A Time-Inhomogeneous Markov Model for Resource Availability under Sparse Observations

TL;DR

The paper tackles predicting smart-city resource availability under sparse observations by introducing a cyclic time-inhomogeneous Markov process with cycle-specific transition matrices $A_x$ and a per-cycle training regime. It develops complete-data and sparse-data parameter estimation, including a modified Baum–Welch algorithm and a fast heuristic, and validates the approach on Canberra and Melbourne parking datasets, showing improvements over time-homogeneous baselines and non-cyclic methods. The combination of short-term information from recent observations with long-term, time-of-day–aware patterns enables accurate short- to medium-term predictions, with practical applicability to routing and augmented reality systems; future work explores hierarchical spatial dependencies. The contribution is a scalable, embedded-systems–friendly framework that can jointly model clusters of spatially related resources while handling missing data effectively.

Abstract

Accurate spatio-temporal information about the current situation is crucial for smart city applications such as modern routing algorithms. Often, this information describes the state of stationary resources, e.g. the availability of parking bays, charging stations or the amount of people waiting for a vehicle to pick them up near a given location. To exploit this kind of information, predicting future states of the monitored resources is often mandatory because a resource might change its state within the time until it is needed. To train an accurate predictive model, it is often not possible to obtain a continuous time series on the state of the resource. For example, the information might be collected from traveling agents visiting the resource with an irregular frequency. Thus, it is necessary to develop methods which work on sparse observations for training and prediction. In this paper, we propose time-inhomogeneous discrete Markov models to allow accurate prediction even when the frequency of observation is very rare. Our new model is able to blend recent observations with historic data and also provide useful probabilistic estimates for future states. Since resources availability in a city is typically time-dependent, our Markov model is time-inhomogeneous and cyclic within a predefined time interval. To train our model, we propose a modified Baum-Welch algorithm. Evaluations on real-world datasets of parking bay availability show that our new method indeed yields good results compared to methods being trained on complete data and non-cyclic variants.

Figures (5)

• Figure 1: Exemplary visual representation of a cyclic Markov chain with period length $p=5$ and two states. $S_{t,i}$ is used as a shorter notation for $q_t = S_i$. The edges denote the respective transition probabilities. Note that not all edges are labeled as the picture becomes unreadable otherwise. The edges between $S_{i,1}$ and $S_{i+1,2}$ should be labeled with $a_{i,1,2}$, the edges between $S_{i,2}$ and $S_{i+1,1}$ with $a_{i,2,1}$.
• Figure 2: Example of determining edge counts for the training algorithm described in \ref{['subsec:estimate_heuristic']}. $S_{t,i}$ is used as a shorter notation for $q_t = S_i$. Assume we have observed state $S_1$ at $t=1$ and $S_2$ at $t=4$. There exist four distinct paths from $S_{1,1}$ to $S_{4,2}$. Each edge is annotated with the number of paths visiting it. These counts will be determined for each pair of adjacent observations, combined and normalized to obtain a state transition probability distribution.
• Figure 3: Predictions and true parking availability for the next 10 hours after a measurement in the morning, for one resource in the Melbourne testing set. Colors of models are the same in all graphs. The algorithms HEUR and STD yield identical models when trained on complete data, thus the graph of HEUR is completely covered by the graph of STD.
• Figure 4: Mean absolute errors (MAE) of the different models for resource clusters containing 5 to 20 parking bays (Melbourne dataset). The MAE is normalized by the number of bays, so a value of 0.1 corresponds to an absolute mean error of 2.0 for a cluster containing 20 bays. These graphs show that the algorithm BW is able to take advantage of knowing the recent measurement at short prediction intervals, but also is able to predict longer intervals, especially if data is not too sparse.
• Figure 5: Average runtimes of algorithms for the evaluations shown in Figure \ref{['fig:results1']}. Note that the vertical axis is logarithmic.