Chemo-inspired Genetic Algorithm for Optimizing the Piecewise

Aggregate Approximation

Muhammad Marwan Muhammad Fuad

Forskningsparken 3, Institutt for kjemi, NorStruct

The University of Tromsø - The Arctic University of Norway, NO-9037 Tromsø, Norway

Keywords: Bacterial Foraging, Chemotaxis, Genetic Algorithm, Hybridization, Piecewise Aggregate Approximation,

Time Series.

Abstract: In a previous work we presented DEWPAA: an improved version of the piecewise aggregate approximation

representation method of time series. DEWPAA uses differential evolution to set weights to different

segments of the time series according to their information content. In this paper we use a hybrid of bacterial

foraging and genetic algorithm (CGA) to set the weights of the different segments in our improved

piecewise aggregate approximation. Our experiments show that the new hybrid gives better results in time

series classification.

1 INTRODUCTION

In the last two decades there has been an increasing

interest in temporal data, namely, time series. A time

series is a chronological collection of observations.

This data type is encountered in many scientific and

financial applications. The main feature of time

series is their high-dimensionality. One of the

common approaches to handle the problem of high-

dimensionality of time series is to transform them

into another domain with a lower-dimensionality

followed by an indexing mechanism, called a

dimensionality reduction technique or a

representation method, applied to this lower-

dimensional data.

Several representation methods have been

proposed to reduce the dimensionality of time series

data. Of those we mention Discrete Fourier

Transformation (DFT) (Faloutsos et al., 1994),

Discrete Wavelet Transformation (DWT) (Chan and

Fu 1999), Chebyshev Polynomials (CHEB) (Cai and

Ng, 2004), Symbolic Aggregate approXimation

(SAX) (Lin et al., 2003), Piecewise Linear

Approximation (PLA) (Morinaka et al, 2001).

The Piecewise Aggregate Approximation (PAA)

(Keogh et al., 2000), (Yi and Faloutsos, 2000) is a

time series representation method that has been used

extensively for its simplicity and its low

computational complexity.

In (Muhammad Fuad, M.M., 2012) we applied

differential evolution; a popular bio-inspired

optimization algorithm, to set weights to different

segments of PAA-represented time series, which

reflect the information content of each segment. The

weights were determined through an optimization

process whose output is the weights that maximize

the information content of the PAA representation.

We showed how this modification improves the

performance of PAA.

Differential evolution, however, may suffer from

stagnation; i.e. the inability of progressing towards

global optima. It may also suffer from premature

convergence.

In this paper we propose solving the

aforementioned optimization problem by applying

an alternative optimizer which is a hybrid of two

bio-inspired optimization algorithms: genetic

algorithm and bacterial foraging. We show how this

hybrid yields better results than those obtained when

using differential evolution.

The rest of this paper is organized as follows:

Section 2 presents related work. The hybrid

algorithm is introduced in Section 3, and the

comparison with the previous method is conducted

in Section 4. We conclude this paper in Section 5.

205

Muhammad Fuad M..

Chemo-inspired Genetic Algorithm for Optimizing the Piecewise Aggregate Approximation.

DOI: 10.5220/0005277302050210

In Proceedings of the International Conference on Operations Research and Enterprise Systems (ICORES-2015), pages 205-210

ISBN: 978-989-758-075-8

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

2 RELATED WORK

PAA is a simple, yet efficient, representation method

of time series. PAA reduces the dimensionality of a

time series S from n to N dimensions by segmenting

the time series into equal-sized frames and mapping

each segment to a point that represents the mean of

the points that constitute that segment. The

similarity measure given in the following equation:









TSd

(1)

is defined on the lower-dimensional space. This

similarity measure is a lower bound of the Euclidean

distance defined on the original space. In Figure 1

we show an example of applying PAA to reduce the

dimensionality of a time series from n=12 to N=3.

The main drawback of PAA is that it is unable to

faithfully represent the original time series due to the

information loss that results from the technique PAA

uses to reduce the dimensionality, as we showed in

(Muhammad Fuad, M.M., 2012). In that paper we

proposed weighting different segments of the PAA

representation differently according to their

information content. We proposed using an

alternative similarity measure for PAA which we

called the Weighted Piecewise Aggregate

Approximation Distance (WPAAD):









iii

tsw

TSWPAAD

(2)



1,0

The challenge here is to find an objective measure to

determine the values of

w .

In (Muhammad Fuad, M.M., 2012) we

formulated the above problem as an optimization

problem where the fitness function is the

classification error and the optimization algorithm

seeks to set the values of

w that minimize the

classification error.

The optimizer we used in (Muhammad Fuad,

M.M., 2012) was the differential evolution, and we

called our method the Differential Evolutionary

Weighted Piecewise Aggregate Approximation

(DEWPAA).

Figure 1: PAA dimensionality reduction.

Differential Evolution (DE): DE is an evolutionary

optimization algorithm that has the same

computational steps as any other evolutionary

algorithm. Its particularity is in the way the

population members are perturbed, as DE uses a

scaled difference of two other members added to a

third to perform this operation.

DE is particularly adapted to solve continuous

optimization problems, and it has been successfully

used to solve real-life optimization problems.

DE starts by initializing a random population of

vectors. After initialization DE, and for each

member



(the target vector), creates a donor

vector



as a weighted difference of two other

vectors in the population added to a third one. In the

next step DE generates a trial vector



through a

crossover operation.

In the next step DE selects which of the trial

vector or the target vector will survive in the next

generation and which will die out. This selection is

based on which of



and



yields a better value of

the fitness function.

The above steps repeat for a number of

generations or until a stopping criterion terminates

the algorithm.

3 CHEMO-INSPIRED GENETIC

ALGORITHM

Hybridization of different optimization algorithms

has been extensively used to solve different

optimization problems. In time series data mining

different hybrid methods have been successfully

used (Muhammad Fuad, 2014a, 2014b, 2014c). The

main advantage that hybridization offers is that the

resulting hybrid method benefits from the strengths

of the two methods, or it avoids their weaknesses.

In this work we use a hybrid of genetic algorithm

and bacterial foraging proposed by (Das and Mishra,

2013) to solve the problem we presented in Section

3.1 The Genetic Algorithm

The Genetic Algorithm (GA) is a famous

evolutionary algorithm that has been applied to solve

a variety of optimization problems. GA is a

population-based global optimization algorithm

which mimics the rules of Darwinian selection in

that weaker individuals have less chance of

surviving the evolution process than stronger ones.

ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems

206

GA captures this concept by adopting a mechanism

that preserves the “good” features during the

optimization process.

In GA a population of candidate solutions (also

called chromosomes) explores the search space and

exploits this by sharing information. These

chromosomes evolve using genetic operations

(selection, crossover, mutation, and replacement).

GA starts by randomly initializing a population

of chromosomes inside the search space. The fitness

function of these chromosomes is evaluated.

According to the values of the fitness function new

offspring chromosomes are generated through the

aforementioned genetic operations. The above steps

repeat for a number of generations or until a

predefined stopping condition terminates the GA.

3.2 Bacterial Foraging

The foraging bahavior of the Escherichia coli (E.

coli) bacteria has inspired a nature-inspired

optimization algorithm called Bacterial Foraging

(BF). The motivation behind this optimizing

algorithm is that in order to perform social foraging,

an animal needs communication capabilities. Over a

period of time this animal gains advantages which

exploit the sensing capabilities of the whole group.

This helps the group to predate on a larger prey, or it

enables the individuals to get better protection

against predators (Kim et al., 2007).

The basis of BF is that animals with poor

foraging strategies tend to be eliminated by natural

selection and they are either replaced by other

individuals with better foraging strategies or they

are shaped into ones which have these desirable

strategies (Passino, 2002). BF formulates this

process as an optimization problem.

The E. coli bacterium moves by means of a set of

flagella, each driven as a biological motor. The two

types of movements the E. coli bacteria perform are

swimming and tumbling. The former takes place

when the flagella rotate in the counterclockwise

direction whereas the latter is achieved by rotating

the flagella in the clockwise direction. Figure 1

shows these two movement types. Together they are

known as chemotaxis (which we will define more

formally later in this section). The aim of

chemotaxis is to help the bacterium approach or

avoid nutrient or noxious substance gradients. This

chemotaxis progress can be destroyed by sudden

environmental changes which cause the elimination

and dispersal of a group of bacteria.

Flagellarotatingcounterclockwise:swimming



Flagellarotatingclockwise:tumbling



Figure 2: The swimming and tumbling movements.

BF finds the minimum of a function





nbp





;

(nbp is the number of parameters)

by applying four mechanisms; chemotaxis,

swarming, reproduction, and elimination-dispersal.

The position of each member of the population

of N

bacteria at the j

chemotactic step, k

reproduction step, and l

elimination-dispersal event

is denoted by













N,...,2,1i| l,k,jk,j,iP 



We now describe the four mechanisms we

mentioned earlier in this section:

 Chemotaxis: Let





l,k,j



be the i

bacterium at

the j

chemotactic step, k

reproduction step, and

elimination-dispersal event, then the

movement of the bacterium can be represented

by:





 

iCl,k,jl,k,1j









(3)

where Δ is a vector in the random direction whose

elements lie in the interval [-1, 1].

 Swarming: E. coli bacteria demonstrate a

swarming behavior as they travel in rings which

move up the nutrient medium when they are

placed in the center of a semisolid matrix with a

single nutrient chemo-effecter. When simulated

Table 1: The symbols used in the description of bacterial

foraging.

The number of bacteria in the population

The number of chemotactic steps

The swimming length

The number of reproduction steps

The number of elimination-dispersal events

The probability of elimination-dispersal

C(i)

The size of the step taken in the random

direction determined by the tumble

Chemo-inspiredGeneticAlgorithmforOptimizingthePiecewiseAggregateApproximation

207

by a high level of succinate the bacteria release

an attractant aspartate which helps them

aggregate into groups and thus move as a swarm.

The cell-to-cell signal in the swam can be

represented by the following function:

 



































































nbp

mmrepellantrepellant

nbp

cccc

lkjflkjPf

attractantattractant

exp.

),,,,,,





(4)

where the coefficients d

attractant

, ω

attractant

, h

repellant

are control parameters.

The objective function



l,k,jP,f



is added

to the original objective function to represent a

time varying objective function in that if many

cells come close together there will be a high

amount of attractant and hence an increasing

likelihood that other cells will move towards the

group. This produces the swarming effect

(Passino, 2002).

 Reproduction: Through this process the least

healthy bacteria die out and the healthier ones

will replicate themselves. This guarantees that the

size of the bacterial swam will remain constant.

 Elimination and dispersal: There might be a

gradual or sudden change in the environment

where the bacteria live. As a result, a small

percentage of the bacteria in a certain region will

be liquidated or a group might be dispersed into

another location. This has two effects on

chemotaxis: the first is destroying the

chemotactic progress, the second is that the new

bacteria might be placed at locations with a better

food source, thus assisting chemotaxis.

3.3 CGA

GA has the advantage of quickly locating high

performance regions of vast and complex search

spaces, but they are not well suited for fine-tuning

solutions (Gendreau and Potvin, 2005), (Kazarlis et

al., 2001).

There have been several attempts to hybridize

GA with other optimization algorithms. In (Mahfoud

and Goldberg, 1995) the authors present the Parallel

Recombinative Simulated Annealing (PRSA) which

combines elements from the simulated annealing

algorithm with others from GA. Another hybrid was

presented in (Lee and Lee, 2005). This method

hybridizes GA with Ant Colony Optimization

(ACO).

On the other hand, BF possesses a poor

convergence behavior over multi-modal and rough

fitness landscapes. Its performance is also heavily

affected with the growth of problem dimensionality

(Biswas et al., 2007).

To take advantage of the two optimizers, (Kim et

al., 2007) proposed a hybrid of GA and BF (called

GA-BF). They validated their method on several test

functions.

In another paper (Das and Mishra, 2013)

proposed another hybrid of GA and BF which they

called the Chemo-inspired Genetic Algorithm

(CGA). Their motivation is that chemotaxis actually

contributes the most in the search process, so instead

of taking the whole BF to hybridize with GA, they

only integrate the chemotaxis step in the hybrid with

GA. CGA has five major steps: selection, crossover,

mutation, elitism and chemotaxis. In addition, CGA

employs three mechanisms: a) adaptive step size b)

squeezed search space c) fitness function criterion.

4 EXPERIMENTS

We conducted intensive experiments to compare

CGA with DEWPAA. The experiments were the

same as those conducted in (Muhammad Fuad,

M.M., 2012) and (Muhammad Fuad, M.M., 2013);

i.e. classification task experiments of time series.

The aim of the experiments is to show that using

CGA in the optimization process of determining the

weights of the segments in equation (2) will result in

a lower classification error than that of DEWPAA.

As in (Muhammad Fuad, M.M., 2012), we

computed the classification error using WPAAD for

different compression ratios (1:8,1:12,1:16). We

conducted our experiments using the UCR archive

(Keogh et al., 2011), which is the same archive used

to test DEWPAA.

The two methods were tested on a classification

task based on the first nearest-neighbor (1-NN) rule

using leaving-one-out cross validation.

Table 2 shows some of the results of our

experiments. As we can see from the table, CGA

outperforms DEWPAA on almost all the datasets

tested and for the different compression ratios.

5 CONCLUSIONS

In this paper we used a hybrid of genetic algorithm

and bacterial foraging (CGA) to calculate the

weights given to different segments of the time

ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems

208

Table 2: Comparison of the classification error between CGA and DEWPAA for different compression ratios.

Dataset Method

Compression Ratio

1:8 1:12 1:16

Lighting7

CGA 0.329 0.342 0.356

DEWPAA 0.370 0.397 0.427

MedicalImages

CGA 0.308 0.308 0.321

DEWPAA 0.337 0.337 0.378

Gun_Point

CGA 0.06 0.06 0.06

DEWPAA 0.053 0.067 0.087

Coffee

CGA 0.179 0.179 0.179

DEWPAA 0.179 0.179 0.25

Lighting2

CGA 0.164 0.180 0.197

DEWPAA 0.213 0.230 0.246

MALLAT

CGA 0.077 0.082 0.094

DEWPAA 0.094 0.094 0.095

FacesUCR

CGA 0.238 0.238 0.238

DEWPAA 0.238 0.316 0.366

FISH

CGA 0.194 0.194 0.194

DEWPAA 0.194 0.229 0.240

synthetic_control

CGA 0.067 0.067 0.100

DEWPAA 0.110 0.113 0.113

CBF

CGA 0.017 0.021 0.021

DEWPAA 0.017 0.021 0.037

ECG

CGA 0.090 0.100 0.100

DEWPAA 0.110 0.120 0.130

Trace

CGA 0.100 0.120 0.120

DEWPAA 0.140 0.180 0.190

series represented by the piecewise aggregate

approximation representation method. The weights

were obtained through an optimization process using

chemo-inspired genetic algorithm (CGA) as

optimizer and the fitness function is the

classification error. Compared with differential

evolution, another optimizer that we used in a

previous work to solve the same optimization

problem, CGA gives better results.

REFERENCES

Biswas, A., Dasgupta, S., Das, S. and Abraham, A., 2007.

Synergy of PSO and bacterial foraging optimization: a

comparative study on numerical benchmarks. HAIS

2007.

Cai, Y., and Ng, R., 2004. Indexing spatio-temporal

trajectories with chebyshev polynomials. In SIGMOD.

Chan, K., and Wai-chee Fu, A., 1999. Efficient time series

matching by wavelets. In Proc. 15th. Int. Conf. on

Data Engineering.

Das, K. N., and Mishra, R., 2013. Chemo-inspired genetic

algorithm for function optimization. Applied

Mathematics and Computation, 220, 394–404.

Faloutsos, C., Ranganathan, M., and Manolopoulos, Y.,

1994. Fast subsequence matching in time-series

databases. In Proc. ACM SIGMOD Conf.,

Minneapolis.

Gendreau M., Potvin., J. Y,. 2005. Annals of operations

research 140(1)pp189–213.

Kazarlis, S.A., Papadakis. S. E., Theocharis, J.B., Petridis,

V., 2001. IEEE transactions on evolutionary

computation 5(3)pp 204–217.

Keogh, E., Chakrabarti, K., Pazzani, M., and Mehrotra,S.,

2000. Dimensionality reduction for fast similarity

search in large time series databases. J. of Know. and

Inform. Sys.

Kim, D. H., Abraham, A., Cho, J. H., 2007. A hybrid

genetic algorithm and bacterial foraging approach for

global optimization. Information Sciences, Vol. 177

(18), 3918-3937.

Lee, Z .J., and Lee, C. Y., 2005. A hybrid search algorithm

with heuristics for resource allocation problem,

Information Sciences 173.

Keogh, E., Zhu, Q., Hu, B., Hao. Y., Xi, X., Wei, L. &

Ratanamahatana, C.A., 2011. The UCR Time Series

Classification/Clustering Homepage: www.cs.ucr.edu/

~eamonn/time_series_data/

Lin, J., Keogh, E., Lonardi, S., Chiu, B. Y., 2003. A

symbolic representation of time series, with

implications for streaming algorithms. DMKD 2003.

Chemo-inspiredGeneticAlgorithmforOptimizingthePiecewiseAggregateApproximation

209

Mahfoud, S., and Goldberg, D., 1995. Parallel

recombinative simulated annealing: a genetic

algorithm. Parallel Computing, vol. 21, pp. 11-28.

Morinaka, Y., Yoshikawa, M., Amagasa, T., and Uemura,

S., 2001: The L-index: An indexing structure for

efficient subsequence matching in time sequence

databases. In Proc. 5th Pacific Asia Conf. on

Knowledge Discovery and Data Mining.

Muhammad Fuad, M. M., 2014a. A hybrid of bacterial

foraging and differential evolution -based distance of

sequences. In Procedia Computer Science 2014.

Volume 35. pp 101-110.

Muhammad Fuad, M. M., 2013. A pre-initialization stage

of population-based bio-inspired metaheuristics for

handling expensive optimization problems. ADMA

2013, December 14-16, 2013, Zhejiang, China.

Lecture Notes in Computer Science Volume 8347,

2014, pp 396-40.

Muhammad Fuad, M. M., 2014b. A synergy of artificial

bee colony and genetic algorithms to determine the

parameters of the Σ-Gram distance. In DEXA 2014,

Munich, Germany. Lecture Notes in Computer Science

Volume 8645, 2014, pp 147-154.

Muhammad Fuad, M. M., 2014c. A weighted minimum

distance using hybridization of particle swarm

optimization and bacterial foraging. In PRICAI 2014,

Gold Coast, QLD, Australia. Lecture Notes in

Computer Science Volume 8862, 2014, pp 309-319.

Muhammad Fuad, M. M., 2012. Using differential

evolution to set weights to segments with different

information content in the piecewise aggregate

approximation. In KES 2012, San Sebastian, Spain,

(FAIA). IOS Press.

Passino, K. M., 2002. Biomimicry of bacterial foraging for

distributed optimization and control, IEEE Control

Syst. Mag., vol. 22, no. 3, pp. 52–67.

Yi, B. K., and Faloutsos, C., 2000. Fast time sequence

indexing for arbitrary Lp norms. Proceedings of the

26th International Conference on Very Large

Databases, Cairo, Egypt.

ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems

210