On the use of non-Euclidean distance measures in geostatistics
On the use of non-Euclidean distance measures in geostatistics
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DOI：
10.1007/s11004-006-9055-7
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发表时间：
2006-11-01
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影响因子：
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通讯作者：
Curriero, Frank C.
中科院分区：
文献类型：
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作者：
Curriero, Frank C.
在许多科学学科中，直线、欧几里得距离可能不能准确地描述空间数据之间的接近关系。然而，在地质统计学应用中，非欧几里得距离测量必须谨慎使用。提供了一个简单的例子来证明，当与非欧几里得距离度量一起使用时，不能保证现有的协方差和变异函数仍然有效（即正定或有条件地负定）。当与现有的协方差和变异函数一起使用时，存在某些距离度量仍然有效，这是一个探索的问题。介绍了等距嵌入的概念，并将其与正确定性和有条件负确定性的概念联系起来，以展示有效的范数相关的各向同性协方差和变异函数的类别，其中许多结果尚未出现在主流地质统计学文献或应用中。这些函数类通过添加一个参数来定义距离范数来扩展已知的类。在实践中，可以先验地设置这个距离参数来表示，例如，欧几里得距离，或者作为一个参数来允许数据选择度量。最后给出了后一种方法的仿真应用。模拟结果还比较了基于欧几里得距离的克里格预测和基于水度量的克里格预测。
In many scientific disciplines, straight line, Euclidean distances may not accurately describe proximity relationships among spatial data. However, non-Euclidean distance measures must be used with caution in geostatistical applications. A simple example is provided to demonstrate there are no guarantees that existing covariance and variogram functions remain valid (i.e. positive definite or conditionally negative definite) when used with a non-Euclidean distance measure. There are certain distance measures that when used with existing covariance and variogram functions remain valid, an issue that is explored. The concept of isometric embedding is introduced and linked to the concepts of positive and conditionally negative definiteness to demonstrate classes of valid norm dependent isotropic covariance and variogram functions, results many of which have yet to appear in the mainstream geostatistical literature or application. These classes of functions extend the well known classes by adding a parameter to define the distance norm. In practice, this distance parameter can be set a priori to represent, for example, the Euclidean distance, or kept as a parameter to allow the data to choose the metric. A simulated application of the latter is provided for demonstration. Simulation results are also presented comparing kriged predictions based on Euclidean distance to those based on using a water metric.