Measuring complexity of observed time series plays an important role for understanding the characteristics of the system under study. Permutation entropy (PE) is a powerful tool for complexity analysis, but it has some limitations. For example, the amplitude information is discarded; the equalities (i.e., equal values in the analysed signal) are not properly dealt with; and the performance under noisy condition remains to be improved. In this paper, the improved permutation entropy (IPE) is proposed. The presented method combines some advantages of previous modifications of PE. Its effectiveness is validated through both synthetic and experimental analyses. Compared with PE, IPE is capable of detecting spiky features and correctly differentiating heart rate variability (HRV) signals. Moreover, it performs better under noisy condition. Ship classification experiment results demonstrate that IPE achieves 28.66% higher recognition rate than PE at 0dB. Hence, IPE could be used as an alternative of PE for analysing time series under noisy condition.

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Abstract:The classification performance of passive sonar can be improved by extracting the features of ship-radiated noise. Traditional feature extraction methods neglect the nonlinear features in ship-radiated noise, such as entropy. The multiscale sample entropy (MSE) algorithm has been widely used for quantifying the entropy of a signal, but there are still some limitations. To remedy this, the hierarchical cosine similarity entropy (HCSE) is proposed in this paper. Firstly, the hierarchical decomposition is utilized to decompose a time series into some subsequences. Then, the sample entropy (SE) is modified by utilizing Shannon entropy rather than conditional entropy and employing angular distance instead of Chebyshev distance. Finally, the complexity of each subsequence is quantified by the modified SE. Simulation results show that the HCSE method overcomes some limitations in MSE. For example, undefined entropy is not likely to occur in HCSE, and it is more suitable for short time series. Compared with MSE, the experimental results illustrate that the classification accuracy of real ship-radiated noise is significantly improved from 75% to 95.63% by using HCSE. Consequently, the proposed HCSE can be applied in practical applications.

Extracting useful features from ship-radiated noise can improve the performance of passive sonar. The entropy feature is an important supplement to existing technologies for ship classification. However, the existing entropy feature extraction methods for ship-radiated noise are less reliable under noisy conditions because they lack noise reduction procedures or are single-scale based. In order to simultaneously solve these problems, a new feature extraction method is proposed based on improved complementary ensemble empirical mode decomposition with adaptive noise (ICEEMDAN), normalized mutual information (norMI), and multiscale improved permutation entropy (MIPE). Firstly, the ICEEMDAN is utilized to obtain a group of intrinsic mode functions (IMFs) from ship-radiated noise. The noise reduction process is then conducted by identifying and eliminating the noise IMFs. Next, the norMI and MIPE of the signal-dominant IMFs are calculated, respectively; and the norMI is used to weigh the corresponding MIPE result. The multi-scale entropy feature is finally defined as the sum of the weighted MIPE results. Experimental results show that the recognition rate of the proposed method achieves 90.67% and 83%, respectively, under noise free and 5 dB conditions, which is much higher than existing entropy feature extraction algorithms. Hence, the proposed method is more reliable and suitable for feature extraction of ship-radiated noise in practice.