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Publication# The Scrambled Net Filter

Abstract

The standard Kalman filter is a powerful and widely used tool to perform prediction, filtering and smoothing in the fields of linear Gaussian state-space models. In its standard setting it has a simple recursive form which implies high computational efficiency. As the latter is essentially a least squares procedure optimality properties can be derived easily. These characteristics of the standard Kalman filter depend strongly on distributional and linearity assumptions of the model. If we consider nonlinear non-Gaussian state-space models all these properties and characteristics are no longer valid. Consequently there are different approaches on the robustification of the Kalman filter. One is based on the the ideas of minimax problems and influence curves. Others use numerical integration and Monte Carlo methods. Herein we propose a new filter by implementing a method of numerical integration, called scrambled net quadrature, which consists of a mixture of Monte Carlo methods and quasi-Monte Carlo methods, providing an integration error of order of magnitude $N^{-3/2}\log(N)^{(r-1)/2}$ in probability, where $r$ denotes dimension. We show that the point-wise bias of the posterior density estimate is of order of magnitude $N^{-3}\log(N)^{r-1}$ but grows linearly with time.

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Related publications (1)

Related concepts (7)

Kalman filter

For statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory.

Numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for numerical integration, especially as applied to one-dimensional integrals.

Monte Carlo method

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.

The standard Kalman filter is a powerful and widely used tool to perform prediction, filtering and smoothing in the fields of linear Gaussian state-space models. In its standard setting it has a simpl

2002