorithm.
Input: Population size N, Number of dimensions n, Maximum
number of iterations epoch.
Result: Best solution P .best
begin
Initialize the population X PPN, the direction vector set
0 , the maximum number of iterations epoch =
= " 1 f
D ddN
= " f
1
t 1= ;
,, ,
1000, the state count phase = 1, and the current iteration
number
repeat
if phase 1== then
.,
c 08=
a 08= ., b 09= ., H ., Dend epoch 05) .;
=09
if phase == 2 then
c 04= ., a 02= ., b 05= ., H ., Dend epoch 09) .;
if phase == 3 then
=02
=00
=
c 01= ., a 00= ., b 00= ., H ., Dend epoch ;
for ,,
=
tt1 Dend tt 1== + do
Evaluate the fitness of the population
() (),, () ;
FX fP fPN ,
= "
1
f
Set the individual with the best fitness as P ;best
/* Perform direction vector operations */
Calculate the new direction vector set Dt
Use Dt
with Eq. (6);
Update each individual Pn
with Eq. (7);
/* Perform collision operations */
Calculate the threshold r using Eq. (8);
phase phase +
until phase 32 ;
=
Calculate the distance between each pair of points; if the
distance is less than r, exchange the direction vectors of
the points, as indicated in Eq. (9);
/* Perform random behavior */
Draw a random number RA in the range of [0,1]; if RA is
less than H, execute random behavior with Eq. (10);
1 ;
return the best solution P .best
to calculate the velocity vector set V vvN,
with Eq. (5);
= " 1 f
,,
n
()=-/ ii
=1
HX Px Px
i
()log (),
Pxi
(13)
where X denotes discrete random variables, () is the probability
of the occurrence of event x, and n is the total number
of states x. ()
HX Y;
n
=represents
the conditional information
entropy and can be expressed as
m
HXYPyP xy Px y
j
ji ji j
log
() / / () () (),;; ;
i =1 =1
Pyj
Px yij
(14)
where m represents the total number of states y, () is the
;
the conditional probability of event x occurring under the
condition of the occurrence of event y. The MI entropy,
I(X, Y ), can be calculated as
IXYHXHYHXY=+ -
56 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2021
(, )( )( )( ,),
(15)
probability of event y occurring alone, and () denotes
=
,, ,
been calculated using a certain approach, a group of new
vectors can be constructed as follows:
X =
J
L
K
K
K
K
x1
x
x+1
x -+
11
x2
x
x+2
x -+
f
f
h
x
x
Nm
Nm
-- -
-- -
x
x
xx f x -N 1
hh h
() () 2mm 1
Tx ,, ,,
= xx()
() xx12 13mm N
-+ -+
f
()
()
11
12
N
P
O
O
O
O
,
(11)
^h (12)
[44]. For a chaotic t ime series
, under the assumption that the time
where X and T are the input signals of the neural model and
the target outputs, respectively. The matrix X can efficiently
describe the original dynamical system if the appropriate values
of the time delay x and embedding dimensionality m are
selected. The operation of constructing X from a time series
()
" xi iN; 12= ,, ...,
, is termed phase space reconstruction.
The core problem in performing this operation is to obtain
the appropriate values for x and m. Takens only proved the existence
of the time delay and embedding dimensionality in theory
and did not provide a method for determining their values. In
general, a time series always contains finite sequences with
noise. There is no single method that can accurately obtain the
time delay and embedding dimensionality for every time series;
instead, the appropriate algorithm must be chosen depending
on the actual situation. In this study, the FNN algorithm is
utilized to calculate the embedding dimensionality [48], and
the MI algorithm is employed to obtain the time delay [49].
The details of these two methods are presented in the two
subsequent subsections.
B. Mutual Information Algorithm
The MI algorithm is one of the most widely used methods for calculating
the time delay of a time series and is based on the theory
of mutual information. The information entropy, H(X), which is
used to represent the degree of uncertainty of X, can be expressed
as follows:
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