## 4 Fingerprinting-Based Positioning with Spatial Side Information
### Pattern-Matching Methods
#### 4.4.4 Pattern-Matching Method 4-MultFingIn-ClassOut
It is under debate whether the architecture of CNNs alone, the introduction of multiple convolutional layers, the introduction of pooling layers, data augmentation, or a combination of the former contributes to the translation invariance property. In [210], translation invariance is discussed in terms of a local translation, enabled by pooling functions, for small translations in the input. In [211], it is concluded that for the processing of images, the invariance increases with the depth of the CNN. In [212], it is concluded that CNNs alone are not translation-invariant but that the main contribution to the invariance is the training with the help of data augmentation. It is argued that CNNs are not shift-invariant due to the presence of downsampling, introduced by layers like strided max-pooling [213, 214]. We implemented a CNN without pooling and without data augmentation in the translation domain (meaning that we augmented data by introducing a random shadow fading component into each sub-tile, but the base path loss component remained constant for the position of each sub-tile, and thus, there was no augmentation through translation in the fixed tiles used for training). Thus, according to the claim of some works in line with those cited above, our CNN does not benefit from the translation invariance property. Yet as a matter of fact, the CNN implemented manages to find with a high probability fixed tiles that resemble moving tiles. Given that the tile-classes from the fixed tiles present a variation in a pattern that can be considered almost continuous, it is unclear to us whether the good performance observed is because the fixed tiles are identified by moving tiles with a close resemblance and/or because the good performance is enabled by the translation invariance. A reasoning in line with the possible contribution of the translation invariance is based on the assumption that the pattern described by a fixed tile, notwithstanding the noise component introduced by the shadow fading, can be found in the parts of a moving tile regardless of the position of the pattern in the tile. Parts of these patterns can be at different positions in the moving tile compared with in a fixed tile. Then, if translation invariance exists, we would expect to observe that the parts of a moving tile can be identified with a higher probability in fixed tiles around the moving tile. \(\circ\): Tile-class Probabilities and Data Aggregation Domains It is open to discussion whether the processing of the tile-class probabilities, along with the known tile center positions, is regarded as positioning data aggregation in the position domain. The input fingerprints are aggregated in the fingerprint domain by the CNN; therefore, there is not a one-to-one mapping from an individual fingerprint to each tile-class probability. Therefore, we do not treat this step as an aggregation in the position domain, in as much as we do not treat as aggregation in the position domain the last layers in the pattern-matching method 3-MultFingIn-PosOut, which ultimately aggregates the NN data to calculate the final output position estimate.