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Local Linear Radial Basis Function Neural Networks for Classification of Breast Cancer
Data
M.R Senapatia , S.P Dasb, P.K.Champatic, P.K.Routrayd,
a
Department of computer science and engineering
Centurion Institute of Technology
Centurion University of Technology and management, 752050, India
+919437132010
E-mail: [email protected]
b
Department of Computer Science and Engineering
Templecity Institute of Technology & Engineering
Biju Pattnaik University of Technology
Rourkela-752002, India
E-mail: [email protected]
c
Department of Computer Science and Engineering
Ajay Binay Institute of Technology
Biju Pattnaik University of Technology
Rourkela-752002, India
E-mail: [email protected]
d
Department of Computer Science and Engineering
NM Institute of Engineering and Technology
Biju Pattnaik University of Technology
Rourkela-752002, India
E-mail: [email protected]
Abstract: - Breast cancer is the major cause of cancer deaths in women today and it is the most common type of
cancer in women. This paper presents some experiments for classification of breast cancer tumor and proposes the
use of local liner radial basis function neural network (LLRBFNN) for classification and recognition of breast
cancer. The experiments were conducted on extracted breast cancer data from University of Wisconsin Hospital,
Madison. The neural network is trained with breast cancer data by using feed forward neural network model and
back propagation learning algorithm with momentum and variable learning rate. The performance of the network is
evaluated. The result has been compared with a wide range of classifiers to evaluate its performance. The
evaluations show that the proposed approach is very robust, effective and gives better correct classification as
compared to other classifiers.
Keywords:- Local liner radial basis function neural network (LLRBFNN), Radial basis function neural network
(RBFNN), Multi layer perceptron (MLP), Wisconsin breast cancer (WBC).
1. Introduction
The high incidence of breast cancer in women especially in developed countries has increased significantly in the
last few years. The causes for this is not very much clear and neither the reason for the increases in number. At
present there is no method to prevent and avoid breast cancer; only early detection can increase the survival rate.
There are various methods to detect breast cancer and mammography is considered the most reliable method. But is
has got a major drawback that is the volume of mammograms to be rated by the Physician decrease the accuracy.
Multiple reading of mammograms (consecutive reading by two physicians) will increase accuracy but at a higher
cost. That is why computer aided classification techniques are necessary to achieve the accuracy and increase the
chance of survival [1, 2].
Digital mammograms are among the most difficult medical images to be read due to their low contrast and
difference in the type of tissues. Important visual clues of breast cancer include preliminary signs of masses and
classification clusters. Unfortunately, in the early stages of breast cancer, those signs are subtle and vary in
appearance, making it difficult for the specialist to diagnose. This is the main reason for the development of
classification system to assist the specialist in the medical institution. Due to the importance and significance much
research in the breast cancer classification has been done recently. [3-6].
Early diagnose requires an accurate and reliable diagnosis procedure that allows physicians to distinguish breast
tumors from malignant ones. Thus finding an accurate and effective diagnosis method is very important.
Many methods of AI have shown better results than obtained by the experimental methods; for example, in 1997,
Burke et al. [7] compared the accuracy of TNM staging system with the accuracy of a multilayer back propagation
Artificial Neural Network (ANN) for predicting the 5-year survival of patients with breast carcinoma.
ANN increased the prediction capacity by 10 % obtaining the final result of 54 %. They used the following
parameters: tumor size, number of positive regional lymph nodes, and distant metastasis. Domingos [8] used a breast
cancer database from UCI repository for classifying survival of patients using the unification of two widely used
empirical approaches: rule induction and instance-based learning. In 2000, Boros et al. [9] used the logical analysis
of data method to predict the nature of the tumor: malignant or benign. Breast Cancer (Wisconsin) database was
used for this purpose. The classification capacity was 97.2 %. This database was used by Street and Kim [10] who
combined several classifiers to create a high-scale classifier. Also, it was used by Wang and Witten [11]; they
presented a general- modeling method for optimal probability prediction over future observations, and they obtained
the 96.7 % of classification. Huang et al. [12] construct a classifier with the mini max probability machine (MPM),
which provides a worst-case bound on the probability of misclassification of future data points based on reliable
estimates of means and covariance matrices of the classes from the training data points. They used the same
database utilized by Domingos. The classification capacity was of 82.5 %. The classification capacity was of 82.5
%. In 2012 Senapati et. Al [13] used K-PSO to predict nature of the tumor: malignant or benign. The classification
accuracy in this case is 96.43.
The method proposed is to classify the breast cancer data, which has been downloaded from University of
Wisconsin Hospital, Madison. Basically the objective of this prediction technique is to assign patient to either a
“benign” group that does not have breast cancer or to a “malignant” group that has strong evidence of breast cancer.
This paper a local linear radial basis function neural network (LLRBFNN) is proposed for breast cancer
detection, in which the connection weights between the hidden layer units and output units are replaced by a local
linear model and the parameters of the network are updated using back propagation. Simulation results for Breast
cancer Pattern Classification problem was compared with some common classification techniques. The result thus
derived shows the effectiveness of the proposed method.
The rest of the paper is organized as follows. The LLRBFNN is described in Sec. 2. Radial basis function is
explained in Sec.3. MLP algorithm for training LLRBFNN is presented in Sec.4. A short discussion as well as
experimental results obtained on pattern classification for Wisconsin Breast Cancer (WBC) problem is given in Sec.
5.Finally, concluding remarks is derived in the last section i.e. Sec. 6.
2. Local linear Radial Basis Function neural network
Local liner radial basis function neural network in fact is a modification of RBFNN. Because in RBFNN if we
use the local liner summation in the last layer, learning parameters will be increased but mapping ability of neural
network will be improved i.e. Performance (accuracy) will be increased. And it will create a local linear model.
RBFNN with local linear model in the hidden layer is called LLRBFNN. The motivations for introducing the local
linear models into a RBFNN are as follows: local linear models should provide a more parsimonious interpolation in
high-dimension spaces when modeling samples and sparse and converges faster than RBFNN with fewer epochs.
The local linear model parameters are randomly initialized at the beginning and are optimized by multilayer
perceptron (MLP) algorithm discussed in the section 4.
LLRBFNN is characterized by localized activation of the hidden layer units. The connecting weights associated
with the hidden nodes can be viewed as locally accurate linear model. The mapping function of the network with
Gaussian activation function and weighted linear summation in output neuron is given by:
𝑛
𝑓(𝑥) = ∑ 𝑣𝑖 𝑧𝑖 (𝑥)
(1)
𝑖=1
where 𝑥 = [𝑥1 , 𝑥2 , … … … . 𝑥𝑛 ]
is the no. of inputs
𝑣𝑖 = 𝜔𝑖0 + 𝜔𝑖1 𝑥1 + … … … + 𝜔𝑖𝑛 𝑥𝑛
𝜔𝑖
represents weights between hidden to output.
−||𝜒− 𝐶𝑖 ||2
𝑧𝑖 (𝑥) = 𝑒𝑥𝑝 (
2𝜎𝑖2
)
is the activation function of ith hidden neurons
𝑐𝑖 is centers of ith activation function;
𝜎𝑖 is the parameter for controlling the smoothness of the activation function named as spread.
||𝑥 − 𝑐𝑖 || indicates the Euclidean distance between the inputs and the centers.
𝑥1
𝑥2
𝑣1
𝑣2
𝑣3
𝒇(𝒙)
𝑥3
𝑣𝑛
𝑥𝑛
Fig-1 A local linear radial basis function neural network
Eq-(1) is a family of functions generated from one single function 𝑓(𝑥) by the operation of centers and weights,
which is localized. The activities of the linear models 𝜈𝑖 (i=1,2…..n) are determined by the associated locally
significant .
Obviously, the localization of the ith units of the hidden layer is determined by the spread parameter 𝜎𝑖 and the
center parameter 𝑐𝑖 . According to the previous researchers, the two parameters can be either is predetermined based
on the radial basis function transformation theory or to the determined by a training algorithm.
The computational steps involved in implementing the LLRBFNN for classification are:
1.
For untrained inputs, initialize the centers(c, weights (w) and spreads (𝜎).
𝑐 = 𝑐𝑖𝑛𝑖𝑡 , 𝜔 = 𝜔𝑖𝑛𝑖𝑡 , 𝑎𝑛𝑑 𝜎 = 𝜎𝑖𝑛𝑖𝑡 (Initialization)
2. Update the LLRBFNN centers and weights using Error back propagation.
3. Calculate error mean ,
4. Convergence test for centers and weights.
The centers and weights are updated one by one in iteration, i.e. by new training input to the LLRBFNN. The
spread of LLRBFNN is taken as 1, it is not updated throughout the program.
An intrinsic feature of the radial basis function networks is the localized activation of the hidden layer units, so
that the connection weights associated with the units can be viewed as locally accurate piecewise constant models
whose validity for a given input is indicated by the activation functions. Compared to the multilayer perceptron
neural network, this local capacity provides some advantages such as learning efficiency and the structure
transparency. However, the problem of basis function networks is also led by it. Due to the crudeness of the local
approximation, a large number of basis function units have to be employed to approximate a given system. A
shortcoming of the LLRBFNN is that for higher dimensional problems many hidden layer units are needed.
3. Radial Basis Function neural network
𝑥1
𝑥2
𝑥3
𝒄𝟏
𝜙1
𝒄𝟐
𝜙2
𝑥4
𝑤1
𝑤2
y
𝑤ℎ
𝜙ℎ
𝒄𝒉
𝑥𝑛
Input Layer
Layer
Hidden Layer
Output
Fig- 2 Radial basis function neural network
The above figure of radial basis function describes the followings:



Input layer: no activation function (no calculation at the level of this layer)
Hidden layer: Gaussian function
Output layer: linear function
The hidden neurons give a set of functions that constitute an arbitrary basis for the input patterns.
Hidden
neurons are known as radial centers and represented by the vectors𝑐1 , 𝑐2 𝑐3 … 𝑐ℎ Dimension of the center is
depending upon number of inputs and number hidden neurons. Number of hidden neurons is not fixed (ranged
between numbers of output neurons to the number of inputs) . Transformation from input space to hidden unit space
is nonlinear whereas transformation from hidden unit space to output space is linear. The radial basis function in the
hidden layer produces a significant non-zero response only when the input falls within a small localized region of
the input space. An input vector 𝑥𝑖 which lies in the receptive field for center 𝑐𝑖 would activate cj and by proper
choice of weights the target output is obtained. The output is given as
𝑦 = ∑𝑛𝑖=1 𝜙𝑖 𝜔𝑖
(2)
Where 𝑤𝑖 : Weight of ith center, 𝜙: some radial basis function (Gaussian function)
2
2
Gaussian Radial Function: 𝜙(𝑧) = 𝑒 −𝑧 ⁄2𝜎
Where,
And
𝑧 = (||𝑥 − 𝑐𝑖 ||)
||𝑥 − 𝑐𝑖 || = √∑𝑛𝑖(𝑥𝑖 − 𝑐𝑖 )2
(3)
(4)
(5)
𝜎 = constant.
Fig-3 Information about Spreads
As shown in fig. 3 spread is the distance or width of the curve. Spread should not be so large that each neuron is
effectively responding in the same large area of the input space.
3.1 LEARNING IN RBFNN
Training in RBFN requires optimal selection of the parameters 𝑐𝑖 and 𝑤𝑖 , where i=1, 2, 3… h. There are several
techniques used to update the weights and centers (Pseudo-Inverse technique, Hybrid Learning and Gradient
Descent Learning). Here Gradient Descent Learning technique is used. The technique is a supervised training by
error correcting term.
The learning steps are as follows:
1.
2.
The width(𝜎) is fixed according to the spread of centers
From fig.2 𝜙 = [𝜙1, 𝜙2, 𝜙3,….. , 𝜙ℎ ]
𝜔 = [𝜔1, 𝜔2, 𝜔3,….. , 𝜔ℎ ]
𝜙𝜔 = 𝑦 𝑑
3.
4.
5.
6.
is the desired output
The update rule for center learning is :
𝜕𝐸
𝑐𝑖𝑗 (𝑡 + 1) = 𝑐𝑖𝑗 (𝑡) − 𝜂1
, 𝑖, 𝑗 = 1 𝑡𝑜 ℎ
𝜕𝑐𝑖𝑗
The weight update law is:
𝜕𝐸
𝜔𝑖 (𝑡 + 1) = 𝜔𝑖 (𝑡) − 𝜂2
𝜕𝜔𝑖
8.
(7)
1
Cost function is : 𝐸 = ∑(𝑦 𝑑 − 𝑦)2
2
Differencing E w.r.t 𝑐𝑖𝑗 , we get
𝜕𝐸
𝜕𝐸
𝜕𝑦
𝜕𝜙𝑖
=
Χ
Χ
𝜕𝑐𝑖𝑗
𝜕𝑦
𝜕𝜙𝑖
𝜕𝑐𝑖𝑗
= −(𝑦 𝑑 − 𝑦) Χ 𝜔𝑖 Χ
7.
(6)
Now,
𝜕𝜙𝑖
𝜕𝑧𝑖
= −
And,
𝜕𝑧𝑖
𝜕𝑐𝑖𝑗
=
𝑧𝑖
𝜎2
𝜕
𝜕𝑐𝑖𝑗
𝜕𝜙𝑖
𝜕𝑧𝑖
(8)
Χ
𝜕𝑧𝑖
𝜕𝑐𝑖𝑗
𝜙𝑖
2
1⁄
2
(∑(𝑥𝑗 − 𝑐𝑖𝑗 ) )
= − (𝑥𝑖 − 𝑐𝑖𝑗 )⁄𝑧𝑖
After simplification , the update rule for center learning is :
𝜙𝑖
𝑐𝑖𝑗 (𝑡 + 1) = 𝑐𝑖𝑗 (𝑡) − 𝜂1 (𝑦 𝑑 − 𝑦)𝜔𝑖 2 (𝑥𝑖 − 𝑐𝑖𝑗 )
𝜎
Where 𝜂1 = center learning rate.
The update rule for linear weight is :
𝜔𝑖 (𝑡 + 1) = 𝜔𝑖 (𝑡) − 𝜂2 (𝑦 𝑑 − 𝑦)𝜙𝑖
Where 𝜂2 = weight learning rate
(9)
(10)
Where 𝜙𝑖 is the radial basis activation function of ith unit of the hidden layer and 𝜔𝑖 is the weight connecting the ith
unit of the hidden layer to the output layer unit.
4. Multi Layer Perceptron(MLP) Fundamentals
Multi layer perceptron (MLP) is a recent nature inspired technique that has been used for solving non-linear
optimization problems.
A multilayer perceptron (MLP) is a kind of feed forward artificial neural network, which is a mathematical
model inspired by the biological neural network. The multilayer perceptron can be used for various machine
learning tasks such as classification. The size of input layer and output layer determines what kind of data a MLP
can accept. Specifically, the number of neurons in the input layer determines the dimensions of the input feature; the
number of neurons in the output layer determines the dimension of the output labels. After a lower most input layer
there are usually any numbers of intermediate, or hidden, layers followed by an output layer at the top. Weights
measure the degree of correlation between the activities levels of neurons that they connect.
In this paper, Error back propagation,, which is described for numerical optimization problems, is applied to
optimize the parameters of LLRBFNN.
4.1 Multi layer perceptron algorithm
The basic component of a multilayer perceptron is the neuron. In this case the neurons are aligned in layers and
in any two adjacent layers the neurons are connected in pairs with weighted edges. Multilayer perceptron consists of
at least three layers of neurons, including one input layer, one or more hidden layers, and one output layer. In hidden
layer, the number of neurons is a design issue. If the neurons are too few, the model will not be able to learn
complex decision boundaries. On the contrary, too many neurons will decrease the generalization of the model.
MLP is used feeding the input features to the input layer and get the result from the output layer. The results are
calculated in a feed-forward approach, from the input layer to the output layer.
One step of feed-forward is illustrated in the following figure.
𝑿𝟏
1
ℎ1
Inputs
𝑿𝟐
ℎ2
𝑓(𝑥)
𝑿𝟑
∑ 𝑊𝑖 𝐻𝑖
𝑓(𝑥)
Output
ℎ𝑛
𝑿𝒏
Hidden Layer
Fig-4 Network architecture of MLPs
For each layer except the input layer, the value of the current neuron is calculated by taking the linear
combination of the values output by the neurons of the previous layer, where the weight determines the contribution
of a neuron in the previous layer to current neuron. Obtaining the linear combination result, a non-linear function is
used to constrain the output into a restricted range .Typically, sigmoid function or tanh ( ) functions are used.
- Input layer: no activation function (no calculation at the level of this layer)
- Hidden layer: simplified hyperbolic tanh( ) function.
H= tanh(weights*inputs)
𝑛
𝐻𝑗 = 𝑡𝑎𝑛ℎ (∑ 𝜔𝑖𝑗 𝑥𝑖 )
(11)
𝑗=1
Output layer: simplified tanh function
O= tanh (Bias+∑ (Weights * Hidden values))
𝑛
𝑂𝑖 = 𝑡𝑎𝑛ℎ (𝜔0 + ∑ 𝐻𝑖 𝜔𝑗 )
(12)
𝑗=1
Where 𝜔𝑖 = weight, 𝑥𝑖= input , 𝜔0 = bias
The proposed model takes training set data after normalization it passes through the network model then after
several epochs (iterations) when the MSE drop towards zero the training is finished then the test dataset is passed
through the network to check the model validity and finally the classified and misclassified classes are obtained
including the classification graph
A neural network learns by continuously adjusting the synaptic weights at the connections between layers of
neurons until a satisfactory response is produced. In the present work, the MLP network was applied to estimate
outputs based on an analysis of the data captured by the inputs. The weight readjustment method employed was back
propagation, which consists of propagating the mean squared error .generated by each layer of neurons, readjusting
the weights of the connections so as to reduce the error in the next iteration.
4.2 The weights updating equation
The equations are different for the weights used in input layer to hidden layer and hidden layer to output layer.
The rule to update the weights from input layer to hidden layer is given by:
𝑑𝐸
𝑑(𝑑 − 𝑜𝑢𝑡)2
=
𝑑𝜔𝑖
𝑑𝜔𝑖
After derivation the generalized equation to update the weights of input to hidden layer is given by the equation:
𝜔(𝑡+1) = 𝜔𝑡 + 2𝜇 𝑒(1 − 𝑜𝑢𝑡 2 )ℎ𝑖
(13)
The rule to update the weights from hidden layer to output layer is given by:
𝑑𝐸
= −2 𝑒 (1 − 𝑜𝑢𝑡 2 )(1 − ℎ𝑖 2 )𝜔𝑖
𝑑𝜔𝑖𝑗
After derivation the generalized equation to update the weights of hidden layer to output layer is given by:
𝜔𝑖𝑗(𝑡+1) = 𝜔𝑖𝑗 + 2 𝜇 (1 − 𝑜𝑢𝑡 2 )(1 − ℎ𝑖 )𝜔𝑖
(14)
Although for approximating non-linear input-output mappings, the RBF networks can be trained much faster, MLPs
may require a smaller number of parameters.
5. Discussion
In order to evaluate the performance of the algorithms, Wisconsin dataset is used. First the training data are fed
to the neural network models (LLRBFNN, RBFNN and MLP) and back propagation technique is used for
optimization of the parameters. The initial values of the centers are initialized to random values and weights of the
network are initialized to ones in RBFNN and LLRBFNN, but in MLP weights are taken randomly. During the
training process the centers and weights are updated once after application of each input pattern. The cost function is
taken to be error function. The training of the networks is continued to 200 epochs. During the training process at
some point both center and weights are converged. The performance of the network is judged by mean square error
method.
5.1 Normalization
We apply the local linear radial basis function neural network explained in Sec. 2 to Wisconsin Breast Cancer
(WBC) databases and compare its performances to the most common classification methods in both computer
science and statistics literatures. All the computations are implemented using MATLAB v 6.5, under dual core
personal computer with a clock speed of 2.4GHz. As commonly done, we normalize the input variables to make sure
that they are independent of measurement units. Thus the predictors are normalized in the interval of (0:1) using the
formula:
𝑥 𝑛𝑒𝑤 =
𝜒𝑜𝑙𝑑 − 𝜒𝑚𝑖𝑛
𝜒𝑚𝑎𝑥 −𝜒𝑚𝑖𝑛
(15)
We have taken take 70% of the total patterns at a random from 699 patterns for training purpose. The new
technique LLRBFNN is compared with two other classifiers to evaluate its performance with respect to correct
classification rate and the time it takes to get trained.
5.2 Wisconsin breast cancer (WBC)
The data set were obtained from University of Wisconsin Hospital; Madison WBC is a nine-dimensional data
set with the following features: (i) Clump thickness;(ii) Uniformality of cell size;(iii) Uniformality of cell shape;(iv)
Marginal adhesion;(v) Single epithelial cell size;(vi) Bare nuclei;(vii) Bland chromatin;(viii) Normal nucleoli; and
(ix) Mitoses. For our classification purpose 489 exemplars were used for training and the rest 210 exemplars for
testing.
5.3 Results and Comparison
Tables 1, 2 and 3 of confusion matrix and Figures 5, 6 and 7 of convergence of center and weight, Figures 8, 9,
10 of mean square error, and Figures 11, 12, 13 of classification results obtained by local linear radial basis function
neural network (LLRBFNN), radial basis function neural network (RBFNN) and multi layer perceptron (MLP)
respectively, we observe the followings:
1.
2.
3.
4.
5.
LLRBFNN method both center and weight convergence is far better than RBFNN method
MLP and Radial basis function neural network shows worst accuracy.
The LLRBFNN is giving the highest values of the accuracy with same number of iterations executions.
The accuracy of the proposed technique for malignant type cancer is almost 99.59%.
The overall percentage of accuracy of LLRBFNN is 98.00.
We draw our conclusion in Sec. 6 by utilizing the useful information shown in tables 1, 2 and 3.
Table.1 The confusion matrix obtained from LLRBFNN
Benign
Benign
445
Malignant
17
%of classification
97.16
Overall classification percentage is
Malignant
8
240
99.59
98.00%
Table.2 The confusion matrix obtained from RBFNN
Benign
Malignant
Benign
443
16
Malignant
30
%of classification
96.72
Overall classification percentage is
240
99.59
97.71%
Table.3 The confusion matrix obtained from MLP
Benign
Benign
Malignant
%of classification
Overall classification percentage is 97.00%
Malignant
442
5
16
236
96.51
97.93
Fig-5(a) Convergence of Centers in LLRBFNN
Fig-5(b) Convergence of weights in LLRBFNN
Fig-6(a) Convergence of centers in RBFNN
Fig-6(b) Convergence of weight in RBFNN
Fig-7(a) Convergence of weights from input to
hidden layer in MLP
Fig-7(b) Convergence of weight from hidden to
output layer in MLP
Fig- 8 Mean square error of LLRBFNN
Fig-9 Mean square error of RBFNN
Fig-10 Mean square error of MLP
Fig-11 Classification (LLRBFNN)
Fig-12 Classification (RBFNN)
Fig-13 Classification (MLP)
5.4 Comparative study with other techniques
The proposed technique is compared with RBFNN and MLP technique. The comparison is depicted in Fig-14.
The result of the comparison shows that the proposed technique gives better classification as compared to some of
the existing techniques.
Percentage of Classification
98.2
98
97.8
97.6
97.4
97.2
97
96.8
96.6
96.4
LLRBFNN
RBFNN
Techniques
MLP
Fig-14 Percentage of classification obtained from different techniques
6. Conclusion
Even though mammography is one of the best techniques for breast cancer detection, but in some cases, despite
their experience radiologist can’t detect cancers. There computer aided methods like the one presented in this paper
could assist medical staff and increase the accuracy of detection. Statistics shows that only 20% - 30% breast cancer
cases are cancerous. In case of a false negative detection the cancer remains undetected that could lead to higher cost
or even to the cost of a human life. Here is the trade of that motivated us to develop a classification system.
In this paper we presented a technique for breast cancer classification using LLRBFNN. The technique provides
an overall classification of 98.00%. The classification accuracy for malignant cancer is 99.59%, which is
encouraging but accuracy for benign cancer is 97.16%, i.e. the false positive detection is high, which may cause an
unnecessary hardship for the patient.
The technique was compared with different methods already developed. We could empirically say that the
proposed approach has better performance, high quality and generalization of common existing approaches.
However more work is needed in evaluating the performance of the proposed method on other medical and / or other
science or business databases.
It is well known that data mining techniques are more suitable to larger databases. We intend to use larger
database, from medical science and / or business sector to evaluate the performance of the technique. Also the
technique needs to be evaluated using time series data to validate the findings.
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