## 3 Data Analysis
### Statistical Analysis  
To validate the feasibility and scope of channel state prediction, we use ray-tracing data to analyze channel changes from one trajectory step to another Theoretically, the channel state can change over distances of a half wavelength. In practice, however, noticeable changes should occur at larger scales in most cases. Furthermore, we expect the process of spatial channel state evolution to retain some memory. Thus, in this section, we study at what distances between trajectory steps the memory is preserved to allow for prediction and interpolation. As we use frame-like data presentation, to analyze the communication channel's state evolution, we utilize and compare the following three different distance measures suitable for multivariate data:  
* The L2, or Euclidean, distance between \(V_{X}(s_{1})\) and \(V_{X}(s_{2})\), given by \[d_{\text{L2}}(s_{1},s_{2})=\sqrt{\sum\limits_{m=1}^{K_{\theta}}\sum\limits_{n= 1}^{K_{\theta}}(v_{m,n}(s_{1})-v_{m,n}(s_{2}))^{2}}.\] (6)
* The normalized cross-correlation (NCC) distance, employed to evaluate the similarity between images [35], which is computed via the Pearson correlation coefficient between \(V_{X}(s_{1})\) and \(V_{X}(s_{2})\) viewed as series of entries: \[d_{\text{NCC}}(s_{1},s_{2})=1-\frac{\sum\limits_{m=1}^{K_{\theta}}\sum\limits_{n =1}^{K_{\theta}}(v_{m,n}(s_{1})-\overline{v}(s_{1}))(v_{m,n}(s_{2})-\overline{ v}(s_{2}))}{\sqrt{\sum\limits_{m=1}^{K_{\theta}}\sum\limits_{n=1}^{K_{\theta}}(v_{ m,n}(s_{1})-\overline{v}(s_{1}))^{2}}\sqrt{\sum\limits_{m=1}^{K_{\theta}}\sum \limits_{n=1}^{K_{\theta}}(v_{m,n}(s_{2})-\overline{v}(s_{2}))^{2}}},\] (7) where \[\overline{v}(s)=\frac{1}{K_{\theta}K_{\varphi}}\sum\limits_{m=1}^{K_{\theta}} \sum\limits_{n=1}^{K_{\varphi}}v_{m,n}(s),\,s=1,\dots,S.\] (8)
* The image Euclidean distance (IMED), proposed in [36] specifically for images and takes into account spatial relationships of pixels, which is calculated as \[d_{\text{IMED}}(s_{1},s_{2})=\\ \sqrt{\frac{1}{2\pi}\sum\limits_{m,m^{\prime}=1}^{K_{\theta}}\sum \limits_{n,m^{\prime}=1}^{K_{\theta}}e^{-\frac{(m-m^{\prime})^{2}+(m-m^{ \prime})^{2}}{2}}(v_{m,n}(s_{1})-v_{m,n}(s_{2}))(v_{m^{\prime},n^{\prime}}(s_{ 1})-v_{m^{\prime},n^{\prime}}(s_{2}))},\] (9) with summation over all pixel coordinates \((m,n)\) and \((m^{\prime},n^{\prime})\). Figures 3-6 show distance or dissimilarity matrices for the studied scenarios. Here, distances are computed using the previous measures pairwise between all steps of a trajectory. The diagonal elements are zero as they represent the distance between a frame and itself. By analyzing the presented data, we can observe that AoA yields a substantially blurrier picture than AoD. This can be explained by the shift in the coordinates at Rx due to its movement and the corresponding shift of the reference point. As could be expected, the image-oriented distance measures better cope with such a shift compared to the L2 distance (cf., e.g., Figure 2(a),d vs. Figure 2(b),e, or Figures 4(a) and 4(b) vs. Figures 4(b) and 4(b)). It can be deduced that for channel state prediction in the context of AoA, more specific loss functions could be worth considering instead of the MSE closely related to the L2 distance. Let us now look in more detail at Figures 3 and 4. The plotted distance matrices confirm that the _SceLoS_ scenario is characterized by considerably more pronounced dynamics compared to _SceNLoS_, as can be seen from Figure 1. The sharp diagonal blocks in Figure 2(d)-f can be explained by beam clustering: it takes several trajectory steps for the strongest direct beam, in particular, to shift from one cluster to another, and this numbergrows as the angular distance between consecutive trajectory points diminishes. We also see that the distance matrices of both trajectories are characterized by pronounced diagonal blocks indicating trajectory segments with very similar data. However, sharp boundaries between the blocks may indicate difficulties in prediction when transitioning from one segment to another.