## Materials and Methods
### Mathematical Models and Methods
#### 2.2.1 Modelling TS Occurrences within a Given Urban Road Environment  
As a first step, we are going to model the TS placements, occurrences, and detections--along a random route and within a certain urban road environment--jointly as events of a continuous-variable homogeneous marked Poisson process (HMPP).  
In the literature dealing with Poisson processes usually the mentioned continuous variable is the time. Although, with the notations used herein, as well as with the verbal expressions describing relations between values, we are going to comply with this \"temporal\" convention, and it should be emphasized that the path-length--that has been covered by the ego-car--is chosen to be the continuous spatial variable.  
Let \(\{T_{n},k_{n}\}\), where \(k_{n}\in\{1,2,\ldots,K\}\) and \(K\in\mathbb{N}\), be a marked Poisson process with counting measures \(N^{k}(\cdot)\). These are defined as \(N^{k}(A)=\#\{n:T_{n}\in A\}\), where \(A\) is typically an interval. Let the rates associated with the marked Poisson process be \(\lambda_{k}\), assuming spatial--and particularly along-the-route--homogeneity, and let the corresponding reference rates of the Poisson process be \(\lambda_{k}^{0}\). Then, the negative logarithm of the likelihood-ratio of an observation sequence \((\{T_{n},k_{n}\})\), \(T_{n}<T\) is given by  
\[D_{T}^{N}\triangleq T\cdot\sum_{k}\Bigl{(}\lambda_{k}-\lambda_{k}^{0}\Bigr{)} -\sum_{k}N^{k}(T)\cdot\log\left(\frac{\lambda_{k}}{\lambda_{k}^{0}}\right) \tag{1}\]  
where \(N^{k}(T)\) is the number of events of type \(k\) prior to \(T\).  
Assuming now that \(\lambda^{0}=\bigl{\{}\lambda_{k}^{0}\bigr{\}}\) is the true set of process parameters, furthermore assuming that \(\lambda=\{\lambda_{k}\}\) is a set of tentative parameter values, and writing \(D_{T}^{N}=D_{T}^{N}(\lambda)\), we have the following inequality:  
\[E\Bigl{\{}D_{T}^{N}(\lambda)\Bigr{\}}\geq 0. \tag{2}\]  
The left-hand side is simply the Kullback-Leibler (KL) divergence of the true distribution from the estimated one. Using the common notation of the KL divergence, the left-hand-side of the above inequality can be rewritten as  
\[E\Bigl{\{}D_{T}^{N}(\lambda)\Bigr{\}}=D_{KL}\Bigl{(}\text{mPois}\Bigl{(} \lambda^{0}T\Bigr{)}\parallel\text{mPois}(\lambda T)\Bigr{)}, \tag{3}\]  
where \(\text{mPois}(\cdot)\) represents the distribution corresponding to the marked Poisson process, while \(D_{KL}\bigl{(}\text{mPois}\bigl{(}\lambda^{0}T\bigr{)}\parallel\text{mPois}( \lambda T)\bigr{)}\) is the expected number of extra nats--NB: not bits, but nats, as the natural logarithm is used in Equation (1), not \(\log_{2}\)--required to encode the observation sequence from the distribution \(\text{mPois}\bigl{(}\lambda^{0}T\bigr{)}\) using a code optimized for the distribution \(\text{mPois}(\lambda T)\) rather than using the code optimized for \(\text{mPois}\bigl{(}\lambda^{0}T\bigr{)}\).  
Associated with \(D_{T}^{N}(\lambda)\) is the computable quantity  
\[L_{T}(\lambda)\triangleq T\cdot\sum_{k}\lambda_{k}-\sum_{k}N^{k}(T)\cdot\log( \lambda_{k}T) \tag{4}\]  
\(L_{T}(\lambda)\) can be interpreted as the approximate length of an optimal code encoding the observation sequence, were \(\lambda\) the true set of the process parameters. Considering, however, that \(L_{T}(\lambda)\) is dependent on \(\lambda^{0}\)--i.e., on the \"real\" true set of process parameters--via \(N^{k}\), we can write \(L_{T}(\lambda)=L_{T}\bigl{(}\lambda^{0},\lambda\bigr{)}\), and similarly, we can indicate the same kind of dependency for \(D_{T}^{N}\), i.e., \(D_{T}^{N}=D_{T}^{N}(\lambda)=D_{T}^{N}\bigl{(}\lambda^{0},\lambda\bigr{)}\).