Symbolic Analysis of Time Series Signals Using Generalized Hilbert Transform ⋆ Soumik Sarkar Kushal Mukherjee Asok Ray szs200@psu.edu kum162@psu.edu axr2@psu.edu Department of Mechanical Engineering The Pennsylvania State University University Park, PA 16802, USA Keywords: Hilbert Transform; Symbolic Time Series Analysis; D-Markov Machines Abstract—A recent publication has shown a Hilbert- transform-based partitioning method, called analytic signal space partitioning (ASSP). When used in conjunction with D- Markov machines, also reported in recent literature, ASSP provides a fast tool for pattern recognition. However, Hilbert transform does not specifically address the issue of noise reduc- tion and the usage of D-Markov machines with a small depth D could potentially lead to information loss for noisy signals. On the other hand, a large D tends to make execution of pattern recognition computationally less efficient due to an increased number of machine states. This paper explores generalization of Hilbert transform that addresses symbolic analysis of noise- corrupted dynamical systems. In this context, theoretical results are derived based on the concepts of information theory. These results are validated on time series data, generated from a laboratory apparatus of nonlinear electronic systems. 1. I NTRODUCTION H ILBERT transform and the associated concept of an- alytic signals, introduced by Gabor [1], have been widely adopted for time-frequency analysis in diverse ap- plications of signal processing. Hilbert transform [2] of a real-valued signal x(t) is defined as:  x(t)  H[x](t)= 1 π  R x(τ ) t - τ dτ (1) That is,  x(t) is the convolution of x(t) with 1 πt over R  (-∞, ∞), which is represented in the Fourier domain as:   x(ω)= -i sgn(ω)  x(ω) (2) where  x(ω)  F [x](ω) and sgn(ω)   +1 if ω> 0 -1 if ω< 0 Given the Hilbert transform of a real-valued signal x(t), the complex-valued analytic signal [2] is defined as: X (t)  x(t)+ i  x(t) (3) ⋆ This work has been supported in part by the U.S. Army Research Office (ARO) under Grant No. W911NF-07-1-0376, by NASA under Cooperative Agreement No. NNX07AK49A, and by the U.S. Office of Naval Research under Grant No. N00014-08-1-380. Any opinions, findings and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the sponsoring agencies. and the (real-valued) transfer function with input  x(ω) and output  X (ω) is formulated as: G(ω)   X (ω)  x(ω) =1+ sgn(ω) (4) Recently, Subbu and Ray [3] have reported an application of Hilbert transform for symbolic time series analysis of dynamical systems where the space of analytic signals, derived from real-valued time-series data, is partitioned for symbol sequence generation. This method, called analytic signal space partitioning (ASSP), is comparable or superior to other partitioning techniques, such as symbolic false nearest neighbor partitioning (SFNNP) [4] and wavelet-space partitioning (WSP) [5], in terms of performance, complexity and computation time. A major shortcoming of SFNNP is that the symbolic false neighbors rapidly grow in number for noisy data and may erroneously require a large symbol alphabet to capture pertinent information on the system dynamics. The wavelet transform largely alleviates these shortcomings and thus WSP is particulary effective for noisy data from high-dimensional dynamical systems. However, WSP has several other shortcomings such as identification of an appropriate basis function, selection of appropriate scales, and non-unique and lossy conversion of the two-dimensional scale-shift wavelet domain to a one-dimensional domain of scale-series sequences [5]. When applied to symbolic analysis in dynamical systems, ASSP is used to formulate a probabilistic finite-state model, called the D-Markov model [6], where the machine states are symbol blocks of depth D. For noisy systems, it is expected that modeling with a large D in the D-Markov machine would result in higher gain in information on the system dy- namics. However, a large D increases the number of machine states, which in turn degrades computation efficiency (e.g., increased execution time and memory requirements) [7]. This paper introduces a generalization of the classical Hilbert transform to modify ASSP for application to noisy systems. The objective here is to partition the transformed signal space such that D-Markov machines can be con- structed with a small D without significant loss of infor- mation for noisy signals. The key idea is to provide a 2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009 FrC08.3 978-1-4244-4524-0/09/$25.00 ©2009 AACC 5422