Koyama Shinsuke, Pérez-Bolde Lucia Castellanos, Shalizi Cosma Rohilla, Kass Robert E
Department of Statistics and Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh, PA 15213 (
J Am Stat Assoc. 2010 Mar;105(489):170-180. doi: 10.1198/jasa.2009.tm08326.
State-space models provide an important body of techniques for analyzing time-series, but their use requires estimating unobserved states. The optimal estimate of the state is its conditional expectation given the observation histories, and computing this expectation is hard when there are nonlinearities. Existing filtering methods, including sequential Monte Carlo, tend to be either inaccurate or slow. In this paper, we study a nonlinear filter for nonlinear/non-Gaussian state-space models, which uses Laplace's method, an asymptotic series expansion, to approximate the state's conditional mean and variance, together with a Gaussian conditional distribution. This Laplace-Gaussian filter (LGF) gives fast, recursive, deterministic state estimates, with an error which is set by the stochastic characteristics of the model and is, we show, stable over time. We illustrate the estimation ability of the LGF by applying it to the problem of neural decoding and compare it to sequential Monte Carlo both in simulations and with real data. We find that the LGF can deliver superior results in a small fraction of the computing time.
状态空间模型为分析时间序列提供了一系列重要技术，但使用这些模型需要估计未观测到的状态。状态的最优估计是给定观测历史时它的条件期望，而当存在非线性时，计算这个期望很困难。现有的滤波方法，包括序贯蒙特卡罗方法，往往要么不准确要么速度慢。在本文中，我们研究了一种用于非线性/非高斯状态空间模型的非线性滤波器，它使用拉普拉斯方法（一种渐近级数展开）来近似状态的条件均值和方差，并结合高斯条件分布。这种拉普拉斯 - 高斯滤波器（LGF）能给出快速、递归、确定性的状态估计，其误差由模型的随机特性决定，并且我们证明它随时间是稳定的。我们通过将LGF应用于神经解码问题来说明其估计能力，并在模拟和实际数据中与序贯蒙特卡罗方法进行比较。我们发现LGF能在一小部分计算时间内给出更优的结果。
