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Human interaction discovery in smartphone proximity networks

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Abstract

Since humans are fundamentally social beings and interact frequently with others in their daily life, understanding social context is of primary importance in building context-aware applications. In this paper, using smartphone Bluetooth as a proximity sensor to create social networks, we present a probabilistic approach to mine human interaction types in real life. Our analysis is conducted on Bluetooth data continuously sensed with smartphones for over one year from 40 individuals who are professionally or personally related. The results show that the model can automatically discover a variety of social contexts. We objectively validated our model by studying its predictive and retrieval performance.

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Acknowledgments

This work was funded by Nokia Research Center Lausanne (NRC) through the LS-CONTEXT project.

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Authors and Affiliations

  1. Idiap Research Institute, Martigny, Switzerland

    Trinh Minh Tri Do & Daniel Gatica-Perez

  2. École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Daniel Gatica-Perez

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  1. Trinh Minh Tri Do
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  2. Daniel Gatica-Perez
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Correspondence to Trinh Minh Tri Do.

Additional information

Based on “GroupUs: Smartphone Proximity Data and Human Interaction Type Mining”, by Trinh Minh Tri Do and Daniel Gatica-Perez which appeared in the Proceedings of the International Symposium on Wearable Computers, San Francisco, California, June 2011. ©2011 IEEE.

Appendix: mathematical derivation of GroupUs learning algorithm

Appendix: mathematical derivation of GroupUs learning algorithm

Begin with the joint distribution

$$ \begin{aligned} &P({\bf u},{\bf v},{\bf c},{\bf s},{\bf t};\varvec{\alpha},\varvec{\beta}) \\ &\quad =\int\limits_\theta P({\bf t}\vert\theta) P(\theta;\varvec{\alpha}) \partial \theta \\ &\qquad\times \int\limits_{\phi} P({\bf u}\vert {\bf t},\phi_1) P({\bf v}\vert {\bf t},\phi_2) P({\bf c}\vert {\bf t},\phi_3) P(\phi;\varvec{\beta}) \partial \phi \\ &\quad = \int\limits_\theta \left( \prod_s \prod_t \theta_{st}^{n_{st}} \prod_s \frac{\prod_t \theta_{st}^{\alpha-1}}{B(\varvec{\alpha})} \right) \partial \theta\\ &\qquad \times \int\limits_{\phi_1} \left( \prod_t \prod_u \phi_{1tu}^{m_{tu}} \prod_t \frac{\prod_u \phi_{1tu}^{\beta-1}}{B(\varvec{\beta})} \right) \partial \phi_1 \\ &\qquad \times \int\limits_{\phi_2} \left( \prod_t \prod_v \phi_{2tv}^{m_{tv}} \prod_t \frac{\prod_v \phi_{2tv}^{\beta-1}}{B(\varvec{\beta})} \right) \partial \phi_2\\ &\qquad \times \int\limits_{\phi_3} \left( \prod_t \prod_c \phi_{3tc}^{m_{tc}} \prod_t \frac{\prod_c \phi_{3tc}^{\beta-1}}{B(\varvec{\beta})} \right) \partial \phi_3 \\ &\quad = \prod_s \int\limits_{\theta_s} \frac{\prod_t \theta_{st}^{\alpha+n_{st}-1}}{B(\varvec{\alpha})} \partial \theta_s \times \prod_t \int\limits_{\phi_{1t}} \frac{\prod_u \phi_{1tu}^{\beta+m_{tu}-1}}{B(\varvec{\beta})} \partial \phi_{1t} \\ &\qquad \times \prod_t \int\limits_{\phi_{2t}} \frac{\prod_v \phi_{2tv}^{\beta+p_{tv}-1}}{B(\varvec{\beta})} \partial \phi_{2t} \times \prod_t \int\limits_{\phi_{3t}} \frac{\prod_c (\phi_{3tc})^{\beta+q_{tc}-1}}{B(\varvec{\beta})} \partial \phi_{3t} \end{aligned} $$

where the term inside integration has similar form as Dirichlet distribution. Note that \( \int_x \frac{\prod_i x_i^{\alpha-1}}{B(\varvec{\alpha})}\partial x = 1, \) we have:

$$ \begin{aligned} &P({\bf u},{\bf v},{\bf c},{\bf s},{\bf t};\varvec{\alpha},\varvec{\beta})\\ &\quad = \prod_s \frac{B(\varvec{\alpha} + {{\bf n}}_s)}{B(\varvec{\alpha})} \times \prod_t \frac{B(\varvec{\beta} + {{\bf m}}_t)}{B(\varvec{\beta})} \times \prod_t \frac{B(\varvec{\beta} + {{\bf p}}_t)}{B(\varvec{\beta})} \times \prod_t \frac{B(\varvec{\beta} + {{\bf q}}_t)}{B(\varvec{\beta})} \\ &\quad = \prod_s \frac{B(\varvec{\alpha} + {{\bf n}}_s)}{B(\varvec{\alpha})} \times \prod_t \frac{B(\varvec{\beta} + {{\bf m}}_t)}{B(\varvec{\beta})} \frac{B(\varvec{\beta} + {{\bf p}}_t)}{B(\varvec{\beta})} \frac{B(\varvec{\beta} + {{\bf q}}_t)}{B(\varvec{\beta})} \end{aligned} $$

The conditional probability can be computed efficiently based on the fact that they are proportional to the joint probability:

$$ \begin{aligned} P(t_i \vert {\bf u},{\bf v},{\bf c},{\bf t}_{\neg i};\alpha,\beta) \\ & = \frac{P({\bf u},{\bf v},{\bf c},{\bf t};\alpha,\beta)}{P({\bf u},{\bf v},{\bf c},{\bf t}_{\neg i};\alpha,\beta)} \\ & \propto \frac{P({\bf u},{\bf v},{\bf c},{\bf t};\alpha,\beta)}{P({\bf u}_{\neg i},{\bf v}_{\neg i},{\bf c}_{\neg i},{\bf t}_{\neg i};\alpha,\beta)} \\ & \propto \frac{\prod_s \frac{B(\varvec{\alpha} + {{\bf n}}_s)}{B(\varvec{\alpha})} \times \prod_t \frac{B(\varvec{\beta} + {{\bf m}}_t)}{B(\varvec{\beta})} \frac{B(\varvec{\beta} + {{\bf p}}_t)}{B(\varvec{\beta})} \frac{B(\varvec{\beta} + {{\bf q}}_t)}{B(\varvec{\beta})}}{\prod_s \frac{B(\varvec{\alpha} + {{\bf n}}^{\neg i}_s)}{B(\varvec{\alpha})} \times \prod_t \frac{B(\varvec{\beta} + {{\bf m}}^{\neg i}_t)}{B(\varvec{\beta})} \frac{B(\varvec{\beta} + {{\bf p}}^{\neg i}_t)}{B(\varvec{\beta})} \frac{B(\varvec{\beta} + {{\bf q}}^{\neg i}_t)}{B(\varvec{\beta})}} \\ & \propto \frac{B(\varvec{\alpha} + {{\bf n}}_{s_i})} {B(\varvec{\alpha} + {{\bf n}}^{\neg i}_{s_i})} \times \frac{B(\varvec{\beta} + {{\bf m}}_{t_i})} {B(\varvec{\beta} + {{\bf m}}^{\neg i}_{t_i})} \times \frac{B(\varvec{\beta} + {{\bf p}}_{t_i})} {B(\varvec{\beta} + {{\bf p}}^{\neg i}_{t_i})} \times \frac{B(\varvec{\beta} + {{\bf q}}_{t_i})} {B(\varvec{\beta} + {{\bf q}}^{\neg i}_{t_i}) } \\ & \propto \frac{\alpha + n^{\neg i}_{s_it_i}}{\sum_t (\alpha + n^{\neg i}_{s_it})} \frac{\beta + m^{\neg i}_{t_iu_i}}{\sum_u (\beta + m^{\neg i}_{t_iu})} \frac{\beta + p^{\neg i}_{t_iv_i}}{\sum_v (\beta + p^{\neg i}_{t_iv})} \frac{\beta + q^{\neg i}_{t_ic_i}}{\sum_c (\beta + q^{\neg i}_{t_ic})} \end{aligned} $$

Since the denominator \(\sum_t (\alpha + n^{\neg i}_{s_it})\) is invariant for any value of t i , we obtain the final sampling equation :

$$ \begin{aligned} P(t_i \vert {\bf u},{\bf v},{\bf c},{\bf t}_{\neg i};\alpha,\beta) &\propto (\alpha + n^{\neg i}_{s_it_i}) \frac{\beta + m^{\neg i}_{t_iu_i}}{\sum_u (\beta + m^{\neg i}_{t_iu})} \\ & \times \frac{\beta + p^{\neg i}_{t_iv_i}}{\sum_v (\beta + p^{\neg i}_{t_iv})} \frac{\beta + q^{\neg i}_{t_ic_i}}{\sum_c (\beta + q^{\neg i}_{t_ic})} \end{aligned} $$

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Do, T.M.T., Gatica-Perez, D. Human interaction discovery in smartphone proximity networks. Pers Ubiquit Comput 17, 413–431 (2013). https://doi.org/10.1007/s00779-011-0489-7

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