Data Classification with the Radial Basis Function Network Based on a Novel Kernel Density Estimation

Algorithm

Yen-Jen OyangDepartment of Computer Science and Informatio

n EngineeringNational Taiwan University

An Example of Data Classification

Data Class Data Class Data Class

（ 15,33）

O （ 18,28）

× （ 16,31）

O

（ 9 ,23）

× （ 15,35）

O （ 9 ,32）

×

（ 8 ,15）

× （ 17,34）

O （ 11,38）

×

（ 11,31）

O （ 18,39）

× （ 13,34）

O

（ 13,37）

× （ 14,32）

O （ 19,36）

×

（ 18,32）

O （ 25,18）

× （ 10,34）

×

（ 16,38）

× （ 23,33）

× （ 15,30）

O

（ 12,33）

O （ 21,28）

× （ 13,22）

×

Distribution of the Data Set

。。

10 15 20

30

。。。 。。

。 。。

××

××

×

×

×

×

×

×

××

×

×

Rule Based on Observation

.

0

30

253015 22

Xclass

else

class

, thenand y

yxIf

Rule Generated by the Proposed RBF(Radial Basis Function)

Network Based Learning Algorithm

Let and

If

then prediction=“O”.

Otherwise prediction=“X”.

2o

2o

210

12o

o 2

1)( i

icv

i i

evf

.

2

1)(

2

214

12x

x

2x

x

j

jcv

j j

evf

),()( xx

oo vf

S

Svf

S

S

(15,33)

(11,31)

(18,32)

(12,33)

(15,35)

(17,34)

(14,32)

(16,31)

(13,34)

(15,30)

1.723 2.745 2.327 1.794 1.973 2.045 1.794 1.794 1.794 2.027

ico

io

(9,23) (8,15)(13,37)

(16,38)

(18,28)

(18,39)

(25,18)

(23,33)

(21,28)

(9,32)(11,38)

(19,36)

(10,34)

(13,22)

6.458 10.08 2.939 2.745 5.451 3.287 10.86 5.322 5.070 4.562 3.463 3.587 3.232 6.260

jcx

jx

Identifying Boundary of Different Classes of Objects

Boundary Identified

The Vector Space Model

• In the vector space model, each object is described by a number of numerical attributes.

• For example, the outlook of a man is described by his height, weight, and age.

Transformation of Categorical Attributes into Numerical

Attributes

• Represent the attribute values of the object in a binary table form as exemplified in the following:

10003

00112

01001School Graduate

Education

College

Education

School High

Education

Female

Male/Objects

• Assign appropriate weight to each column.

• Treat the weighted vector of each row as the feature vector of the corresponding object.

4

21

3

0003

002

0001School Graduate

Education

College

Education

School High

Education

Female

Male/Objects

Application of Data Classification in Bioinformatics

• Data classification has been applied to predict the function and tertiary structure of a protein sequence.

Basics of Protein Structures

• A typical protein consists of hundreds to thousands of amino acids.

• There are 20 basic amino acids, each of which is denoted by one English character.

Three-dimensional Structure of Myoglobin

Source: Lectures of BioInfo by yukijuan

Prediction of Protein Functions and Tertiary Structures

• Given a protein sequence, biochemists are interested in its functions and its tertiary structure.

• The PDB database, which collects proteins with verified tertiary structures, contains ~19,000 proteins.

• The SWISSPROT database, which collects proteins with verified functions, contains ~110,000 proteins.

• The PIR-PSD database, which collects proteins with verified functions, contains ~280,000 proteins.

• The PIR-NREF database, which collects all protein sequences, contains ~1,060,000 proteins.

Problem Definition of Kernel Smoothing

• Given the values of function at a set of samples . We want to find a set of symmetric kernel functions and the corresponding weights such that

msssS ,...,, 21)(f

),;( ii bcK

iw

).(),;()(ˆ fbcKwf iii

i

Kernel Smoothing with the Spherical Gaussian Functions

• Hartman et al. showed that a linear combination of spherical Gaussian functions can approximate any function with arbitrarily small error.

• “Layered neural networks with Gaussian hidden units as universal approximations”, Neural Computation, Vol. 2, No. 2, 1990.

• With the Gaussian kernel functions, we want to find such that

).(2

exp)(ˆ2

2

fwfi

i

ii

iiiw and ,,

Problem Definition of Kernel Density Estimation

• Assume that we are given a set of samples taken from a probability distribution in a d-dimensional vector space. The problem now is how to find a linear combination of kernel functions that approximate the probability density function of the distribution?

)(f

• The value of the probability density function at a vector can be estimated as follows:

where n is the total number of samples, is the distance between vector and its k-th nearest samples, and

is the volume of a sphere with radius =

in a d-dimensional vector space.

v

,)1

2(

)()(

1

2

dvR

n

kvf

dd

)(vRv

)12

(

)( 2

d

vRd

d )(vR

A 1-D Example of Kernel Smoothing with the Spherical

Gaussian Functions

).(2

exp)(ˆ2

2

fwfi

i

ii

2

2

2exp

i

iiw

The Existing Approaches for Kernel Smoothing with Spherical Gauss

ian Functions

• One conventional approach is to place one Gaussian function at each sample. As a result, the problem becomes how to find

for each sample such that

).(2

exp)(ˆ2

2

f

swf

i

i

ii

iiw and is

• The most widely-used objective is to minimize

where are test samples and S is the set of training samples.

• The conventional approach suffers high time complexity, approaching , due to the need to compute the inverse of a matrix.

,))()(ˆ(1

2

1

2

S

iii

n

j

wff jj vv

nvvv ,...,, 21

)(3

SO

• M. Orr proposed a number of approaches to reduce the number of units in the hidden layer of the RBF network.

• Beatson et. al. proposed O(nlogn) learning algorithms using polyharmonic spline functions.

An O(n) Algorithm for Kernel Smoothing

• In the proposed learning algorithm, we assume uniform sampling. That is, samples are located at the crosses of an evenly-spaced grid in the d-dimensional vector space. Let denote the distance between two adjacent samples.

• If the assumption of uniform sampling does not hold, then some sort of interpolation can be conducted to obtain the approximate function values at the crosses of the grid.

A 2-D Example of Uniform Sampling

The Basic Idea of the O(n) Kernel Smoothing Algorithm

• Under the assumption that the sampling density is sufficiently high, i.e. , we have the function values at a sample and its k nearest samples, , are virtually equal. That is, .

• In other words, is virtually a constant function equal to in the proximity of

0

hs

ksss ,..., 21

)(...)()()( 21 kh sfsfsfsf

.hs

)(xf

)( hsf

• Accordingly, we can expect that

.......

;...

21

21

hk

hk wwww

A 1-D Example

)(xf

)()( hfxf

2

kh )1( h h )1( h

2

kh

h

• In the 1-D example, samples at located at

, where i is an integer.

• Under the assumption that , we have

and

• The issue now is to find appropriate and

such that

.......

;......

)2/(1)2/()2/(

)2/(1)2/()2/(

khhkhkh

khhkhkh wwww

hw

h

).(2

)(exp

)2/(

)2/(2

2

hfix

wkh

khi hh

0 hxhfxf for )()(

i

• If we set ,then we have h

large.ly sufficient is if,5066.2)2/(

)2/(

2

12

2

kekh

khi

ix

2

2

2

1

ix

e

)( hf

• Therefore, with , we can set

and obtain for

,5066.2

)( hfwh

).(2

)(exp)(

)2/(

)2/(2

2

hfix

wxgkh

khi hh

hx

h

• In fact, it can be shown that with ,

is bounded by

• Therefore, we have the following function approximator:

h

i h

ix2

2

2

)(exp

.1035.15066282745.2 8

.2

)(exp)(

5066282745.2

1)(ˆ

])2

1(,)

2

1((number reala for

)2/(

)2/(2

2

kh

khi

ixifxf

hhx

Generalization of the 1-D Kernel Smoothing Function

• We can generalize the result by setting , where is a real number.

• The table on the next page shows the bounds of

with various values.

n...21

j

jx22

2

2

)(exp

Bounds of

0.5

1.0

1.5

j

jx22

2

2

)(exp

2108.1253.1 81034.15066282745.2

111094.23397599424119.3

An Example of the Effect of Different Setting of β

2

2

5.02

1

jx

e

j

jx2

2

5.02

)(exp

The Smoothing Effect

• The kernel smoothing function is actually a weighted average of the sampled function values. Therefore, selecting a larger value implies that the smoothing effect will be more significant.

• Our suggestion is set

.1

An Example of the Smoothing Effect

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1targetfit

The smoothing effect Elimination of the smoothing effect with a compensation procedure

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1targetfit

The General Form of a Kernel Smoothing Function in the Multi-

Dimensional Vector Space

• Under the assumption that the sampling density is sufficiently high, i.e. , we have the function values at a sample and its k nearest samples, , are virtually equal. That is, .

hs

ksss ,..., 21

)(...)()()( 21 kh sfsfsfsf

0

• As a result, we can expect that

where are the weights and bandwidths of the Gaussian functions located at , respectively.

,... and ... 2121 khkh wwww

kkwww ,...,, and ,...,, 2121

ksss ,..., 21

• Since the influence of a Gaussian function decreases exponentially as the distance increases, we can set k to a value such that, for a vector in the proximity of sample , we have

hsv

.2

exp2

exp

2exp)(ˆ

12

2

2

2

2

2

k

j j

j

jh

hh

i

i

Ssi

sw

sw

swf

mi

• Since we have

our objective is to find and such that

,... and ... 2121 khkh wwww

hw h

).(

2exp

2exp

2exp

2exp

2exp)(ˆ

12

2

2

2

12

2

2

2

2

2

vf

sw

sw

sw

sw

swf

k

j h

jh

h

hh

k

j j

jj

h

hh

i

i

Ssi

mi

• Let

Then, we have

.

2

,...,,exp...)(

2

2

21

21

dj h

d

jjh

jjjwg

).(ˆ2

exp2

exp)(1

2

2

2

2

vfs

ws

wvgk

j h

j

hh

hh

.101.35452.50662827by bounded is

2

,...,,exp... have wey,Accordingl

.101.35452.50662827by bounded is 2

)(exp

then, toset weIf

).,...,( where

,2

)(exp...

2

)(exp

2

,...,,exp...

have We

d8-

2

2

21

8-2

211

21

2

2

2

211

2

2

21

21

1

1

21

d

d

d

j h

d

jj

j h

h

d

j h

dd

j h

j h

d

jj

jjj

jx

σ

xxxv

jxjx

jjj

• Therefore, with ,

is virtually a constant function and

• Accordingly, we want to set

h

)(vg

. ofproximity in the for

,5066282745.2)()(ˆ

h

hd

sv

wvgvf

dh

h

sfw

5066282745.2

)(

• Finally, by setting uniformly to , we obtain the following kernel smoothing function that approximates f(v):

.at locatedobject the

of samplesnearest theare ,..,, where

,2

exp)5066282745.2(

)()(ˆ

21

ˆ2

2

v

ksss

ssff

k

s

id

i

i

,,...,2,1 , nii

• Generally speaking, if we set uniformly to , we will obtain

.2

exp

and at locatedobject the

of samplesnearest theare ,..,, where

,2

exp)(1

)(ˆ

2

2

21

22

2

j

k

s

iid

j

v

ksss

ssff

i

,,...,2,1 , nii

Application in Data Classification

• One of the applications of the RBF network is data classification.

• However, recent development in data classification focuses on the support vector machines (SVM), due to accuracy concern.

• In this paper, we propose a RBF network based data classifier that can delivers the same level of accuracy as the SVM and enjoys some advantages.

The Proposed RBF Network Based Classifier

• The proposed algorithm constructs one RBF network for approximating the probability density function of one class of objects based on the kernel smoothing algorithm that we just presented.

The Proposed Kernel Density Estimation Algorithm for Data

Classification• Classification of a new object is

conducted based on the likelihood function:

objects. -class of

functiondensity y probabilit eapproximat theis )(ˆ and

ly,respective classes, all of samples trainingofnumber total theand

class of samples trainingofnumber theare and where

),(ˆ)(

m

f

mSS

vfS

SvL

m

m

mm

m

• Let us adopt the following estimation of the value of the probability density function at each training sample:

space. vector theof dimension theis (3)

class;

same theof samples ainingnearest tr its and

sample trainingbetween distance theis )( (2)

objects; -class

of functiondensity y probabilit theis )( (1)

where,)1

2(

)(1)(

th

1

2

d

ks

sR

m

f

dsR

S

ksf

i

ik

m

dd

ik

mim

• In the kernel smoothing problem, we set the bandwidth of each Gaussian function uniformly to , where is the distance between two adjacent training samples.

• In the kernel density estimation problem,

for each training sample, we need to determine the average distance between two adjacent training samples of the same class in the local region.

• In the d-dimensional vector space, if the average distance between samples is ,

then the number of samples in a subspace

of volume V is approximately equal to

• Accordingly, we can estimate by

i

.di

V

i

.

)12

()1(

)(ˆ

d

iki

dk

sR

• Accordingly, with the kernel smoothing function that we obtain earlier, we have the following approximate probability density function for class-m objects:

.2

exp and,

)12

()1(

)(

,at locatedobject theof samples training

-classnearest theare ,...,, where

,2

exp1

)(ˆ

2

2

21

2

2

1

jd

ikii

k

i

i

dk

i imm

j

dk

sR

v

mksss

s

Sf

• An interesting observation is that, regardless of the value of ,

we have .

• If the observation holds generally, then

i

i

2

.2

exp2

11)(ˆ

2

2

1

i

i

dk

i imm

s

Sf

• In the discussion above, is defined to be the distance between sample and its nearest training sample.

• However, this definition depends on only one single sample and tends to be unreliable, if the data set is noisy.

• We can replace with)( ik sR

is thk

. as class same theof

samples ainingnearest tr theare ,...,, where

,11

)(

21

1

2

i

k

k

jijik

s

ksss

sskd

dsR

)( ik sR

Parameter Tuning

• The discussions so far are based on the assumption that the sampling density is sufficiently high, which may not hold for some real data sets.

• Three parameters, namely d’, k’, k”,are incorporated in the learning algorithm:

.2

exp2

11)(ˆ

2

2

1

i

i

dk

i imm

s

Sf

. as class same theof

samples ainingnearest tr theare ,...,, where

,11

)(

21

1

2

i

k

k

jijik

s

ksss

sskd

dsR

• One may wonder how should be set.

• According to our experimental results, the value of has essentially no effect, as long as is set to a value within .

i

i

]2,6.0[

Time Complexity

• The average time complexity to construct a RBF network is

if the k-d tree structure is employed, where n is the number of training samples.

• The time complexity to classify c new objects with unknown class is

),loglog( nnkndnO

).loglog( nckndnO

Comparison of Classification Accuracy of the 6 Smaller Data

SetsData sets classification algorithms

proposed algorithm SVM 1NN 3NN

1. iris (150) 97.33 (k’ = 24, k” = 14, d’ = 5, = 0.7)

97.33 94.0 94.67

2. wine (178) 99.44 (k’ = 3, k” = 16, d’ = 1, = 0.7) 99.44 96.08 94.97

3. vowel (528) 99.62 (k’ = 15, k” = 1, d’ = 1, = 0.7) 99.05 99.43 97.16

4. segment (2310) 97.27 (k’ = 25, k” = 1, d’ = 1, = 0.7) 97.40 96.84 95.98

Avg. 1-4 98.42 98.31 96.59 95.70

5. glass (214) 75.74 (k’ = 9, k” = 3, d’ = 2, = 0.7) 71.50 69.65 72.45

6. vehicle (846) 73.53 (k’ = 13, k” = 8, d’ = 2, = 0.7) 86.64 70.45 71.98

Avg. 1-6 90.49 91.89 87.74 87.87

Data sets classification algorithms

proposed algorithm SVM 1NN 3NN

7. satimage(4435,2000)

92.30 (k’ = 6, k” = 26, d’ = 1, = 0.7) 91.30 89.35 90.6

8. letter(15000,5000)

97.12 (k’ = 28, k” = 28, d’ = 2, = 0.7)

97.98 95.26 95.46

9. shuttle(43500,14500)

99.94 (k’ = 18, k” = 1, d’ = 3, = 0.7) 99.92 99.91 99.92

Avg. 7-9 96.45 96.40 94.84 95.33

Comparison of Classification Accuracy of the 3 Larger Data

Sets

Data Reduction

• As the proposed learning algorithm is instance-based, removal of redundant training samples will lower the complexity of the RBF network.

• The effect of a naïve data reduction mechanism was studied.

• The naïve mechanism removes a training sample, if all of its 10 nearest samples belong to the same class as this particular sample.

Effect of Data Reductionsatimage letter shuttle

# of training samples in the original data set

4435 15000 43500

# of training samples after data reduction is applied

1815 7794 627

% of training samples remaining

40.92% 51.96% 1.44%

Classification accuracy after data reduction is applied

92.15% 96.18% 99.32%

Degradation of accuracy due to data reduction

-0.15% -0.94% -0.62%

# of training samples after data reduction is applied

# of support vectors identified by LIBSVM

satimage 1815 1689

letter 7794 8931

shuttle 627 287

Execution Times(in seconds)

Proposed algorithm without data reduction

Proposed algorithm with data reduction

SVM

Cross validation satimage 670 265 64622

letter 2825 1724 386814

shuttle 96795 59.9 467825

Make classifier satimage 5.91 0.85 21.66

letter 17.05 6.48 282.05

shuttle 1745 0.69 129.84

Test satimage 21.3 7.4 11.53

letter 128.6 51.74 94.91

shuttle 996.1 5.85 2.13

Conclusions

• A novel learning algorithm for data classification with the RBF network is proposed.

• The proposed RBF network based data classification algorithm delivers the same level of accuracy as the SVM.

• The time complexity for constructing a RBF network based on the proposed algorithm is

, which is much lower than that required by the SVM.

• The proposed RBF network based classifier can handle data sets with multiple classes directly.

• It is of interest to develop some sort of data reduction mechanisms.

)loglog( nnkndnO

• The powerpoint file of this presentation can be downloaded from

syslab.csie.ntu.edu.tw

• An extended version of the presented paper can be downloaded at the same address.