Echo state network (ESN), a type of special recurrent neural network with a large-scale randomly fixed hidden layer (called a reservoir) and an adaptable linear output layer, has been widely employed in the field of time series analysis and modeling. However, when tackling the problem of multidimensional chaotic time series prediction, due to the randomly generated rules for input and reservoir weights, not only the representation of valuable variables is enriched but also redundant and irrelevant information is accumulated inevitably. To remove the redundant components, reduce the approximate collinearity among echo-state information, and improve the generalization and stability, a new method called hierarchical ESN with sparse learning (HESN-SL) is proposed. The HESN-SL mines and captures the latent evolution patterns hidden from the dynamic system by means of layer-by-layer processing in stacked reservoirs, and leverage monotone accelerated proximal gradient algorithm to train a sparse output layer with variable selection capability. Meanwhile, we further prove that the HESN-SL satisfies the echo state property, which guarantees the stability and convergence of the proposed model when applied to time series prediction. Experimental results on two synthetic chaotic systems and a real-world meteorological dataset illustrate the proposed HESN-SL outperforms both original ESN and existing hierarchical ESN-based models for multidimensional chaotic time series prediction.

Multivariate time-series prediction is a challenging research topic in the field of time-series analysis and modeling, and is continually under research. The echo state network (ESN), a type of efficient recurrent neural network, has been widely used in time-series prediction, but when using ESN, two crucial problems have to be confronted: 1) how to select the optimal subset of input features and 2) how to set the suitable parameters of the model. To solve this problem, the modified biogeography-based optimization ESN (MBBO-ESN) system is proposed for system modeling and multivariate time-series prediction, which can simultaneously achieve feature subset selection and model parameter optimization. The proposed MBBO algorithm is an improved evolutionary algorithm based on biogeography-based optimization (BBO), which utilizes an S-type population migration rate model, a covariance matrix migration strategy, and a Lévy distribution mutation strategy to enhance the rotation invariance and exploration ability. Furthermore, the MBBO algorithm cannot only optimize the key parameters of the ESN model but also uses a hybrid-metric feature selection method to remove the redundancies and distinguish the importance of the input features. Compared with the traditional methods, the proposed MBBO-ESN system can discover the relationship between the input features and the model parameters automatically and make the prediction more accurate. The experimental results on the benchmark and real-world datasets demonstrate that MBBO outperforms the other traditional evolutionary algorithms, and the MBBO-ESN system is more competitive in multivariate time-series prediction than other classic machine-learning models.

Rail transit station areas (RSAs) are heralded as a transformative approach to urban planning, emphasizing the integration of transportation, housing, and commercial development to foster sustainable and inclusive cities. This study presents a comprehensive exploration of the interplay between transit-oriented development (TOD) economic performance and social equity in RSAs, employing advanced methodologies, like eXtreme Gradient Boosting (XGBoost) and SHapley Additive exPlanations (SHAPs), to decipher the complex relationships between TOD characteristics and social equity outcomes. Focused on Dalian’s urban center, this study integrates diverse datasets, including mobile location, geospatial, and economic price data, to construct a nuanced analysis framework within the NPE (node–place–economic) model. The results indicate that economic factors significantly impact overall social equity, particularly influencing key variables, such as weekday and weekend commuter population densities. Local explanatory plots reveal that economic performance variables associated with transportation development exhibit a broad non-linear impact on social equity in RSAs. This study advances equitable urban development through TOD by stressing the importance of factoring in multiple variables in RSA planning. This approach is vital for creating synergistic effects, fostering equitable spatial planning, and has both theoretical and practical benefits for improving residents’ well-being.