Analytical investigation of intersection based range-free localization

Abstract

The localization of mobile devices is essential for the provisioning of location-based services, e.g., to locate people facing an accident or to provide relevant information to device users, depending on their current whereabouts. Several localization mechanisms have been developed using estimates of absolute distances or angles between the devices and the base stations of the networks. These mechanisms often require expensive enhancements of the existing base stations or mobile devices. In recent years, so-called range-free approaches have been proposed, which limit the possible positions of a device to the coverage areas of radio network cells, without relying on precise distances or angles. The accuracy of the corresponding information can be refined by computing the intersection area of all cells that cover the current position of the device. However, the computation of this intersection area, e.g., by the location server of a network carrier, can be a complex task. To avoid unnecessary workload, one would like to preestimate the possible reduction of location uncertainty, i.e., the information gain that can be achieved. The contribution of this paper is an analytical and numerical investigation of the problem. Several approaches are presented for the computation of the information gain, based on stochastic geometry and on a Monte-Carlo method. We show that simple scaling arguments can be used to estimate the order of magnitude of the average information gain, while more complex approximations based on Voronoi cells lead to relatively good results.

Appendix

Let us now give some brief explanations, as well as a few references concerning translation invariance and ergodicity for random tessellations of the plane. Recall that a Poisson–Voronoi tessellation of the plane may be defined by considering a planar Poisson point process $\left\lbrace X_i\right\rbrace$ and attaching a convex polygonal cell C _i to each of the random points X _i in a very simple fashion: C _i consists of all points Y in the plane lying closer to X _i than to any of the other points X _j in the Poisson cloud (see Fig. 5). Such a construction may certainly be carried out for any Poisson point process in the plane; however, the resulting tessellation will be translation-invariant only if the underlying Poisson random cloud has constant intensity λ. The translation-invariance property we have in mind is the following: denoting by T _V :Y↦Y + V the translation along an arbitrarily fixed planar vector V, one asks for the statistical properties of the random tessellation $\left\lbrace C_i\right\rbrace$ to be left unchanged when applying T _V to each of the cells, i.e., changing $\left\lbrace C_i\right\rbrace$ into $\left\lbrace T_V\left( C_i\right) \right\rbrace$. A more precise statement of this property requires considering an arbitrary bounded measurable functional Ψ of the random tessellation $\left\lbrace C_i\right\rbrace$, as well as an arbitrary planar translation T _V , and requiring that the mean value of $\Psi\left[ \left\lbrace C_i\right\rbrace\right]$ be left unchanged when composing Ψ with T _V :

$$\mathbb{E}\left( \Psi\left[ \left\lbrace T_V\left( C_i\right)\right\rbrace\right]\right) = \mathbb{E}\left( \Psi\left[\left\lbrace C_i\right\rbrace\right]\right)$$

(one may think of Ψ as a geometric characteristic of the polygonal cell covering the origin O in the plane, e.g., number of vertices, perimeter, or area, even though such reasonable functionals are not bounded). A Poisson–Voronoi tessellation of the plane satisfying such translation-invariance property is said to be stationary.

Of course, one might be interested in the stationarity of some other kind of random tessellation, for example, the Poisson line tessellations mentioned in Section 5.2, where a straight line Δ _i is drawn through each point X _i of the random cloud, orthogonally to the segment joining O and X _i (see Fig. 6). In the latter case, however, the intensity of the underlying Poisson random cloud should be taken proportional to $d\rho d\theta=\frac{dxdy}{\sqrt{x^2+y^2}}$ in order for the resulting line tessellation to be stationary. Indeed, assuming that the lines in such random tessellation are being translated along some fixed vector V = (V ₁;V ₂), consider a particular line Δ and its translate Δ′, as well as the point X on Δ lying closest to the origin O and the point X′ on Δ′ lying closest to O. Denoting by (ρ;θ) the polar coordinates of X and by (ρ′;θ′) those of X′, one obviously has θ′ = θ or θ′ = θ±π (Δ′ is parallel to Δ), whereas

$$\rho'=\rho+(V_{\!1}\cos\theta + V_2\sin\theta)=\rho+\left\langle V;U_{\theta}\right\rangle .$$

(see Fig. 16). Thus, for an arbitrary bounded measurable function (ρ;θ)↦F(ρ;θ), one certainly has

$$\int\int F(\rho';\theta')d\rho'd\theta'=\int\int F(\rho;\theta)d\rho d\theta,$$

because the corresponding Jacobian is unitary, but this identity breaks down when replacing the differential dρ dθ, e.g., by ρd ρd θ = dx dy.

Fig. 16

The effect of a translation on a Poisson line tessellation

One of the main consequences of such translation-invariance property lies in the validity of Wiener’s ergodic theorem: assuming that $\left\lbrace C_i\right\rbrace$ is a stationary random tessellation of the plane and that Ψ is a square integrable measurable functional of $\left\lbrace C_i\right\rbrace$ (i.e., such that $\mathbb{E}\left( \left\vert\Psi\left[\left\lbrace C_i\right\rbrace\right]\right\vert^2\right)$ be finite), one may assert the almost sure existence of the limit¹

$$\lim_{R\rightarrow +\infty} \frac{1}{|B_R|}\underset{B_R}{\int\int} \Psi\left[ \left\lbrace T_V\left( C_i\right)\right\rbrace\right] dV =L(\Psi),$$

where B _R stands for the Euclidian ball of radius R centered at the origin, and |B _R | for the corresponding area (see [20] for a series of precise statements of such theorems and their proofs). At this stage, some further condition on the random tessellation $\left\lbrace C_i\right\rbrace$ is needed to make sure that the limit L(Ψ) coincides (almost surely) with the stochastic mean value $\mathbb{E}\left( \Psi\left[\left\lbrace C_i\right\rbrace\right]\right)$. A very natural sufficient condition of this kind is the following: the stationary random tessellation $\left\lbrace C_i\right\rbrace$ is said to satisfy a mixing condition whenever

$$\lim_{R\rightarrow +\infty} \mathbb{E}\left( \Phi\left[\left\lbrace C_i\right\rbrace\right] \cdot\Psi\left[ \left\lbrace T_{R.V}\left( C_i\right)\right\rbrace\right]\right) = \mathbb{E}\left( \Phi\left[\left\lbrace C_i\right\rbrace\right]\right) \cdot\mathbb{E}\left( \Psi\left[\left\lbrace C_i\right\rbrace\right]\right),$$

holds true for any unit vector V and bounded measurable functionals Φ, Ψ. Roughly speaking, this means that the geometric characteristics of two parts of the random tessellation located far from each other are nearly uncorrelated. Such property holds true for any of the stationary random tessellations $\left\lbrace C_i\right\rbrace$ we have been considering so far, and as a consequence, one may assert the validity of the identity

$$L(\Psi)\!=\!\! \lim_{R\rightarrow +\infty} \frac{1}{|B_R|}\underset{B_R}{\int\!\int} \!\Psi\left[ \left\lbrace T_V\left( C_i\right)\right\rbrace\right] dV \!=\! \mathbb{E}\left( \Psi\left[\left\lbrace C_i\right\rbrace\right]\right),$$

almost surely in the realizations of $\left\lbrace C_i\right\rbrace$. Readers interested in rigorous applications of Wiener’s or Birkhoff’s ergodic theorems to stochastic geometry should certainly consult [21] or [22]. Some of the main applications of this kind go under the name of “mean-value relationships” for stationary random tessellations. Here is an important example of such mean value relationships (many more may be found in Section 10.3 of [23]): denoting by λ ₀ (resp. λ ₂) the mean number of vertices (resp. cells) of the tessellation $\left\lbrace C_i\right\rbrace$ to be found per unit area, and by $\overline{n}_{0,2}$ the mean number of cells touching a vertex in this tessellation, one has

$$\overline{n}_{0,2}=2+\frac{2\lambda_2}{\lambda_0} .$$

In the simple situation where $\left\lbrace C^A_i\right\rbrace$ is the Poisson–Voronoi tessellation attached to a Poisson random cloud of constant intensity λ, one has λ ₀ = 2λ and λ ₂ = λ, so that $\overline{n}_{0,2}=3$. In fact, in such a simple situation, there are exactly three polygons meeting at each of the vertices, almost surely in the realizations of $\left\lbrace C^A_i\right\rbrace$. Let us consider a further Poisson–Voronoi tessellation $\big\{ C^B_j\big\}$ attached to a second Poisson random cloud of constant intensity μ, independent of the first one. In the compound tessellation $\big\lbrace C^{Comp}_k\big\rbrace$ obtained by superimposing the cells in $\big\lbrace C^B_j\big\rbrace$ to those of $\left\lbrace C^A_i\right\rbrace$, there are three kinds of vertices:

• Vertices already present in the tessellation $\left\lbrace C^A_i\right\rbrace$: these come up with intensity 2λ and are adjacent to three polygons in the compound tessellation.

• Vertices already present in the tessellation $\big\lbrace C^B_j\big\rbrace$: these come up with intensity 2μ and are adjacent to three polygons in the compound tessellation.

• Vertices appearing at the intersection of an edge in $\left\lbrace C^A_i\right\rbrace$ with an edge in $\left\lbrace C^B_j\right\rbrace$: these come up with intensity $\frac{8}{\pi}\sqrt{\lambda\mu}$ and are adjacent to four polygons in the compound tessellation. (The original paper [24] should be consulted for more details on this third intensity).

As a result of the preceding considerations, one may state that the mean number of polygons adjacent to a vertex in the compound tessellation is given by

$$\overline{n}^{Comp}_{0,2}=\frac{6(\lambda+\mu)}{2\lambda + 2\mu+\frac{8}{\pi}\sqrt{\lambda\mu}} +\frac{\frac{32}{\pi}\sqrt{\lambda\mu}}{2\lambda + 2\mu+\frac{8}{\pi}\sqrt{\lambda\mu}} .$$

Using the above stated mean-value relationship then leads to the following value Λ^Comp for the cell intensity in the compound tessellation:

$$\begin{array}{rll}\Lambda^{Comp}(\lambda,\mu)&=&\left(2\lambda + 2\mu+\frac{8}{\pi}\sqrt{\lambda\mu}\right)\cdot\left(\frac{\overline{n}^{Comp}_{0,2}}{2}-1\right)\\ &=&\lambda+\mu+\frac{8}{\pi}\sqrt{\lambda\mu}. \end{array}$$

The very same method may be used to determine the cell intensity of a compound tessellation where a third Poisson–Voronoi tessellation built upon a new Poisson cloud of intensity ν is superimposed to the preceding compound tessellation: the resulting cell intensity is then

$$\Lambda^{Comp}(\lambda,\mu,\nu)\!=\! \lambda\!+\!\mu\!+\!\nu\!+\!\frac{8}{\pi}\left(\sqrt{\lambda\mu}\!+\!\sqrt{\lambda\nu}\!+\!\sqrt{\mu\nu}\right).$$

More generally, superimposing J Voronoi tessellations built upon independent homogeneous Poisson clouds having the intensities λ ₁, λ ₂, ..., λ _J yields a stationary random tessellation whose cell intensity is given by

$$\Lambda^{Comp}(\lambda_1, \lambda_2, \ldots, \lambda_J)=\sum_{j=1}^J\lambda_j +\frac{8}{\pi}\sum_{1\leq j<k\leq J}\sqrt{\lambda_j\lambda_k}.$$

Let us finally give a brief description of some of the main results contained in [17]. { X _i } denoting a homogeneous Poisson cloud of unit intensity λ = 1 on the plane, recall that the Palm cell $\mathcal{C}=\mathcal{C}\left( \{ X_i\}\right)$ attached to { X _i } may be defined as the convex polygon consisting of all points X lying closer to the origin O than to any point X _i in the cloud (see Fig. 17). In other words, $\mathcal{C}$ is the Voronoi cell about O appearing when adding this point to the cloud. Letting $N_0(\mathcal{C})$ denote the number of vertices (or, equivalently, the number of sides) appearing in the random convex polygon $\mathcal{C}$, it turns out that a proper use of Slyvniak’s formula (Proposition 4.1.1 in [18]) enables one to compute $\mathbb{P}\left\lbrace N_0(\mathcal{C})=k\right\rbrace$ explicitly for arbitrary k ≥ 3, and one has:

$$\begin{array}{l} \mathbb{P}\left\lbrace N_0(\mathcal{C})=k\right\rbrace \\[6pt] \qquad =\displaystyle{\frac{(2\pi)^k}{k!} \int_{\Sigma_k}}d\sigma_k(\delta_1,\ldots,\delta_k)\displaystyle{\int_{\mathbb{R}_+^k}}dp_1\ldots dp_k\\[6pt] \qquad \quad \displaystyle{\prod_{i=1}^k} p_i e^{-H(\delta_i,p_i,p_{i+1})}\cdot \boldsymbol{1}_B(p_{i-1},p_i,p_{i+1},\delta_{i-1},\delta_i), \end{array}$$

(15)

where dσ _k (δ ₁,...,δ _k ) stands for the uniform probability measure on the (k − 1)-dimensional simplex

$$\Sigma_k=\left\lbrace (\delta_1,\ldots,\delta_k)\in [0;2\pi]^k:\sum_{i=1}^k\delta_i=2\pi\right\rbrace ,$$

whereas H is the function given by

$$\begin{array}{rll}H(\delta,p,q)\!&=&\! \frac{1}{2\sin(\delta)^2} \left\{\frac{\delta}{2}(p^2\!+\!q^2\!-\!2pq\cos\delta)\right. \\&& \left.\qquad\quad\; +pq\sin\delta\!-\!\frac{p^2}{4}\sin2\delta\!-\!\frac{q^2}{4}\sin2\delta\right\} \end{array}$$

and $\boldsymbol{1}_B$ stands for the indicator function of

$$\begin{array}{rll}B&=&\Big\{ (p,q,r,\alpha,\beta)\in (\mathbb{R}_+)^3\times(0;\pi)^2: p\sin\beta\\&&\ \ +r\sin\alpha\geq q\sin(\alpha+\beta)\Big\}. \end{array}$$

Fig. 17

Palm cell about the origin for a given cloud.

In the multiple integral appearing in Eq. 15, δ ₁,..., δ _k are angular variables, whereas p ₁,...,p _k stand for distances to the origin, and these variables are labelled cyclically, so that p ₀ = p _k , p _k + 1 = p ₁ and δ ₀ = δ _k .

Following [17], we further denote by (P _i ,Θ _i ) the polar coordinates of the vertices of the random cell $\mathcal{C}$, for $i=1,2,\ldots,N_0(\mathcal{C})$. Fixing again a natural number k ≥ 3, it turns out that, conditionally to $\left\lbrace N_0(\mathcal{C})=k\right\rbrace$, the joint distribution of the vector (P ₁,...,P _k ,Θ₂ − Θ₁,...,Θ _k  − Θ _k − 1,2π + Θ₁ − Θ _k ) may be explicitly described as the measure ν _k on $\mathbb{R}_+^k\times[0;2\pi]^k$ given through

$$\begin{array}{l} d\nu_k(p_1,\ldots,p_k,\delta_1,\ldots,\delta_k)\\[6pt] {\kern10pt} =\varphi_k(p_1,\ldots,p_k,\delta_1,\ldots,\delta_k)\\[6pt] {\kern21pt} \; \cdot dp_1\ldots dp_kd\sigma_k(\delta_1,\ldots,\delta_k), \end{array}$$

where dσ _k (δ ₁,...,δ _k ) again stands for the uniform pro bability measure on the (k − 1)-dimensional simplex Σ _k , whereas

$$\begin{array}{l} \!\varphi_k(p_1,\ldots,p_k,\delta_1,\ldots,\delta_k)\\ {\kern9pt} =\displaystyle{\frac{(2\pi)^k}{k!\cdot\mathbb{P}\left\lbrace N_0(\mathcal{C})=k\right\rbrace} \prod_{i=1}^k} p_i e^{-H(\delta_i,p_i,p_{i+1})}\\[6pt] {\kern21pt} \; \cdot\boldsymbol{1}_B(p_{i-1},p_i,p_{i+1},\delta_{i-1},\delta_i) \end{array}$$

In other words, conditionally to the event $\left\lbrace N_0(\mathcal{C})=k\right\rbrace$, the probability for (P ₁,...,P _k , Θ₂ − Θ₁,...,Θ _k  − Θ _k − 1, 2π + Θ₁ − Θ _k ) to realize any event $E\subset\mathbb{R}_+^k\times[0;2\pi]^k$ is given as the quotient of the integral

$$\underset{E}{\int \int}dp_1\ldots dp_k d\sigma_k(\delta_1,\ldots,\delta_k) \prod_{i=1}^k p_i e^{-H(\delta_i,p_i,p_{i+1})} \cdot \boldsymbol{1}_B(p_{i-1},p_i,p_{i+1},\delta_{i-1},\delta_i)$$

through a similar integral computed over the whole of $\mathbb{R}_+^k\times[0;2\pi]^k$.

This provides us with an explicit description of the behavior of the random cell $\mathcal{C}$. From there on, mean values such as $\mathbb{E}\left[ |\mathcal{C}|\right]$ or $\mathbb{E}\left[ |\mathcal{C}|\cdot\log|\mathcal{C}|\right]$ may readily be computed.