适用于大型异方差数据集的可扩展异方差高斯过程模型
Scalable Heteroskedastic Gaussian Process Models for Large Inhomogeneous Datasets
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中文总结 AI 辅助
该研究提出HetNV框架，结合Vecchia似然近似与残差非参数方差估计，用于大型异方差数据集的高斯过程回归，在保持均值预测精度的同时提升不确定性估计，且可应用于航天器等离子体测量的信号检测。
中文摘要 AI 辅助
我们提出了异方差归一化Vecchia高斯过程（HetNV），这是一种用于输入相关观测噪声的高斯过程回归的可扩展框架。HetNV将归一化输入上的Vecchia似然近似与基于残差的非参数方差估计相结合。潜在均值通过具有观测特定 nugget 方差的Vecchia高斯过程估计，而对数噪声方差则通过对稳定化的对数平方残差伪响应进行平滑得到，这些伪响应考虑了克里金不确定性和当前nugget估计值，在一维中使用LOESS，在二维中使用薄板样条广义加性模型。该方法在均值和方差更新之间交替进行，避免了方差过程的潜在变量推断。对于固定邻域大小（$m$）和观测数（$n$），每次迭代的主导成本是Vecchia更新，其缩放为$O(nm^2)$。模拟研究表明，与同方差Vecchia模型相比，该方法能更好地恢复输入相关的不确定性，同时保持有竞争力的均值预测精度，在二维均值估计中还有额外提升。对航天器等离子体测量的应用表明，在大型、噪声大、异方差的场景中，局部自适应不确定性估计如何影响下游信号检测决策。
英文摘要
We introduce Heteroskedastic Normalized Vecchia Gaussian Processes (HetNV), a scalable framework for Gaussian process regression with input-dependent observation noise. HetNV combines Vecchia likelihood approximations on normalized inputs with residual-based nonparametric variance estimation. The latent mean is estimated via a Vecchia GP with observation-specific nugget variances, while the log noise variance is obtained by smoothing stabilized log-squared residual pseudo-responses that account for kriging uncertainty and current nugget estimates, using LOESS in one dimension and thin plate spline generalized additive models in two dimensions. The method alternates between mean and variance updates, avoiding latent-variable inference for the variance process. For fixed neighborhood size ($m$) and number of observations ($n$), the dominant per-iteration cost is the Vecchia update, scaling as $O(nm^2)$. Simulation studies show improved recovery of input-dependent uncertainty relative to homoskedastic Vecchia models while maintaining competitive mean prediction accuracy, with additional gains in two-dimensional mean estimation. An application to spacecraft plasma measurements demonstrates how locally adaptive uncertainty estimates influence downstream signal-detection decisions in large, noisy, heteroskedastic settings.
