Non-Stationary Models for Spatial Statistics and Bayesian Image Analysis

空间统计和贝叶斯图像分析的非平稳模型。摘要本研究开发了全贝叶斯推理的各向异性空间模型，并将其应用于空间和时间相关的数据。采取了两个截然不同的方向。首先，在之前研究的基础上，考虑了栅格的马尔可夫随机场先验，用于偶有大水平变化的过程建模。其次，使用建设性卷积方法，开发了连续模型，允许非平稳性和相关结构随空间（和/或时间）而变化。使用这种模型进行推理需要考虑到这种模型的灵活性或变化程度的不确定性。这项研究的一个关键部分是开发将这种不确定性来源纳入统计分析的方法。在医学成像、环境监测和探测全球气候随时间变化等领域，对随空间和可能随时间变化的现象进行统计建模至关重要。统计学中的大多数空间模型假设，在给定区域内，空间（可能还有时间）模式的基本属性在该区域内保持不变。虽然这在有限的区域内是一个有用的简化，但在更大的范围内，这样的假设通常会导致一个不令人满意的模型。这项研究将开发统计模型来解释空间现象性质的突变。此外，将开发模型，允许空间过程的属性随着空间位置的变化而变化。该研究将为大尺度时空现象提供更真实的模型。将这些模型拟合到大型数据集将需要大量的计算工作，并且可能依赖于高性能计算资源。将考虑在全球海洋气候变化、农业、遗传学和医学成像方面的应用。

NSF DMS-9704425 Non-Stationary Models for Spatial Statistics and Bayesian Image Analysis. David Higdon Duke University Abstract: This research develops anisotropic spatial models for fully Bayesian inference with applications to data which are correlated over space and time. Two distinct directions are taken. First, in continuation of previous research, Markov random field priors for lattices are considered for modeling processes with occasional large shifts in level. Second, using a constructive convolution approach, continuous models are developed that allow non-stationarity and correlation structure that can vary over space (and/or time). Using such models for inference requires that one account for uncertainty about just how flexible or varying such models are. A key part of the research is developing methods which incorporate this source of uncertainty in the statistical analysis. Statistical modeling of phenomena that evolve over space and possibly time is critical in areas such as medical imaging, environmental monitoring, and detecting changes over time in global climate. Most spatial models in statistics assume that over a given area, the underlying properties of a spatial (and possibly temporal as well) pattern remain unchanged over the region. Though this is a useful simplification over limited regions, on a larger scale such an assumption often results in an unsatisfactory model. This research will develop statistical models that account for abrupt changes in the nature of the spatial phenomenon. Also, models will be developed that allow properties of the spatial process to evolve over spatial location. This research will lead to more realistic models for large scale spatial and spatio-temporal phenomena. Fitting these models to large datasets will require a substantial computing effort and will likely rely high performance computing resources. Applications in global ocean climate change, agriculture, genetics, and medical imaging will be considered.