## 2 Basic Concepts and Work Scope
### Fingerprinting-based Positioning
As all of the pattern-matching methods considered in this article are based on NN models, the testing phase is equivalent to the inference phase used in the deep learning field and its literature. In this phase, in practice, when fingerprinting-based positioning is used as the primary positioning method, there is  
Figure 4: Distinction between the training and testing phases (left and right subfigures, respectively) in fingerprinting-based positioning, with a pattern-matching method based on supervised learning using a neural network (NN). Figure 3: Example of fingerprints from three sources for positioning in 2D. The fingerprints from each source are represented as a 3D continuous surface, where the fingerprint measure is a function of the position coordinates. A position is represented by an ideally unique combination of fingerprints. no need for a secondary positioning system to produce a ground truth reference position. However, for research purposes, the ground truth reference position is collected anyway in order to measure the performance of the pattern-matching method in use in terms of the positioning error. The mapping from a set of fingerprints to a position is carried out using a _pattern-matching method_. Figure 4 (right) and Figure 5 show the mapping from a set or vector of fingerprints provided as the input to an output position estimate. Figure 5 shows this concept, in which a set of input fingerprints are, as an example, arranged into a vector of fingerprints. The fingerprints belong to a _fingerprint domain_. The pattern-matching method maps the input fingerprints to a position estimate in the _position domain_. Many pattern-matching methods and associated algorithms have been developed for estimating a position from a set or vector of fingerprints. Some examples of these are look-up tables [36], databases [37], \(k\)-Nearest Neighbors (\(k\)NN) [38], fuzzy logic [39], decision trees [40], probabilistic methods [41], genetic algorithms [42], support vector machines (SVMs) [43], and artificial neural networks (NNs). Look-up tables, also known as databases, map a sampled set of fingerprints to a ground truth reference position. These are implemented in the form of tables in databases, matrices, or any suitable data structures and are combined with a given general-purpose algorithm (e.g., a greedy algorithm or least-squares minimization) to estimate the position by finding the closest set of fingerprints that matches a given input. Another way to map a set of fingerprints to a position is to train a suitable model such that it takes a set of fingerprints as the input and returns a set of position coordinates as the output. Ultimately, the goal is to generalize the model that maps the input set of fingerprints to a position estimate such that it returns as its output a position estimate with a statistically minimal positioning error. An overview of the machine learning methods used in fingerprinting-based positioning and the motivation for using pattern-matching methods based on machine learning is discussed in [44; 45; 46; 47; 48]. In fingerprinting-based positioning, the typical use of an FFNN with supervised learning and a regression-based model consists of providing as input to the FFNN a set or vector of \(N\) fingerprints from \(N\) fingerprint sources. The training is carried out to match labeled data. The labeled data consist of 2D or 3D ground truth position coordinates associated with the input fingerprints. Then, in the testing phase, an estimate of the position is obtained from the output of the FFNN for a given input set or vector of fingerprints. In the context of this article, we work only in the Euclidean plane; hence, we use as the positioning error the _error distance_. The error distance is the Euclidean distance between the ground truth reference position and the estimated position.