How Kernel density estimation works part2(Machine Learning)
- Learning Transfer Operators by Kernel Density Estimation(arXiv)
Author : Sudam Surasinghe, Jeremie Fish, Erik M. Bollt
Abstract : Inference of transfer operators from data is often formulated as a classical problem that hinges on the Ulam method. The usual description, which we will call the Ulam-Galerkin method, is in terms of projection onto basis functions that are characteristic functions supported over a fine grid of rectangles. In these terms, the usual Ulam-Galerkin approach can be understood as density estimation by the histogram method. Here we show that the problem can be recast in statistical density estimation formalism. This recasting of the classical problem, is a perspective that allows for an explicit and rigorous analysis of bias and variance, and therefore toward a discussion of the mean square error. Keywords: Transfer Operators; Frobenius-Perron operator; probability density estimation; Ulam-Galerkin method;Kernel Density Estimation.
2. 2D Density Control of Micro-Particles using Kernel Density Estimation(arXiv)
Author : Ion Matei, Johan de Kleer, Maksym Zhenirovskyy
Abstract : We address the problem of 2D particle density control. The particles are immersed in dielectric fluid and acted upon by manipulating an electric field. The electric field is controlled by an array of electrodes and used to bring the particle density to a desired pattern using dielectrophoretic forces. We use a lumped, 2D, capacitive-based, nonlinear model describing the motion of a particle. The spatial dependency of the capacitances is estimated using electrostatic COMSOL simulations. We formulate an optimal control problem, where the loss function is defined in terms of the error between the particle density at some final time and a target density. We use a kernel density estimator (KDE) as a proxy for the true particle density. The KDE is computed using the particle positions that are changed by varying the electrode potentials. We showcase our approach through numerical simulations, where we demonstrate how the particle positions and the electrode potentials vary when shaping the particle positions from a uniform to a Gaussian distribution.
