Intuitively, there is a relation between measures of spatial dependence and
information theoretical measures of entropy. For instance, we can provide an
intuition of why spatial data is special by stating that, on average, spatial
data samples contain less than expected information. Similarly, spatial data,
e.g., remotely sensed imagery, that is easy to compress is also likely to show
significant spatial autocorrelation. Formulating our (highly specific) core
concepts of spatial information theory in the widely used language of
information theory opens new perspectives on their differences and similarities
and also fosters cross-disciplinary collaboration, e.g., with the broader AI/ML
communities. Interestingly, however, this intuitive relation is challenging to
formalize and generalize, leading prior work to rely mostly on experimental
results, e.g., for describing landscape patterns. In this work, we will explore
the information theoretical roots of spatial autocorrelation, more specifically
Moran's I, through the lens of self-information (also known as surprisal) and
provide both formal proofs and experiments.

通过自信息（也称为惊讶值）的视角，我们将研究空间自相关的信息理论基础，具体而言是莫兰 I，提供形式化证明和实验。

探究空间相关性度量的信息论根源

Probing the Information Theoretical Roots of Spatial Dependence Measures

In machine learning one often assumes the data are independent when
evaluating model performance. However, this rarely holds in practise.
Geographic information data sets are an example where the data points have
stronger dependencies among each other the closer they are geographically. This
phenomenon known as spatial autocorrelation (SAC) causes the standard cross
validation (CV) methods to produce optimistically biased prediction performance
estimates for spatial models, which can result in increased costs and accidents
in practical applications. To overcome this problem we propose a modified
version of the CV method called spatial k-fold cross validation (SKCV), which
provides a useful estimate for model prediction performance without optimistic
bias due to SAC. We test SKCV with three real world cases involving open
natural data showing that the estimates produced by the ordinary CV are up to
40% more optimistic than those of SKCV. Both regression and classification
cases are considered in our experiments. In addition, we will show how the SKCV
method can be applied as a criterion for selecting data sampling density for
new research area.

本文研究机器学习中的空间自相关问题，提出了基于空间 k 折交叉验证的方法，并通过三个真实开放数据集的实验验证了该方法的有效性和应用性。

通过空间 k 折交叉验证估计空间模型的预测性能

Estimating the Prediction Performance of Spatial Models via Spatial  k-Fold Cross Validation

Machine learning algorithms find frequent application in spatial prediction
of biotic and abiotic environmental variables. However, the characteristics of
spatial data, especially spatial autocorrelation, are widely ignored. We
hypothesize that this is problematic and results in models that can reproduce
training data but are unable to make spatial predictions beyond the locations
of the training samples. We assume that not only spatial validation strategies
but also spatial variable selection is essential for reliable spatial
predictions. We introduce two case studies that use remote sensing to predict
land cover and the leaf area index for the "Marburg Open Forest", an open
research and education site of Marburg University, Germany. We use the machine
learning algorithm Random Forests to train models using non-spatial and spatial
cross-validation strategies to understand how spatial variable selection
affects the predictions. Our findings confirm that spatial cross-validation is
essential in preventing overoptimistic model performance. We further show that
highly autocorrelated predictors (such as geolocation variables, e.g. latitude,
longitude) can lead to considerable overfitting and result in models that can
reproduce the training data but fail in making spatial predictions. The problem
becomes apparent in the visual assessment of the spatial predictions that show
clear artefacts that can be traced back to a misinterpretation of the spatially
autocorrelated predictors by the algorithm. Spatial variable selection could
automatically detect and remove such variables that lead to overfitting,
resulting in reliable spatial prediction patterns and improved statistical
spatial model performance. We conclude that in addition to spatial validation,
a spatial variable selection must be considered in spatial predictions of
ecological data to produce reliable predictions.

本文证明了在生态数据的空间预测中，除了空间验证之外，空间变量选择也必须考虑，以产生可靠的预测结果。使用 “随机森林” 机器学习算法，作者通过两个案例研究了非空间和空间交叉验证策略，并发现高度自相关的预测变量可能导致过拟合，从而产生错误的空间预测模式。