Novel Block Diagonalization for Reducing Features and Computations in Medical Diagnosis

Abstract

Diagonalization is an "age old" technique from Linear Algebra, and it has had significant applications in Pattern Recognition (PR) and data pre-processing. By using the eigenvectors of the covariance matrix of a single class as the basis vectors describing the feature space, the transformed data can be rendered to have a diagonal covariance matrix. If the covariance matrices of two classes are utilized, the covariance matrix of transformed data of the first class can be the made the Identity, while that of the second can be diagonal, implying independence in the case of Normally distributed data 1 . In all of the cases reported in the literature, the entire covarince matrix is diagonalized, which is, computationally, a very tedious and cumbersome process. In this paper, we propose a radically different paradigm where we opt to render the transformed data to be block diagonalized. In other words, the covariance of the transformed data is made up of a predetermined number of block matrices, implying that these corresponding features are assumed to be correlated, while the others are assumed independent. Regression is now done by getting the best value based on each of these sub-blocks and averaging between them. This is essentially an ensemble machine, where the sub-blocks lead to their own respective regression values, which are then averaged to obtain the overall solution. This technique has been used to analyze the survival rate of cancer patients depends on the type of cancer, the treatments that the patient has undergone, and the severity of the cancer when the treatment was initiated. In our prima facie study, we consider adenocarcinoma, a type of lung cancer detected in chest Computed Tomography (CT) scans on the entire lung, and images that are "sliced" versions of the scans as one progresses along the thoracic region. The results that we have obtained using such a block diagonalization are quite amazing. Indeed, they surpass the results obtained from some of the well-established feature selection/reduction strategies.