## 2 Gaussian Process Data Modelling and Maximum Likelihood-Based Data Fusion Method
### Data Modelling and Data Fusion Principle from the View of Dimensional Measurement Science  
From a statistical point of view, the measurement dataset that has a smaller measurement uncertainty has a larger weighting in relation to the fused measurement value and vice versa. It is intuitively understood that there is a bias towards the measurement data which is more accurate. Equation (19) also shows that the fused uncertainty is smaller than any other uncertainty from the original measurement data. The fusion of two Gaussian distributions is illustrated in Figure 2. The horizontal axis denotes the measurement value at a specific position and the vertical axis denotes the probability of obtaining that measurement value. The two measurement datasets are of Gaussian distribution and are denoted as Data 1 and Data 2 in Figure 2. Data 2 is more accurate than Data 1 since Data 2 has a smaller standard deviation. The fused data has an even smaller standard deviation which illustrates that it has improved accuracy. Due to the nature of dimensional measurement, every measurement result has associated uncertainty and the true value is never known. Moreover, there are systematic errors and random errors contained in the measurement result. The systematic error denotes the bias from the estimated mean value to the true value while the random error complies with the Gaussian distribution of the measurement result which represents the measurement uncertainty. In the procedure of maximum likelihood-based multi-sensor data fusion, the fusion algorithm takes into account both the systematic error and random error and calculates the fused result in a statistical manner. It is easy to understand that the output uncertainty is reduced in the fusion result since the fusion procedure itself can be  
Figure 2: Fusion of two Gaussian distributions. considered as a multiple (double) measurement process which can reduce the measurement uncertainty compared to single measurement in the final result [22]. It is interesting to note that the weights for each dataset are determined by the associated uncertainties, i.e., the random errors. If a measurement has large systematic error but small random error while another measurement has small systematic error but large random error, the fused result will have a large bias to the measurement with small random error, which will introduce a large systematic error in the fused result. This issue also exists in other fusion methods such as weighted least square fusion [23]. This is especially true for state-of-the-art technology where the measurement instruments have high repeatability with very low measurement noise. However, the systematic error is sometimes large as compared to the random error, which means that the repeatability is high but the accuracy is not ensured (may deviate from the true value) [24]. On the other hand, with advanced technology such as the precision calibration method, the systematic error of CMMs is controlled well, which can reduce the influence of this effect. The influence of the systematic error and random error on the fused result under different situations is shown in Figure 3. With two different measurement datasets Data 1 and Data 2, it is assumed that Data 2 has smaller uncertainty than Data 1, the fused result Df has smaller uncertainty than both Data 1 and Data 2, and the mean of Df has a bias towards Data 2, which is shown in Figure 3. It is interesting to note that the true value (short as \(Tv\)) of the measurement is unknown and it can be considered under the following four situations: (a) \(Tv<M1<M2\); (b) \(M1<Tv<M2\) (bias to \(M1\)); (c) \(Tv<M1<M2\) (bias to \(M2\)); and (d) \(M1<M2<Tv\), where \(M1\) and \(M2\) are the mean values of Data 1 and Data 2. The case shown in Figure 3c is the ideal case, which has the optimal result that the fused result has the smallest bias to the true value. The case shown in Figure 3a is the worst case in which the fused result has the largest bias to the true value.