## 1 Introduction  
The Kalman filter is one of the algorithms most commonly used for the localization of a vehicle. Localization of a self-driving car is a sensor fusion process based on a global positioning system (GPS), on-board sensor, inertial measurement unit (IMU) and range sensors like those used for light detection and ranging (LiDAR) and those for radio detection and ranging (RADAR). Different sensors have different noise characteristics. To process the different sensors with different noise perception characteristics, the Kalman filter processes the uncertainty based on the variance numerically. There are different kinds of applications of the Kalman filter family under investigation [1; 2; 3; 4; 5; 6; 7; 8; 9]. Most researchers involved defined the variance of prediction and measurement empirically, meaning not only that a lot of time is needed to tune the variance parameters, but also that it is unknown if the result generated is optimal for a given situation or not. [PERSON] provided an accurate self-localization method based on an extended Kalman filter (EKF) by using radio frequency identification (RFID) [10]. Like most Kalman filter literature, they presented a novel prediction model, a measurement model and supposed the variance of the model with experience. Therefore, the final result could be improved with a parameter training algorithm and the training process of variance could be automated. Some of the literature has presented a training method based on fuzzy logic in the localization field [11; 12]. Researchers constructed the fuzzy logic with rules geared to learning the pattern of the variance. However, the rules are made by researchers manually and according to their experience. The final estimation result is directly affected by the experiences of the researchers. In fuzzy logic, there is no ground truth data and evaluation parameters are used, so it is hard to verify the performance of the filter when it is training. Since the neural network is an online algorithm, it needs additional computation resources. The deep Kalman filter is a recent parameter learning method [13]. The recurrent neural network is used to enhance the Kalman filter with arbitrarily complex transition dynamics and emission distributions. However, like all deep neural networks, the operation flow of the algorithm is a black box which means it is not reliable. Some researchers have implemented the neural network to the Kalman filter to fuse the IMU error [14]. The recurrent neural network is used to generate the error term and variance of the prediction model and measurement model. However, this method adds to the vehicle computing unit's online computational burden. In previous research, we proposed the extended particle-aided unscented Kalman filter (PAUKF) for localizing a self-driving car based on a pre-defined map [15; 16]. The basic PAUKF and extended PAUKF improved the performance by combining the advantages of the particle filter (PF) with those of the unscented Kalman filter (UKF). To achieve a better result, we took plenty of time to tune the parameters of the extended PAUKF manually. In the extended PAUKF filter framework, the particle filter is used for matching the position of the features to the ground truth (GT) position on the map. The estimated x, y position and yaw angle of the PF become the more precise measurement input to the UKF. The uncertainty of the estimated result from the PF is decided by using the variances. However, it is hard to measure the measurement variances of the PAUKF. In this paper, to determine the measurement variances of the filter, a discriminative PAUKF training algorithm is proposed based on previous research [17; 18]. The purpose of the discriminative training method is to find the optimal measurement variances of the PAUKF. The coordinate descent algorithm is used as a training algorithm to find the optimal value of the measurement variances.