# RSS Indoor Localization Based on a Single Access Point
## 2 Related Work
### Fingerprint-Based Positioning Methodologies  
Since the proposed approach primarily focuses on indoor RSS-based positioning, the main localization methods corresponding to such types of systems are presented below.  
Two main types of localization algorithms are used in RSS-based indoor positioning platforms--deterministic and probabilistic. The _deterministic_ algorithms estimate location \(\widehat{\ell}\) as a convex combination of the \(K\) reference locations [17]. \(\widehat{\ell}\) is calculated as the shortest distance between \(\overline{r}_{i}\) and \(s\) in the \(n\)-dimensional space by using the following equation:  
\[\widehat{\ell}=\sum_{i=1}^{K}\left(\frac{w_{i}}{\sum_{j=1}^{K}w_{j}}\ell_{i}^{ \prime}\right) \tag{1}\]  
The data set \(\{\ell_{1}^{\prime},\ldots,\ell_{l}^{\prime}\}\) denotes the reference locations between the respective fingerprint \(\overline{r}_{i}\) and the observed measurement during positioning \(s\), with respect to an increasing distance--that is, \(\|\overline{r}_{i}-s\|\). The distance can be calculated using standard norms, such as the Manhattan (1-norm) [18], the Euclidean (2-norm) [19], or the Mahalanobis norm [20].  
In its simplest form, the prescribed algorithm can assume \(K=1\), resulting in the simple Nearest Neighbour (NN) method [19; 21]. If several candidate locations are taken into consideration, but with equal weight factor \(w_{i}\), the formula represents the \(K\)-Nearest Neighbour (KNN) method. In more complex environments, for improved accuracy, the full implementation of the Weighted \(K\)-Nearest Neighbour (WKNN) [18] is employed by setting the non-negative weight factor \(w_{i}\) in Equation (1) as the inverse of \(\|\overline{r}_{i}-s\|\).  
In probabilistic methods, location \(\ell\) is estimated by calculating and maximising the conditional posterior probabilities \(p(\ell_{i}|s),\ i=1,\ldots,l\) given an observed fingerprint \(s\) and a fingerprint database. In order to estimate the expected value of \(\ell\), the method may implement either the Maximum A Posteriori (MAP) [22] or the Minimum Mean Square Error (MMSE) approach [23].  
The posterior probability \(p(\ell_{i}|s)\) is obtained by applying [PERSON] rule:  
\[p(\ell_{i}|s)=\frac{p(s|\ell_{i})p(\ell_{i})}{\sum_{i=1}^{l}p(s|\ell_{i})p( \ell_{i})}, \tag{2}\]  
where \(p(s|\ell_{i})\) is a conditional probability calculated through statistics at the survey stage, and \(p(\ell_{i})\) is the a priori probability. The a priori probability is a weighting factor based on the probability distribution of the target over the reference position candidates that exist in the fingerprint data set. Similarly to the deterministic method, the formula can be simplified if no prior knowledge is assumed. In such a case, this _prior_ can be assumed to be a unity, meaning that an equal a priori probability exists for all fingerprint candidates.  
Both deterministic and probabilistic methods are implemented in more sophisticated fingerprint-based hybrid solutions, where data and information are fused in an effort to improve localization accuracy. Such approaches may combine different wireless technologies, such as Wi-Fi and Bluetooth Low Energy (BLE) beacons [4; 5], Wi-Fi with Visual Light Positioning (VLP) [6], and VLP with ultra sound [7]. Hybrid solutions are also presented, including utilization of map information [8], inertial data, or other sensors' input [9].