Human activity spatio-temporal indicators using mobile phone data
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In the context of Smart Cities, monitoring the dynamic of the presence of people is a crucial aspect for the well-being of an urban area. We use mobile phone data as a proxy for the total number of people (Carpita & Simonetto 2014), with the specific aim of computing spatio-temporal region specific indicators. Telecom Italia Mobile (TIM), which is the largest operator in Italy, thanks to a research agreement with the Statistical Office of the Municipality of Brescia, provided to us about two years (April 2014 to June 2016) of High-Frequency Daily Mobile Phone Density Profiles (DMPDPs) in the form of a regular grid polygon each 15 minutes. Densities have to be rescaled in order to express the total amount of people rather than just TIM users. Separately for selected regions in the province of Brescia, characterized by being either working or residential areas, we group similar DMPDPs and we characterize groups by their spatial and temporal components. In doing so, we propose a mixed-approach procedure.
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Human activity spatio-temporal indicators using mobile phone data
- 1. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material Humam Activity Spatio-Temporal Indicators by using Mobile Phone Data Rodolfo Metulini, Maurizio Carpita Data Methods and Systems Statistical Laboratory - Department of Economics and Management, University of Brescia 1/15
- 2. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material Context & Objective • This work use mobile phone data provided by Telecom Italia Mobile (TIM), which is currently the largest operator in Italy in this sector (∼ 1/3 of the market). Similar data used by Carpita & Simonetto (2014) Secchi et al. (2017), Zanini et al. (2016), Manfredini et al. (2015), Finazzi, Paci (2017, 2018). • Data are characterized by a 2-D spatial component (i.e. a raster made of nxn cells) and by a temporal component (i.e. each cell has repeated values in time, one each 15 minutes). • The aim is to to find reference daily profiles by clustering similar days in terms of the spatial and the temporal structure. 2/15
- 3. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material The application • We select the grid of the city of Brescia (lat/long [10.18,10.245,45.516,45.564] made of 39 x 39 150 m2 square cells, • at 15-minutes intervals (quarters) over the period September 1st, 2015 - August 10th, 2016. • We input missing quarters and remove the full day when they are too many, • ending up with a number of 330 days. 3/15
- 4. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material The Approach Step Action Aim Method Using .. 1 group days find similar raster images histogram of ori- ented gradients (HOG) HOG features 2 group groups of days find similar densities functional model-based clustering daily density profiles 3 characterize groups find reference daily profiles functional box plots daily density profiles 4/15
- 5. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material Histogram of Oriented Gradients I From a nxn raster data .... 93 124 77 ... ... 217 55 94 ... ... 24 77 109 ... ... ... ... ... ... ... ... ... ... ... ... ...to Xt , a matrix representing the number of people in that cell at time t 1 Define Zt = Xt /max(Xt ) *100; 2 split Zt in 3x3 = 9 matrices Zt,c ; 3 for each Zt,c compute the matrices of gradients Gx and Gy using the sobel operator; 4 define each element of the direction matrix as g = arctan gy gx ; 5 define each element of the magnitude matrix as θ = g2 x + g2 y ; 6 assign each value of the direction matrix to one of the 6 bins of the histogram using its magnitude as weight. 5/15
- 6. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material Histogram of Oriented Gradients II 1 Stack into a vector the features of the 96 quarters of the same day, producing the matrix H˜t ; 2 K-mean cluster to group days (H’s columns) in terms of the HOG features (H’s rows); 3 6 groups, by looking to the decreasing of the within deviance total deviance by increasing the clusters. quart. feat. day1 day2 ... day ˜T 1 1 h11,1 h21,1 ... h ˜T1,1 1 2 h11,2 h21,2 ... h ˜T1,2 1 ... ... ... ... ... 1 k h11,k h21,k ... h ˜T1,k ... ... ... ... ... ... 96 k h196,k h296,k ... h ˜T96,k q q q q q q q q q q q q q q q 2 4 6 8 10 12 14 0.40.50.60.70.80.91.0 Number of Clusters Withingroups/Totalsumofsquares Advantages: • It permits to pass from a 2D raster data to a 1D vector by preserving the “spatial” structure of the data. • It reduces dimensionality: we describe a raster of 1521 values with 54 HOG features, with a dimensionality reduction of order 1521/54 = 28.17. 6/15
- 7. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material Cluster HOG features: results Red: n= 41 weekends from September to Xmas holidays Orange: n = 90 week days from September to Xmas holidays Yellow: n = 47 week days of Summer Green: n =35 Saturdays from January to August Light blue: n = 81 week days from January to May Blue: n=36 Sundays (except September & October) 7/15
- 8. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material Clustering HOG: results II Daily density profiles: Week days of summer, yellow cluster Daily density profiles: Weekends September - Xmas holidays, red cluster 04:00 0 20 40 60 80 100 120 140 160 12:00 0 20 40 60 80 100 120 140 160 20:00 0 20 40 60 80 100 120 140 160 04:00 0 20 40 60 80 100 120 140 160 12:00 0 20 40 60 80 100 120 140 160 20:00 0 20 40 60 80 100 120 140 160 8/15
- 9. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material Step 2. Daily curves clustering 1 Separately for each group, we remove abnormal curves using functional data outlier detection by likelihood ratio test (LRT) (fda package), as proposed by Febrero-Bande et al. (2008); 2 we apply the cluster method developed by Bouveyron et al. (2015) along with funFEM package in R: • curves are modelled by smoothing a Fourier basis (basis=9); • the command automatically choose for the best model among alternatives applying constraints on the parameters of the matrix Σk (var-cov matrix of the latent expansion coefficients of the curves); • number of groups using BIC over the range [2:7]; • random initial values for the prior probability πk . Why a Model-Based Function Data Analysis approach? • It is more flexible. • Each group corresponds to a distribution with specific parameters. 9/15
- 10. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material Daily curves clustering: results Estimated daily profiles by group (left) and groups’ centroid (right): Week days of summer, yellow cluster (2 outliers removed with LRT) 0 20 40 60 80 300000400000500000 time value 0 20 40 60 80 350000450000 time value Estimated daily profiles by group (left) and groups’ centroid (right): Weekends from September to Xmas, red cluster (4 outliers removed with LRT) 0 20 40 60 80 200000300000400000500000 time value 0 20 40 60 80 250000350000450000 time value 10/15
- 11. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material Step 3. Confidence Intervals with Functional Box Plots • The analog of the traditional box plot for curves, proposed by Sun & Genton (2011) • Curves are ordered using the concept of “band depth” • A curve is an outlier if it exceed 1.5 * envelope in at least one point. 11/15
- 12. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material Functional Box Plots: results Functional box plot for daily profiles (in thousands of people). Week days of summer, yellow cluster. 0 20 40 60 80 303540455055 June 0 20 40 60 80 303540455055 July 0 20 40 60 80 303540455055 August 12/15
- 13. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material Future directions • To Increase the sample size by adding repeated measure of the same day, in order to have more robust functional box plots • To apply a coefficient for TIM’s market share to estimate the total number of people 13/15
- 14. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material References 1 Bouveyron, C., Come, E., & Jacques, J. (2015). The discriminative functional mixture model for a comparative analysis of bike sharing systems. The Annals of Applied Statistics, 9(4), 1726-1760. 2 Carpita M., Simonetto A. (2014). Big Data to Monitor Big Social Events: Analysing the mobile phone signals in the Brescia Smart City. Electronic Journal of Applied Statistical Analysis: Decision Support Systems, Volume 5, Issue 1, pp 31-41, DOI: 10.1285/i2037-3627v5n1p31 3 Finazzi, F., & Paci, L. (2017, June). Space-time clustering for identifying population patterns from smartphone data. In SIS 2017 Statistics and Data Science: new challenges, new generations (pp. 423-428). Firenze University Press. 4 Finazzi, F., & Paci, L. (2018). A comparison of statistical methods for estimating individual location densities from smartphone data. In ITISE 2018-International conference on Time Series and Forecasting (pp. 1471-1482). Godel Impresiones Digitales SL. 5 Febrero, M., Galeano, P., & Gonzalez-Manteiga, W. (2008). Outlier detection in functional data by depth measures, with application to identify abnormal NOx levels. Environmetrics: The official journal of the International Environmetrics Society, 19(4), 331-345. 6 Manfredini, F., Pucci, P., Secchi, P., Tagliolato, P., Vantini, S., & Vitelli, V. (2015). Treelet decomposition of mobile phone data for deriving city usage and mobility pattern in the Milan urban region. In Advances in complex data modeling and computational methods in statistics (pp. 133-147). Springer, Cham. 7 Secchi, P., Vantini, S., & Zanini, P. (2017). Analysis of Mobile Phone Data for Deriving City Mobility Patterns. In Electric Vehicle Sharing Services for Smarter Cities (pp. 37-58). Springer, Cham. 8 Sun, Y., & Genton, M. G. (2011). Functional boxplots. Journal of Computational and Graphical Statistics, 20(2), 316-334. 9 Tomasi, C. (2012). Histograms of oriented gradients. Computer Vision Sampler, 1-6. 10 Zanini, P., Shen, H., & Truong, Y. (2016). Understanding resident mobility in Milan through independent component analysis of Telecom Italia mobile usage data. The Annals of Applied Statistics, 10(2), 812-833. 14/15
- 15. Humam Activity Indicators Metulini Carpita Context & Objective The Approach Step 1: Cluster HOG features Step 2. Daily curves clustering Step 3. Functional Box Plots Conclusions References Supplementary material FD Model-Based Clustering by “Bouveyron et al., 2015” We know values xi on a finite set of ordered times but we do not know the functional expressions of the observed curves. We define X(t) = p j=1 γj (X)ψj (t) to be a p basis expansion of X. The aim is to predict the value zi = (zi1, ..., ziK ) of the unobserved random variable Z = (Z1, ..., ZK ) for each observed curve xi . The marginal distribution on Γ reads as a mixture of Gaussians: p(λ) = K k=1 πk φ(γ; Uµk , Ut Σk Ut + Ξ) φ is the Gaussian density function and πk is the prior probability of the k-th group. Γ = UΛ + . Λ = λ1, ..., λn is the latent expansion coefficients of the curves x1, ..., xn and, conditionally on Z, it distributes as a multivariate Gaussian density ∼ N(µk , Σk ), where µk and Σk are the mean and the covariance matrix of the k-th group. 15/15