Permutation Entropy Based on Non-Uniform Embedding

Slope Entropy Normalisation by Means of Analytical and Heuristic Reference Values

Slope Entropy (SlpEn) is a very recently proposed entropy calculation method. It is based on the differences between consecutive values in a time series and two new input thresholds to assign a symbol to each resulting difference interval. As the histogram normalisation value, SlpEn uses the actual number of unique patterns found instead of the theoretically expected value. This maximises the information captured by the method but, as a consequence, SlpEn results do not usually fall within the classical [0,1] interval. Although this interval is not necessary at all for time series classification purposes, it is a convenient and common reference framework when entropy analyses take place. This paper describes a method to keep SlpEn results within this interval, and improves the interpretability and comparability of this measure in a similar way as for other methods. It is based on a max-min normalisation scheme, described in two steps. First, an analytic normalisation is proposed using known but very conservative bounds. Afterwards, these bounds are refined using heuristics about the behaviour of the number of patterns found in deterministic and random time series. The results confirm the suitability of the approach proposed, using a mixture of the two methods.

On the automatic parameter selection for permutation entropy

Permutation Entropy (PE) is a cost effective tool for summarizing the complexity of a time series. It has been used in many applications including damage detection, disease forecasting, detection of dynamical changes, and financial volatility analysis. However, to successfully use PE, an accurate selection of two parameters is needed: the permutation dimension n and embedding delay τ. These parameters are often suggested by experts based on a heuristic or by a trial and error approach. Both of these methods can be time-consuming and lead to inaccurate results. In this work, we investigate multiple schemes for automatically selecting these parameters with only the corresponding time series as the input. Specifically, we develop a frequency-domain approach based on the least median of squares and the Fourier spectrum, as well as extend two existing methods: Permutation Auto-Mutual Information Function and Multi-scale Permutation Entropy (MPE) for determining τ. We then compare our methods as well as current methods in the literature for obtaining both τ and n against expert-suggested values in published works. We show that the success of any method in automatically generating the correct PE parameters depends on the category of the studied system. Specifically, for the delay parameter τ, we show that our frequency approach provides accurate suggestions for periodic systems, nonlinear difference equations, and electrocardiogram/electroencephalogram data, while the mutual information function computed using adaptive partitions provides the most accurate results for chaotic differential equations. For the permutation dimension n, both False Nearest Neighbors and MPE provide accurate values for n for most of the systems with a value of n=5 being suitable in most cases.

Permutation Entropy: Enhancing Discriminating Power by Using Relative Frequencies Vector of Ordinal Patterns Instead of Their Shannon Entropy

Many measures to quantify the nonlinear dynamics of a time series are based on estimating the probability of certain features from their relative frequencies. Once a normalised histogram of events is computed, a single result is usually derived. This process can be broadly viewed as a nonlinear IRn mapping into IR, where n is the number of bins in the histogram. However, this mapping might entail a loss of information that could be critical for time series classification purposes. In this respect, the present study assessed such impact using permutation entropy (PE) and a diverse set of time series. We first devised a method of generating synthetic sequences of ordinal patterns using hidden Markov models. This way, it was possible to control the histogram distribution and quantify its influence on classification results. Next, real body temperature records are also used to illustrate the same phenomenon. The experiments results confirmed the improved classification accuracy achieved using raw histogram data instead of the PE final values. Thus, this study can provide a very valuable guidance for the improvement of the discriminating capability not only of PE, but of many similar histogram-based measures.

Identification of Denatured Biological Tissues Based on Time-Frequency Entropy and Refined Composite Multi-Scale Weighted Permutation Entropy during HIFU Treatment

Identification of denatured biological tissue is crucial to high intensity focused ultrasound (HIFU) treatment. It is not easy for intercepting ultrasonic scattered echo signals from HIFU treatment region. Therefore, this paper employed time-frequency entropy based on generalized S-transform (GST) to intercept ultrasonic echo signals. First, the time-frequency spectra of ultrasonic echo signal is obtained by GST, which is concentrated around the real instantaneous frequency of the signal. Then the time-frequency entropy is calculated based on time-frequency spectra. The experimental results indicate that the time-frequency entropy of ultrasonic echo signal will be abnormally high when ultrasonic signal travels across the boundary between normal region and treatment region in tissues. Ultrasonic scattered echo signals from treatment region can be intercepted by time-frequency entropy. In addition, the refined composite multi-scale weighted permutation entropy (RCMWPE) is proposed to evaluate the complexity of nonlinear time series. Comparing with multi-scale permutation entropy (MPE) and multi-scale weighted permutation entropy (MWPE), RCMWPE not only measures complexity of signal including amplitude information, but also improves the stability and reliability of multi-scale entropy. The RCMWPE and MPE are applied to 300 cases of actual ultrasonic scattered echo signals (including 150 cases in normal status and 150 cases in denatured status). It is found that the RCMWPE and MPE values of denatured tissues are higher than those of the normal tissues. Both RCMWPE and MPE can be used to distinguish normal tissues and denatured tissues. However, there are fewer feature points in the overlap region between RCMWPE of denatured tissues and normal tissues compared with MPE. The intra-class distance and the inter-class distance of RCMWPE are less and greater respectively than MPE. The difference between denatured tissues and normal tissues is more obvious when RCMWPE is used as the characteristic parameter. The results of this study will be helpful to guide doctors to obtain more accurate assessment of treatment effect during HIFU treatment.