J Supercomput

DOI 10.1007/s11227-013-0943-6

Solving symbolic regression problems with uniform

design-aided 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 UniformDesign-Aided Gene Expression Programming (UGEP).UGEP uses

(1) a mixed-level 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

e-mail:

chendan@pmail.ntu.edu.sg

S.U.Khan (

)

Department of Electrical and Computer Engineering,North Dakota State University,Fargo,ND,

USA

e-mail:

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 pre-speciﬁed 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 multi-parent 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 mixed-level 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 “best-so-far” solution is returned.

2.1 Chromosome encoding

Each chromosome is a character string in ﬁxed-length,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 K-expression 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 three-gene

chromosome and its sub-ETs

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

roulette-wheel 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 well-distributed 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 cross-hybridization,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 cross-hybridization,use the adap-

tive multi-parent 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 mixed-level 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 mixed-level

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 mixed-level uniform table can be constructed by

Algorithm1,which is shown in Fig.4.

Algorithm1 (Construct the mixed-level 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 fractional-addition in [21]).We can get the mixed-level 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 direct-product 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 mixed-level 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 multi-parent 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 multi-parent 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 mixed-level 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 ∈S|f(x) =f

∗

}.

Deﬁnition 2 The ε-ﬁeld set of the optimal solutions is deﬁned as ∀ε >0;M

(x −ε)

=

{x ∈ S|f(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 ∈ P|i/∈ V

m

}) is C

N−m

N

• μ

,where μ

=

|V

m

∩S|.Therefore,a new m-dimension 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 multi-parent 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].

R-square is calculated to evaluate the ﬁtting accuracy by comparing the ﬁtting func-

tion to the experimental function.A larger R-square 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 R-Square is 0.995039

and the best value of R-Square is 0.998652.The standard deviation of R-Square 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 R-Square 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 R-Square 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;

(R-Square =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 R-Square 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)

;(R-Square =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 in-depth 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 SA-MGEP

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 SA-MGEP 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

(R-Square,execution time,and success rate) are shown in Fig.5.

As shown in Fig.7,UGEP performs better than SA-MGEP 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,SA-MGEP and

OGEP increase signiﬁcantly.UGEP maintains a success rate higher than 90 %while

that using SA-MGEP drops from 92 % to 76 % and that using OGEP drops from

94 %to 64 %.The R-Square values also indicate that UGEP can ensure an accurate

prediction.It is also noticeable that UGEP executes even faster than SA-MGEP 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 SA-MGEP 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,SA-MGEP and OGEP using SWPM

Y.Chen et al.

Table 4 The standard deviation with each dimension

Dimension 6 Dimension 10 Dimension 12

R-Square (UGEP) 0.00186 0.00127 0.00248

R-Square (SA-MGEP) 0.00172 0.00189 0.00283

R-Square (OGEP) 0.00242 0.00367 0.00642

Time (UGEP) 1.47 s 2.23 s 2.12 s

Time (SA-MGEP) 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 (SA-MGEP) 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

;(R-Square =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

)

;(R-Square =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

)

;

(R-Square =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 high-order to a low-order,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 SA-MGEP

and original OGEP do.Table 5 indicates that the UGEP algorithmperforms stably in

DEPMexperiments.The ﬁtting performance is the best when the 2-order 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)

;(R-Square =0.9891)

where “u” represents the actual observed time,“x” represents the actual observed

value,“z” represents the 1-order derivative.The performance of UGEP slightly drops

in spite of the high computing complexity in 3-order.Even in this case,the value of

R-Square can still reach 0.9688 and the success rate is 98 %,which are signiﬁcantly

higher than those obtained with SA-MGEP and OGEP.UGEP again executes faster

than SA-MGEP 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 SA-MGEP using DEPM

Solving symbolic regression problems

Table 5 The standard deviation for the results

1-order 2-order 3-order

R-Square (UGEP) 0.00207 0.00213 0.00234

R-Square (SA-MGEP) 0.00312 0.00243 0.00313

R-Square (OGEP) 0.00325 0.003187 0.00347

Time (UGEP) 1.14 s 1.67 s 2.34 s

Time (SA-MGEP) 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 (SA-MGEP) 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 mixed-level 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 in-depth 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 R-Square 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 SA-MGEP and OGEP,the

R-Square values obtained using UGEP are always higher (e.g.,with the dimension

=12,0.9814 (UGEP) VS.0.8677 (SA-MGEP) VS.08842 (OGEP).Notwithstand-

ing,the convergence speeds of UGEP are always higher than those of SA-MGEP and

Y.Chen et al.

OGEP (e.g.,with the dimension =12,111.8 s (UGEP) VS.131.2 s (SA-MGEP) VS.

148.6 s (OGEP)).UGEP also stands a higher rate to achieve the global optimal solu-

tion than SA-MGEP and OGEP (e.g.,with dimension =12,91 %(UGEP) VS.76 %

(SA-MGEP) 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.NCET-11-0722),and Wuhan Chenguang Project (2013070104010019).The authors would also like

to thank Dr.Siwei Jiang for the source code of SA-MGEP [20].

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