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ны технологические и программно-аппаратные особенности их реализации. Предложенная структура системы отражает общую концепцию, планируется дальнейшая проработка вопросов, связанных с реализацией как всей системы в целом, так и ее отдельных компонентов.
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2. Федотов А.А., Акулов С.А. Измерительные преобразователи биомедицинских сигналов систем клинического мониторинга: учебное пособие. -М.: Радио и связь, 2013. - 250 с.
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COMPARATIVE STUDY OF NAIVE BAYES CLASSIFIERS IN BREAST CANCER DATABASE
© Ruzibayev O.B.1, Yaxshibaev D.S.2
Tashkent university of information technologies, Uzbekistan
Breast cancer is known as the most common invasive cancer type among women, therefore automatic breast cancer detection systems are in demand. Currently pattern recognition is one of the most important methods in a
1 Ассистент кафедры «Программное обеспечение информационных технологий».
2 Старший преподаватель кафедры «Высшая математика».
number of different practice areas in solving such a long-standing worries. Pattern recognition is one of the widely used classification methods in this field. This paper describes comparative results in breast cancer database, using two Naïve Bayesian (NB) methods. The experiments were realized with 10-fold cross validation test. Obtained evaluation values according to the experiments, comparative study the weighted NB and Naïve Bayesian are described in this article.
Keywords: Naïve Bayesian, cross validation, breast cancer, accuracy, Sensitivity, Specificity.
1. INTRODUCTION
In the supervised classification problems, Naive Bayesian classifier [1] is a simple and efficient probabilistic model based on the Bayesian theory. This paper discusses two well known classification methods Naïve Bayesian (NB) and weighted Naïve Bayesian (WNB) methods, using MATLAB implements them on selected breast cancer database and analyses the experimental results.
2. NAIVE BAYES CLASSIFIER
Let us consider a dataset x = [x1, x2, ..., xn] is a data sample, which has n attributes and its category is unknown. Suppose that the decision attributes takes values from C = (c1, c2, ..., cm} into Cj classes. Bayes classifier is to predict the class Cj for a new instance x, according to the evidence provided by a set of training instances for which both the attribute and class values are known. An independence assumption is embodied in the Naïve Bayesian classifier which assumes that each attribute xt is conditionally independent of all other attributes, given the class C, this yields
n
P(x\cj) = P(x,,x2,...,Xn \c )nP(X \cj). (1)
i=i
Given the conditional independence assumptions made by Naive Bayesian Classifier and P(x,|Cj) is constant to each category, we can simplify (1) and the result is:
n
CNB(X) = argmaxnP(x \cj)P(c.). (2)
i=i
2.1. Weighted naive Bayesian classifier
Naive Bayes conditional independence assumption is difficult to meet, can be assigned to different attributes of different weights, then Naive Bayes can be extended The formula of weighted NBC as follow:
n
CWNB (X) = arg max n wkP(xk \c. )P(c,.). (3)
i=i
Where wk denotes the weight of attribute xk. The larger the weight, the rasp attribute has the effect on the classification.
3. DATA NORMALIZATION
To deal with this kind of dilemma the values of the parameters are normalized (standardized) and then distances are computed. Conversion can be achieved using this formula below [2]:
x — x
ij i min ...
zv =—-• (4)
x
i max
Here will be in the range of, i.e. 0 < zy < 1. This method of conversion is used by most mathematicians and programmers. But in our software we used the statistic method of conversion given below:
z = ^ (5)
a
Here, is the mean (average) of, and is the standard deviation from mean.
n
Us
n
U xi rn-
/ = ——, a = U (x — /)2, n - number of points.
a
After standardization the average of the vector will be equal to (zero), and standard deviation of the vector will be equal to.
4. CROSS VALIDATION
Cross Validation is based on the principle that testing the algorithm on a new set of data yields a better estimate of its performance [3]. Most real applications have a limited amount of data. Because of this the dataset is split into the training sample and the validation sample.
4.1. Pseudo code module classifier using k-fold cross validation
INPUT: .Split Lists OUTPUT: Accuracy of Classifier 1: BEGIN 3: {
4: SET no Of Splits 5: FOR no Of Splits=k 6: {
7: Set ccorrect Classification Count to 0 Splitlists 8: Call Set Training Set and Test Set with
9: FOREACH Test Instance Tt in Test Set 10: {
11 : CALL Train Classification with Training Set
12: CALL Classify Test Data with Set Ti
13: IF classification = Ti Class THEN
14: INCREMENT correct Classification Count
15: ENDIF
16: }
5. EXPERIMENTAL RESULTS
An implementation diagram of breast cancer database diagram is shown in Figure 1, this overall diagram can be breast cancer database Using Naïve Bayes classifiers viewed as three stages.
Fig. 1. Block diagram of overall Naïve Bayes classifiers
Step 1 : The data database used for experimental purpose is downloaded repository web site http://archive.ics.uci.edu. The database is composed 699 objects, 9 features, and is composed of two classes. The database represents two classes. These database presented in table 1.
Step 2: These classifiers are divided into two different categories: (1) Naive Bayesian algorithm (2) weight Naive Bayesian algorithm.
Step 3: Performance measures are used to evaluate the classification efficiency, evaluated in terms of correct classification rate (%), sensitivity and specificity.
The comparative performance of the NB and weight NB classification methods are evaluated with sensitivity, specificity and accuracy tests. True Negative (TN) is the number of instances correctly identified as benign; FN (False Negative) is the number of instances incorrectly identified as benign; TP (True Positive) is the number of instances correctly identified as malignant; False Positive (FP) is the number of instances incorrectly identified as malignant. Classification accuracy, sensitivity and specificity value can be defined by using the confusion matrix is shown in Table 2. A confusion matrix describes a number of correct and incorrect predictions identified by a classification system. Sensitivity is defined as the percentage of actual positives which are correctly classified. Specificity is defined as the percentage of actual negatives which are correctly classified. Accuracy is calculated using sensitivity and specificity.
Table 1
Attributes and values of cancer
№ Attribute (Value)
1 Clump thickness 1-10
2 Uniformity of cell size 1-10
3 Uniformity of cell shape 1-10
4 Single Epithelial cell size 1-10
5 Single Epithelial Cell Size 1-10
6 Bare Chromatin 1-10
7 Bland Chromatin 1-10
8 Normal Nucleoli 1-10
9 Mitoses 1-10
10 Class 2 (Benign) or 4 (Maligant)
TP
Sensitivity (%) =--100,
TP + FN
Specificity (%) =—TN--100,
TN + FP
TP + TN
Accuracy (%) =--100.
TP + FN + TN + FN
Table 2
Confusion matrix for obtained results
Classified Actual
Actual Positive Negative
Positive (TP) (FP)
Negative (FN) (TN)
The accuracy is obtained using the data for the training and validation of an algorithm of stratified 10-fold cross-validation. Table 3 shows the accuracy comparison. The constructed confusion matrix is given in Table 2 and the calculated accuracy values for each fold are given in Table 3.
Table 3
Accuracy comparison results
Classifiers TP FP TN FN Senst Spec Acc
Naïve Bayes 431 13 232 7 98.4 94.69 97.07
Weighted Naïve Bayes 223 8 450 2 99.11 98.25 98.54
Acc - Accuracy (%), Senst - Sensitivity (%), Spec - Specificity (%).
6. CONCLUSIONS
As a conclusion, I can give there some statistical information that is compared the error of the prediction data from the website of the University of California archive.ics.uci.edu. This paper, comparative to Naïve Bayes algorithms: Naïve Bayesian and Weighted Naïve Bayesian algorithms are presented for the breast cancer database with table results (table 3.). In this case, of the algorithm Weighted Naïve Bayesian the results can be considered better than others.
References:
1. Panechnikov VA Non-parametric estimation of a multidimensional probability density // Probability theory and its application. 1969. T. 14, No 1. P. 156-161.
2. Gaydyshev I.P. Analysis and data processing: a special directory // I.P. Gaydyshev. - SPb.: Peter, 2001. - 762 p.
3. Kohavi R. A study of cross-validation and bootstrap for accuracy estimation and model selection // The Fourteenth International Joint Conference on Artificial Intelligence: proceedin g s. San M ateo, CA, 1995. No 2. P. 1137-1143.
4. Orlov A.I. Non-numerical statistics. M.: MZ-Press, 2004.
