## 1 Introduction  
Errors of machine tools can be classified into quasi-static errors and dynamic errors [1]. Quasi-static errors include geometry errors [2], load-induced errors [3], etc., of which thermal error is also an very important part, greatly influencing the accuracy of the machine tools [4; 5], and may account for up to 40%-70% of the total error [6; 7]. Thermal error compensation is considered as one of the most effective methods to alleviate the influence of the thermal error [8; 9], in which the temperature field and the thermal error of the machine tool are firstly measured then, according to the measured data, thermal error model [10; 11; 12] is established to predict the thermal error of the machine tool and the prediction is transmitted into the numerical control (NC) system to complete the compensation. Among the whole procedure, temperature sensor location selection is considered an important step. Although a large number of temperature sensors are placed in the machine tool to obtain the whole temperature field containing detailed temperature information, not every one of them is required [13; 14; 15] in thermal error modeling. There are some reasons: firstly, more temperature sensors lead to a more complex thermal error model, greatly increasing the cost of software and hardware in the establishment of a compensation system; secondly, some sensors may contain similar information that would reduce the stability and accuracy of the model, and a bad selection of them would lead to a poor prediction of the thermal error model. Thus, proper temperature sensors are needed to be selected. A \"sensors-near-heat source\" method was firstly used for selection in which the selected sensors are near heat sources. It is obvious that the \"sensors-near-heat source\" method is not a scientific method and it may ignore some key sensors that not close to heat sources. Thus, some mathematical methods were introduced by researchers to guide the selection of proper temperature sensors [16; 17; 18; 19; 20]. [PERSON] [21] established an optimization objective function by a modified model adequacy criterion based on the Mallow's \(C_{p}\) statistic and used a discrete domain search method for the selection. [PERSON] [22] and [PERSON] [23] introduced grey correlation theory to select proper sensors. [PERSON] [24] used the Optimal Brain Surgeon (OBS) method to eliminate sensors with insignificant information. In recent years, clustering methods have also been used to select proper temperature sensors. The core concept of a clustering method is to cluster sensors into clusters (groups). Then, one most representative sensor of each cluster is selected to form the final selection. This kind of method has been the most widely used one since it can reduce redundant temperature information. Many clustering algorithms have been used by researchers, such as K-means clustering [7], fuzzy C-means clustering [25; 26; 27], etc. Additionally, new measuring instruments have also been used for the selection recently. [PERSON] [28] used a thermal imaging camera to obtain the thermal images and then key sensors were identified. In summary, all of the above methods can effectively select proper temperature sensors and also guarantee a high-accuracy thermal error model. It can be concluded from the above references that one common thing in these methods is deriving an indicator that describes the difference between the temperatures of the sensors. The indicator that can more effectively describe the real difference between sensors leads to a better selection. Therefore, in this paper, a new indicator was firstly derived. Considering the Euclidean distance between the value of temperature sensors can describe their physical distance on the machine tool, and the correlation coefficient depicts the similarity of the variation of the value of temperature sensors, they were both included in the new indicator.