GWR (Geographical Weighted Regression) is a widely accepted regression method under spatial dependency. Since the calibration of GWR is computationally intensive, some efficient methods for calibration were proposed. However, these methods require extensive computation environments or limit the class of kernel functions, limiting the applicability of GWR. To improve the applicability, we propose sgGWR (stochastic gradient GWR), an optimization approach for GWR based on stochastic gradient, which stochastically approximates cross-validation errors and applies gradient-based optimization methods. To achieve this, we show the analytical derivate of the GWR cross-validation error. sgGWR can handle a broader class of kernels than that by the existing scalable method, and we can benefit from it even high-performance computers cannot be accessed. Therefore, sgGWR fills in the gap that existing scalable methods do not cover. We examine the performances of sgGWR and the existing methods by simulation studies. Additionally, we apply sgGWR for the land price analysis for Tokyo, Japan. As a result, a spatio-temporal version of GWR has the best prediction performance, and it captures the spatio-temporal heterogeneity of regression coefficients.

We propose a scalable regression model with spatially and temporally varying coefficients based on Moran’s eigenvectors and efficient computation algorithms. Regression models that consider spatiotemporal non-stationarity are important because many real-world datasets, such as housing prices, are tied to geographical and temporal locations. Although geographically weighted regression (GWR) and its variants are widely used to model spatially varying coefficients, they cannot handle large datasets. We employ an alternative modelling method of spatially varying coefficients based on Moran’s eigenvectors and extend it to handle large spatiotemporal datasets. Additionally, we introduce a scalable learning algorithm that exploits the model structures based on the Kalman filter and the expectation–maximisation algorithm. Our scalable algorithm is efficient even for large datasets that cannot be handled by GWR. To evaluate the performance of the proposed model, we applied it to a housing market dataset collected in Tokyo, Japan. The results show that the predictive performance of the proposed model is comparable to that of GWR while increasing the computational speed. Moreover, larger datasets can accelerate the algorithm convergence.

In the COVID-19 pandemic, keeping a two-metre physical distance (called social distancing) from others is a critical action to limit the amount of social contact among individuals. Thus, regulating the number of persons in a space is one means to limit social contact and ensure physical distancing among people. In addition to the privacy concerns of individuals’ locations, the lack of elevation information can result in false positive cases for people who are horizontally close but vertically apart beyond the social distance. Therefore, we develop a normative model of the statistical distribution of physical distances to estimate the number of persons not social distancing in a three-dimensional complex built space from the current number of persons in the space (i.e. population density) present on an hourly basis. We conduct an empirical case study in the districts around the central stations of 13 of Japan’s prefectures. To determine the number of people not social distancing in these districts, the proposed model uses open data such as the number of persons before the national emergency declaration, the ratio of the number of persons after the declaration to that before, and the floor area ratio specified by land use regulations in these districts.

The size distribution of urban blocks is important for the characterisation of urban block patterns and is known to follow several parametric statistical distributions. However, it has not previously been analysed on the basis of a normative plane tessellation and in terms of urban block density and mean road width. In this article, we formulate the size distribution of Voronoi cells using the gamma distribution estimated by urban block density and mean road width. We found that (1) both log-normal and gamma distributions can be good candidates for the size distribution of urban blocks at the scale of a region that consists of regular urban blocks and that has a uniform road width; and (2) the size distribution of urban blocks depends on the balance between pattern and width improvement effects. Based on one study region in Tokyo, if the pattern improvement effect is more prominent than the width improvement effect, the mode of urban block sizes tends to be larger than if it is not. These findings are expected to provide scientific support for urban planning (e.g. land readjustment projects).