## 2 Basic Principles and Methodologies
### The Coefficient of Heating Effect
#### 2.3.1 RBF Approximation and the Adaptive Hybrid Kernel Method
Luckily, the integration of hybrid kernel methods has been demonstrated to enhance the overall reconstruction performance to a limited extent by adjusting the ratio of different basis functions in the hybrid kernel [28; 29]. The fundamental principle of the hybrid kernel method resides in the integration of multiple kernel functions, thereby synthesizing their complementary advantages to establish a versatile framework for addressing complex data challenges. For example, [PERSON] [29] applied a combination of the Gaussian and cubic (\(\Phi(r)=r^{3}\)) functions. [PERSON] [29] proposed a hybrid function using a combination of the cubic kernel and the exponential correlation function (\(\Phi(r)=e^{-\varepsilon r}\)). In interpolation scenarios where datasets concurrently exhibit localized nonlinearities and global trends, hybrid kernels demonstrate superior capability in balancing these competing fitting requirements. It is observed that both the cubic kernel and the exponential correlation function have their own disadvantages for accomplishing the reconstruction task. While the cubic function is finitely smooth and unhampered by shape parameter, using the cubic kernel may lead to the singular problem at some scattered locations. The exponential kernel does not fully balance the distances and randomness among the data points. Consequently, an improved hybrid kernel using a different basis function combination was proposed. As the most commonly used RBF, G-RBF has sharp spatial localization characteristics that enhance its suitability for reconstructing complex temperature fields in the furnace with less measured data [36; 38]. MQ-RBF is a reliable method for thermal image distribution reconstruction [26]. As shown in Equation (8), the value range of MQ-RBF is significantly broader than that of G-RBF. Furthermore, MQ-RBF is not sensitive to the change of shape parameters [36; 38], which leads to a wide optimal range of shape parameters and good global characterization. Therefore, an adaptive hybrid kernel method (AHK) based on MQ and G radial basis functions was proposed, as shown in Equation (11) and Figure 2. The purpose of this method is to dynamically adjust kernel weights based on the data, with a view to further enhancing generalization capabilities. \[\phi_{AHK}(r)=\kappa\sqrt{1+\varepsilon_{1}^{2}r^{2}}+\exp(-\varepsilon_{2}^{2 }r^{2}) \tag{11}\]where \(\epsilon_{1}\) and \(\epsilon_{2}\) are the shape parameters associated with the RBFs, which can be determined through parametric investigation. And \(\kappa\) is a weight coefficient that controls the contribution of the G and MQ kernel function in the improved hybrid kernel. This combination has the advantage that the involvement of the Gaussian kernel in the improved hybrid method helps to regulate when the weight coefficient \(\kappa\) is very small. The flow chart of the AHK method is shown in Figure 2. Due to the relationship between the value range of MQ and G, the \(\kappa\) in Equation (11) is initially set to 1 and the temperature field is first reconstructed mainly by the global properties of MQ. The grid-scale C\({}_{\text{HE}}\) is calculated and the complexity of the reconstructed temperature field is initially assessed. Subsequently, the proportion of G in the hybrid kernel is augmented by decreasing the \(\kappa\) value to accommodate the localized nonlinearities of the temperature phantom. It is imperative to repeatedly adjust the \(\kappa\) value until the calculated C\({}_{\text{HE}}\) value converges, thus enabling the determination of the finalized adaptive hybrid kernel. The finalized adaptive hybrid kernel from the AHK method combines the sharp spatial localization characteristics of G with the global characterization of MQ.