Trajectory data of floating cars form an important data source for the studies on transportation, while map matching is always one essential step for them. Most map matching algorithms perform better with trajectory data of high sampling rates than with those of low sampling rates, but the latter can be commonly accessed because of their low cost. In this article, we proposed a map matching algorithm based on then hidden Markov Model. In this algorithm, we concerned both position and direction information for calculating observation and transition probabilities and solved the labelling problem with the Viterbi algorithm by maximizing the state sequence probabilities. We carried out a case study with the GPS trajectory data of floating cars and road network data of Wuhan. The results show that this algorithm can effectively match trajectory data of low sampling rates with the road network with good topology, and the correct rate can reach up to 86% within an acceptable time cost. In particular, it performs well even in some error-prone scenarios, such as two-way multiple parallel lanes, intersections, overpasses and roundabouts. Furthermore, we also discussed factors that might affect the accuracy and efficiency of this algorithm, particularly investigating the effect of topology correctness of the road network. INDEX TERMS Map matching, hidden Markov model, low sampling rate, direction, road network.

As an established spatial analytical tool, Geographically Weighted Regression (GWR) has been applied across a variety of disciplines. However, its usage can be challenging for large datasets, which are increasingly prevalent in today's digital world. In this study, we propose two highperformance R solutions for GWR via Multi-core Parallel (MP) and Compute Unified Device Architecture (CUDA) techniques, respectively GWR-MP and GWR-CUDA. We compared GWR-MP and GWR-CUDA with three existing solutions available in Geographically Weighted Models (GWmodel), Multi-scale GWR (MGWR) and Fast GWR (FastGWR). Results showed that all five solutions perform differently across varying sample sizes, with no single solution a clear winner in terms of computational efficiency. Specifically, solutions given in GWmodel and MGWR provided acceptable computational costs for GWR studies with a relatively small sample size. For a large sample size, GWR-MP and FastGWR provided coherent solutions on a Personal Computer (PC) with a common multi-core configuration, GWR-MP provided more efficient computing capacity for each core or thread than FastGWR. For cases when the sample size was very large, and for these cases only, GWR-CUDA provided the most efficient solution, but should note its I/O cost with small samples. In summary, GWR-MP and GWR-CUDA provided complementary high-performance R solutions to existing ones, where for certain data-rich GWR studies, they should be preferred.

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Spatial heterogeneity is important for exploring data relationships between real estate price and its influential factors. The geographically weighted regression (GWR) technique has been frequently adopted for this purpose. In this study, we collected a second-hand real estate house price data set of Wuhan, in which each property is located the same as the community it belongs to. Thus, this data set possesses a typical characteristic, that is, dozens or even hundreds of observations could be allocated to one pair of coordinates, but vary in their attributes. This specific feature might lead to serious problems with bandwidth optimisations and coefficient estimates for calibrating the GWR model. We then proposed an extension by combining the hierarchical linear model (HLM) and GWR, namely HLM-GWR to cope with these problems. Results show that the HLM-GWR performs much better than the conventional GWR and HLM technique in terms of bandwidth optimisation, coefficient estimates. With a controlled simulation test, we again validated the advantage of the HLM-GWR model in comparison to both the HLM and GWR in handling this specific scenario. Overall, HLM-GWR is workable and should be recommended in this case or other scenarios with observations of similar spatial distributions.