PHYSICAL REVIEW E 87, 022911 (2013) Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information Bilal Fadlallah, 1,* Badong Chen, 2,† Andreas Keil, 3 and Jos´ e Pr´ ıncipe 1,* 1 Computational NeuroEngineering Laboratory, Department of Electrical and Computer Engineering, University of Florida, Gainesville, Florida 32611, USA 2 Institute of Artificial Intelligence and Robotics, Xi’an Jiaotong University, Xi’an 710049, China 3 NIMH Center for the Study of Emotion and Attention, Department of Psychology, University of Florida, Gainesville, Florida 32611, USA (Received 6 March 2012; revised manuscript received 5 December 2012; published 20 February 2013) Permutation entropy (PE) has been recently suggested as a novel measure to characterize the complexity of nonlinear time series. In this paper, we propose a simple method to address some of PE’s limitations, mainly its inability to differentiate between distinct patterns of a certain motif and the sensitivity of patterns close to the noise floor. The method relies on the fact that patterns may be too disparate in amplitudes and variances and proceeds by assigning weights for each extracted vector when computing the relative frequencies associated with every motif. Simulations were conducted over synthetic and real data for a weighting scheme inspired by the variance of each pattern. Results show better robustness and stability in the presence of higher levels of noise, in addition to a distinctive ability to extract complexity information from data with spiky features or having abrupt changes in magnitude. DOI: 10.1103/PhysRevE.87.022911 PACS number(s): 05.45.Tp, 05.40.−a, 02.50.−r I. INTRODUCTION There is little consensus on the definition of a signal’s complexity. Among the different approaches, entropy-based ones are inspired by either nonlinear dynamics [1] or symbolic dynamics [2,3]. Permutation entropy (PE) has been recently suggested as a complexity measure based on comparing neighboring values of each point and mapping them to ordinal patterns [2]. Using ordinal descriptors is helpful in the sense that it adds immunity to large artifacts occurring with low frequencies. PE is applicable for regular, chaotic, noisy, or real-world time series and has been employed in the context of neural [4], electroencephalographic (EEG) [5–8], electrocardiographic (ECG) [9,10], and stock market time series [11]. In this paper, we suggest a modification that alters the way PE handles the patterns extracted from a given signal by incorporating amplitude information. For many time series of interest, the new scheme better tracks abrupt changes in the signal and assigns less complexity to segments that exhibit regularity or are subject to noise effects. Examples include any time series containing amplitude-coded information. For such signals, the suggested method has the advantage of providing immunity to degradation by noise and (linear) distortion. The paper is organized as follows. In Secs. II and III, we briefly introduce permutation entropy and formulate weighted- permutation entropy. Simulations details are presented in Sec. IV, respectively, on synthetic, single channel and dense- array EEG, and epileptic data. Section V offers discussion and concluding remarks. II. PERMUTATION ENTROPY Consider the time series {x t } T t =1 and its time-delay embed- ding representation X m,τ j ={x j ,x j +τ ,...,x j +(m−1)τ } for j = * {bhf, principe}@cnel.ufl.edu † chenbd@mail.xjtu.edu.cn 1,2,...,T − (m − 1)τ , where m and τ denote, respectively, the embedding dimension and time delay. To compute PE, each of the N = T − (m − 1)τ subvectors is assigned a single motif out of m! possible ones (representing all unique orderings of m different real numbers). PE is then defined as the Shannon entropy of the m! distinct symbols {π m,τ i } m! i =1 , denoted as : H (m,τ ) =−  i :π m,τ i ∈ p ( π m,τ i ) ln p ( π m,τ i ) . (1) p(π m,τ i ) is defined as p ( π m,τ i ) = ‖  j : j  N, type ( X m,τ j ) = π m,τ i  ‖ N , (2) where type(.) denotes the map from pattern space to symbol space and ‖.‖ denotes the cardinality of a set. An alternative way of writing p(π m,τ i ) is p(π m,τ i ) = ∑ j N 1 u:type(u)=π i ( X m,τ j ) ∑ j N 1 u:type(u)∈ ( X m,τ j ) , (3) where 1 A (u) denotes the indicator function of set A defined as 1 A (u) = 1 if u ∈ A and 1 A (u) = 0 if u/ ∈ A. PE assumes values between in the range [0, ln m!] and is invariant under nonlinear monotonic transformations. The main shortcoming in the above definition of PE resides in the fact that no information besides the order structure is retained when extracting the ordinal patterns for each time series. This may be inconvenient for the following reasons: (i) most time series have information in the amplitude that might be lost when solely extracting the ordinal structure; (ii) ordinal patterns where the amplitude differences between the time series points are greater than others should not contribute similarly to the final PE value; and (iii) ordinal patterns resulting from small fluctuations in the time series can be due to the effect of noise and should not be weighted 022911-1 1539-3755/2013/87(2)/022911(7) ©2013 American Physical Society