J Supercomput
DOI 10.1007/s1122701309436
Solving symbolic regression problems with uniform
designaided gene expression programming
Yunliang Chen
·
Dan Chen
·
Samee U.Khan
·
Jianzhong Huang
·
Changsheng Xie
©Springer Science+Business Media New York 2013
Abstract
Gene Expression Programming (GEP) signiﬁcantly surpasses traditional
evolutionary approaches to solving symbolic regression problems.However,exist
ing GEP algorithms still suffer from premature convergence and slow evolution
in anaphase.Aiming at these pitfalls,we designed a novel evolutionary algorithm,
namely UniformDesignAided Gene Expression Programming (UGEP).UGEP uses
(1) a mixedlevel uniform table for generating initial population and (2) multiparent
crossover operators by taking advantages of the dispersibility of uniform design.In
addition to a theoretic analysis,we compared UGEP to existing GEP variants via
a number of experiments in dealing with symbolic regression problems including
function ﬁtting and chaotic time series prediction.Experimental results indicate that
UGEP excels in terms of both the capability of achieving the global optimumand the
convergence speed in solving symbolic regression problems.
Keywords
Gene expression programming
·
Uniformdesign
·
Symbolic regression
problem
·
Function ﬁtting
·
Time series prediction
Y.Chen
·
J.Huang
·
C.Xie
School of Computer Science,Huazhong University of Science &Technology,Wuhan,Hubei
430074,China
Y.Chen
·
D.Chen (
)
School of Computer Science,China University of Geosciences,Wuhan 430074,China
email:
chendan@pmail.ntu.edu.sg
S.U.Khan (
)
Department of Electrical and Computer Engineering,North Dakota State University,Fargo,ND,
USA
email:
samee.khan@ndsu.edu
Y.Chen et al.
1 Introduction
Evolutionary algorithms have been widely adopted in handling symbolic regression
problems in modern sciences and engineering such as function ﬁtting and time series
prediction.To solve a symbolic regression problem,we normally need to establish a
mathematical expression,which ﬁts a number of discrete data points.The goal is to
minimize the errors between the values computed with the expression and the actual
values of the data points.
Existing Genetic Algorithms (GAs) and Genetic Programming (GP) meth
ods [1,2] have achieved many successes in dealing with these problems [3–5].In
GAs,individuals are expressed as linear strings with ﬁxed length (chromosome)
through all evolution steps.This makes GAs not applicable to ﬁtting some very
complex functions [6],while in GP methods,individuals are expressed as nonlin
ear objects with different sizes and shapes.A GP method is normally capable of
characterizing very complex functions,but the variety in object sizes often hampers
the evolutionary procedure to achieve the optimal solution [7].
In contrast,Gene Expression Programming (GEP) is a salient approach in creating
computer programs denoting the learned models and/or discovered knowledge [8,9].
GEP is similar to Genetic Algorithms (GAs) and Genetic Programming (GP),and
it differs from these evolutionary approaches mainly in chromosome encoding.GEP
encodes individuals as chromosomes and implement themas linear stings with ﬁxed
lengths [9,10].The separation of genotype and phenotype has endowed GEP with
more ﬂexibility and power of exploring the entire search space.The chromosomes of
GEP are simple and linear.It can be operated by the genetic process easily,and it has
the capability to handle complex problems [7,11–16].
GEP exhibits signiﬁcant advantages over its counterparts:for example,compar
ing to GAs,GEP performs almost 2–4 orders of magnitude faster in solving general
problems because of its special individual expression [6,9].GEP can ﬁnd a brand
new function,which is much better than GAs do [6,9].Zuo et al.presented a typical
symbolic regression example in which some subjects may be satisﬁed with a formula,
y =ax
3
+bx
2
+cx +d,in which the coefﬁcients a,b,c,and d can be found by GA;
while a much better and more complex formula,y =sin(ax
3.5
+bx
2.5
+dx),can be
found using GEP [10].
The original GEP algorithm begins with randomly generating linear ﬁxed chro
mosomes for individuals within the initial population [9].Each individual is judged
by a ﬁtness function for each evolution generation.The individuals are then reserved
by ﬁtness values to reproduce the modiﬁcation.The new individuals are subjected to
the same process.The evolution process will continue until it reaches a prespeciﬁed
number of generations or a solution is found.The original GEP still suffers frompre
mature convergence and slow evolution in anaphase (e.g.,evolution in a long period
of time from the 300
th
generation to the 500
th
generation as the prespeciﬁed termi
nation condition) [7,12,17].Attempts have been made to overcome these pitfalls by
redesigning some operations within the evolution process,such as forming solutions,
individual initialization according to speciﬁc problemand sampling fromparents for
propagation,and the method of crossover operators [8,18–20].For example,the hy
brid GEP parallel algorithmintroduced in [20] achieves a higher stability and search
ability by combining simulated annealing and genetic mechanism.
Solving symbolic regression problems
Although these methods have some successes,there is a critical point ignored,
which we consider the key to solving the symbolic regression problems with GEP:
whether the elements of initial population can properly represent the whole element
sets.Without a proper scheme,some key elements may be lost when the size of the
initial population is small;some key elements may be repeated,and the others may
still be lost when the size is large.
In GEP,keeping the diversity of chromosome plays an important role in the evo
lution process.Initialization of the population is the premise in the evolution process
and the quality of the population will affect the diversity of chromosome for each
evolution generation.If the elements of the initial population are sampled uniformly
from element sets,the key elements will not be missed.In this study,we proposed a
novel Gene Expression Programming aided by uniformdesign (referred to as UGEP)
for initializing the population.According to the properties of uniform design,it not
only makes each sampling more representative but also decreases the number of ex
periment times comparing to using orthogonal design [21,22].Given mfactors each
having n levels,when exhaustively performing the experiments,there are n
m
experi
ments to be executed in total;when using orthogonal design,the number is n
2
;when
using uniformdesign,the number dramatically drops to n.
In GEP,the individuals of the initial population are generated randomly from the
elements set.In contrast,UGEP adopts uniform tables generated from the elements
set to initialize the population,and this makes the individuals well distributed.In
addition,UGEP uses adaptive multiparent crossover operators as genetic operators,
which can play an important part in evolution process.
Function ﬁtting and chaotic time series (e.g.,sunspots) prediction problems have
been tested to check whether a UGEP algorithm is capable of solving symbolic re
gression problems.In the function ﬁtting test,an optimal parameter setting has been
obtained and a performance comparison has been made between UGEP and the origi
nal GEP.In experiments on real data sets for chaotic time series (sunspots) prediction,
two methods based on UGEP are applied,i.e.,the slide window prediction method
and differential equation prediction method.Results show the proposed algorithm is
efﬁcient in making prediction on chaotic time series.
The remainder of this paper is organized as follows:Sect.2 recaps the existing
GEP algorithms and motivates this study.Section 3 presents the UGEP algorithmfor
symbolic regression problems,which covers (1) the construction method of initial
population based on mixedlevel uniformtable and (2) the adaptive crossover uniform
operator.In Sect.4,we analyzed the capability to achieve global convergence and the
convergence speed of UGEP in theory.Section 5 presents the experiments and results
of using UGEP to dealing with function ﬁtting and chaotic time series prediction.We
concluded the paper and present the future work in Sect.6.
2 Basics of gene expression programming
This section recaps the basics of Gene Expression Programming (GEP).GEP is a
powerful evolutionary method derived fromGenetic Programming (GP) to overcome
the common drawbacks of GA and GP [9].Similar to GA and GP,GEP follows the
Y.Chen et al.
Fig.1 An example of
chromosome (single gene) and
its decoding in GEP
Darwinian principle of the survival of the ﬁttest and uses populations of candidate
solutions to a given problemin order to evolve newones.The difference among GEP,
GA,and GP is the way in which individuals of a population of solutions are repre
sented [9].Although GEP has a simple and linear form,it is ﬂexible and powerful in
solving complex problems [6].
In GEP,an individual (chromosome) is represented by a genotype,constituted by
one or more genes.Achromosome is a linear and compact entity,which can be easily
manipulated with genetic operators such as mutation,crossover,and transposition.
When using GEP to solve a problem,there are ﬁve components that should be
speciﬁed:the function set,the terminal set,the ﬁtness function,GEP control param
eters,and the stop condition.
Generation of the initial population of solutions is the ﬁrst step.This can be done
by using a random process.The individuals are then expressed as expression trees
(ETs,an example is given in Fig.1),which can be evaluated according to a ﬁtness
function that determines howgood a solution is in the problemdomain.According to
the value of each chromosome evaluated by the ﬁtness function,the operator on the
selected chromosomes will be applied such as crossover,mutation,and rotation.If a
solution of satisfactory quality is found,or a predetermined number of generations
are reached,the evolution stops and the “bestsofar” solution is returned.
2.1 Chromosome encoding
Each chromosome is a character string in ﬁxedlength,which can be composed of
any element fromthe function set or the terminal set.Each gene has a head and a tail.
The size of the head (h) is deﬁned by the user,but the size of the tail (t ) is obtained as
a function of h and a parameter n (the number of elements of the function set).The
tail size can be calculated by the following equation:
t =h
∗
(n −1) +1 (1)
Each gene is referred to as a Karva notation or Kexpression and can be mapped
into an expression tree (ET).In the case of multigenic chromosomes,all ETs are con
nected by their root node using a linking function such as Boolean function,function
“+”,etc.Functions,terminals,and constants are allowed in the heads,whereas only
terminals or constants in the tails.For example,the ET shown in Fig.1 corresponds
Solving symbolic regression problems
Fig.2 A threegene
chromosome and its subETs
Table 1 Statistic result of the initial population
Element
Gene
0 1 2 3 4 5 6 7 8 9
+ 11 7 4 3 8 12 0 0 0 0
− 13 4 9 6 7 5 0 0 0 0
* 8 10 9 8 8 6 0 0 0 0
/8 12 8 8 16 9 0 0 0 0
Q 2 4 7 7 5 6 0 0 0 0
E 6 10 7 11 5 3 0 0 0 0
S 2 12 5 9 7 12 0 0 0 0
T 2 3 5 8 9 11 0 0 0 0
C 1 10 6 6 1 2 0 0 0 0
A 7 5 5 7 7 10 17 18 19 20
B 9 5 8 7 6 7 16 23 21 20
C 8 7 5 8 9 4 22 17 26 16
D 9 7 11 8 9 7 19 24 22 18
E 14 4 11 4 3 6 26 18 12 26
to a sample chromosome,and can be interpreted in a mathematical form.The ET
shown in Fig.2 is a multigenic chromosome.It is constructed by three genes and
can be connected by a function.As shown in Fig.2,the ﬁrst gene is constructed by
thirteen elements,i.e.,“Q +aaa/babbaba.” The highlighted substring “babbaba” rep
resents the tail of the gene while the substring “Q + aaa/” is the head.According to
the rules of ET,the elements “a/babbaba” are useless.
2.2 Population initialization
The population initialization is the ﬁrst step of evolution procedure.The quality of
the chromosomes of the ﬁrst generation plays an important role in the convergence
process.The initial population needs to have as many different individuals as possible
in order to efﬁciently explore the search space in further generations [19].The origi
nal GEP generates the initial population at random.For instance,Table 1 presents a
statistic result of the initial population of a GEP programintroduced in [23].
In this GEP program,the number of population (p) is 100;the head (h) is 6;
function set ={+,−,∗,/,Q,E,S,T,C};terminal set ={a,b,c,d,e}.From Table 1,
Y.Chen et al.
we can see that generating the initial population randomly cannot improve the diver
sity of chromosomes.For instance,function “/”appears 16 times in the No.4 gene
while function “C” appears only once.Especially when the number of the initial
population is small,some key elements may be lost.Even the number is big,some
key elements may be repeated,and the others may be yet lost.We need an effective
method to ensure all key elements can be generated in the ﬁrst generation.In order
to keep the diversity of chromosomes,it is necessary to make sure that samples are
well distributed.In [21],a uniform design method is proposed for experimenting in
industry.The method uses a uniform table to keep samples taken from the solution
space well distributed.In GEP,we may also adopt a uniform table to achieve this
when generating the initial population at the ﬁrst step of the evolution procedure.
2.3 Genetic operators and selection method
GEP uses genetic operators,i.e.,mutation,transposition,and crossover,to create vari
ations for evolution.Amutation operator introduces a randomchange into symbols at
any position in a chromosome [9].A transposition operator transports the sequence
elements of gene to another place [17].The crossover operator chooses and pairs
two chromosomes to exchange some elements between them[19].Howto efﬁciently
create variation depends on the nature of the complex problemunder investigation.
Generally,after applying genetic operators to create variation in each generation,
GEP selects some individuals and copies those into the next generation based on their
ﬁtness,such as simple elitism [7] and cloning of the best individual.Typically,the
roulettewheel method [9] is used in many GA [1] and GP algorithms [24].
3 Uniformdesign genetic expression programming
Although GEP has showed its superiority compared to its counterparts,it has suffered
from pitfalls such as high probability of premature convergence and slow evolution
speed in evolution anaphase.This means that while solving symbolic regression prob
lems the ﬁnal results may fall into local optimum or otherwise it takes an extremely
long time to ﬁnd the global optimal solution.The diversity of chromosomes is a key
factor in the GEP evolution procedure,and welldistributed samples are the premise
to keep the diversity of chromosomes.To ensure that the key elements can be in
volved on initialization,we designed a newGEP algorithmbased on uniformdesign,
namely UGEP,to tackle the pitfalls of the original GEP centering on population ini
tialization.In addition,UGEP uses the uniform optimization method (see Sect.3.3)
instead of stochastic evolution.
3.1 The ﬂow of the UGEP algorithm
A UGEP process can be separated into several parts including:(1) population ini
tialization;(2) genetic operation,selection and reserving;(3) revealing the global
solution.The UGEP algorithmﬂow can be illustrated in Fig.3.
Solving symbolic regression problems
Fig.3 The UGEP algorithm
ﬂow
Block 1:Set the parameters:chromosome’s head length;the probabilities of three
evolutionary operators (multiple crosshybridization,gene recombination,Dc do
main recombination);function set and terminal set;
Block 2:Initialize the population P:generate the uniform tables from the elements
of function set and terminal set,then initialize the individuals according to the
uniformtables (see Algorithm1);
Block 3:According to the probabilities of multiple crosshybridization,use the adap
tive multiparent crossover operator to get offspring (see Algorithm 2),and then
apply gene recombination and Dc domain recombination for all offspring.Rank
all offspring according to their ﬁtness value;
Block 4:Adopt elitismstrategy for all individuals:keep the current optimal solution
and eliminate the worst individuals of the parent population at a preset ratio;
Block 5:If the global optimal solution is found or the preset maximum number of
generations is reached,end the evolution process.Otherwise,go to Step 3.
3.2 Initializing population upon a mixedlevel uniformtable
The uniform design [21] is one of the space ﬁlling designs and has been widely
used in experimenting.Its main objective is to sample a small set of elements from
Y.Chen et al.
Fig.4 The algorithmof
construction of the mixedlevel
uniformtable
a given set,such that the sampled elements are uniformly scattered and maintain
the characteristics of the whole set.A uniform table can be expressed by a matrix
U
M
(Q
S
),where S is the factors and Q is the levels,M is the selecting samples of
combinations fromthe whole space Q
S
[25].
In UGEP,the initial population can be generated in accordance with the uniform
table.The matrix is ﬁlled with the elements of chromosomes or genes,and each
row represents an individual.The mixedlevel uniform table can be constructed by
Algorithm1,which is shown in Fig.4.
Algorithm1 (Construct the mixedlevel uniformtable for initializing the population)
Block 1:This step set the parameters of chromosome and gene:the length of head
h,function set f(x),the number of elements f,terminal set var(x),the number
of elements v,the length of tail t =h ∗ (n −1) +1,the number of polygene g,
the number of chromosomes n∗(f +v +1),a constant integer n.Then give each
element in f(x) and var(x) a tag number as the primer members of the matrix;
Block 2:Construct a matrix U
n∗k
((n ∗ k)
h+t
) and ﬁll the elements using generation
vectors method [18].Here h+t is the number of the columns,n∗k is the number
Solving symbolic regression problems
of the rows and also the levels of each factors,k ∈(v +1,f +v +1) is generated
randomly;
Block 3:In GEP,the number of the levels of the head is not equal to that of the tail.
This step adjust the uniformtable U
n∗k
((n∗k)
h+t
) [21],separate the columns into
two classes (the elements of the head and the tail) by the formulary 2 as follows:
u
∗
ij
=
u
ij
mod (f +v +1),{i ∈(1,n ×k),j ∈(1,h)},u
ij
←u
∗
ij
u
ij
mod (v +1),{i ∈(1,n ×k),j ∈(h +1,h +t)},u
ij
←u
∗
ij
(2)
Then the uniformtable U
n∗k
((n ∗ k)
h
• (v +1)
t
) is constructed.
Block 4:Let U
n∗k
((n∗k)
h
•(v+1)
t
) be the basic table.The matrix U
n∗(f+v+1−k)
(v+
1)
h+t
is the empty table.Then circularly ﬁll the empty table with the elements of
the basic table (see fractionaladdition in [21]).We can get the mixedlevel uni
formtable for one gene under processing U
n∗(f+v+1)
((f +v +1)
h
• (v +1)
t
);
Block 5:If g =1,then go to step 6.Otherwise,update k,g =g −1,go to step 2;
Block 6:The matrix U
n∗(f+v+1)
(g • (f + v + 1)
h
• (v + 1)
t
) can be given by g
matrices using directproduct method [21].
In UGEP,each factor of the head has more levels to choose than the tail does.
In Algorithm 1,after constructing the matrix U
n∗k
((n ∗ k)
h+t
),UGEP adjusts the
matrix conforming to formulary 2 to balance the number of levels of the head and the
number of levels of the tail.Assume that a chromosome is constructed by g genes;
the mixedlevel uniformtable can be obtained fromg matrices,which are referred to
when initializing the population with each row representing an individual.
3.3 Adaptive crossover operator
Genetic operation is the strategy applied in evolution procedure for ﬁnding the global
optimal solution.We use a crossover operator based on a multiparent method which
is also empowered by uniform design.The crossover operator is designed using a
uniformoptimization method (see Algorithm2) instead of stochastic evolution.
Given m individuals (i.e.,chromosomes) in a generation during evolution,each
chromosome is divided into n exclusive genes.A uniform table is designed here to
sample n genes fromthose to forman offspring.
The crossover operator is adaptive.The scale of hybridization is controlled by par
ents’ current state:if the distance amongst the parents’ ﬁtness values f
p
and the cur
rent best ﬁtness values f
max
becomes larger,it can enable communications amongst
more parents thus to increase the chances for accommodating excellent gene seg
ments by constructing a uniform table of a larger scale for hybridization.If the dis
tance becomes shorter,it may avoid more excessive mutations from excellent gene
segments to have a uniformtable of a smaller scale.The number of parents in uniform
table can be determined as follows:
m
i
=
1 −
f
p
f
max
+δ
×m
i−1
(3)
where δ ∈(0,1),the parameter i is the current generation.The algorithmis illustrated
in Fig.5.
Y.Chen et al.
Fig.5 The algorithmof the
adaptive multiparent crossover
operation
Algorithm2 (Adaptive multiparent crossover operation)
Block 1:Divide randomly the chromosome (L) into n disjoint subsets L
i
(i ∈(1,n)),
where L
i
denotes a subset,L
i
∩L
j
=Φ(i =j),
n
i=1
L
i
=L;select parents for
hybridizing and add theminto a “matching pool”;
Block 2:Calculate the number of parents (m) of the current generation using formu
lary (3),and randomly select m−1 parents fromthe population and add theminto
the matching pool;
Block 3:Construct the crossover uniform table U
m
(m
n
).Each row presents a
new offspring.Calculate the ﬁtness value of offspring G
i
(L
1
,L
2
,...,L
n
),i ∈
1,2,...,m;
Block 4:Reserve the chromosomes as offspring generated by hybridization between
parents,which have the best ﬁtness value among offspring G
i
(L
1
,L
2
,...,L
n
).
4 A theoretical analysis on UGEP
In this section,we analyze a UGEP algorithm’s performance of global convergence
and the computational complexity.The algorithm’s population initialization is imple
Solving symbolic regression problems
mented using the mixedlevel uniformtable and the multiparent crossover operator is
adopted.
4.1 Global convergence
Deﬁnition 1 Let f
∗
=max(f(x)) be the global maximumin the search space S and
the set of the global optimal solutions is deﬁned as M(x
∗
) ={x ∈Sf(x) =f
∗
}.
Deﬁnition 2 The εﬁeld set of the optimal solutions is deﬁned as ∀ε >0;M
(x −ε)
=
{x ∈ Sf(x) ≥f
∗
−ε},where m(M
(x−ε)
) >0 and the function m(A) represents the
Lebesgue measure of A [21].
Theorem 1 For the UGEP algorithm,after a limited number of generations,the
population P eventually covers the set M
(x−ε)
with the probability of convergence
p >0.
Proof Let us assume a subspace V
m
,which is formed by randomly selecting m
parents from the population P.After applying the multiparent crossover opera
tor,their offspring obeys the uniform distribution in the space V
m
.If the condi
tion M
(x−ε)
⊆ V
m
holds,the probability of ﬁnding M
(x−ε)
in the next generation
is
s
μ
• ε >0,where μ =M
(x−ε)
∩ V
m
 and s represents the number of population
in V
m
.
Since the individuals are uniformly distributed in V
m
,if M
(x−ε)
⊂V
m
,then the
probability we have an individual i = {i ∈ Pi/∈ V
m
}) is C
N−m
N
• μ
,where μ
=
V
m
∩S.Therefore,a new mdimension subspace V
m
(V
m
=V
m
) can be generated
by two randomly selected individuals i and j,where i/∈V
m
and j ∈P.
If M
(x−ε)
⊆V
m
,we have p >0.Otherwise,a new space V
m
will be generated.
After a limited number of generations for extending the new subspace,based on the
concave associativity,the probability that the population covers M
(x−ε)
can be for
malized as
p =min
s
μ
• ε,
N
i=m
C
N−i
N
• μ
>0.
Theorem 2 The UGEP algorithm converges to the global optimum solution set
M(x
∗
) with the probability of 1.In other words,∀ω,we always have lim
k→∞
P(f
∗
−
f
k
(x) <ω) =1,where f
k
(x) represents the optimum solution in the population at
the kth generation.
Proof According to Theorem1,the probability of generating an individual i satisfy
ing i ∈ M
(x−ε)
at the kth generation is p.Otherwise,the failure probability for the
optimal solution is p
k
≤1 −p,where p
k
represents the probability to generate the
offspring i (i/∈M
(x−ε)
) at the kth generation.
Since the elite replacement strategy is applied in the UGEP algorithm,we have the
inequality P(f
∗
−f
k
(x) <ω) ≤1 −p{(i/∈ M
(x−ε)
) at the kth generation} ≤1 −
(1 −p)
k
holds for any ω.Furthermore,according to the theorem of inﬁnite product,
we have lim
k→∞
P(f
∗
−f
k
(x) <ω) =1 −
∞
k=1
(1 −p)
k
=1,which means the
UGEP algorithmconverges to the optimal solution with a probability of 1.
Y.Chen et al.
4.2 Convergence speed
Let H
∗
be the template in the population of the kth generation,the set Generation
(H
∗
,k) is the template set from the population Generation(k),where k denotes the
kth generation.N(H
∗
,k) is the number of individuals that hold the template H
∗
,that
is,N(H
∗
,k) =Generation(H
∗
,k).The number of individuals that hold the template
H
∗
in Generation(k +1) [26] can be expected from the formula:E(H
∗
,k +1) =
f(H
∗
,k) ×N(H
∗
,k).
f(H
∗
,k) =
N(H
∗
,k)
g=1
ﬁtness(g)
N(H
∗
,k)
g∈Generation(k)
g=1
ﬁtness(g)
Generation(k)
(4)
Furthermore,let M be the set of the individuals involved in genetic operations.
According to formulary (4),the following relationship holds with the original GEP:
f
O
(H
∗
,k) =
Generation(k)
N(H
∗
,k)
×
g∈{Generation(H
∗
,k)\M}
g=1
ﬁtness(g) +
g∈M
g=1
ﬁtness
O
(g)
g∈{Generation(k)\M}
g=1
ﬁtness(g) +
g∈M
g=1
ﬁtness
O
(g)
(5)
similarly,for UGEP the following relationship holds:
f
U
(H
∗
,k) =
Generation(k)
N(H
∗
,k)
×
g∈{Generation(H
∗
,k)\M}
g=1
ﬁtness(g) +
g∈M
g=1
ﬁtness
U
(g)
g∈{Generation(k)\M}
g=1
ﬁtness(g) +
g∈M
g=1
ﬁtness
U
(g)
(6)
As concluded in [27],the optimal results generated from the set of uniform tests
are better than the
N
/(N +1) solutions obtained in exhaustive tests,where N is the
number of uniformtests.For example,giving a uniformtable U
8
(2
7
),the optimal so
lutions fromeight uniformtests are not worse than the top ﬁfteen (2
7
∗1/9) solutions
of 2
7
tests.In other words,under the same condition,the individuals generated from
the set of uniform test are better than that of the random test as a whole.In addition,
due to the property of the elite replacement strategy,the sum of individuals’ ﬁtness
values in uniformset is always larger:
g∈M
g=1
ﬁtness
U
(g) ≥
g∈M
g=1
ﬁtness
O
(g) (7)
Furthermore,we have
g∈{Generation(H
∗
,k)\M}
g=1
ﬁtness(g) ≤
g∈{Generation(k)\M}
g=1
ﬁtness(g) (8)
Solving symbolic regression problems
According to formulas (6),(7),and (8),we can have the formulary (9):
f
O
(H
∗
,k) ≤f
U
(H
∗
,k)
E
O
(H
∗
,k +1) ≤E
U
(H
∗
,k +1)
(9)
where E
O
and E
U
denotes the expected value of f
O
and f
U
in the (k +1)th gener
ation.
In summary,the number of optimal offspring at the subsequent generation in
UGEP increases more quickly than in the original GEP.The UGEP algorithm tends
to explore better solutions than the original GEP algorithmdoes.
5 Performance evaluation
Two types of symbolic regression experiments have been performed to evaluate the
performance of UGEP.We ﬁrst explored the optimal parameter settings for the UGEP
algorithm through the study of a function ﬁtting problem.After that,we compared
the performance of UGEP on addressing the function ﬁtting problemwith the original
GEP.We used UGEP with the optimal parameters in an application for prediction of
Sun Spot Time Series with two alternative prediction methods.All experiments were
executed over a desktop computer with conﬁgurations:CPU (AMD AM2 Athlon 64
X2 5000,2.6 GHz);RAM (2 GB),Operating System (Windows XP Professional,
Service Pack2).
5.1 Parameter setting in a function ﬁtting problem
The key parameters of the UGEP algorithminclude:the head length (H),the number
of genes (M),the probability of gene recombination (Pr),the population scale (S),
evolution generation (G),the rate of multiparent crossover (Pc),and the probabil
ity of Dc domain recombination (Pd).Using the method introduced in [28,29],we
treat the seven parameters as seven factors to construct the uniform table U
10
(5
7
),
which consists of 10 rows each representing a set of parameters (see the ﬁrst eight
columns of Table 2).The parameters were speciﬁed with empirical values as sug
gested in [9,10,20].
In this set of experiments,we selected an experimental function conforming to
expression y =10
(1−e
(−0.38×x)
×cos(0.7×x))
,where x is a variable of ﬂoat type in [0,6].
Rsquare is calculated to evaluate the ﬁtting accuracy by comparing the ﬁtting func
tion to the experimental function.A larger Rsquare represents a higher degree of
model ﬁtting.We repeated 100 independent trials of experiments for each parameter
setting.Table 2 presents the averaged experimental results and their standard devia
tion for each combination of parameters.
The experimental results indicate that the UGEP algorithm achieves a very high
accuracy with the 7
th
parameter setting.The averaged value of RSquare is 0.995039
and the best value of RSquare is 0.998652.The standard deviation of RSquare for
each parameter combination is less than 0.006,which indicates the UGEP algorithm
constantly performs well in dealing with these function ﬁtting problems.
Y.Chen et al.
Table 2 The experimental results of function ﬁtting using UGEP
H M Pr S G Pc Pd RSquare Standard
deviation
1 6 2 0.16 40 200 1.0 0.05 0.925626 0.004378
2 6 2 0.04 10 500 0.6 0.05 0.900774 0.002158
3 2 1 0.20 20 300 0.6 0.25 0.781383 0.005264
4 8 4 0.08 10 100 0.8 0.20 0.915682 0.003829
5 4 3 0.08 50 400 1.0 0.20 0.893454 0.001617
6 4 4 0.16 20 200 0.2 0.10 0.959686 0.003374
7 8 3 0.12 30 500 0.2 0.25 0.995039 0.002418
8 10 1 0.12 50 100 0.4 0.15 0.820848 0.001901
9 10 5 0.20 30 400 0.8 0.10 0.987813 0.004519
10 2 5 0.04 40 300 0.4 0.15 0.965160 0.003827
Although there are a number of factors that contribute to the performance of the
UGEP,we trust that the number of genes (M) has a signiﬁcant impact on the evolu
tion process through the experiments.As indicated in Table 2,the results of the 100
trials with the 3
rd
parameter settings (M=1) all fall into a local value 0.781383 with
out exception;the averaged result of the experiments with the 8
th
parameter settings
(M=1) is also low compared to the others.It can be observed that in general better
results can be obtained when the value M increases.However,the best result we have
is with the 7
th
parameter settings (M=3).
It can also be observed that the results with the 7
th
and 9
th
parameter settings are
the best when the scale of the population S is 30.This means that in UGEP,a suitable
scale of the population is needed to achieve the optimal solution.In the original GEP,
usually a large scale of population is needed to extend the searching space.In UGEP,
we trust that the adaptive crossover operator can enable communications among more
parents thus to increase the chances for accommodating excellent gene segments.
From above,we can get the range of empirical parameter settings suitable for the
UGEP algorithmas follows:
The head length H [4,8],the number of genes M [3,5],the probability of gene
recombination Pr [0.1,0.2],the population scale S [30,40],evolution generation G
[400,500],the rate of multiparent crossover Pc [0.1,0.3],and the probability of Dc
domain recombination Pd [0.05,0.1].
We eventually identiﬁed a set of parameters:H(6),M(5),Pr(0.16),S(30),
G(500),Pc(0.23),and Pd(0.75).After executing 100 trials with this parameter set
ting,the averaged value of RSquare is 0.998333,the best value is 0.999795,and the
standard deviation is 0.002368.The resulted function is as the follows:
Y
∗
=
cos(x)sqrt
abs(0.713614)
∗
cos(x)x
+cos
sqrt
abs
log
abs
10ˆ
cos
cos(0.755058)
+x
+cos
sqrt
abs
0.198157 ∗ 0.518540
+sin
cos(x)
+x;
(RSquare =0.999795)
Solving symbolic regression problems
Fig.6 The ﬁtness convergence curves of UGEP vs.the original GEP
The ﬁnal parameter setting has been applied in the subsequent experiments.To
evaluate the performance of UGEP,the original GEP has been used for the same
function ﬁtting problem.After 100 independent trials of experiments parameter set
ting,an optimal parameter setting is found for OGEP (H(6),M(5),Pr(0.20),S(50),
G(500),Pc(0.23),and Pd(0.75)).Each experiment with the optimal parameter set
ting has been repeated for ten times and the averaged results are presented in Fig.6.
The best ﬁtness value at each generation is recorded.The two convergence curves
denote how these averaged values change with the generations for both UGEP and
the original GEP (referred to as OGEP).
Figure 6 shows that at the early stage (generations 0–50),the best ﬁtness values of
both evolutionary processes increase quickly and UGEP can always achieve higher
best ﬁtness value.Fromgeneration 50 onward,the superiority of UGEP to the original
GEP becomes more and more signiﬁcant (from 16 at the 50
th
generation to 41 at the
500
th
generation).
The original GEP has an averaged RSquare value of 0.984667 and the best value
is 0.986652 among ten runs.The resulted ﬁtness function is as the follows:
Y =sin(x) +
sin
sin
(0.046175)
+
(0.056429) +x
∗ x
+tan
cos
cos
(0.479415)
∗ sin
sqrt
abs(x)
+
s ∗ (0.628590)
∗
x
∗
0.868160
∗ sin
(0.868160 −x)
+cos
cos
cos
x/(0.253029)
/(−0.217811)
;(RSquare =0.986652)
Table 3 presents the overall execution times of UGEP and the original GEP and
highlights the times for population initialization.Although it takes a longer time for
UGEP to initialize the population than the original GEP does,UGEP signiﬁcantly
excels in terms of runtime performance at the price of negligible overhead incurred
by the construction of the uniformtable.
Y.Chen et al.
Table 3 The average time of
population initialization and the
whole execute time of UGEP
and GEP
Time UGEP The original GEP
Population initialization (sec) 0.048 0.025
Overall execution time (sec) 18.32 26.47
5.2 Sun spot predication via slide window prediction method (SWPM)
The Sun Spot Time Series prediction is a classic benchmark to evaluate algorithms
for chaotic time series prediction [30,31].We used the real sun spot data in order to
have an indepth examination on the UGEP algorithm’s performance.
The ﬁrst round of experiments used a Slide Window Prediction Method (SWPM)
[32].The method aims to predict an element’s value in a time series based on the
history,which operates in this manner:Given the length of a sliding window (h,i.e.,
the dimension) and the values of the elements of a time series (x
i
,1 ≤i ≤n) covered
by the window,the method ﬁnds a function f,such that for any m(n −h +1 ≤m≤
n),to predict the value of x
m
:
_x
m
=f(x
m−h
,x
m−h+1
,...,x
m−2
,x
m−1
),(h <m≤n) (10)
Obviously,the difference between the _x
m
and x
m
should be as small as possible
to ensure accuracy.
In the experiments of time series prediction,we compared UGEP with SAMGEP
and OGEP.The same parameter setting described in Sect.5.1 applies to OGEP.The
test data (Wolfer Sun Spot Time Series) and the parameter setting for SAMGEP are
available in [20].The time series contains 100 data elements.Assume dimension is
n,we can use (100 −n) data sets in the experiments.Amongst the (100 −n) data
sets,the ﬁrst half and the second half are used for training and evaluation purposes
respectively.
The conﬁgurations of the experiments are:dimension as 6,10,and 12;delay is
1;function set F ={+,−,∗,/,ˆ,sin,cos,exp,ln} (xˆn means x
n
,0 <n <5);ter
minal set T = {a,?} (?denotes a random constant,a is an independent variable).
For each conﬁguration,100 trials of experiments were executed.The average results
(RSquare,execution time,and success rate) are shown in Fig.5.
As shown in Fig.7,UGEP performs better than SAMGEP and the original GEP
do in chaotic time prediction using SWPMunder all conﬁgurations.With the increas
ing dimension of the sliding window,the differences amongst UGEP,SAMGEP and
OGEP increase signiﬁcantly.UGEP maintains a success rate higher than 90 %while
that using SAMGEP drops from 92 % to 76 % and that using OGEP drops from
94 %to 64 %.The RSquare values also indicate that UGEP can ensure an accurate
prediction.It is also noticeable that UGEP executes even faster than SAMGEP and
OGEP do when achieving the above successes.
The standard deviation with each dimension is presented in Table 4.The results
indicate that the UGEP algorithmperforms as stably as SAMGEP does in Sun Sport
time series prediction.The best functions obtained using UGEP can be written as the
following expressions:
Solving symbolic regression problems
Fig.7 Experimental results with UGEP,SAMGEP and OGEP using SWPM
Y.Chen et al.
Table 4 The standard deviation with each dimension
Dimension 6 Dimension 10 Dimension 12
RSquare (UGEP) 0.00186 0.00127 0.00248
RSquare (SAMGEP) 0.00172 0.00189 0.00283
RSquare (OGEP) 0.00242 0.00367 0.00642
Time (UGEP) 1.47 s 2.23 s 2.12 s
Time (SAMGEP) 1.32 s 2.43 s 2.37 s
Time (OGEP) 1.74 s 2.26 s 2.38 s
Success rate (UGEP) 0.023 0.031 0.027
Success rate (SAMGEP) 0.023 0.036 0.032
Success rate (OGEP) 0.024 0.022 0.014
(1) Dimension is 6:
X
m
=
X
m−1
/exp(X
m−4
)
−sqrt
abs(X
m−4
)
+
X
m−1
/(X
m−5
−X
m−3
)
∗ sqrt
abs(X
m−3
)
+10ˆ
sqrt
abs
sqrt
abs(0.999756)
/(X
m−1
−X
m−4
)
+
X
m−2
/exp(X
m−5
)
+X
m−1
;(RSquare =0.9587)
(2) Dimension is 10:
X
m
=exp
sqrt
abs(X
m−7
)
/X
m−6
−cos(X
m−7
)
+
X
m−10
/sqrt
abs(X
m−7
)
−log
abs
tan(X
m−6
)
+log
abs(X
m−10
)
+cos
sqrt
abs
10 ˆ(X
m−10
)/X
m−8
+(X
m−10
−X
m−6
)
+tan
X
m−8
−sqrt
abs
log
abs(X
m−6
)
∗ X
m−9
+sqrt
abs
X
m−4
−(−0.358257)
∗ X
m−1
−
X
m−1
/(−0.117832)
+
sqrt
abs
X
m−4
/sin(X
m−1
)
−X
m−4
+10ˆ
sqrt
abs
sin(X
m−5
)
+
10ˆ
cos
(X
m−1
−X
m−5
)
+sqrt
abs
sqrt
abs(X
m−4
)
+sqrt
abs
X
m−1
∗ sin
log
abs(X
m−5
)
;(RSquare =0.9690);
(3) Dimension is 12:
X
m
=log
abs
tan
tan
(X
m−9
−0.744011) ∗
X
m−11
∗ (−0.385296)
+tan
sin
log
abs
log
abs
sin
(X
m−12
/X
m−10
)
+
log
abs
log
abs
X
m−12
∗ X
m−11
∗ sqrt
abs(X
m−11
)
+log
abs
tan
cos
(X
m−10
−X
m−11
)
+log
abs
tan(X
m−11
) +X
m−11
Solving symbolic regression problems
+10 ˆ
exp
cos
log
abs
log
abs
(X
m−5
+X
m−7
)
+tan
log
abs
sin
log
abs
sqrt
abs
X
m−8
/(−0.202551)
+log
abs(X
m−8
)
+log
abs
log
abs
sin
log
abs
X
m−7
∗ X
m−5
+log
abs
log
abs
sin
log
abs
log
abs(X
m−5
)
+sqrt
abs
X
m−1
−
X
m−1
∗ (−0.544969)
∗ sqrt
abs(X
m−1
)
+tan
log
abs
X
m−2
+(−0.956297)
/sqrt
abs
log
abs(X
m−4
)
+
tan
log
abs(X
m−2
)
∗
X
m−1
./cos
(−0.800043)
+tan
tan(X
m−4
)
+log
abs
tan
sqrt
abs
sqrt
abs(X
m−2
)
−(X
m−2
−X
m−4
)
;
(RSquare =0.9814).
5.3 Sun spot predication via Differential Equation Prediction Method (DEPM)
The second round of experiments used a Differential Equation Prediction Method
(DEMP) [33].The method ﬁrst analyzes the whole test data (e.g.,a time series).It
then constructs a differential equation to predict the future evolvement of the data
using the equation.Consider space partial derivative discretization and the way of
transforming a highorder to a loworder,and this study only involves ordinary differ
ential equations less than 3 orders.We use Differential by Microscope Interpolation
(DMI),which incurs a relatively lowerror and noise [11].For each conﬁguration,100
trials of experiments were executed.The averaged experimental results are presented
in Fig.8.The standard deviation of the results for each order is presented in Table 5.
Experimental results indicate that UGEP always performs better than SAMGEP
and original OGEP do.Table 5 indicates that the UGEP algorithmperforms stably in
DEPMexperiments.The ﬁtting performance is the best when the 2order differential
equation is adopted.The best model derived using UGEP is:
Y =tan
sin
x −sin
exp(z)
+tan
sin
exp(0.278970) +u
∗ sin(x)
+
log
abs(u)
−tan
cos
(x +u)
+
cos
cos
(0.871639 +x)
+tan
log
abs(u)
+tan
sin(z) +(x +u)
;(RSquare =0.9891)
where “u” represents the actual observed time,“x” represents the actual observed
value,“z” represents the 1order derivative.The performance of UGEP slightly drops
in spite of the high computing complexity in 3order.Even in this case,the value of
RSquare can still reach 0.9688 and the success rate is 98 %,which are signiﬁcantly
higher than those obtained with SAMGEP and OGEP.UGEP again executes faster
than SAMGEP and OGEP do.
6 Conclusions and future work
In this study,we examined the feasibility and effectiveness of a Uniform Design
aided Gene Expression Programming (GEP) approach to solving symbolic regression
Y.Chen et al.
Fig.8 Experimental results with UGEP and SAMGEP using DEPM
Solving symbolic regression problems
Table 5 The standard deviation for the results
1order 2order 3order
RSquare (UGEP) 0.00207 0.00213 0.00234
RSquare (SAMGEP) 0.00312 0.00243 0.00313
RSquare (OGEP) 0.00325 0.003187 0.00347
Time (UGEP) 1.14 s 1.67 s 2.34 s
Time (SAMGEP) 1.45 s 2.37 s 2.42 s
Time (OGEP) 1.224 s 1.47 s 2.44 s
Success rate (UGEP) 0.017 0.026 0.023
Success rate (SAMGEP) 0.021 0.034 0.038
Success rate (OGEP) 0.015 0.027 0.022
problems.GEP emerged as a salient variant of evolutionary computing approaches,
which signiﬁcantly surpasses its counterparts such as GPs and GAs in dealing with
these problems.However,existing GEP algorithms still suffer from premature con
vergence and slow evolution in anaphase.
We trust that the key to address these problems lies with how to maximize the
diversity of chromosome,which in turn demands an appropriate population initial
ization approach to achieve this goal.Based on this hypothesis,we developed a novel
GEP algorithm,namely uniformdesign GEP (UGEP).When initializing the popula
tion,UGEP uses a mixedlevel uniformtable to ensure that the samples are represen
tative and well distributed.The size of the initial population (the set of samples) has
also been minimized to make sure the cost for locating the optimal solution is toler
able.Furthermore,we developed a multiparent approach instead of using stochastic
evolution in the design of the cross operator.The approach thoroughly hybridizes
multiple parents to increase the chance to obtain offspring with high ﬁtness values.
We performed a theoretical analysis on UGEP to examine its performance.It has
been mathematically proved that UGEP can always converge to the global optimal
solution.In terms of convergence speed,the number of optimal offspring at the sub
sequent generation increases more quickly in UGEP than that in the original GEP.
Aseries of experiments have been carried out to have an indepth investigation on
the performance of the proposed UGEP against existing GEP variants.The symbolic
regression problems under investigation include function ﬁtting and chaotic time se
ries prediction.For the function ﬁtting problem,ten sets of experiments have ﬁrst
been performed to search for the optimal parameter setting for the UGEP algorithm.
The RSquare value obtained using UGEP with the best parameter setting can reach
0.999795 while the best value is 0.986652 using the original GEP;and the execu
tion time of UGEP and OGEP are 18.32 s and 26.47 s,respectively.We then applied
the UGEP algorithm with the parameter setting in sun spot predication using alter
native methods,namely Slide Window Prediction Method (SWPM) and Differential
Equation Prediction Method (DEPM).In comparison with SAMGEP and OGEP,the
RSquare values obtained using UGEP are always higher (e.g.,with the dimension
=12,0.9814 (UGEP) VS.0.8677 (SAMGEP) VS.08842 (OGEP).Notwithstand
ing,the convergence speeds of UGEP are always higher than those of SAMGEP and
Y.Chen et al.
OGEP (e.g.,with the dimension =12,111.8 s (UGEP) VS.131.2 s (SAMGEP) VS.
148.6 s (OGEP)).UGEP also stands a higher rate to achieve the global optimal solu
tion than SAMGEP and OGEP (e.g.,with dimension =12,91 %(UGEP) VS.76 %
(SAMGEP) VS.64 %(OGEP)).
Both theoretic analysis and experimental results indicate that UGEP excels in
terms of both the capability of achieving the global optimum and the convergence
speed when dealing with symbolic regression problems.
For future work,we will consider the interactions among the parameters.Another
interesting work is to use various uniform tables for population initialization and for
the crossover operator.
Acknowledgement This work is sponsored in part by the National Basic Research Program of China
(973 Program) under Grant No.2011CB302303,the National Natural Science Foundation of China (Grant
Nos.61272314,60933002),National High Technology Research and Development Program of China
(863 Program) under Grant No.2013AA013203,the Specialized Research Fund for the Doctoral Program
of Higher Education (Grant No.20110145110010),the Excellent Youth Foundation of Hubei Scientiﬁc
Committee (Grant No.2012FFA025),the Programfor NewCentury Excellent Talents in University (Grant
No.NCET110722),and Wuhan Chenguang Project (2013070104010019).The authors would also like
to thank Dr.Siwei Jiang for the source code of SAMGEP [20].
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