## 3 Hybrid Stochastic Approach to Location Estimation
### Stochastic Approach for Location Estimation  
The location estimation problem can be statistically stated as follows. For simplicity, the true distribution \(\Pr(X=x)\) and \(\Pr(X=x\mid Y=y)\) are denoted as \(\Pr(x)\) and \(\Pr(y)\). The model parameters are denoted by \(\p(\mid)\).  
Let \(b\) be the number of base-stations. We denote \(o\) as an observation. The observation variable is a \(b\)-dimensional vector; one for each signal strength from each base-station. We denote as \(o^{j}\) the signal strength from base-station \(j\) for \(j\in\{1,\ldots,b\}\). We have a location \(l\) associated to each observation. In this work we use bi-dimensional locations for simplicity, but it can be used in three dimensions easily.  
The methodology used is based on the definition of a function \(\Pr(l\o)\) that returns the probability of the location \(l\), given the observation \(o\). This nomenclature has been used in other proposals [22, 28, 29]. Once this function is estimated, the problem can be formulated to find the location \(l\) that maximizes the probability \(\Pr(l\o)\) for a given observation \(o\). Using [PERSON] theorem, we can write:  
\begin{table}
\begin{tabular}{c l l l} \hline \hline
**Parameter** & **Description** & **Parameter** & **Description** \\ \hline \(l\) & found location & \(l\) & location of base-station \(j\) \\ \(o\) & current observation & \(d_{o}\) & reference distance for signal strength \\  & & & measured \\ \(b\) & number of base station & \(o_{d}\) & mean signal strength measured to \(d_{0}\) \\ \(o^{j}\) & signal strength from base-station \(j\) & \(d^{j}\) & Euclidian distance between \(l\) and \(l\) \\  & of \(o;j\!\in\!\{1,\ldots,b\}\) & & \\ \(T\) & set of training data & \(n\) & attenuation variation index \\ \(t\) & number of training samples & \(L_{w}^{j}\) & attenuation caused by the obstacles \\ \(T_{i}\) & training sample \(i\); \(i\!\in\!\{1,\ldots,t\}\); \(T_{i}=\) & \(w\) & number of wall crossed \\  & (\(l_{i}\), \(o_{i}\)) & & \\ \(l_{i}\) & location of training sample \(i\) & \(L_{\theta}\) & wall average attenuation \\ \(o_{i}\) & observation of training sample \(i\) & \(X_{\sigma}\) & zero-mean normal distributed random \\ \(o_{i}^{j}\) & signal strength of training sample & \(d_{i}^{j}\) & variable with standard deviation \(\sigma\). \\  & \(i\) from base-station \(j\) & & distance from \(l_{i}\) to \(l\) \\ \(B^{j}\) & base-station \(j\); \(j\!\in\!\{1,\ldots,b\}\); \(B^{j}=\) & \(L_{w}^{j}\) & wall attenuation obtained from \(l_{i}\) to \(l\) \\  & (\(l^{j}\), \(o_{i}^{j}\)) & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Variables.  
\[\Pr(l\mid o)=\frac{\Pr(l)\Pr(o\mid l)}{\Pr(o)} \tag{1}\]  
The denominator in equation (1) does not depend on the location variable \(l\). And, therefore, the location estimation problem can be presented as:  
\[l^{\prime}=\operatorname*{arg\,}_{l}\operatorname*{arg\,}_{l}\operatorname*{ arg\,}\Pr(l)\Pr(o\mid l) \tag{2}\]  
where Pr(_l_) is the _prior probability_ of the location \(l\), knowing the observation. This probability can be used to incorporate information such a more training locations [22] or tracking [29] to our statistical model. The tracking information will not be taken into account in this work, so, for the prior probability, we use the uniform distribution.  
In equation (1), \(\Pr(o|l)\) is the so called _likelihood function_. It estimates the probability of one observation given a location. In the literature, we find two main approaches to estimate this function: inductive approach and deductive approach. The next two subsections explain these methods analytically in order to propose our approach.