Dimension estimation using weighted correlation dimension method

Abstract

Dimension reduction is an important tool for feature extraction and has been widely used in many fields including image processing, discrete-time systems, and fault diagnosis. As a key parameter of the dimension reduction, intrinsic dimension represents the smallest number of variables which is used to describe a complete dataset. Among all the dimension estimation methods, correlation dimension (CD) method is one of the most popular ones, which always assumes that the effect of every point on the intrinsic dimension estimation is identical. However, it is different when the distribution of a dataset is nonuniform. Intrinsic dimension estimated by the high density area is more reliable than the ones estimated by the low density or boundary area. In this paper, a novel weighted correlation dimension (WCD) approach is proposed. The vertex degree of an undirected graph is invoked to measure the contribution of each point to the intrinsic dimension estimation. In order to improve the adaptability of WCD estimation, k-means clustering algorithm is adopted to adaptively select the linear portion of the log-log sequence (logδk,logC(n,δk)). Various factors that affect the performance of WCD are studied. Experiments on synthetic and real datasets show the validity and the advantages of the development of technique.