Similarly, according to Equation (2), angular diameter distances can be calculated from observed angular sizes as
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and we also treated the length scale lm as a free parameter. In order to test the CDDR using different samples one should use a redshift matching criterion Δz  0.005 (Li et al. 2011; Liao et al. 2016). However, it turned out difficult to fulfill this criterion in order to compare distances derived from QSO [XUV] and QSO [CRS] directly. Therefore, proceeding in a similar manner as Li & Lin (2018), we reconstruct QSO [CRS] angular size as a function of redshift from the binned data. The angular size of intermediate-luminosity quasars was grouped into 20 redshift bins of width Δz = 0.1 starting from the smallest redshift of this sample. Median values of angular size plotted against the mean redshift in each bin are shown in Figure 4. The python package GaPP based on Gaussian Processes (Seikel et al. 2012) was used for the reconstruction process that depends on the mean function and the covariance function 





. In order to discuss the influence of the choice of these two prior functions, we studied four mean functions and three covariance functions. The prior mean functions that we discussed are the following: zero mean function, the theoretical function of angular size calculated from the angular diameter distance under the assumption of three cosmological models: flat ΛCDM with Ωm = 0.27, so-called Rh = ct universe (Melia & Shevchuk 2012), and a Mirage model with Ωm = 0.27, w0 = −0.7 and w1 = −1.09 (Shafieloo et al. 2012). The linear size scaling factor lm = 11.42 pc was assumed as calibrated with SN Ia in Cao et al. (2017a). These functions (except the zero mean function) are shown in Figure 2. There are many possible covariance functions and we studied the three most popular ones. They comprise: the squared exponential function
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the 







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and Cauchy covariance function
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where σf and ℓ are hyperparameters that control the amplitude of deviation from the mean function and the typical length scale in x-direction, respectively. It is instructive to discuss the effects of the mean function and covariance function selection (prior assumptions) on the reconstruction. In order to show the impact of covariance function choice, we fixed the zero mean and preformed reconstruction with three different covariance functions mentioned above. The result obtained under the assumption of squared exponential covariance function is illustrated in Figure 4, where the green solid line represents the reconstructed θ(z) relation and the green region around it represents a 1σ uncertainty band. The blue dashed line and black dashed–dotted line represent the reconstructed θ(z) relation with the Matérn and Cauchy covariance functions, respectively. Their uncertainty bands are not shown in order to not blur the picture since they are similar to the one displayed. One can see that differences between reconstructions performed with different choices of covariance function are insignificant. Similarly, we checked sensitivity of reconstructions with respect to the choice of the mean function fixing the covariance as a squared exponential one and using three main functions mentioned above. It turned out that the impact of mean function choice on the reconstruction was even smaller than that of the covariance function. Therefore for further calculations we assumed the zero mean function and the squared exponential covariance function to get the reconstructed θ(z) function (i.e., the green line and region in Figure 4). Using this reconstructed relation we were able to have a one-to-one matching between the QSO [CRS] angular diameter distance and the QSO [XUV] luminosity distance at the same redshift.
Citation: Melia & Shevchuk 2012)
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