Context

Interpreting spatial autocorrelation is complicated by differences in data type, spatial conformation, and contiguity definitions. Though lacking consistent meaning, Moran’s I is commonly reported, compared, and interpreted based on conceptual ideals. To provide consistent, logical, and intuitive meaning and enable broader synthetic work, a new approach to I is needed.

Objectives

We sought to standardize I and true it to conceptual ideals and existing intuition regarding regular correlations. We also wished to test performance of transformed metrics over a diversity of designed and empirical datasets.

Methods

We developed two means to rectify I. Both fit null distributions from data permutation to a target frame of [− 1, 0, 1], followed by projection of original I into this conformation. One method used three-point registration employing the distribution median and select tail percentiles. The other directly projected all I based on theory or cumulative frequencies reflecting the distribution of regular correlations. Repeatability and sensitivity of results were examined for varied permutation replication and framing parameter choices. Empirical and designed datasets were used to compare rectified to traditional metrics.

Results

Both rectification methods improved distributional characteristics of I. Three-point registration produced overly broad distributions with discontinuous peaks. Continuous projection fit the distribution for regular correlations precisely. Diverse case studies demonstrated failings of I and the clarity gained by rectification.

Conclusions

Rectified I enabled meaningful comparisons of spatial patterns for diverse data and landscape conditions. Preserving the intuitive value of Moran’s I while providing a theoretically sound and consistent approach for standardizing its values should foster sustained use.

中文翻译：

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修正 I：空间自相关度量 Moran's I 的三点连续拟合至理想形式

语境

由于数据类型、空间构造和邻接定义的差异，解释空间自相关很复杂。尽管缺乏一致的含义，Moran's  I 通常被基于概念理想进行报道、比较和解释。为了提供一致、合乎逻辑和直观的含义并实现更广泛的综合工作，需要一种新的I方法  。

目标

我们试图将I标准化 ， 并使其符合关于规则相关性的概念理想和现有直觉。我们还希望在各种设计和经验数据集上测试转换指标的性能。

方法

我们开发了两种方法来纠正 I。两者都将数据排列的空分布拟合到 [− 1, 0, 1] 的目标帧，然后将原始I投影  到该构象中。一种方法使用三点配准，采用分布中位数和选择尾部百分位数。另一个直接 基于理论或反映规则相关性分布的累积频率预测所有 I。针对不同的排列复制和框架参数选择，检查了结果的可重复性和敏感性。经验和设计的数据集用于比较校正与传统指标。

结果

两种整流方法都改善了I 的分布特性 。三点配准产生具有不连续峰的过于宽泛的分布。连续投影精确地拟合正则相关性的分布。不同的案例研究证明了I 的失败  以及通过整改获得的清晰度。

结论

纠正 我 启用了对不同数据和景观条件的空间模式的有意义的比较。保留 Moran's I的直观价值，  同时提供一种理论上合理且一致的方法来标准化其价值，应该会促进持续使用。