Uniform Stability of a Particle Approximation of the Optimal Filter Derivative

Sequential Monte Carlo methods, also known as particle methods, are a widely used set of computational tools for inference in non-linear non-Gaussian state-space models. In many applications it may be necessary to compute the sensitivity, or derivative, of the optimal filter with respect to the static parameters of the state-space model; for instance, in order to obtain maximum likelihood model parameters of interest, or to compute the optimal controller in an optimal control problem. In Poyiadjis et al. [2011] an original particle algorithm to compute the filter derivative was proposed and it was shown using numerical examples that the particle estimate was numerically stable in the sense that it did not deteriorate over time. In this paper we substantiate this claim with a detailed theoretical study. Lp bounds and a central limit theorem for this particle approximation of the filter derivative are presented. It is further shown that under mixing conditions these Lp bounds and the asymptotic variance characterized by the central limit theorem are uniformly bounded with respect to the time index. We demon- strate the performance predicted by theory with several numerical examples. We also use the particle approximation of the filter derivative to perform online maximum likelihood parameter estimation for a stochastic volatility model

Uniform stability of a particle approximation of the optimal filter derivative

Particle methods, also known as Sequential Monte Carlo methods, are a principled set of algorithms to approximate numerically the optimal lter in non-linear non-Gaussian state-space models. However, when performing maximum likelihood parameter inference in state-space models, it is also necessary to approximate the derivative of the optimal lter with respect to the parameter of the model. Poyiadjis et al. [2005, 2011] present an original particle method to apoproximate this derivative and it was shown in numerical examples to be numerically stable in the sense that it did not deteriorate over time. In this paper we theoretically substantiate this claim. Lp bounds and a central limit theorem for this particle approximation are presented. Under mixing conditions these Lp bounds and the asymptotic variance are uniformly bounded with respect to the time index.This is the final version. It first appeared at http://epubs.siam.org/doi/abs/10.1137/140993703

Bias of Particle Approximations to Optimal Filter Derivative

In many applications, a state-space model depends on a parameter which needs to be inferred from a data set. Quite often, it is necessary to perform the parameter inference online. In the maximum likelihood approach, this can be done using stochastic gradient search and the optimal filter derivative. However, the optimal filter and its derivative are not analytically tractable for a non-linear state-space model and need to be approximated numerically. In [Poyiadjis, Doucet and Singh, Biometrika 2011], a particle approximation to the optimal filter derivative has been proposed, while the corresponding $L_{p}$ error bonds and the central limit theorem have been provided in [Del Moral, Doucet and Singh, SIAM Journal on Control and Optimization 2015]. Here, the bias of this particle approximation is analyzed. We derive (relatively) tight bonds on the bias in terms of the number of particles. Under (strong) mixing conditions, the bounds are uniform in time and inversely proportional to the number of particles. The obtained results apply to a (relatively) broad class of state-space models met in practice

A multi-domain hybrid method for head-on collision of black holes in particle limit

A hybrid method is developed based on the spectral and finite-difference methods for solving the inhomogeneous Zerilli equation in time-domain. The developed hybrid method decomposes the domain into the spectral and finite-difference domains. The singular source term is located in the spectral domain while the solution in the region without the singular term is approximated by the higher-order finite-difference method. The spectral domain is also split into multi-domains and the finite-difference domain is placed as the boundary domain. Due to the global nature of the spectral method, a multi-domain method composed of the spectral domains only does not yield the proper power-law decay unless the range of the computational domain is large. The finite-difference domain helps reduce boundary effects due to the truncation of the computational domain. The multi-domain approach with the finite-difference boundary domain method reduces the computational costs significantly and also yields the proper power-law decay. Stable and accurate interface conditions between the finite-difference and spectral domains and the spectral and spectral domains are derived. For the singular source term, we use both the Gaussian model with various values of full width at half maximum and a localized discrete $\delta$-function. The discrete $\delta$-function was generalized to adopt the Gauss-Lobatto collocation points of the spectral domain. The gravitational waveforms are measured. Numerical results show that the developed hybrid method accurately yields the quasi-normal modes and the power-law decay profile. The numerical results also show that the power-law decay profile is less sensitive to the shape of the regularized $\delta$-function for the Gaussian model than expected. The Gaussian model also yields better results than the localized discrete $\delta$-function.Comment: 25 pages; published version (IJMPC

Analysis of error propagation in particle filters with approximation

This paper examines the impact of approximation steps that become necessary when particle filters are implemented on resource-constrained platforms. We consider particle filters that perform intermittent approximation, either by subsampling the particles or by generating a parametric approximation. For such algorithms, we derive time-uniform bounds on the weak-sense $L_p$ error and present associated exponential inequalities. We motivate the theoretical analysis by considering the leader node particle filter and present numerical experiments exploring its performance and the relationship to the error bounds.Comment: Published in at http://dx.doi.org/10.1214/11-AAP760 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org