Features extraction using random matrix theory

Representing the complex data in a concise and accurate way is a special stage in data mining methodology. Redundant and noisy data affects generalization power of any classification algorithm, undermines the results of any clustering algorithm and finally encumbers the monitoring of large dynamic systems. This work provides several efficient approaches to all aforementioned sides of the analysis. We established, that notable difference can be made, if the results from the theory of ensembles of random matrices are employed. Particularly important result of our study is a discovered family of methods based on projecting the data set on different subsets of the correlation spectrum. Generally, we start with traditional correlation matrix of a given data set. We perform singular value decomposition, and establish boundaries between essential and unimportant eigen-components of the spectrum. Then, depending on the nature of the problem at hand we either use former or later part for the projection purpose. Projecting the spectrum of interest is a common technique in linear and non-linear spectral methods such as Principal Component Analysis, Independent Component Analysis and Kernel Principal Component Analysis. Usually the part of the spectrum to project is defined by the amount of variance of overall data or feature space in non-linear case. The applicability of these spectral methods is limited by the assumption that larger variance has important dynamics, i.e. if the data has a high signal-to-noise ratio. If it is true, projection of principal components targets two problems in data mining, reduction in the number of features and selection of more important features. Our methodology does not make an assumption of high signal-to-noise ratio, instead, using the rigorous instruments of Random Matrix Theory (RNIT) it identifies the presence of noise and establishes its boundaries. The knowledge of the structure of the spectrum gives us possibility to make more insightful projections. For instance, in the application to router network traffic, the reconstruction error procedure for anomaly detection is based on the projection of noisy part of the spectrum. Whereas, in bioinformatics application of clustering the different types of leukemia, implicit denoising of the correlation matrix is achieved by decomposing the spectrum to random and non-random parts. For temporal high dimensional data, spectrum and eigenvectors of its correlation matrix is another representation of the data. Thus, eigenvalues, components of the eigenvectors, inverse participation ratio of eigenvector components and other operators of eigen analysis are spectral features of dynamic system. In our work we proposed to extract spectral features using the RMT. We demonstrated that with extracted spectral features we can monitor the changing dynamics of network traffic. Experimenting with the delayed correlation matrices of network traffic and extracting its spectral features, we visualized the delayed processes in the system. We demonstrated in our work that broad range of applications in feature extraction can benefit from the novel RMT based approach to the spectral representation of the data.