In-stream mobility and speed estimation of mobile devices from mobile network data

Abstract

The cellular network is now nearly an almost ubiquitous and real-time sensor with coverage anywhere and anytime for any device. Mobile network data is a rich source for official statistics, such as human mobility. However, unlike GPS tracks, each mobile device in this data is described without precise knowledge of its spatial characteristics. Furthermore, there is no information about the device’s mobility status (i.e., whether it is moving or not) or speed which are important for behavioral analysis. Common mobility and speed estimations rely on precise location and do not consider privacy leakage risk. In this work, we propose two probabilistic approaches that estimate respectively devices’ mobility and devices’ speed from cellular data and connection likelihood maps for each network cell. Every estimation is computed in a short time and with a short history of data (for speed and for mobility). This constraint may be helpful with the most stringent legal frameworks for mobile operators including the combination of ePrivacy Directive and General Data Protection Regulation (GDPR) in Europe. The proposed approaches are the first we are aware of that allows for both mobility and speed estimation in this context. We experimented on two datasets, obtained from a mobile network operator’s signaling data and the associated GPS tracks of many consenting users. Our speed estimations are over 20% more accurate than common ones based on mobile sites and we provide confidence intervals for each estimation. Mainly due to mobile network uncertainty, our approach for speed estimation are relatively inaccurate at low speeds and the movement detection could remain unclear. However our approach for mobility estimation fills this gap.

Appendices

Appendix A: Proof of Eq. 3

To demonstrate Eq. (A), we rely rigourously on Garnier et al. (2019, pages 123–127).

Let us consider $\begin{array}{lrcl} g_0 : &{} {\mathbb {R}}^4 &{} \longrightarrow &{} {\mathbb {R}}_+ \\ &{} (x_1,y_1, x_2, y_2) &{} \longmapsto &{} \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \\ \end{array}$

which is measurable.

Therefore,

$$\begin{aligned} \begin{aligned} D_{12}&= \sqrt{(X_2-X_1)^2 + (Y_2-Y_1)^2}\\&= g(X_1, Y_1, X_2, Y_2) \end{aligned} \end{aligned}$$

(19)

We introduce now $\begin{array}{lrcl} g' : &{} {\mathbb {R}}^4 &{} \longrightarrow &{} {\mathbb {R}}^3 \times {\mathbb {R}}_+ \\ &{} (x_1,y_1, x_2, y_2) &{} \longmapsto &{} (x_1,y_1, x_2, \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}) \\ \end{array}$

The function $g'$ is clearly not bijective. However, we can partition ${\mathbb {R}}^4$ such that

$$\begin{aligned} \begin{aligned} {\mathbb {R}}^4&= \{(x_1, y_2, x_2, y_2) \in {\mathbb {R}}^4 / y_2 < y_1\}\\&\quad \cup \{(x_1, y_1, x_2, y_2) \in {\mathbb {R}}^4 / y_2 > y_1\}\\&\quad \cup \{(x_1, y_1, x_2, y_2) \in {\mathbb {R}}^4 / y_2 = y_1\}\\ \end{aligned} \end{aligned}$$

(20)

and note these 3 sets respectively $E_1$, $E_2$, and $E_3$ so ${\mathbb {R}}^4 = E_1 \cup E_2 \cup E_3$. Let us remark that $E_1$ and $E_2$ are open sets of ${\mathbb {R}}^4$ for the Euclidean norm. This can be shown by saying that $E_1$ (resp. $E_2$) are the reciprocal image of the open space $]0, +\infty [$ (resp. $]-\infty , 0[$) by the continuous application $\begin{array}{lrcl} k : &{} {\mathbb {R}}^4 &{} \longrightarrow &{} {\mathbb {R}} \\ &{} (x_1,y_1, x_2, y_2) &{} \longmapsto &{} y_2-y_1 \\ \end{array}$

We can also notice that $E_3$ is a linear subset of ${\mathbb {R}}^4$ which is a normed linear subset. $E_3$ being not equal to ${\mathbb {R}}^4$, its interior is empty and its measure is null.

Let us note h a bounded and continuous function defined from ${\mathbb {R}}^4$ to ${\mathbb {R}}$. Therefore,

$$\begin{aligned} \begin{aligned} {\mathbb {E}}[h \circ g'(X_1, Y_1, X_2, Y_2)]&= \int _{E_1}h \circ g'_1(x_1, x_2, y_1, y_2)f_1(x_1,y_1)f_2(x_2, y_2)\\&\quad + \int _{E_2}h \circ g'_2(x_1, x_2, y_1, y_2)f_1(x_1,y_1)f_2(x_2, y_2)\\&\quad + \int _{E_3}h \circ g'_3(x_1, x_2, y_1, y_2)f_1(x_1,y_1)f_2(x_2, y_2) \end{aligned} \end{aligned}$$

(21)

thanks to the assumption of independence of $(X_1, Y_1)$ and $(X_2, Y_2)$ and omitting the integrands in each integral. The third integral (over $E_3$) equals to 0 because the interior of $E_3$ is empty, and we want to do the change of variable $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ in the two other.

Then, we consider $\begin{array}{lrcl} g'_{1} : &{} E_1 &{} \longrightarrow &{} F \\ &{} (x_1,y_1, x_2, y_2) &{} \longmapsto &{} (x_1,y_1, x_2, \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}) \\ \end{array}$

with $F = \{(x_1, y_2, x_2, d) \in {\mathbb {R}}^3 \times {\mathbb {R}}^*_+ / d > |x_2-x_1|\}$.

The function $g'_{1}$ is a bijection continuously differentiable, therefore its inverse function can be defined as $\begin{array}{lrcl} g'^{(-1)}_{E_1} : &{} F &{} \longrightarrow &{} E_1 \\ &{} (x_1,y_1, x_2, d) &{} \longmapsto &{} (x_1,y_1, x_2, y_1 - \sqrt{d^2 - (x_2-x_1)^2}) \\ \end{array}$

The associated Jacobian matrix is therefore

$$\begin{aligned} J_1 = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ -\frac{x_2-x_1}{\sqrt{d^2 - (x_2-x_1)^2})} &{}\quad 1 &{}\quad \frac{x_2-x_1}{\sqrt{d^2 - (x_2-x_1)^2})} &{}\quad -\frac{d}{\sqrt{d^2 - (x_2-x_1)^2})} \end{pmatrix} \end{aligned}$$

(22)

and the absolute value of its determinant is

$$\begin{aligned} |det(J_1)| = \frac{d}{\sqrt{d^2 - (x_2-x_1)^2}} \end{aligned}$$

(23)

We can do the same with $E_2$ and define the bijective continuously differentiable function $\begin{array}{lrcl} g'_{2} : &{} E_2 &{} \longrightarrow &{} F \\ &{} (x_1,y_1, x_2, y_2) &{} \longmapsto &{} (x_1,y_1, x_2, \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}) \\ \end{array}$

and therefore obtain

$$\begin{aligned} |det(J_2)| = \frac{d}{\sqrt{d^2 - (x_2-x_1)^2}} \end{aligned}$$

(24)

Thus, we can show that

$$\begin{aligned} {\mathbb {E}}[h \circ g'(X_1, Y_1, X_2, Y_2)] = \int _{{\mathbb {R}}^3}\int _{{\mathbb {R}}_+} h(x_1, x_2, y_1, d)f_1(x_1, y_1)g(d, x_1, x_2, y_1) \end{aligned}$$

(25)

with

$$\begin{aligned} \begin{aligned} g(d, x_1, x_2, y_1)&= \sum _{\pm }f_2[x_2,y_1 \pm \sigma (d, x_1, x_2)]\frac{d{\varvec{1}}_{d>|x_2-x_1|}}{\sigma (d, x_1, x_2)} \\ \sigma (d, x_1, x_2)&= \sqrt{d^2 - (x_2 - x_1)^2} \end{aligned} \end{aligned}$$

and ${\varvec{1}}$ being the indicator function.

Finally, we can obtain

$$\begin{aligned} f_{D_{12}}(d) = \int _{{\mathbb {R}}^3}f_1(x_1,y_1)g(d, x_1, x_2, y_1)dx_1dx_2dy_1 \end{aligned}$$

(26)

Appendix B: Direction estimation model

We consider a device making two events as in Sect. Main assumptions, and we define,

$$\begin{aligned} \Theta _{12} = \arctan (\frac{Y_2-Y_1}{X_2-X_1}) \end{aligned}$$

(27)

the random variable representing the direction of movement between $e_1$ and $e_2$. This is based on the third assumption made in Sect. Main assumptions, and because $(X_i, Y_i)$ are random variables giving respectively Lambert II coordinates (plane projection). We keep noting $f_i$ the spatial probability density of presence related to $(X_i, Y_i)$ and $C_i$ at time $t_i$, which can be obtain as in Sect. Cell coverage modelling. The second assumption made in Sect. Main assumptions (independence) is here necessary to show, in the same way as in Appendix A, that $\Theta _{12}$ has density $f_{\Theta _{12}}$, and for all $\theta \in [0, 2\pi [$,

$$\begin{aligned} f_{\Theta _{12}}(\theta ) = \int _{{\mathbb {R}}^3}f_1(x_1,y_1)g(\theta , x_1, x_2, y_1)dx_1dx_2dy_1 \end{aligned}$$

(28)

with

$$\begin{aligned} g(\theta , x_1, x_2, y_1)= & {} f_2[x_2, y_1 + (x_2 - x_1)\tan (\theta )]\sigma (\theta , x_1, x_2) \\ \sigma (\theta , x_1, x_2)= & {} (x_2-x_1)[1+\tan ^2(\theta )]{\varvec{1}}_{x_2 \ne x_1}{\varvec{1}}_{\theta \ne (k+\frac{1}{2})\pi , k \in {\mathbb {Z}}} \end{aligned}$$

and ${\varvec{1}}$ being the indicator function.

Integrating the density (28) over a direction interval $[\theta _a, \theta _b]$ included in

$$\begin{aligned}{}[0, 2\pi [ ~\setminus ~ \{(k+\frac{1}{2})\pi , k \in {\mathbb {Z}}\} \end{aligned}$$

and noting

$$\begin{aligned} E(\theta _a, \theta _b) = \{(x, y, x', y') \in {\mathbb {R}}^4 / y' \in [y + (x'-x)\tan (\theta _a), (x'-x)\tan (\theta _b)]\} \end{aligned}$$

(i.e., all the location pairs corresponding to an angle of more than $\theta _a$ and less than $\theta _b$), one obtains, after inversion and with the change of variable $y_2 = y_1 + (x_2-x_1)\tan (\theta )$, a much more convenient form for numerical calculation:

$$\begin{aligned} \begin{aligned} \int _{\theta _a}^{\theta _b} f_{\Theta _{12}}(\theta )d\theta&= \int _{{\mathbb {R}}^4}f(x_1, y_1, x_2, y_2)dx_1dx_2dy_1dy_2 \\&= P(\theta _a \le \Theta _{12} \le \theta _b) \end{aligned} \end{aligned}$$

(29)

with

$$\begin{aligned} \begin{aligned} f(x_1, y_1, x_2, y_2)&= f_1(x_1, y_1)h(x_1, y_1, x_2, y_2) \\ h(x_1, y_1, x_2, y_2)&= f_2(x_2, y_2){\varvec{1}}_{E(\theta _a, \theta _b)}(x_1, y_1, x_2, y_2) \end{aligned} \end{aligned}$$

(30)

At this point, we can do the same remarks as in Sect. Speed density: two network events but considering direction (angle) instead of distance or speed. Then we can follow the method defined in Sect. Oscillation phenomenon mitigation and Sect. Speed density: multiple network events with only slight changes in order to deduce a direction probability density (of a variable $\Theta$).

Then, computing a mean direction and a confidence interval requires a bit of caution. A direction probability density p is a circular distribution and its mean direction is ${\mathbb {E}}(\Theta ) = \arg (m)$ with $m = \int _{0}^{2\pi } p(\theta )e^{i\theta }d\theta$, i being the imaginary unit.

To the best of our knowledge, there is no computation of confidence intervals for circular distributions, but it is possible to estimate standard deviations. One possible estimation is the one propose in Yamartino (1984), $\sigma _{\Theta } = \arcsin (\epsilon )(1+(\frac{2}{\sqrt{3}} - 1)\epsilon ^{3})$ where $\epsilon = \sqrt{1-(Re(m)^2 + Im(m)^2)}$ where Re(m) and Im(m) are the real part and imaginary part of m.

However, this proposed variant to compute a direction of movement does not give as satisfactory results as the proposed speed estimation model. The direction estimated is accurate qualitatively (i.e., looking at some examples on a map), but it is not much more accurate than naive approaches (i.e., based on mobile sites and a rough direction estimation). Therefore, we left this as future work and only present in this paper the experimentation made for speed and mobility estimation (see Sect. Experimentation).

It is also possible to propose the same kind of approach for mobility estimation but based on direction (angle) densities, computed as in this “Appendix”. The intuition is that when a movement occurs, the direction density of a mobile device gets tight whereas it is wide when the mobile device is static (mainly because of the oscillation phenomenon and the relatively wide covered areas). We intend to investigate further this approach in future work.