J. Biomedical Science and Engineering, 2009, 2, 323-335
doi: 10.4236/jbise.2009.25048 Published Online September 2009 (http://www.SciRP.org/journal/jbise/
Published Online September 2009 in SciRes. http://www.scirp.org/journal/jbise
CANFISa computer aided diagnostic tool for cancer
Latha Parthiban1, R. Subramanian2
1Research Scholar in Pondicherry University and working in Aarupadai Veedu Institute of Technology, Chennai, India; 2Department
of Computer Science, Pondicherry University, Puducherry, India.
Email: 1lathaparthiban@yahoo.com, 2subbur@yahoo.com
Received 5 November 2008; revised 20 May 2009; accepted 25 May 2009.
In this investigation, an approach using Coac-
tive Neuro-Fuzzy Inference System (CANFIS) as
diagnosis system for breast cancer has been
proposed on Wisconsin Breast Cancer Data
(WBCD). It is occasionally difficult to attain the
ultimate diagnosis even for medical experts due
to the complexity and non-linearity of the rela-
tionships between the large measured factors,
which can be possibly resolved with a human
like decision-making process using Artificial
Intelligence (AI) algorithms. CANFIS is an AI
algorithm which has the advantages of both
fuzzy inference system and neural networks and
can deal with ambiguous data and learn from
the past data by itself. The Multi Layer Percep-
tron Neural Network (MLPNN), Probabilistic
Neural Network (PNN) Principal Component
Analysis (PCA), Support Vector Machine (SVM)
and Self Organizing Map (SOM) were also tested
and benchmarked for their performance on the
classification of the WBCD.
Keywords: Neural Network; Coactive Neuro-Fuzzy
Inference Systems; Probabilistic Neural Network;
Principal Component Analysis; Stern Series; Wis-
consin Breast Cancer Data
Breast cancer is the most common tumor-related disease
among women throughout the world, and the mortality
rate caused by breast cancer is dramatically increasing.
The etiologies of breast cancer remain unclear and no
single dominant cause has emerged [1]. Preventive way
is still a mystery and the only way to help patients to
survive is by early detection. If the cancerous cells are
detected before spreading to other organs, the survival
rate for patient is more than 97% [25] which is the mo-
tivation factor to develop this automated diagnostic tool.
Again, a major class of problems in medical science in-
volves the diagnosis of disease, based upon several tests
performed upon the patient and this has given rise, over
the past few decades, to computerized diagnostic tools,
intended to aid the physician in making sense out of the
confusing data [2].
There have been substantial previous research works
with WBCD database to achieve an automatic ultimate
diagnostic system. Genetic Algorithm (GA) [2,13],
Fuzzy Inference Systems(FIS) [3,4], Neural Networks
(NN) [5,12], Adaptive Boosting (AdaBoost) [6,10,11]
and Neuro-Fuzzy Hybrid Models [4,8] have been ap-
plied to this problem. The performances of each infer-
ence system were evaluated by calculating the degree of
correctness in predicted results against diagnosed results
represented as PPV (Positive Predicted Value). Each
system shows the PPV within the range from less than
60% (AdaBoost) up to over 95% (Neuro-Fuzzy Hybrid
Models). Among those algorithms, Neuro-Fuzzy Hybrid
models provide relatively remarkable performances in
diagnosis. Those models are the combination of Neural
Networks and Fuzzy Inference Systems encouraging the
advantages and resolving the drawbacks of both NNs
and FIS models.
For our experiments, after preprocessing and cleaning
of clinical data from mammogram, a modified method of
using CANFIS [8,9] was applied to attain the ultimate
diagnosis as being either benign or malignant. In order
to find the neural network model with the highest accu-
racy for classification of the WBCD, we implemented
five types of classifiers: Multi Layer Perceptron Neural
Network (MLPNN), Probabilistic Neural Network
(PNN), Principal Component Analysis (PCA), Support
Vector Machine (SVM) and Self Organizing Map
(SOM). To achieve fast training, the weights of these
classifiers were initialized with Stern series. In applying
CANFIS and other artificial intelligent algorithm, the
required system size is changed in proportion to the size
324 L. Parthiban et al. / J. Biomedical Science and Engineering 2 (2009) 323-335
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of the inference system such as the number of inputs, the
number of internal nodes and the number of learning
iteration. Among those critical factors of the inference
system, the number of internal nodes and learning itera-
tion are changeable only in the process of designing the
system. Therefore, methods for reducing the number of
input factors within range of not losing the accuracy of
diagnosis were considered. For this purpose, we used
two methods, which are decision tree [14] and the corre-
lation coefficient between the individual inputs and the
test diagnosis results. The overall process flow chart is
presented in Figure 1.
The rest of this paper is organized as follows: In
Section 2, we briefly describe the WBCD data, data
cleaning and feature extraction, dominant input selec-
tion methods, a review of the classifiers that are con-
sidered and initializing the weight of neural networks
with Stern series are presented. In Section 3, we pre-
sent the diagnostic system using CANFIS. In Section 4,
experimental results of the classifiers trained on com-
posite features of the WBCD data is compared in terms
of positive predicted value and discussion of the pre-
sented results is provided in the light of existing stud-
ies in the literature. In Section 5, we highlight the re-
sults of the paper and finally, in Section 6, we con-
clude the paper.
2.1. Wisconsin Breast Cancer Database
The Wisconsin Breast Cancer Diagnosis (WBCD) data-
base is the result of efforts provided by the University of
Wisconsin Hospital based on microscopic examination
of breast masses with fine needle aspirate tests. Fine
needle aspiration of breast masses is a common non-
invasive diagnostic test that obtains information needed
to evaluate malignancy [3,15].
Figure 1. Overall processes flowchart.
Masses often characterize as early breast cancer be-
fore it is palpable by a woman or a physician [1]. It has
its own characteristics and may be used as a clue to clas-
sify them. Masses can be circumscribed, speculated (sat-
ellite), lobulated or ill-defined (irregular) and radiologist
will need to take a good look at its texture information,
statistical descriptions as well as the background tissue
from mammography images. Therefore, in order to clas-
sify a mass, the characteristics or attributes recorded
were used as an input features. There are nine criteria
recorded for masses in this dataset represented as a 1-10
integer value. (Table 1).
The database (Table 2) itself contains 699 cases, with
65.5% classified as benign and 34.5% as malignant. The
diagnostics do not provide any information about the
degree of benignity or malignancy. In considering the
relationship between the measured values and the diag-
nostics, there are almost no relationships which stand out.
Therefore, there is no convenient and effective method
to attain the ultimate diagnostics with this original data
even for the specialists.
In addition to that, there may be the possibility that
one or more of the measured pieces do not affect the
diagnosis result. These are the reasons that artificial in-
telligent system can be used as an expert to assist the
specialist in diagnosing the disease correctly.
2.2. Data Cleaning and Feature Extraction
Preliminary examination on dataset chosen is compul-
sory before cleaning process takes place. By knowing
the relationship between attributes, and how strongly
they depend on each other, data quality and evaluation
can be found easily. The discovery of data relationship
Table 1. Dataset features.
Criteria Integer Value
Clump Thickness X1
Uniformity of Cell Size X2
Uniformity of Cell Shape X3
Marginal Adhesion X4
Single Epithelial Cell Size X5
Bare Nuclei X6
Bland Chromatin X7
Normal Nucleoli X8
Mitosis X9
Clinical data from
Mammogram Preprocess data
Table 2. WBCD database.
Case X1X2X3……... X9 Diagnostics
1 5 1 1 …….. 1 Benign
2 5 4 4 …….. 1 Benign
… .. .. .. …….. .. …
… .. .. .. …….. .. …
6994 8 8 …….. 1 Malignant
All attributes
Neural Network
model for data
Dominant input
selection methods
Classification Result
L. Parthiban et al. / J. Biomedical Science and Engineering 2 (2009) 323-335 325
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might lead to data cleaning rules, and therefore it can
suggest improvements to its constraints. These are car-
ried out by analyzing patterns between the attributes,
using statistical tools like bivariate or multivariate
Therefore, two different cleaning processes have been
carried out on the dataset. Dataset named Set A will only
eliminate records with missing value and outliers, with
the hypothesis that medical data are best not to be tam-
pered or changed. While data in Set B will undergo
normal statistical cleaning process where all attributes
must be distributed normally. Therefore, data in Set B
has been changed many times to fulfill the normal dis-
tribution functions
Using Neural Connection 2.0, 180 simulations was
carried out for both dataset to test which data set is the
best. As shown in Figure 2, set A gives 100% as the
highest accuracy percentage (AP) and the smallest root
mean squared (RMS) error is only 0.02751. As com-
pared to set B, the highest AP is only 83.36% with
smallest RMS error is 0.21002. It is proven that our hy-
pothesis is true and therefore, set A will be used as an
input database.
After data cleaning, the data in the WBCD database
was divided into two sets: training and testing datasets.
There are 444 benign and 238 malignant cases in the
database. The training dataset constitutes 50% of WBCD
database taken in order and the remaining is testing
dataset. The training dataset was also used to figure out
the most effective and dominant inputs of the inference
system and the result with dominant inputs using deci-
sion tree and correlation coefficient computation was
tested for correctness verification of the output.
2.3. Dominant Input Selection Methods
2.3.1. Decision Tree
The input recommender used in our experiment was a
decision tree learning algorithm using SAS9™ package.
The decision tree construction algorithms generate deci-
sion trees from a set D of cases. These algorithms parti-
tion the data set D into subsets D1, D2, ……DM by a set
of tests X with mutually outcomes X1, X2 ,……. XM,
where Dv contains those cases that have outcome Xi. A
decision tree (DT) is known as a good classifier of huge
data. It classifies input data by partitioning example
spaces with entropy calculation. DT is especially useful
when dataset is represented by attribute-value pairs and
the target function has discrete output value. In our ex-
periment, a binary decision tree was constructed to select
dominant inputs. In each node of the tree, the most use-
ful attribute for classifying whole data is selected by
calculating the information gain (measure for deciding
the relevance of an attribute) G (D, X) of an attribute X,
Figure 2. Simulations results of data Set A and Set B.
326 L. Parthiban et al. / J. Biomedical Science and Engineering 2 (2009) 323-335
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Figure 3. DT dominant input selection.
relative to a collection of examples X, with following
where pi denotes the proportion of classes in D that belong
to the ith class, Va l u e s ( X ) is the set of all possible values
for attribute X, Dv is a subset of D for which attribute X
has a value v and. |Dv| represents cardinality of Dv
dataset. (i.e., D={d D | X(d) = v}). The first term in
the equation for Gain is just the entropy of the original
collection D and the second term is the expected value of
the entropy after D is partitioned using attribute X. The
expected entropy described by this second term is simply
the sum of the entropies of each subset Dv, weighted by
the fraction of examples |Dv|/|D| that belong to Dv. Gain
(D,X) is therefore the expected reduction in entropy
caused by knowing the value of attribute X. Alternatively,
Gain(D,X) is the information provided about the target
attribute value, given the value of some other attribute X.
The DT is constructed in a way to reduce the entropy
with the attribute which has the highest gain value at
each node. Through this way, the final DT model has the
most useful measured data on the top node, next useful
one on the right node of the top node and so on. The
input selection process derived by DT is presented in
Figure 3.
2.3.2. Correlation Coefficient Computation
An efficient method for dominant input selecting process
is calculating correlation coefficients between each
measured input data and the diagnosis results. The Cor-
relation Coefficient is a numerical measure of the degree
of the linear relationship between two variables. The
value of the correlation coefficient always falls between
–1 and 1. A positive correlation coefficient indicates a
direct relationship, and a negative correlation coefficient
indicates an inverse relationship between two variables.
In the calculation, first it is possible to assume that all
the correlation coefficients by calculating with data in
WBCD should be positive. Then we selected four meas-
ured input data from the one which has the highest cor-
relation coefficient for diagnosis system. The correlation
coefficient indicates the degree of linear relationship
between two variables. The correlation coefficient al-
ways lies between -1 and +1. -1 indicates perfect linear
negative relationship between two variables, +1 indi-
cates perfect positive linear relationship and 0 indicates
lack of any linear relationship The correlation coeffi-
cient(ρij ) can be calculated by the following formula.
)(cov ,
ij XX
 (1)
)(,)( 22
where Cij is the covariance of Xi and Xj,
is the
mean and σ(Xi) is the standard deviation of Xi and σ(Xj)
is the standard deviation of Xj .
Each result by the correlation coefficient calculation
between each measured input and the correct output
indicates the degree of linear relationship between
them. In this procedure, the selected input features are
possibly said to have more linear relationships so that
they affects the results. Therefore, the inputs highly
correlated with the output were selected as dominant
inputs in our experiment. The result of dominant input
selection by the correlation coefficient calculation is
given in Figure 4.
L. Parthiban et al. / J. Biomedical Science and Engineering 2 (2009) 323-335 327
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Figure 4. Correlation coefficient output.
2.4. Brief Review of Different Classifiers
1) Multilayer Perceptron Neural Network
The MLPNNs are the most commonly used neural-
network architectures since they have features such as
the ability to learn and generalize smaller training-set
requirements, fast operation, and ease of implementation.
One major property of these networks is their ability to
find nonlinear surfaces separating the underlying pat-
terns, which is generally considered as an improvement
on conventional methods. The MLPNN is a nonparamet-
ric technique for performing a wide variety of detection
and estimation tasks [12,17,18,19]. Figure 5 shows the
architecture of MLPNN. There is one neuron in the input
layer for each predictor variable. In the case of categori-
cal variables, N-1 neurons are used to represent the N
categories of the variable.
Input Layer—A vector of predictor variable values
(x1...xp) is presented to the input layer. The input layer
(or processing before the input layer) standardizes these
values so that the range of each variable is -1 to 1. The
input layer distributes the values to each of the neurons
in the hidden layer. In addition to the predictor variables,
there is a constant input of 1.0, called the bias that is fed
to each of the hidden layers; the bias is multiplied by a
weight and added to the sum going into the neuron.
Hidden Layer—Arriving at a neuron in the hidden
layer, the value from each input neuron is multiplied by
a weight (wji), and the resulting weighted values are
added together producing a combined value uj. The
weighted sum (uj) is fed into a transfer function, σ,
which outputs a value hj. The outputs from the hidden
layer are distributed to the output layer.
Output Layer-Arriving at a neuron in the output layer,
the value from each hidden layer neuron is multiplied by
a weight (wkj), and the resulting weighted values are
added together producing a combined value vj. The
weighted sum (vj) is fed into a transfer function, σ,
which outputs a value yk. The y values are the outputs of
the network.
The algorithm for the MLPNN is given below. It re-
quires the units to have thresholding non linear functions
that are continuously differentiable, i.e., smooth every-
where. A sigmoid function f (net)=1/(1+e-knet), is used,
since it has a simple derivative. All training and testing
data were normalized.
Initialize weights and thresholds:
Set all weights and thresholds to small random vari-
Present input and desired output:
Present input: XP = x1, x2,…………… xp
Target output: YP = y1,………… ym
where, p is the number of input nodes and m is the num-
ber of output nodes. Set w0 to be -θ, the bias, and x1 to be
always 1.
For pattern association, XP and YP represent the pat-
terns to be associated. For classification, YP is set to zero
except for one element set to 1 that corresponds to the
class the XP is in.
Calculation of actual output:
Each layer calculates and passes it
as input to the next layer. The final layer outputs values
Op j and passes that as input to the next layer.
xipi i
Adapt weights (start from control layer, and work
wi j(t + 1) = wi j(t) + η δpjOp j
where, wi j (t) represents the weights from node i to node
j at time t, η is a gain term, and δp j is an error term for
pattern p on node j.
For output units: δ p j = k Op j (1 -Op j) (tp j -Op j)
For hidden units: δp j = kOp j (1 -Op j) δ Pk wjk
where, the sum is over the k nodes in the layer above
node j. The stopping condition may be weight change,
number of epochs, and so on.
The main issues involved in designing and training a
MLPNN are selecting how many hidden layers to use in
the network, deciding how many neurons to use in each
hidden layer, finding a globally optimal solution that
avoids local minima, converging to an optimal solution
in a reasonable period of time and validating the neural
network to test for overfitting.
2) Probabilistic Neural Network
The PNN introduced by Specht [20] is essentially
based on the well-known Bayesian classifier technique
commonly used in many classical pattern-recognition
problems. Consider a pattern vector
with m dimen-
sions that belongs to one of two categories K1 and K2.
Let F1(x) and F2(x) be the probability density functions
(pdf) for the classification categories K1 and K2, respect
328 L. Parthiban et al. / J. Biomedical Science and Engineering 2 (2009) 323-335
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Figure 5. Architecture of MLPNN.
Figure 6. Architecture of PNN.
tively. From Bayes’ discriminant decision rule,
longs to K1 if
xF (3)
belongs to K2 if
xF (4)
where L1 is the loss or cost function associated with
misclassifying the vector as belonging to category K1
while it belongs to category K2, L2 is the loss function
associated with misclassifying the vector as belonging to
category K2 while it belongs to category K1, P1 is the
prior probability of occurrence of category K1, and P2 is
the prior probability of occurrence of category K2. In
many situations, the loss functions and the prior prob-
abilities can be considered equal. Hence the key to using
the decision rules given by Eq.3 and 4 is to estimate the
probability density functions from the training patterns.
The PNN architecture (Figure 6) is composed of
many interconnected processing units or neurons organ-
ized in successive layers. The input layer unit does not
perform any computation and simply distributes the in-
put to the neurons in the pattern layer. On receiving a
from the input layer, the neuron xij of the
pattern layer computes its output using
L. Parthiban et al. / J. Biomedical Science and Engineering 2 (2009) 323-335 329
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/2 2
() (2 )2
ij ij
ij dd
xx xx
x exp
 
where d denotes the dimension of the pattern vector x, σ
is the smoothing parameter, and xij is the neuron vector
For two class problem, if input units is assumed to be x1
to xn, Pattern units be Class A (ZA1 to ZAj) and Class B (ZB1
to ZBj), Summation units be fA and fB and output unit y, then
the training algorithm for the probabilistic neural net is
Step 1: For each training input pattern, x(p), p=1,……P
perform Steps 2-3.
Step 2: Create pattern unit Zp: Weight vector for unit Zp:
wp=x(p) (unit Zp is either a ZA unit or ZB unit)
Step 3: Connect the pattern unit to summation unit. If
x(p) belongs to Class A, connect pattern unit Zp
to summation unit SA. Else, connect pattern unit
Zp to summation unit SB.
The application algorithm for classifying is given as
Step 1: Initialize weights from training algorithm.
Step 2: For input pattern to be classified, do Steps 3–5.
Step 3: Patterns units:
Calculate net input,
Zinj = x.wj=xTwj
Calculate the output
Step 4: Summation units
The weights used by the summation unit for
Class B is,
Step 5: Output unit:
It sums the signals from fA and fB. Input vector is classi-
fied as Class A if the total input to decision unit is positive.
The main advantage of PNN compared to MLPNN is
faster to train a PNN network than a MLPNN.
more accurate than MLPNN.
insensitive to outliers (wild points).
generate accurate predicted target probability
approach Bayes optimal classification.
slower than MLPNN at classifying new cases.
require more memory space to store the model
3) Support Vector Machine
The SVM proposed by Vapnik [22] has been studied
extensively for classification, regression, and density
estimation. The SVM is a binary classifier and it maps
the input patterns into a higher dimensional feature space
through some nonlinear mapping chosen a priori. A lin-
ear decision surface is then constructed in this high-di-
mensional-feature space. Thus, SVM is a linear classifier
in the parameter space, but it becomes a nonlinear clas-
sifier as a result of the nonlinear mapping of the space of
the input patterns into the high-dimensional feature
space. Training the SVM is a quadratic-optimization
problem. SVM has been shown to provide high- gener-
alization ability. A proper kernel function for a certain
problem is dependent on the specific data and till now
there is no good method on how to choose a kernel func-
tion [22,23]. In this paper, the choice of the kernel func-
tions was studied empirically and optimal results were
achieved using radial-basis function (RBF) kernel function.
SVMs are free of optimization headaches of neural
networks because they present a convex programming
problem, and guarantee finding a global solution. They
are much faster to evaluate than density estimators, be-
cause they make use of only relevant data points, rather
than looping over each point regardless of its relevance
to the decision boundary.
4) Self Organizing Maps
Self-organizing maps learn to classify input vectors
according to how they are grouped in the input space.
Feature maps allocate more neurons to recognize parts of
the input space where many input vectors occur and al-
locate fewer neurons to parts of the input space where
few input vectors occur. SOM also learn the topology of
their input vectors. Neurons next to each other in the
network learn to respond to similar vectors. The layer of
neurons can be imagined to be a rubber net that is
stretched over the regions in the input space where input
vectors occur. SOM allow neurons that are neighbors to
the winning neuron to output values. Thus the transition
of output vectors is much smoother than that obtained
with competitive layers, where only one neuron has an
output at a time.
Initially, the weight and learning rate are set. The in-
put vectors to be clustered are presented to the network.
Once the input vectors are given, based on initial
weights, the winner unit is calculated either by Euclid-
ean distance method or sum of products method. An
epoch is said to be completed once all the input vectors
are presented to the network. By updating the learning
rate, several epochs of training may be performed. The
training algorithm for SOM is as below
Step 1: Set topological neighborhood parameters
Set learning rate, initialize weights.
Step 2: While stopping condition is false do Steps 3-9
Step 3: For each input vector x, do Steps 4-6.
Step 4: For each j, compute squared Euclidean distance.
D(j)=(Wij-Xi)2 i =1 to n and j = 1 to m
Step 5: Find index J, when D(j) is minimum.
Step 6: For all units J with the specified neighbourhood
of J, for all i, update the weights.
Wij(new) = Wij(old) + α[xi-Wij(old)]
Step 7: Update the learning rate.
Step 8: Reduce the radius of topological neighborhood at
specified times.
Step 9: Test the stopping condition.
330 L. Parthiban et al. / J. Biomedical Science and Engineering 2 (2009) 323-335
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The map formation occurs in two phases: required training patterns and a reduction in the training
times of the classifier. Problem with linear PCA net-
works is evident when the input data contains outliers.
Outliers are individual pieces of data that are far re-
moved from the data clusters (i.e., noise). They tend to
distort the estimation of the eigenvectors and create
skewed data projections.
a) Initial formation of perfect (correct) order
b) Final convergence.
The second phase takes a longer duration than the first
phase and requires a small value of learning rate. The
learning rate is a slowly decreasing function of time and
the radius of the neighborhood around a cluster unit also
decreases as the clustering process goes on. The initial
weights are assumed with random values. The learning
rate is updated by α(t+1) = 0.5α(t).
5) Principal Component Analysis
PCA network combine unsupervised and supervised
learning in the same topology. PCA is an unsupervised
linear procedure that finds a set of uncorrelated features,
principal components, from the input. A MLP is super-
vised to perform the nonlinear classification from these
components. PCA is a technique that finds an orthogonal
set of directions in the input space and provides a way to
find the projections into these directions in an ordered
fashion. The orthogonal directions are called eigen vec-
tors of the correlation matrix of the input vector and the
projections of the corresponding eigen values.
a) Ordering of the principal components
PCA must transform the input samples into a new space
(the feature space) such that the information about the
samples is kept, but the dimensionality is reduced. From
the input space, it finds an orthogonal set of P directions
where the input data has the largest energy, and extracts P
projections from these directions in an ordered fashion.
The first principal component is the projection, which has
the largest value (think of the projections as the shadow of
the data clusters in each direction as in Figure 7), while
the Pth principal component has the smallest value. If the
largest projections are extracted, then the most significant
information about the input data is kept. This segment of
the network computes the eigenvectors of the input’s cor-
relation function without ever computing the correlation
function itself. The outputs of the PCA layer are therefore
related to the eigenvalues and can be used as input fea-
tures to the supervised segment for classification. Since
many of these eigenvalues are usually small, only the M
(M<P) largest values need to be kept. This speeds up
training even more.
The importance of PCA analysis is that the number of
inputs for the MLP classifier can be significantly re-
duced. This results in a reduction of the number of
Figure 7. Ordering of the principal components.
6) Initializing Neural Networks with Stern Series
A Calkin-Wilf tree is a special type of binary tree ob-
tained by starting with the fraction 11 and iteratively
adding )( baa
and bba )(
below each fraction
ba . The Stern-Brocot tree is closely related, putting
)( baa
and )( bab
below each fractionba . Both
trees generate every rational number. Writing out the
terms in sequence gives 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1,
1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4/1,… as shown in Fig-
ure 8.
The sequence has the property that each denominator
is the next numerator [26] and is known as Stern's dia-
tomic series represented mathematically as
a(0) = 0, a(1) = 1; for n >= 0, a(2n) = a(n),
a(2n+1) = a(n) + a(n+1).
As an array the terms are:
1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5 and so on .
Finding 1/ [a(n)*a(n+1)] for each row
R=1 ½, ½
R=2 1/3,1/6,1/6,1/3
R=3 1/4,1/12,1/15,1/10,1/10,1/15,1/12,1/4 and so on.
Depending on the importance of a specific attribute
chosen, we can initialize the weight of neural network
with stern series for quick training. A tree showing the
designed stern series for weight initialization is shown
below in Figure 9.
The main impact of initializing the neural network
weight with stern series is quick training period .The
code for generating Stern series is given below.
Figure 8. Stern-brocot tree.
L. Parthiban et al. / J. Biomedical Science and Engineering 2 (2009) 323-335 331
SciRes Copyright © 2009
Figure 9. Weight initialization using stern sequence.
static int ans=0; //STERN'S RESULT
void stern(int n)
if(n==0||n==1) //STERN'S ASSUMTION
{ ans+=n; }
else if(n%2==0) //STERN'S EVEN
{ n=n/2;
{ stern(n); }
{ ans+=n;}
{ n=n/2;
void main()
{ textcolor(WHITE);
int n;
CANFIS combines Classification and Regression Trees
(CART) and the Neuro-Fuzzy Inference System (NFIS)
in a two step procedure. CART is a tree-based algorithm
used to optimize the process of selecting relevant pre-
dictors from a large pool of potential predictors. Using
the selected predictors, NFIS builds a model for con-
tinuous output of the predictand. In this sense, CANFIS
migrates various degrees of neuro-fuzzy spectrum be-
tween the two extremes: a completely understandable
FIS and a black-box NN, which is at the other end of
interpretability spectrum. Neuro-fuzzy models can be
characterized by neuro-fuzzy spectrum, in light of lin-
guistic transparency and input-output mapping precision.
cprintf("STERN'S SERIES\n\n");
cprintf("VALUE N: ");
for(int i=1;i<=n;i++)
{ ans=0;
7) Experiments for Implementation of Classifiers
The key design decisions for the neural networks used
in the classification are the architecture and the training
process. The adequate functioning of neural networks
depends on the sizes of the training and the testing set.
To comparatively evaluate the performance of the classi-
fiers, all the classifiers presented in this paper were
trained by the same training data set and tested with the
evaluation data set. In order to compare the performance
of the different classifiers for the same classification
problem, in addition to CANFIS, we also implemented
the MLPNN, PNN, PCA, SVM, and SOM. We per-
formed different experiments during implementation of
the classifiers and the number of hidden neurons was
determined by taking into consideration the classifica-
tion accuracies. In the hidden layers and the output lay-
ers, the activation function used was the sigmoidal func-
tion. The sigmoidal function with the range between
zero and one introduces two important properties. First,
the sigmoid is nonlinear, allowing the network to per-
form complex mappings of input to output vector spaces,
and secondly, it is continuous and differentiable, which
allows the gradient of the error to be used in updating
the weights. The training algorithm for different classifi-
ers is based on adjusting all the weights between the
neurons to minimize the mean square error of all the
training patterns. The Levenberg-Marquardt algorithm is
used for training the classifiers as it combines the best
features of Gauss-Newton technique and steep-
est-descent algorithm and does not suffer from slow
convergence [19].
CANFIS powerful capability stems from pattern-de-
pendent weights between consequent layer and fuzzy
association layer. Membership values correspond to
those dynamically changeable weights that depend on
input patterns. CANFIS bears a close relationship to the
computational paradigms of radial basis function (RBF)
networks and modular networks.
The fundamental component for CANFIS is a fuzzy
neuron that applies membership functions (MFs) to the
inputs. Two membership functions commonly used are
general Bell and Gaussian. The network also contains a
332 L. Parthiban et al. / J. Biomedical Science and Engineering 2 (2009) 323-335
SciRes Copyright © 2009
normalization axon to expand the output into a range of
0 to 1. The second major component in this type of
CANFIS is a modular network that applies functional
rules to the inputs. The number of modular networks
matches the number of network outputs, and the number
of processing elements in each network corresponds to
the number of MFs. CANFIS also has a combiner axon
that applies the MFs outputs to the modular network
outputs. Finally, the combined outputs are channeled
through a final output layer and the error is
back-propagated to both the MFs and the modular net-
The function of each layer is described as follows.
Each node in Layer 1 is the membership grade of a fuzzy
set (A, B, C, or D) and specifies the degree to which the
given input belongs to one of the fuzzy sets. The fuzzy
sets are defined by three membership functions. Layer 2
receives input in the form of the product of all output
pairs from the first layer. The third layer has two com-
ponents. The upper component applies the membership
functions to each of the inputs, while the lower compo-
nent is a representation of the modular network that
computes, for each output, the sum of all the firing
strengths. The fourth layer calculates the weight nor-
malization of the output of the two components from the
third layer and produces the final output of the network.
The architecture of CANFIS network is presented in
Figure 10.
igure 10. CANFIS network topology.
Layer 4
Layer 2
Layer 3
Layer 1
L. Parthiban et al. / J. Biomedical Science and Engineering 2 (2009) 323-335 333
SciRes Copyright © 2009 JBiSE
One disadvantage of CANFIS is that it should not be
used to predict values outside the extreme contained in
the learning database. This limitation becomes less
relevant with increased database size. Another disad-
vantage is that sufficient data base volume is required
to build the model. As such it is not capable of direct
prediction for sites which have a lack of archived ob-
The simulations were realized by using MATLAB 6.0
Neural Network Toolbox and Neurosolution software.
Six different neural network structure, Multi layer per-
ceptron, Probabilistic neural network, Principal compo-
nent analysis, Radial basis function, Support vector ma-
chine and Self organizing map neural network were ap-
plied to WBCD database to show the performance of
these neural networks on breast cancer data. To evaluate
the correctness of the proposed system, PPV (positive
predicted value) was computed in each case.
PPV is computed as:
100 resultsAll
Table 3 gives the recommended inputs by each input
recommenders, Decision tree and Correlation coeffi-
Table 4 gives the results citied in the literature on
WBCD dataset and Table 5 gives our results. Figure 11
shows the CANFIS networks learning curve using Neuro
Solution software on WBCD database. Figure 12 shows
the output vs. desired plot for CANFIS network on
WBCD dataset and the obtained Mean Square Error is
only 0.020588.
Table 3. Recommended Inputs by each input recommender.
Decision Tree input X6 X
3 X
7 X
Correlation coefficient
input X2 X
7 X
3 X
Table 4. Experimental results of previous work on WBCD
Experiment PPV (percent) Reference
Fuzzy-Genetic 97.07 [2]
ILFN 97.23 [7]
Fuzzy 96.71 [7]
ILFN &Fuzzy 98.13 [7]
SANFIS 96.07~96.3 [4]
NNs 97.95 [24]
Table 5. Experimental results of our works on WBCD dataset.
Experiment All Inputs
PPV (%)
PPV (%)
PPV (%)
CANFIS 98.82 98.53 97.94
PCA 98.53 98.24 97.65
SOM 97.94 96.77 97.94
SVM 97.65 95.30 95.89
PNN 97.06 97.65 97.65
MLP 97.65 98.24 97.36
The reduced input dataset shows almost the same per-
formances or better performances with the same learning
iteration number and shows better/similar performance
against the results of previous works. Since the result
derived by the reduced input dataset shows better per-
formance and it has significantly higher advantage in
computation, it would be a better method to be imple-
mented in real situations. Therefore, the proposed meth-
ods-combined algorithm with dominant input recom-
menders, can be appropriate methods of inference sys-
tem for the problem of breast cancer diagnosis.
Figure 11. CANFIS network learning curve on WBCD database.
Figure 12. Output vs desired plot for CANFIS on WBCD
334 L. Parthiban et al. / J. Biomedical Science and Engineering 2 (2009) 323-335
SciRes Copyright © 2009 JBiSE
Based on the results of the present paper, we would like
to highlight the following.
1) The high classification accuracies of CANFIS with
full data give insights into the nine measures of the
WBCD database. This classification accuracy slightly
decreases with input recommenders.
2) The classification accuracy of PCA, SOM, PNN
and MLP does not change much even after decreasing
the inputs with input recommenders.
3) When we initialized the weight of neural network
using stern series instead of zero as we usually do, the
speed of training was noted to increase.
4) During SVM training, most of the computational
effort is spent on solving the quadratic programming
problem in order to find the support vectors. The SVM
maps the features to higher dimensional space and then
uses an optimal hyperplane in the mapped space. This
implies that though the original features carry adequate
information for good classification, mapping to a higher
dimensional feature space could potentially provide bet-
ter discriminatory clues that are not present in the origi-
nal feature space. The selection of suitable kernel func-
tion appears to be a trial-and-error process. One would
not know the suitability of a kernel function and per-
formance of the SVM until one has tried and tested with
representative data. For training the SVMs with RBF-
kernel functions, one has to predetermine the σ values.
The optimal or near optimal σ values can only be ascer-
tained after trying out several, or even many values.
5) The pattern layer of a PNN often consists of all
training samples of which many could be redundant.
Including redundant samples can potentially lead to a
large network structure, which, in turn, induces two
problems. First, it would result in a higher computational
overhead simply because the amount of computation
necessary to classify an unknown pattern is proportional
to the size of the network. Second, a consequence of a
large network structure is that the classifier tends to be
oversensitive to the training data and is likely to exhibit
poor generalization capabilities to the unseen data.
However, the smoothing parameter also plays a crucial
role in the PNN classifier, and an appropriate smoothing
parameter is often data dependent.
In this work, the performance of various neural network
structures was investigated for breast cancer diagnosis
problem. Initializing Neural network with Stern series
was proposed to speed up training. CANFIS is the best
trade off between neural networks and fuzzy logic pro-
viding smoothness and adaptability. It also gives better
classification accuracy in terms of PPV [Table 5] than
all other neural classifiers analyzed. The performance of
the SVM was not as high as the SOM and PCA. This
may be attributed to several factors including the train-
ing algorithms, estimation of the network parameters,
and the scattered and mixed nature of the features. The
results of the present paper demonstrated that the CAN-
FIS and PCA can be used in the classification of the
WBCD data by taking into consideration the misclassi-
fication rates. This work also indicates that CANFIS can
be effectively used for breast cancer diagnosis to help
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AdaBoost Adaptive Boosting
AP Accuracy Percentage
CAD Computer Aided Diagnosis
CANFIS Coactive Neuro-Fuzzy Inference Systems
CART Classification and Regression Trees
DT Decision Tree
FIS Fuzzy Inference Systems
GA Genetic Algorithm
MLPNN Multi Layer Perceptron Neural Network
NFIS Neuro-Fuzzy Inference System
NN Neural Network
PCA Principal Component Analysis
PNN Probabilistic Neural Network
PPV Positive Predicted Value
RMS Root Mean Square
SOM Self Organizing Map
SVM Support Vector Machine
WBCD Wisconsin Breast Cancer Data