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Some Convex Functions Based Measures of Independence and Their Application to Strange Attractor Reconstruction
School of Information Science and Engineering, Southeast University, Nanjing 210096, China
Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan
Aihara Complexity Modelling Project, ERATO, JST, Saitama 332-0012, Japan
* Author to whom correspondence should be addressed.
Received: 8 February 2011; in revised form: 28 March 2011 / Accepted: 28 March 2011 / Published: 8 April 2011
Abstract: The classical information-theoretic measures such as the entropy and the mutual information (MI) are widely applicable to many areas in science and engineering. Csiszar generalized the entropy and the MI by using the convex functions. Recently, we proposed the grid occupancy (GO) and the quasientropy (QE) as measures of independence. The QE explicitly includes a convex function in its definition, while the expectation of GO is a subclass of QE. In this paper, we study the effect of different convex functions on GO, QE, and Csiszar’s generalized mutual information (GMI). A quality factor (QF) is proposed to quantify the sharpness of their minima. Using the QF, it is shown that these measures can have sharper minima than the classical MI. Besides, a recursive algorithm for computing GMI, which is a generalization of Fraser and Swinney’s algorithm for computing MI, is proposed. Moreover, we apply GO, QE, and GMI to chaotic time series analysis. It is shown that these measures are good criteria for determining the optimum delay in strange attractor reconstruction.
Keywords: entropy; mutual information; convex function; quality factor; strange attractor; delay-coordinate
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Chen, Y.; Aihara, K. Some Convex Functions Based Measures of Independence and Their Application to Strange Attractor Reconstruction. Entropy 2011, 13, 820-840.
Chen Y, Aihara K. Some Convex Functions Based Measures of Independence and Their Application to Strange Attractor Reconstruction. Entropy. 2011; 13(4):820-840.
Chen, Yang; Aihara, Kazuyuki. 2011. "Some Convex Functions Based Measures of Independence and Their Application to Strange Attractor Reconstruction." Entropy 13, no. 4: 820-840.