This paper develops and applies new techniques for the simultaneous detection of boundaries and clusters within a probabilistic framework. The new statistic "little b" (written b(ij)) evaluates boundaries between adjacent areas with different values, as well as links between adjacent areas with similar values. Clusters of high values (hotspots) and low values (coldspots) are then constructed by joining areas abutting locations that are significantly high (e.g., an unusually high disease rate) and that are connected through a "link" such that the values in the adjoining areas are not significantly different. Two techniques are proposed and evaluated for accomplishing cluster construction: "big B" and the "ladder" approach. We compare the statistical power and empirical Type I and Type II error of these approaches to those of wombling and the local Moran test. Significance may be evaluated using distribution theory based on the product of two continuous (e.g., non-discrete) variables. We also provide a "distribution free" algorithm based on resampling of the observed values. The methods are applied to simulated data for which the locations of boundaries and clusters is known, and compared and contrasted with clusters found using the local Moran statistic and with polygon Womble boundaries. The little b approach to boundary detection is comparable to polygon wombling in terms of Type I error, Type II error and empirical statistical power. For cluster detection, both the big B and ladder approaches have lower Type I and Type II error and are more powerful than the local Moran statistic. The new methods are not constrained to find clusters of a pre-specified shape, such as circles, ellipses and donuts, and yield a more accurate description of geographic variation than alternative cluster tests that presuppose a specific cluster shape. We recommend these techniques over existing cluster and boundary detection methods that do not provide such a comprehensive description of spatial pattern.

中文翻译：

---

边界、链接和集群：空间分析的新范式？

本文开发并应用了在概率框架内同时检测边界和集群的新技术。新的统计量“小 b”（写作 b(ij)）评估具有不同值的相邻区域之间的边界，以及具有相似值的相邻区域之间的链接。高值（热点）和低值（冷点）的集群然后通过连接与显着高（例如，异常高的发病率）相邻的区域并通过“链接”连接，使得相邻区域中的值地区差别不大。提出并评估了两种用于完成集群构建的技术：“大 B”和“阶梯”方法。我们将这些方法的统计功效和经验类型 I 和类型 II 误差与摇摆和局部 Moran 检验的方法进行比较。可以使用基于两个连续（例如，非离散）变量的乘积的分布理论来评估显着性。我们还提供了一种基于观测值重采样的“无分布”算法。这些方法应用于边界和聚类位置已知的模拟数据，并与使用局部 Moran 统计和多边形 Womble 边界找到的聚类进行比较和对比。边界检测的小 b 方法在 I 类错误、II 类错误和经验统计功效方面可与多边形摆动相媲美。对于聚类检测，大 B 和阶梯方法都具有较低的 I 类和 II 类错误，并且比局部 Moran 统计更强大。新方法不限于找到预先指定形状的集群，例如圆形、椭圆和甜甜圈，并且比以特定集群形状为前提的替代集群测试产生更准确的地理变化描述。我们推荐这些技术而不是现有的集群和边界检测方法，这些方法不能提供如此全面的空间模式描述。