Entropy has long been a subject that has attracted researchers from a diverse range of fields, including healthcare, finance, and fault detection. Slope entropy (SE) has recently been proposed as a new approach to address the shortcomings of permutation entropy, which ignores magnitude information. However, SE is sensitive to parameters γ and δ, and some information may be lost when segmenting symbols. Moreover, the δ has only a limited gain on the time series classification performance of SE and increases the algorithm complexity. Considering the aforementioned limitations, this study introduces the concept of fuzzification to the SE and eliminates the δ to simplify the parameters, resulting in the proposal of fuzzy slope entropy (FuSE). Furthermore, we incorporate the artificial rabbit optimization algorithm to optimize the parameter γ to enhance the effectiveness of FuSE for time series classification, and finally proposed an optimized fuzzy slope entropy (OFuSE). OFuSE can greatly reduce the information loss in the mapping process and adaptively search for the optimal parameter. The study evaluated FuSE and OFuSE on several synthetic datasets and concluded that FuSE is more sensitive to changes in signal amplitude and frequency, while confirming the advantage of OFuSE in classification. The application of OFuSE on three different real datasets verifies that its classification performance and generalization ability is better than other entropy methods.

In order to accurately identify various types of ships and develop coastal defenses, a single feature extraction method based on slope entropy (SlEn) and a double feature extraction method based on SlEn combined with permutation entropy (SlEn&PE) are proposed. Firstly, SlEn is used for the feature extraction of ship-radiated noise signal (SNS) compared with permutation entropy (PE), dispersion entropy (DE), fluctuation dispersion entropy (FDE), and reverse dispersion entropy (RDE), so that the effectiveness of SlEn is verified, and SlEn has the highest recognition rate calculated by the k-Nearest Neighbor (KNN) algorithm. Secondly, SlEn is combined with PE, DE, FDE, and RDE, respectively, to extract the feature of SNS for a higher recognition rate, and SlEn&PE has the highest recognition rate after the calculation of the KNN algorithm. Lastly, the recognition rates of SlEn and SlEn&PE are compared, and the recognition rates of SlEn&PE are higher than SlEn by 4.22%. Therefore, the double feature extraction method proposed in this paper is more effective in the application of ship type recognition.

To promote the effect of variational mode decomposition (VMD) and further enhance the recognition performances of bearing fault signals, genetic algorithm (GA) is applied to optimize the combination of VMD parameters in this paper, and GA-VMD algorithm is put forward to improve the decomposition accuracy of VMD. In addition, combined with the center frequency, a feature extraction method based on GA-VMD and center frequency is proposed to ameliorate the difficulty of bearing fault feature extraction. Firstly, the bearing signal is decomposed into a series of intrinsic mode components (IMFs) by GA-VMD. Then, the Center Frequency of IMFs is extracted, and the recognition rate is calculated by k-nearest neighbor (KNN) algorithm. Simulation signal experiments state clearly that, compared with manual parameter setting-VMD algorithm and parameter optimization VMD algorithm based on particle swarm optimization (PSO), the decomposition result of GA-VMD has a smaller root mean square error and higher decomposition accuracy, which verifies the effectiveness of GA-VMD. The experimental results demonstrate that, by comparison with the feature extraction method based on envelope entropy, the feature extraction method based on center frequency has better inter class separability and higher mean recognition rate (the highest recognition rate of single feature is 94.5%, and in the case of multiple features, the recognition rate reaches 100% when four features are extracted) and can realize the accurate identification of different bearing fault signals.

This paper investigates a novel robust Kalman filter by incorporating kernel density estimation in the Kalman filtering framework to address the disturbance of measurement outliers on system state estimation. It establishes a logarithmic Gaussian kernel function to approximate the unknown probability density function of abrupt-change measurement noise covariance caused by measurement outliers. Based on the logarithmic Gaussian kernel function, a state estimation equation is derived according to the Bayesian estimation theory in the presence of measurement outliers. Upon the above, a novel robust Kalman filter is established for system state estimation against measurement outliers. Simulation and experiment results demonstrate the superiority of the proposed robust Kalman filter for integrated vehicle navigation in the presence of measurement outliers.

The analysis of ship radiation signals to identify ships is an important research content of underwater acoustic signal processing. The traditional fast Fourier transform (FFT) is not suitable for analyzing non-stationary, non-Gaussian, and nonlinear signal processing. In order to realize the feature extraction and accurate classification of ship radiation signals with higher accuracy, a feature extraction method of ship radiation signals based on wavelet packet decomposition and energy entropy is proposed in this paper. According to wavelet packet decomposition, the ship radiation signal is decomposed into different frequency bands, and its energy entropy feature is extracted. As for comparisons, the center frequency and permutation entropy are also used as features to be extracted, then the k-nearest neighbor is applied to classify and recognize the extracted results. Based on the comparisons of wavelet packet decomposition, the center frequency, permutation entropy, and the k-nearest neighbor are used for classification and recognition. The experimental results present that, when comparing with center frequency and permutation entropy, the method based on energy entropy has the best availability, with the highest average recognition rate for four types of ship radiation signals, up to 98.