## 3 Demonstration of Filter-Wrapper Approach in Evaluating Chronic Disease Datasets
### Description of the Proposed Hybrid Classification Model
Here the K-Means algorithm initializes with K initial cluster means. Then the process is iterated thereby allotting samples to its nearest cluster and the cluster means is recomputed. Basic notations used in the algorithm are listed in Table 12.  
Squared Euclidean distance to the mean is calculated for every sample observation. Every sample is assigned to the nearest cluster. It is followed by solving logistic regression equations represented in Equations (4)-(7). \[\mathrm{Prob}(\mathrm{zi}=\mathrm{M})=\frac{1}{1+\sum_{m=1}^{\mathrm{M=1}} \mathrm{e}^{\mathrm{cam.yi}}} \tag{4}\]  
\[\mathrm{Prob}(\mathrm{zi}=1)=\frac{\mathrm{e}^{\mathrm{\alpha 1.yi}}}{1+\sum_{m=1}^{ \mathrm{M=1}}\mathrm{e}^{\mathrm{cam.yi}}} \tag{5}\]  
\[\mathrm{Prob}(\mathrm{zi}=2)=\frac{\mathrm{e}^{\mathrm{\alpha 2.yi}}}{1+\sum_{m=1}^{ \mathrm{M=1}}\mathrm{e}^{\mathrm{cam.yi}}} \tag{6}\]  
\[\mathrm{Prob}(\mathrm{zi}=\mathrm{M-1})=\frac{\mathrm{e}^{\mathrm{cam-1.yi}}} {1+\sum_{m=1}^{\mathrm{M=1}}\mathrm{e}^{\mathrm{cam.yi}}} \tag{7}\]  
Here the predicted value of regression is used to predict cluster member probability for every sample in all clusters. This probability value is reordered in descending order to determine the proportion of the two highest probabilities. Cluster means is recomputed, and cumulative summation of total distance is determined. Variation in cumulative sum of square distances from predecessor round is measured and compared with the converging factor. After cluster formation, the left-over attributes which do not belong to any cluster are identified as outliers and are eliminated. Then, the Correlation Feature Selection (CFS) method is applied. It computes correlation among a set of attributes by ranking all attributes in descending order. Correlation among attributes is computed using [PERSON]'s linear correlation coefficient, as in Equation (8). A has a values and class labels R with r values where A and R are assumed to be random variables. \[\varphi(\mathrm{A,R})=\
icefrac{{\mathrm{E(A,R)-E(A)E(R)}}}{\sqrt{\mathrm{e}^{ \mathrm{\alpha(A)\sigma^{2}(R)}}}} \tag{8}\]  
\(\varphi\)(A,R) is regarded as 0 when there is no correlation or as 1 when A and R exhibit linear dependency. An error function is used to estimate the correlation between the two attributes as shown in Equation (9). \begin{table}
\begin{tabular}{c c} \hline \hline
**yi** & **It Denotes the i-th Sample Vector (i = 1,... n)** \\ \hline zi & Arrangement of cluster for i-th sample, zi e{1,... M}, i = 1,... n \\ I (I,m) & Returns 1 for zi = m else returns 0 \\ Cm & Mean vector corresponding to the mth cluster (m = 1,... K) \\ CS & Cumulative sum of squared distance \\ \(\beta\) & Converging factor for linear difference in cumulative sum of squared distances \\ CMP & Cluster Member Probability of a sample under consideration \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Parameters used in Integrated Supervised K-Means. \[\mathrm{P}(\mathrm{A}\approx\mathrm{R})=\mathrm{err}(\left|\varphi(\mathrm{A}, \mathrm{R})\right|\sqrt{\mathrm{M}/2}) \tag{9}\]  
The pseudo code highlighting integrated supervised k-means is presented in Figure 13.  
The list of attributes ranked in terms of decreasing order of P (A \(\approx\mathrm{R}\)) serves as an attribute ranking order. Highly correlated and redundant attributes have been detected and eliminated. Furthermore, the non-redundant attributes have been input to the BFS method which explored all attributes and found a reduced optimal attribute subset. The output of the 'Attribute Evaluator' led to the generation of a better and reduced attribute set which constitutes only the relevant attributes that can be considered in the diagnosis process. Subsequently, the classification has been performed using an improved decision tree on the attribute set to determine the risk level of a person being affected by a chronic disease. Finally, performance of the developed model has been assessed according to vital performance parameters, determining the effectiveness of the classification.