# Exploratory Data Analysis and Searching Cliques in Graphs  
[PERSON]  
1  
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2 Institute of Mathematics, University of Pecs, 7622 Pecs, Hungary; [EMAIL_ADDRESS]  
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graphics and visualization [1]. It is the starting point for statistics [4], but it is not an \"initial data analysis\" [5; 6]. It is generally model-free, but sometimes it relies on \"a minimum of\" a priori knowledge [4], e.g., in the case of exploratory structural equation modeling [7] (and the like), or assisting model selection. It is generally qualitative, though some consider the visualization of descriptive statistics to be part of its uses [4]. It shares techniques with similar fields, e.g., cluster analysis within data mining, which is also model-free. An important feature of the method we would like to emphasize is that, in our approach, we not only do not place serious restrictions on the underlying multidimensional probability distributions but rather we replace them with graph theoretical concepts such as cliques in a graph. At a suitable juncture we will describe the bare minimum of terminology we need from graph theory. The data sets we are considering in this work consist of a number objects, each object of which possesses a number of attributes. In this way, the data set is identified by an \(m\)-by-\(n\) matrix, the so-called data matrix. The rows are labeled with the objects. The columns are labeled with the attributes. It is a fact from elementary linear algebra that the dimensions of the row space and the dimensions of the column space in the original \(m\)-by-\(n\) data matrix are equal. When we estimate the essential dimension of the data set we may restrict our attention to the row space. In fact, we will work with the \(m\)-by-\(m\) matrix that contains similarity indices between objects. For the sake of definiteness, the reader my think of the Pearson correlation coefficient of two given objects as a similarity index. In this case, the \(m\)-by-\(m\) matrix is filled with the correlation coefficients. The essential dimension of the row space \(d\) is commonly estimated via the spectral decomposition of the \(m\)-by-\(m\) correlation matrix of the objects. The entries of the correlation matrix are real numbers and the matrix is symmetric with respect to the main diagonal. The associated quadratic form is positive and semi definite. Consequently, the eigenvalues of the correlation matrix are real and non-negative. We list the eigenvalues in non-increasing order starting with the largest and ending with the smallest. The last smallest \(m-d\) eigenvalues should be negligible compared to the first largest \(d\) eigenvalues. The main point is that a large part of the information in the original data set can be condensed into a relatively low-dimensional linear manifold. In other words, the \(d\)-dimensional linear manifold well approximates the original data set [8]. A minute of contemplation will convince the reader that it is possible that the original data are a union of a few 1-dimensional linear manifolds and at the same time the data set cannot be globally approximated by a low-dimensional linear manifold because of the relative position of the 1-dimensional linear manifolds. Putting it differently, it may be the case that a data set can be decomposed into (not necessarily disjoint) parts that can all be well approximated by 1-dimensional linear manifolds while the whole data set cannot be well approximated with a low-dimensional linear manifold. We propose the following related problems. Given a data set, try to locate a subset of objects that can be well approximated by a 1-dimensional linear manifold. Or alternatively, try to decompose the set of objects into parts such that each part can be locally well approximated with 1-dimensional linear manifolds. Instead of similarity indices one may use distances between objects. In this situation an \(m\)-by-\(m\) matrix will be filled with distances. We refer to this matrix as the distance matrix of the objects.