Fig. 2

The main idea of ARNN-LNE. a The frame of the auto-reservoir neural network (ARNN). ARNN is a model-free multistep-ahead prediction approach for a target y. In the architecture of ARNN, the reservoir component consists of a fixed multilayer neural network F with a randomly assigned weight and data input \(I_{t}\). Moreover, \(O_{t}\) is a target vector generated by solving the ARNN-based equation iteratively. b Calculate the ARNN-LNE index. There are four steps to obtaining the early warning signals according to the ARNN-LNE approach. Step 1. Construct a regional network. Step 2. Predict daily new cases of COVID-19 by ARNN. The raw data were processed through window shift where window breadth is set as m + L. The blue part of the data is the training data and the orange part is the predicting data gained by ARNN method. Step 3. Calculate the local network entropy \(H_{k} (t)\) of the local network \(N_{k}\) with (L + 1) members at the time point \(T = t\). Step 4. Gain the early warning signals (ARNN-LNE index)