Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data

基于似然推断的具有复合出生机制的出生-死亡过程

Likelihood-based inference for birth-death processes with composite birth mechanisms

本文提出基于似然的推断方法，用于具有复合出生率的有限状态出生-死亡过程，研究了包含多种机制的总出生强度。通过Doob h-变换Q过程，推导了条件似然并研究了最大似然估计和准最大似然估计，证明了估计量的一致性和渐近正态性。

Anomaly Detection in Trajectory Data with Normalizing Flows

The task of detecting anomalous data patterns is as important in practical applications as challenging. In the context of spatial data, recognition of unexpected trajectories brings additional difficulties, such as high dimensionality and varying pattern lengths. We aim to tackle such a problem from a probability density estimation point of view, since it provides an unsupervised procedure to identify out of distribution samples. More specifically, we pursue an approach based on normalizing flows, a recent framework that enables complex density estimation from data with neural networks. Our pr…

Mean-Field Nonparametric Estimation of Interacting Particle Systems

This paper concerns the nonparametric estimation problem of the distribution-state dependent drift vector field in an interacting $N$-particle system. Observing single-trajectory data for each particle, we derive the mean-field rate of convergence for the maximum likelihood estimator (MLE), which depends on both Gaussian complexity and Rademacher complexity of the function class. In particular, when the function class contains $α$-smooth H{ö}lder functions, our rate of convergence is minimax optimal on the order of $N^{-\fracα{d+2α}}$. Combining with a Fourier analytical deconvolution argument…

The LAN property for McKean-Vlasov models in a mean-field regime

We establish the local asymptotic normality (LAN) property for estimating a multidimensional parameter in the drift of a system of $N$ interacting particles observed over a fixed time horizon in a mean-field regime $N \rightarrow \infty$. By implementing the classical theory of Ibragimov and Hasminski, we obtain in particular sharp results for the maximum likelihood estimator that go beyond its simple asymptotic normality thanks to Hájek's convolution theorem and strong controls of the likelihood process that yield asymptotic minimax optimality (up to constants). Our structural results shed so…

弱相互作用亚椭圆扩散的参数估计

Parameter Estimation for Weakly Interacting Hypoelliptic Diffusions

针对弱相互作用亚椭圆扩散系统，提出局部高斯近似似然法，实现参数估计并证明渐近正态性，支持部分观测，数值实验验证有效性。

The Extended Kalman Filter is a Natural Gradient Descent in Trajectory Space

The extended Kalman filter is perhaps the most standard tool to estimate in real time the state of a dynamical system from noisy measurements of some function of the system, with extensive practical applications (such as position tracking via GPS). While the plain Kalman filter for linear systems is well-understood, the extended Kalman filter relies on linearizations which have been debated. We recover the exact extended Kalman filter equations from first principles in statistical learning: the extended Kalman filter is equal to Amari's online natural gradient, applied in the space of trajecto…

部分可观测扩散过程中基于粒子的似然推断

Particle-based likelihood inference in partially observed diffusion processes using generalised Poisson estimators

本文涉及使用的期望最大化（EM）算法的推理部分观察扩散过程。在这方面，一个众所周知的问题是，除了少数扩散过程缺乏封闭形式的过渡密度的表达式。因此，为了有效地估计EM中间量，我们构造，使用新的技术无偏估计的扩散过渡密度，随机权重固定滞后辅助粒子平滑，避免了众所周知的问题，粒子轨迹退化的平滑模式。理论上证明了该估计器的合理性，并在一个模拟的例子上进行了演示。

Learning Interaction Kernels for Agent Systems on Riemannian Manifolds

Interacting agent and particle systems are extensively used to model complex phenomena in science and engineering. We consider the problem of learning interaction kernels in these dynamical systems constrained to evolve on Riemannian manifolds from given trajectory data. The models we consider are based on interaction kernels depending on pairwise Riemannian distances between agents, with agents interacting locally along the direction of the shortest geodesic connecting them. We show that our estimators converge at a rate that is independent of the dimension of the state space, and derive boun…

离散观测相互作用粒子系统的参数估计

Parameter estimation of discretely observed interacting particle systems

本文针对随机McKean-Vlasov方程及其相互作用粒子系统，提出基于伪似然的对比函数，在离散观测下实现漂移与扩散系数的联合估计，并证明估计量的一致性与渐近正态性。