# A Machine Learning-Based Method for Modeling TEC Regional Temporal-Spatial Map
## 2 Modeling  
### Modeling Method  
Modeling methods of the TEC model can be divided into traditional and intelligent methods. Most traditional methods are based on mathematical principles; that is, different mathematical functions or spatial interpolation methods are used to characterize the temporal-spatial dynamic variations of TEC. However, with the rapid development of artificial intelligence, many intelligent methods are beginning to emerge, the methods based on ML are introduced into the model of TEC and organic combination with traditional methods to promote the leapfrog development of the TEC model. ML is interdisciplinary and builds a probabilistic statistical model based on data and uses the model to predict and analyze data [44]. Compared with the black box algorithm of the NN [45], the data processing ideas of ML are easy to understand, and the determined model parameters can be interpreted [46]. Therefore, ML is widely applied in the modeling of various ionospheric parameters. As shown in Figure 1, ML is a process of determining the three elements (model, algorithm, and strategy) and solving the following four problems:  
1. What kind of data are needed for training and predicting? The premise of ML is that the data used for training models have the same statistical characteristics, and the modeling process is to find the appropriate map to describe such statistical characteristics. This paper selects the monthly mean values of TEC (hereinafter referred to as TEC) from eight ionospheric observation stations in parts of Europe as the modeling dataset. The validation dataset consists of six stations in the same region, of which three are used for modeling, and three are not involved in the modeling process. The details are described in Section 2.2. 2. How to choose a suitable model? This requires us to have an in-depth understanding of the purpose of modeling, the principles and applicable scope of different models, their advantages and disadvantages, and the characteristics of the training dataset before modeling. In this paper, the principal component analysis (PCA) function set is selected as the hypothesis space to construct the temporal and spatial characteristic  
Figure 1: Four problems to be solved in ML and corresponding solutions. map of TEC. PCA is a commonly used data analysis method known as eigenvector analysis [47]. It transforms the original data into a set of linearly independent ordered basis functions and correlation coefficients through a linear transformation. It can extract the main eigen components of the data and is often used for dimensionality reduction in high-dimensional data. In 1901, [PERSON] [48] first proposed PCA as an effective method to establish an empirical model. And then, [PERSON] [49] introduced it into the modeling of the ionospheric parameter empirical model for the first time in 1988. Subsequently, many empirical models of ionospheric parameters, such as foF2 [50], the propagation factor at 3000 km of the F2 layer [51], and TEC [52], were modeled using the PCA method. PCA can not only reduce the number of parameters involved in the modeling process but also reduce the amount of calculation and storage of intermediate data to achieve high accuracy and fast convergence [53]. Therefore, we apply PCA to decompose TEC orthogonally into temporal and spatial variables controlled by independent and unrelated processes so that TEC can be calculated through the orthogonal temporal and spatial map. The flow chart of modeling and validation is shown in Figure 2.  
The nonlinear map of geographical location (latitude and longitude), solar activity, month, and universal time to TEC (unit is TECU) constructed based on PCA theory can be expressed as the harmonic function:  
\[\text{TEC}(\lambda,\varphi,p,r,m,t)=\sum_{n=1}^{N}W_{n}(\lambda,\varphi) \cdot\hat{\text{TEC}}(p,r,m,t), \tag{1}\]  
Figure 2: Flowchart for TEC modeling and validation.