We consider the problem of non-parametric testing of independence of two components of a stationary bivariate spatial process. In particular, we revisit the random shift approach that has become the standard method for testing the independent superposition hypothesis in spatial statistics, and it is widely used in a plethora of practical applications. However, this method has a problem of liberality caused by breaking the marginal spatial correlation structure due to the toroidal correction. This indeed means that the assumption of exchangeability, which is essential for the Monte Carlo test to be exact, is not fulfilled.

We present a number of permutation strategies and show that the random shift with the variance correction brings a suitable improvement compared to the torus correction in the random field case. It reduces the liberality and achieves the largest power from all investigated variants. To obtain the variance for the variance correction method, several approaches were studied. The best results were achieved, for the sample covariance as the test statistics, with the correction factor $\sqrt{1∕n}$. This corresponds to the asymptotic order of the variance of the test statistics.

In the point process case, the problem of deviations from exchangeability is far more complex and we propose an alternative strategy based on the mean cross nearest-neighbor distance and torus correction. It reduces the liberality but achieves slightly lower power than the usual cross $K$-function. Therefore we recommend it, when the point patterns are clustered, where the cross $K$-function achieves liberality.

我们考虑非参数检验平稳双变量空间过程的两个分量的独立性的问题。特别是，我们重新审视了随机移位方法，该方法已成为测试空间统计中独立叠加假设的标准方法，并且已在许多实际应用中广泛使用。然而，该方法具有由于环形校正而破坏边缘空间相关结构而引起的自由度问题。确实，这意味着不能满足对可交换性的假设，这对于准确进行蒙特卡洛检验至关重要。

我们提出了许多置换策略，并表明与随机场情况下的圆环校正相比，采用方差校正的随机移位带来了适当的改进。它降低了自由度，并从所有研究的变体中获得了最大的威力。为了获得方差校正方法的方差，研究了几种方法。使用校正因子作为样本协方差作为检验统计量时，获得了最佳结果$\sqrt{\mathrm{1个}∕ñ}$。这对应于检验统计量方差的渐近顺序。

在点处理的情况下，与可交换性偏离的问题要复杂得多，我们提出了基于平均交叉最近邻居距离和环面校正的替代策略。它降低了自由度，但功率却比通常的十字架低$ķ$-功能。因此，当点模式聚类时，我们建议使用$ķ$功能实现自由。