Compared to the intuitionistic fuzzy sets, the Pythagorean fuzzy sets (PFSs) can provide the decision makers with more freedom to express their evaluation information. There exist some research results on the correlation coefficient between PFSs, but sometimes they fail to deal with the problems of disease diagnosis and cluster analysis. To tackle the drawbacks of the existing correlation coefficients between PFSs, some novel directional correlation coefficients are put forward to compute the relationship between two PFSs by taking four parameters of the PFSs into consideration, which are the membership degree, non-membership degree, strength of commitment, and direction of commitment. Afterwards, two practical examples are given to show the application of the proposed directional correlation coefficient in the disease diagnosis, and the application of the proposed weighted directional correlation coefficient in the cluster analysis. Finally, they are compared with the previous correlation coefficients that have been developed for PFSs.

With the dramatic proliferation of global positioning system (GPS) devices, a rich range of research has been conducted on the analysis of GPS trajectories. Research on trajectory prediction uses historical trajectory data to forecast future positions. The typical method is to use a statistical model based on the Markov chain. However, existing models are inefficient in two aspects. The methods of using lower-order Markov models use only current information and ignore historical information, degrading the prediction accuracy. In contrast, higher-order Markov models can improve the prediction accuracy but incur increased time and space complexity. Here, we propose the kernel variable length Markov model (KVLMM), a variable-order Markov model based on kernel smoothing, which combines sequence analysis with the Markov statistical model. The KVLMM can adaptively train trajectory data and learn rules from the training results. When training a large data sample, the KVLMM can rapidly execute training in linearly complex time and space. Moreover, this model uses kernel smoothing when training fewer data samples. In other words, the KVLMM improves the prediction accuracy and reduces the overhead of the data process. Our experimental results show that KVLMM has a lower algorithm complexity and a higher prediction accuracy.

The outliers in traffic flow represent the anomalies or emergencies in the road. The detection and research of outliers will help to reveal the mechanism of such events. Aiming at the problem of outlier detection in urban traffic flow, this paper innovatively proposes a highly efficient traffic outlier detection framework based on the study of road traffic flow patterns. The main research works are as follows: (1) data pre‐processing, the road traffic flow matrix of the roads is calculated based on the collected GPS data, the non‐negative matrix factorisation algorithm is chosen to reduce the dimension of the matrix. (2) Road traffic flow pattern extraction, the fuzzy C‐means clustering algorithm with the Optimal k‐cluster centre (K‐FCM) is adopted to cluster the roads with the same road traffic flow pattern. (3) Outlier detection model training and evaluation, kernel density estimation is introduced to fit the probability density of roads traffic flow matrices which are used to train the back propagation neural network based on particle swarm optimisation to obtain the outlier detection and evaluation model, and a threshold is introduced to optimise the precision and recall of the model. The experimental results show that: the average precision and recall of the proposed method in this paper are 95.38% and 96.23%, respectively, and the average detection time is 28.4 seconds. The method has high accuracy, high efficiency and good practical significance.

Traffic flow prediction plays a crucial role in intelligent transportation systems as it enables effective control and management of urban traffic. However, existing methods that based on Graph Convolutional Networks (GCNs) primarily utilize local neighborhood information for message passing, resulting in limited perception of global structures. Additionally, it is also a challenge to extract spatial-temporal similarity features due to the constraints of graph structures. To address these issues, we propose a novel traffic flow prediction model based on Dynamic Spatial-Temporal Similarity Pyramid Network (DSTSPYN). Our model employs a spatial-temporal pyramid architecture, which dynamically adjusts the weights of central, edge, and global spatial-temporal features using an enhanced attention mechanism. Furthermore, it captures dynamic temporal dependencies at different scales through pyramid gated convolution. Meanwhile, the spatial similarity features of different time steps can be extracted through the spatial-temporal global similarity (STGS) module. We evaluate our model on four public transportation datasets and demonstrate that the DSTSPYN model outperforms several baseline methods in terms of prediction accuracy. It effectively captures the dynamic spatial-temporal correlations of the road network and edge node features, making it well-suited for long-term traffic flow prediction.