Probabilistic Modelling for Unsupervised Analysis of Human Behaviour in Smart Cities

Abstract

The growth of urban areas in recent years has motivated a large amount of new sensor applications in smart cities. At the centre of many new applications stands the goal of gaining insights into human activity. Scalable monitoring of urban environments can facilitate better informed city planning, efficient security, regular transport and commerce. A large part of monitoring capabilities have already been deployed; however, most rely on expensive motion imagery and privacy invading video cameras. It is possible to use a low-cost sensor alternative, which enables deep understanding of population behaviour such as the Global Positioning System (GPS) data. However, the automated analysis of such low dimensional sensor data, requires new flexible and structured techniques that can describe the generative distribution and time dynamics of the observation data, while accounting for external contextual influences such as time of day or the difference between weekend/weekday trends. In this paper, we propose a novel time series analysis technique that allows for multiple different transition matrices depending on the data's contextual realisations all following shared adaptive observational models that govern the global distribution of the data given a latent sequence. The proposed approach, which we name Adaptive Input Hidden Markov model (AI-HMM) is tested on two datasets from different sensor types: GPS trajectories of taxis and derived vehicle counts in populated areas. We demonstrate that our model can group different categories of behavioural trends and identify time specific anomalies.

Keywords: probabilistic modelling; smart city; time series; trajectory analysis.

Figure 1

iHMM (top) and AR-iHMM (bottom) probabilistic graphical models. Mixing Parameters $\mathit{\beta }$ are used to sample the transition distribution ${\mathit{\pi }}_k$, which the state indicators $\mathbf{z}$ are sampled from. The observations ${\mathbf{y}}_t$ are generated from functions with parameters ${\mathsf{\Theta }}_{z_t}$ which are in turn sampled from base distribution H. The AR-iHMM differs such that there are autoregressive dependence between observations.

Figure 2

PGM of the AI-HMM structure. Observation ${\mathbf{y}}_t$ being dependent on the indicator variable $z_t$ and p previous observations, whereas $z_t$ is dependent on $z_{t-1}$ in addition to the semi-Markov feature ${\tau }_t$. Parameters $\mathit{\beta }$ are used to sample the transition distribution ${\mathit{\pi }}_k$ which the state indicators $\mathbf{z}$ are sampled from. The observations ${\mathbf{y}}_t$ are generated from functions with parameters ${\mathsf{\Theta }}_{z_t}$ which are in turn sampled from base distribution H.

Figure 3

Scatter plots for vehicular density counts on an average game day on a weekday (left) and weekend (right). The first two distinct peaks on the weekday represent the morning and afternoon traffic peaks respectively, whereas the sharp peaks around 23:00 on the weekday and on 16:30 on the weekend represent traffic caused by fans leaving the stadium after a baseball game.

Figure 4

Scatter plots for vehicular density counts. Each colour represents a specific state that the respective observations was clustered into. Morning and evening traffic peaks on weekdays are clustered into state 1 (blue) and state 3 (green), respectively. Traffic caused by evening baseball games near 23:00 (Wednesday, Friday and Saturday) have been clustered into state 6 (brown), whereas traffic peaks caused by afternoon games at about 16:30 (Sunday) were clustered into state 7 (pink).

Figure 5

Figure depicting the first four (PCs with the largest eigen values in the PCA analysis of the Dodgers loop sensor dataset). PC 2 capture the dynamics of the morning peak; PC 3 represents the sharp traffic rise due to the evening baseball game, which commonly occurs on weekdays and Saturdays; and PC 4 captures the dynamics of the weekend light midday traffic.

Figure 6

Results obtained from analysing 10 taxi trajectories using the novel AI-HMM framework (left) and the GRU technique (right). Each data point corresponds to a GPS observation where its colour represents the state to which it was assigned. The plots depict the longitude $\lambda$ against latitude l reconstructions of the movements of the taxi locations on a map.

Figure 7

Prediction of the value of ${\tau }_t$ corresponding to the taxi identity on the test. The top plot shows the sample labels, whereas the bottom plot shows the predicted ${\tau }_t$ values using the AI-HMM with an accuracy of 93%.