We consider the problem of high-dimensional filtering of state-space models (SSMs) at discrete times. This problem is particularly challenging as analytical solutions are typically not available and many numerical approximation methods can have a cost that scales exponentially with the dimension of the hidden state. Inspired by lag-approximation methods for the smoothing problem, we introduce a lagged approximation of the smoothing distribution that is necessarily biased. For certain classes of SSMs, particularly those that forget the initial condition exponentially fast in time, the bias of our approximation is shown to be uniformly controlled in the dimension and exponentially small in time. We develop a sequential Monte Carlo (SMC) method to recursively estimate expectations with respect to our biased filtering distributions. Moreover, we prove for a class of class of SSMs that can contain dependencies amongst coordinates that as the dimension $d\rightarrow\infty$ the cost to achieve a stable mean square error in estimation, for classes of expectations, is of $\mathcal{O}(Nd^2)$ per-unit time, where $N$ is the number of simulated samples in the SMC algorithm. Our methodology is implemented on several challenging high-dimensional examples including the conservative shallow-water model.
翻译:这个问题特别具有挑战性,因为分析解决方案通常不具备,而且许多数字近似方法的成本可能随隐藏状态的维度而指数化地计算出。由于对平滑问题的滞后-协调方法的启发,我们引入了平滑分布偏差的滞后近似值。对于某些类的SMS,尤其是那些在时间上迅速忘记初始状态的,我们近似偏差在尺寸上被统一控制,在时间上被指数性地缩小。我们开发了一个连续的Monte Carlo(SMC)方法,对偏差的过滤分布进行回溯性估计。此外,我们还为一类SMS提供了证据,它能够包含各种坐标之间的依赖性,因为SMS的维度是美元/rightrowr\infty$,对于预期等级来说,在估算中实现稳定平均误差的成本是每单位美元=mathcal{O}(Nd ⁇ 2)美元,而每单位时间的偏差值则被统一控制在维度上。我们开发了一个连续的Monte Caro(SMC)方法,其中的模型样本数为SMC中的高度。我们采用的方法具有挑战性。