[论文解读] Improving particle filter performance with a generalized random field model of observation errors

Gregor Robinson, Ian Grooms+1 more|arXiv (Cornell University)|Nov 17, 2017

Hydrology and Drought Analysis被引用 2

一句话总结

本文提出了一种广义高斯随机场（GRF）模型，用于粒子滤波中的观测误差，以减少集合大小需求并防止滤波崩溃。通过引入随尺度变化的误差方差，弱化小尺度差异，GRF模型在不改变集合成员的情况下平滑后验均值，从而在连续概率评分上将崩溃频率降低最多20%，提升精度。

ABSTRACT

This article shows that generalized random field (GRF) models of additive observation error can reduce the ensemble size required to avoid collapse in particle filtering of spatially-extended dynamics. This kind of random field model has increasing variance at small scales, and realizations appear `jagged.' Particle weights depend on how well a particular ensemble member agrees with the observations, and collapse occurs when a few ensemble members receive most of the weight. The GRF observation error model reduces the incidence of collapse by de-emphasizing small-scale differences between the ensemble members and the observations. This observation error model smooths the posterior mean, though it does not smooth the individual ensemble members. Two options for implementing the observation error model are described. Taking discretized elliptic differential operators as an observation error covariance matrix provides the desired jagged property of process realizations. This choice also introduces structure exploitable by scalable computation techniques, including multigrid solvers and multiresolution approximations to the corresponding integral operator. Alternatively the observations can be smoothed and then assimilated under the assumption of independent errors, which is equivalent to assuming large errors at small scales. The method is demonstrated on a linear stochastic partial differential equation, where it significantly reduces the occurrence of particle filter collapse while maintaining accuracy. It also improves continuous ranked probability scores by as much as 20\%, indicating an improvement in the quality of the probability distribution associated with the particle weights. The method is compatible with other techniques for improving the performance of particle filters.

研究动机与目标

• 解决由于高维观测误差导致的空间扩展系统中粒子滤波崩溃的问题。
• 在不损害精度的前提下，减少稳定粒子滤波所需的集合大小。
• 开发一种观测误差模型，弱化集合成员与观测值之间的小尺度差异。
• 通过由椭圆微分算子导出的结构化协方差矩阵，实现可扩展的计算。
• 通过连续概率评分衡量，提升预测概率分布的质量。

提出的方法

• 使用广义高斯随机场（GRF）建模观测误差，其在小尺度上具有递增的方差，产生锯齿状实现。
• 使用离散化的椭圆微分算子作为观测误差协方差矩阵，以强制实现尺度相关的误差结构。
• 通过多重网格求解器和GRF积分算子的多分辨率近似，实现可扩展计算。
• 另一种方法是先验平滑观测值并假设误差独立，这在数学上等价于小尺度误差较大。
• 将GRF模型应用于线性随机偏微分方程以测试性能。
• 通过连续概率评分和崩溃频率评估性能。

实验结果

研究问题

• RQ1 广义高斯随机场模型能否减少空间扩展系统中粒子滤波的崩溃？
• RQ2 GRF模型在多大程度上减少了稳定粒子滤波所需的集合大小？
• RQ3 GRF模型如何影响后验概率分布的精度和质量？
• RQ4 GRF模型能否通过多重网格求解器等可扩展数值技术高效实现？
• RQ5 与标准粒子滤波相比，GRF模型是否能提升连续概率评分？

主要发现

• GRF模型通过弱化集合成员与观测值之间的小尺度差异，显著降低了粒子滤波崩溃的发生频率。
• 该方法在保持高精度的同时，减少了稳定滤波所需的集合大小。
• 连续概率评分最高提升20%，表明预测分布的校准性更好。
• 使用离散化椭圆微分算子作为协方差矩阵，可通过多重网格和多分辨率方法实现高效计算。
• 先验平滑观测值并假设误差独立，与GRF模型在数学上等价，且性能提升相似。
• GRF模型可与其他粒子滤波改进技术兼容，表明其具有广泛的应用潜力。