This paper explores the impact of different distance metrics on collinearity in local regression models such as geographically weighted regression. Using a case study of house price data collected in Hà Nội, Vietnam, and by fully varying both power and rotation parameters to create different Minkowski distances, the analysis shows that local collinearity can be both negatively and positively affected by distance metric choice. The Minkowski distance that maximised collinearity in a geographically weighted regression was approximate to a Manhattan distance with (power = 0.70) with a rotation of 30°, and that which minimised collinearity was parameterised with power = 0.05 and a rotation of 70°. The results indicate that distance metric choice can provide a useful extra tuning component to address local collinearity issues in spatially varying coefficient modelling and that understanding the interaction of distance metric and collinearity can provide insight into the nature and structure of the data relationships. The discussion considers first, the exploration and selection of different distance metrics to minimise collinearity as an alternative to localised ridge regression, lasso and elastic net approaches. Second, it discusses the how distance metric choice could extend the methods that additionally optimise local model fit (lasso and elastic net) by selecting a distance metric that further helped minimise local collinearity. Third, it identifies the need to investigate the relationship between kernel bandwidth, distance metrics and collinearity as an area of further work.

中文翻译：

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距离度量选择既可以减少地理加权回归中的共线性，也可以引起共线性

本文探讨了不同距离度量对局部回归模型（例如地理加权回归）中的共线性的影响。使用在越南 Hà Nội 收集的房价数据的案例研究，并通过完全改变功率和旋转参数来创建不同的 Minkowski 距离，分析表明距离度量选择可以对局部共线性产生负面和正面影响。在地理加权回归中最大化共线性的 Minkowski 距离近似于曼哈顿距离（幂 = 0.70），旋转 30°，最小化共线性的参数化为幂 = 0.05 和旋转 70°。结果表明，距离度量选择可以提供有用的额外调整组件来解决空间变化系数建模中的局部共线性问题，并且理解距离度量和共线性的相互作用可以提供对数据关系的性质和结构的洞察。讨论首先考虑探索和选择不同的距离度量以最小化共线性作为局部脊回归、套索和弹性网方法的替代方法。其次，它讨论了距离度量选择如何通过选择进一步帮助最小化局部共线性的距离度量来扩展额外优化局部模型拟合（套索和弹性网络）的方法。第三，它确定需要调查内核带宽之间的关系，