Semi-supervised SVM with Fuzzy Controlled Cooperation of Biology

Related Algorithms

Shakhnaz Akhmedova, Eugene Semenkin and Vladimir Stanovov

Department of System Analysis and Operations Research, Reshetnev Siberian State University of Science and Technology,

“Krasnoyarskiy Rabochiy” avenue, 31, Krasnoyarsk, Russia

Keywords: Bio-inspired Algorithms, Fuzzy Controller, Support Vector Machines, Semi-Supervised Learning,

Classification, Constrained Optimization.

Abstract: Due to its wide applicability, the problem of semi-supervised classification is attracting increasing attention

in machine learning. Semi-Supervised Support Vector Machines (SVM) are based on applying the margin

maximization principle to both labelled and unlabelled examples. A new collective bionic algorithm,

namely fuzzy controlled cooperation of biology related algorithms (COBRA-f), which solves constrained

optimization problems, has been developed for semi-supervised SVM design. Firstly, the experimental

results obtained by the two types of fuzzy controlled COBRA are presented and compared and their

usefulness is demonstrated. Then the performance and behaviour of proposed semi-supervised SVMs are

studied under common experimental settings and their workability is established.

1 INTRODUCTION

One of the most important machine learning tasks is

classification that consists in identifying to which of

a set of categories a new instance belongs. If

sufficient labelled training data are given, there

exists a variety of techniques, for example, artificial

neural networks (Bishop, 1996), fuzzy logic

classifiers (Kuncheva, 2000) or Support Vector

Machines (SVM) (Vapnik and Chervonenkis, 1974),

to address such a task. However, labelled data are

often rare in real-world applications. Therefore,

recently semi-supervised learning has attracted

increasing attention among researchers (Zhu and

Goldberg, 2009).

In contrast to supervised methods, the latter class

of techniques takes both labelled and unlabelled data

into account to construct appropriate models. A

well-known concept in this field is semi-supervised

support vector machines (Bennett and Demiriz,

1999), which depict the direct extension of support

vector machines to semi-supervised learning

scenarios.

In this study semi-supervised SVMs generated

by a new collective bionic optimization algorithm,

namely fuzzy controlled cooperation of biology

related algorithms or COBRA-f, are described.

Initially, a meta-heuristic approach called Co-

Operation of Biology Related Algorithms or

COBRA (Akhmedova and Semenkin, 2013 (1)) was

developed for solving unconstrained real-parameter

optimization problems. Its basic idea consists in the

cooperative work of different nature-inspired

algorithms, which were chosen due to the similarity

of their schemes. However, there are still various

algorithms which can be used as components for

COBRA as well as previously conducted

experiments demonstrating that even the bionic

algorithms already chosen can be combined in

various ways.

Thus, to solve the described problem, in this

work COBRA was modified by implementing

controllers based on fuzzy logic (Lee, 1990). The

aim of this was to determine in an automated way

which bionic algorithm should be included in the co-

operative work. The proposed modification also

allows resources to be allocated properly while

solving unconstrained optimization problems. And

finally the obtained modification COBRA-f was

adopted for solving constrained optimization

problems.

Therefore, in this paper firstly a brief description

of the semi-supervised SVM is presented. Then the

COBRA meta-heuristic approach and the fuzzy

controller are described. In the next section, the

Akhmedova, S., Semenkin, E. and Stanovov, V.

Semi-supervised SVM with Fuzzy Controlled Cooperation of Biology Related Algorithms.

DOI: 10.5220/0006417400640071

In Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2017) - Volume 1, pages 64-71

ISBN: 978-989-758-263-9

experimental results obtained by two types of fuzzy

controller are discussed. And after that the

implementation of the best obtained fuzzy controlled

COBRA was applied for solving constrained

optimization problems as well as training the semi-

supervised SVM. For experiments several datasets

have been chosen, among which there are synthetic

and real datasets. In particular, we have used a

popular two moons problem, two datasets from the

UCI repository (namely Breast Cancer Wisconsin

(BCW) and Pima Indian Diabetes (PID)) with only

the 10 labels used available, and the gas turbine

dangerous vibrations detection problem. Finally,

some conclusions are given in the last section.

2 SEMI-SUPERVISED SUPPORT

VECTOR MACHINES

In Support Vector Machines (SVM), the intuition is

to try to create a separating hyperplane between the

instances from different classes (Vapnik and

Chervonenkis, 1974). SVM is based on the

maximization of the distance between the

discriminating hyperplane and the closest examples.

In other words since many choices could exist for

the separating hyperplane, in order to generalize

well on test data, the hyperplane with the largest

margin has to be found.

Suppose

()(){}

yxyxL ,,...,,

= ,

Rx ∈

is a

training set with l examples (instances), each

instance

x has m attributes and is labelled as

y ,

where

li ,1= . Let v be a hyper-plane going through

the origin,

be the margin and

w =

. The margin

maximizing hyperplane can be formulated as a

constrained optimization problem in the following

manner:

min

→w

(1)

()

1≥⋅

xwy

(2)

To solve the given optimization problem, the

proposed fuzzy controlled cooperation of biology

related algorithms or COBRA-f was used.

However, in this study semi-supervised SVMs

were considered. Thus, given the additional set

{}

ull

xxU

= ,...,

of unlabelled training patterns,

semi-supervised support vector machines aim at

finding an optimal prediction function for unseen

data based on both the labelled and the unlabelled

part of the data (Joachims, 1999). For unlabelled

data, it is assumed that the true label is the one

predicted by the model based on what side of the

hyperplane the unlabelled point ends up being.

In this study, self-training was used to learn from

the unlabelled data. Namely, the idea is to design the

model with labelled data and then use the model’s

own predictions as labels for the unlabelled data to

retrain a new model with the original labelled data

and the newly labelled data and then iteratively

repeat this process.

The problem with this method is that considering

its own predictions as true labels can cause the

model to drift away from the correct model if the

predictions were wrong initially. The model would

then continue to mislabel data and use it again and

continue to drift away from where it should be.

Therefore, to prevent this problem the technique

described in (Ravi, 2014) was used. More

specifically, the model’s predictions were used to

label the data only when there is a high confidence

about the predictions.

The notion of confidence used for the SVM

model is the distance from the found hyperplane.

The larger the distance from the hyperplane, the

higher the probability that the instance belongs to

the corresponding side of the separating hyperplane.

Consequently, the following basic steps were

performed:

 Train SVM on the labelled set L by the

proposed meta-heuristic approach COBRA-f;

 Use obtained SVM to classify all unlabelled

instances from U by checking the confidence

criteria from (Ravi, 2014);

 Label instances from the set U if this is

possible;

 Repeat from the first step.

Thus, the simplest semi-supervised learning

method was used for examining the workability of

COBRA-f.

3 CO-OPERATION OF BIOLOGY

RELATED ALGORITHMS

The meta-heuristic approach called Co-Operation of

Biology Related Algorithms or COBRA

(Akhmedova and Semenkin, 2013) was developed

based on five optimization methods, namely Particle

Swarm Optimization (PSO) (Kennedy and Eberhart,

1995), Wolf Pack Search (WPS) (Yang et al., 2007),

the Firefly Algorithm (FFA) (Yang, 2009), the

Cuckoo Search Algorithm (CSA) (Yang and Deb,

Semi-supervised SVM with Fuzzy Controlled Cooperation of Biology Related Algorithms

2009) and the Bat Algorithm (BA) (Yang, 2010)

(hereinafter referred to as “component-algorithms”).

Also, the Fish School Search (FSS) (Bastos and

Lima, 2009) was later added as COBRA’s

component-algorithm.

The main reason for the development of a

cooperative meta-heuristic was the inability to say

which of the above-listed algorithms is the best one

or which algorithm should be used for solving any

given optimization problem (Akhmedova and

Semenkin, 2013). Thus, the idea was to use the

cooperation of these bionic algorithms instead of any

attempts to understand which one is the best for the

problem in hand.

The originally proposed approach consists in

generating five populations, one population for each

bionic algorithm (or generating six populations with

the FSS algorithm added) which are then executed in

parallel, cooperating with each other. The COBRA

algorithm is a self-tuning meta-heuristic, so there is

no need to choose the population size for each

component-algorithm. The number of individuals in

the population of each algorithm can increase or

decrease depending on the fitness values: if the

overall fitness value was not improved during a

given number of iterations, then the size of each

population increased, and vice versa, if the fitness

value was constantly improved during a given

number of iterations, then the size of each

population decreased.

There is also one more rule for population size

adjustment, whereby a population can “grow” by

accepting individuals removed from other

populations. The population “grows” only if its

average fitness value is better than the average

fitness value of all other populations. Therefore, the

“winner algorithm” can be determined as an

algorithm whose population has the best average

fitness value. This can be done at every step. The

described competition among component-algorithms

allows the biggest population size to be allocated to

the most appropriate bionic algorithm on the current

generation.

The main goal of this communication between

all populations is to bring up-to-date information on

the best achievements to all component-algorithms

and prevent their preliminary convergence to their

own local optimum. “Communication” was deter-

mined in the following way: populations exchange

individuals in such a way that a part of the worst

individuals of each population is replaced by the

best individuals of other populations. Thus, the

group performance of all algorithms can be

improved.

The performance of the COBRA algorithm was

evaluated on a set of various benchmark problems

and the experiments showed that COBRA works

successfully and is reliable on different benchmarks

(Akhmedova and Semenkin, 2013). Besides, the

simulations showed that COBRA is superior to its

component-algorithms when the dimension grows or

when complicated problems are solved.

Then COBRA’s modification for solving

constrained optimization problems COBRA-c was

developed (Akhmedova and Semenkin, 2013 (2)).

Three constraint handling methods were used for

this purpose: dynamic penalties (Eiben and Smith,

2003), Deb’s rule (Deb, 2000) and the technique

described in (Liang, Shang and Li, 2010). The

method proposed in (Liang, Shang and Li, 2010)

was implemented in the PSO-component of

COBRA; at the same time other components were

modified by implementing Deb’s rule followed by

calculating function values using dynamic penalties.

The performance of this modification was

evaluated with a set of various test functions. It was

established that COBRA-c works successfully and is

sufficiently reliable. Finally, COBRA’s modification

outperforms all of its component-algorithms.

4 FUZZY CONTROLLER

The size control of the COBRA populations was

performed by the fuzzy controller, which received

algorithms’ success rates as inputs, and returned the

populations’ size modification values. Overall, there

were 7 input variables, i.e. one variable for each of

COBRA’s 6 component-algorithms, showing its

success rate, plus the overall success rate of all

components.

The success rate for all input variables except for

the last one was evaluated as the best fitness value of

its population. The last input variable was

determined as the ratio of the number of iterations,

during which the best-found fitness value was

improved, to the given number of iterations, which

was a constant period.

The number of outputs was equal to the number

of components.

The fuzzy rules had the following form:

: IF x

is A

and … and x

is A

THEN y

and … y

is B

(3)

where R

is the q-th fuzzy rule,

()

xxx ,...,

= are the

input values (components’ success rate) in n-

dimensional space (

7=n in this study),

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

()

yyy ,...,

= is the set of outputs ( 6=k ), A

is the

fuzzy set for the i-th input variable, B

is the fuzzy

set for the j-th output variable. The Mamdani-type

fuzzy inference with a centre of mass calculation

was used as the defuzzification method.

For the purposes of this study two variants of the

fuzzy controller, which differed in the number of

terms for output variables, have been implemented.

All inputs were values in the range

[]

1;0 , so that the

input fuzzy terms were equal for all variables. Also,

3 basis triangular fuzzy terms were used, and in

addition the A

term combining A

and A

, as well as

the “Don’t Care” condition (DC) have been included

to decrease the number of rules. The term shapes are

shown in Figure 1.

Figure 1: Fuzzy sets for inputs.

The first fuzzy controller’s 3 fuzzy terms, which

were used for the output, are demonstrated in Figure

Figure 2: Fuzzy terms for all 6 outputs, first controller.

The adjustable parameters of the first fuzzy

controller are the values encoding positions of

output fuzzy terms, i.e. the position of central term

and side terms. For the example shown in Figure 2,

these values are -20, -10, 10 and 20, i.e. four values

were encoded, so that the terms may appear to be

non-symmetric after optimization.

A part of the rule base for the first controller is

presented in Table 1.

Table 1: Part of the first controller’s rule base.

№ IF THEN

1 X

is A

-X

-Y

2 X

is A

-X

-Y

3 X

is A

-X

-Y

…

19 X

-X

is DC X

is A

is B

20 X

-X

is DC X

is A

is B

21 X

-X

is DC X

is A

is B

The second controller had 7 terms for output

variables instead of 5. Two additional terms were

used for the last part of the rule base, which defined

the influence of the overall success rate, i.e. the 7-th

variable. The fuzzy terms for the second controller

are shown in Figure 3.

Figure 3: Fuzzy terms for all 6 outputs, second controller.

Terms 1 and 5 in the second controller have

different shapes, which allows the size control of

overall populations to be tuned more accurately.

The rule base is also different; it uses different

terms for the output. A part of the rule base is

presented in Table 2.

Table 2: Part of the second controller’s rule base.

№ IF THEN

1 X

is A

-X

is B

-Y

2 X

is A

-X

is B

-Y

3 X

is A

-X

is B

-Y

…

19 X

-X

is DC X

is A

is B

20 X

-X

is DC X

is A

is B

21 X

-X

is DC X

is A

is B

Semi-supervised SVM with Fuzzy Controlled Cooperation of Biology Related Algorithms

The second controller tends to be more flexible,

although it requires the tuning of 6 parameters

instead of 4 for the first one. For the case shown in

Figure 3 the parameter values are -30, -20, -10, 10,

20, 30, but terms may end up non-symmetric

afterwards.

5 EXPERIMENTAL RESULTS

In this section, the methodology employed in this

study to validate the proposed approach is presented.

The next sections describe the techniques used for

comparison purposes, the benchmark functions and

the statistical analysis.

5.1 Constrained Optimization

Problems

In this study 6 benchmark problems taken from

(Whitley, 1995) were used in experiments for

comparing the constrained optimization algorithms.

Optimal solutions for these problems are already

known, thus the algorithm’s reliability was estimated

by the achieved error value.

The given benchmark functions were considered

to evaluate the robustness of the fuzzy controlled

COBRA, which was modified for solving

constrained optimization problems in two ways:

 By using dynamic penalties (Eiben and Smith,

2003);

 By using Deb’s rule (Deb, 2000).

Consequently, firstly test problems were used to

determine the best parameters for the four types of

fuzzy controllers:

 Controller with 4 parameters, constraint

handling technique is dynamic penalties;

 Controller with 4 parameters, constraint

handling technique is Deb’s rule;

 Controller with 6 parameters, constraint

handling technique is dynamic penalties;

 Controller with 6 parameters, constraint

handling technique is Deb’s rule.

The standard Particle Swarm Optimization

algorithm was used for this purpose. Therefore, the

individuals were each represented as parameters of

the fuzzy controlled COBRA, namely the positions

of the output fuzzy terms. The following objective

function was optimized by the PSO algorithm:

() ()



(4)

where

10=T is the total number of program runs

for each benchmark problem listed earlier. Thus, on

each iteration all test problems were solved T times

by a given fuzzy controlled COBRA and then the

obtained results were averaged. Calculations were

stopped on each program run if the number of

function evaluations exceeded

D10000 . The

population size for the PSO algorithm was equal to

50 and the number of iterations was equal to 100;

calculations were stopped on the 100-th iteration for

the PSO heuristic.

Accordingly, the following parameters for the

fuzzy controllers were obtained:



[]

27;5;5;9 −− ;



[]

34;27;10;33 −−

;



[]

29;24;1;0;0;14− ;



[]

21;16;14;3;6;23 −−−

;

On the following step, the obtained parameters

were applied to the fuzzy controlled COBRA and it

was tested on the mentioned benchmark functions.

There were 51 program runs for each constrained

optimization problem, and calculations were stopped

if the number of function evaluations was equal to

D10000 . Also, for example, a change in population

sizes was obtained while testing on the benchmark

problems. This change for the third problem is

presented in Figure 4.

Figure 4: Change in population sizes.

Based on the results received for other functions,

it was concluded that the algorithm with Deb’s rule

exhibits strange behaviour, i.e. it tends to increase

the size of all populations, while dynamic penalties

show more complicated cooperation. More

specifically, for the second fuzzy controller with 6

variables, for example, CSA is one of the worst

algorithms for the first 20000 calculations (it has

around 3 points available), but after 20000 it rapidly

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

increases its population size to 300 and more,

because it shows a much better ability to optimize

the function. At the same time, BA and WPS, who

were winners at the first phase, gradually decrease

their resource.

In Table 3 and Table 4 the results obtained by the

fuzzy controlled COBRA, the controller of which

has 4 parameters, are presented with the best

parameters. The following notations are used: the

best found function value (Best), the function value

averaged by the number of program runs (Mean) and

the standard deviation (STD).

Table 3: Results obtained by the fuzzy controlled COBRA

(4 parameters) with dynamic penalties.

Best Mean STD

1 0.000222883 0.000450223 0.000269495

2 4.44089e-016 1.20453e-006 5.73208e-006

3 0.0168729 0.0861003 0.106644

4 0.00246478 0.0270861 0.0240794

5 0.000349198 0.00205344 0.00184918

6 4.07214e-005 0.00012163 7.28906e-005

Table 4: Results obtained by the fuzzy controlled COBRA

(4 parameters) with Deb’s rule.

Best Mean STD

1 0.000194127 6.48378 12.6967

2 1.37668e-014

0.0508509 0.153545

3 0.241445 1.59201 1.35446

4 0.093186 6.36309 6.95277

5 0.0646682 5.93762 7.14939

6 0.00463816 3.50586 19.3554

Table 5: Results obtained by the fuzzy controlled COBRA

(6 parameters) with dynamic penalties.

Best Mean STD

1 0.000222799 0.0023863 0.00386538

2 4.44089e-016 5.45027e-006 1.14566e-005

3 0.0265527 0.249473 0.645633

4 0.0114249 0.0379335 0.0643213

5 0.000159134 0.0380605 0.125444

6 0.000236098 0.000955171 0.00178648

Table 6: Results obtained by the fuzzy controlled COBRA

(6 parameters) with Deb’s rule.

Best Mean STD

1 1.47682e-009 0.00234639 0.0054576

2 3.10862e-015 0.000577974 0.00155304

3 0.015536

0.055806 0.0984222

4 0.00559601 0.038134 0.113326

5 0.000160994 0.0656134 0.130952

6 6.15548e-006 0.000296266 0.000680345

In Table 5 and Table 6 the results obtained by the

fuzzy controlled COBRA, the controller of which

has 6 parameters, are presented with the best

parameters. The same notations as in the previous

tables are used.

For comparison, in Table 7 the results obtained

by COBRA-c with six component-algorithms with

the standard tuning method are given.

Table 7: Results obtained by COBRA-c with six

component algorithms.

Best Mean STD

1 2.54087e-005 0.00710629 0.0177543

2 2.08722e-014 0.000114461 5.15068e-005

3 6.39815e-005 0.0477759 0.0422946

4 0.0267919 0.0324922 0.00119963

5 0.000341288

0.0678906 0.0331402

6 1.21516e-005 0.000278707 0.000268217

Thus, the comparison demonstrates that the

fuzzy controlled COBRA with dynamic penalties

outperformed the same algorithms with Deb’s rule.

Aside from this, there is no significant difference

between the results obtained by the fuzzy controlled

COBRA with either 4 or 6 parameters. However, the

4-parameter fuzzy controlled COBRA with dynamic

penalties also outperformed the COBRA with six

components without a controller. Therefore, it can

be used for solving the optimization problems

instead of the given algorithm’s versions.

5.2 Classification Performance

Several artificial and real-world data sets described

in Table 8 were considered in this study, namely the

well-known two-dimensional “Moons” data set and

data sets for two medical diagnostic problems (Frank

Semi-supervised SVM with Fuzzy Controlled Cooperation of Biology Related Algorithms

and Asuncion, 2010). Each data set instance was

split into a labelled part and an unlabelled one, and

the different ratios for the particular settings were

used.

Table 8: Data sets considered in the experimental

evaluation, each consisting of n patterns having d features.

Data Set n d

Moons 200 2

Breast Cancer Wisconsin 699 9

Pima Indians Diabetes 768 8

For the sake of exposition, firstly the well-known

“Moons” data set was considered. This choice is

conditioned by the fact that the given data set is a

difficult training instance for semi-supervised

support vector machines due to its non-linear

structure. The “Moons” problem is a classical semi-

supervised problem for testing algorithms. It consists

of two groups of moon-like sets of points, which are

easily recognized as two classes by a human, but

represent significant difficulty for modern

algorithms. In the conducted experiments only 2

labelled points for every class were known, and the

rest of the points were classified using the semi-

supervised SVM described above. The results

obtained on the “Moons” problem are shown in

Figure 5.

Figure 5: Semi-supervised classification of “Moons”.

As can be seen, the algorithm does not recognize

all points correctly, i.e. it builds an almost linear

classification. However, most of the points are in the

right class.

Then two medical diagnostic problems, namely

Breast Cancer Wisconsin (BCW) and Pima Indian

Diabetes (PID), were solved. Both problems are

binary classification tasks. For these data sets, 10

examples were randomly selected to be used as

labelled examples, and the remaining instances were

used as unlabelled data. The experiments are

repeated 10 times and the average accuracies and

standard deviations are recorded. The results are

shown in Table 9. Alternative algorithms (linear

SVMs) for comparison are taken from (Li and Zhou,

2011).

Table 9: Performance comparison of semi-supervised

methods.

BCW PID

TSVM 89.2±8.6 63.4±7.6

S3VM-c 94.2±4.9 63.2±6.8

S3VM-p 93.9±4.9 65.6±4.8

S3VM-us 93.6±5.4 65.2±5.0

This study 95.5±1.8 69.3±1.5

The gas turbine dangerous vibrations problem

includes 11 input variables, which are process

parameters, potentially connected to the vibration

level, and the output is the class number –

dangerous/stable vibration level. The vibration

signal is one of the most important diagnostic

instruments when measuring the turbine wear.

For the experiments with this dataset, we have

used 5%, 10% and 15% of the labelled data for

training, while the rest of the training set was

unlabelled. The total size of the dataset is 1000

instances, 900 were used for training, while 100

instances were left for a test set. In 3 experiments,

the number of labelled examples was 45, 90 and 135

instances. The average classification quality on the

test set obtained after 10 experiments is presented in

Table 10.

Table 10: Performance comparison, gas turbine dataset.

Labelled

COBRA Semi-

supervised

SVM

5% 86.2±1.7

10% 87.8±0.4

15% 88.2±0.6

The classification quality is relatively high even

with only 5% of labelled examples in the training

set. This result provides the possibility to use a vast

amount of available unlabelled data for model

improvements in future.

Consequently, the inference should be drawn that

the suggested algorithm successfully solved all the

problems of designing semi-supervised SVM-based

classifiers with competitive performance. Thus, the

study results can be considered as confirming the

reliability, workability and usefulness of the fuzzy

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

controlled cooperative algorithm in solving real

world optimization problems.

6 CONCLUSIONS

The problem of semi-supervised classification is

important due to the fact that obtaining labelled

examples is often very expensive. However, using

this data during classification may be helpful. In this

paper, the semi-supervised SVM was trained using a

cooperative algorithm, whose components were

automatically adjusted by a fuzzy controller. The

fuzzy controller itself was tuned to deliver better

results for constrained optimization problems. This

tuning of the meta-heuristic allowed better results of

SVM training to be achieved, compared to other

studies. The proposed approach, combining biology-

related algorithms and fuzzy controllers could be

applied to other complex constrained optimization

problems.

ACKNOWLEDGEMENTS

Research is performed with the support of the

Ministry of Education and Science of Russian

Federation within State Assignment project №

2.1680.2017/ПЧ.

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