一类连续时间状态空间模型基于得分的参数估计
Score-Based Parameter Estimation for a Class of Continuous-Time State Space Models
摘要：
考虑一类连续时间状态空间模型的参数估计问题。特别是，我们探讨的情况下，部分观察到的扩散，数据也到达根据扩散过程。基于一个标准的身份的评分函数，我们考虑两个粒子滤波器的方法来估计的得分函数。这两种方法都依赖于$\mathcal{O}(N^2)$成本的得分函数的在线估计算法，其中$N\in\mathbb{N}$是颗粒的数量。第一种方法采用了一个简单的欧拉离散化和标准的粒子平滑，是成本$\mathcal{O}(N^2 + NΔ_l^{-1})$每单位时间，其中$Δ_l=2^{-l}$，$l\in\mathbb{N}_0$，是时间离散化步骤。第二种方法是新的，基于一种新的扩散桥结构。给出了连续时间下得分函数的一个新的向后型Feynman-Kac公式，并给出了一种粒子方法对其进行逼近。考虑时间离散化，成本是每单位时间的$\mathcal{O}(N^2Δ_l^{-1})$。为了提高计算成本，我们考虑多级方法的得分函数。我们说明我们的参数估计方法通过随机梯度方法在几个数值例子。
英文摘要：
We consider the problem of parameter estimation for a class of continuous-time state space models. In particular, we explore the case of a partially observed diffusion, with data also arriving according to a diffusion process. Based upon a standard identity of the score function, we consider two particle filter based methodologies to estimate the score function. Both methods rely on an online estimation algorithm for the score function of $\mathcal{O}(N^2)$ cost, with $N\in\mathbb{N}$ the number of particles. The first approach employs a simple Euler discretization and standard particle smoothers and is of cost $\mathcal{O}(N^2 + NΔ_l^{-1})$ per unit time, where $Δ_l=2^{-l}$, $l\in\mathbb{N}_0$, is the time-discretization step. The second approach is new and based upon a novel diffusion bridge construction. It yields a new backward type Feynman-Kac formula in continuous-time for the score function and is presented along with a particle method for its approximation. Considering a time-discretization, the cost is $\mathcal{O}(N^2Δ_l^{-1})$ per unit time. To improve computational costs, we then consider multilevel methodologies for the score function. We illustrate our parameter estimation method via stochastic gradient approaches in several numerical examples.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
