10,760 research outputs found
Diffusion Maps Kalman Filter for a Class of Systems with Gradient Flows
In this paper, we propose a non-parametric method for state estimation of
high-dimensional nonlinear stochastic dynamical systems, which evolve according
to gradient flows with isotropic diffusion. We combine diffusion maps, a
manifold learning technique, with a linear Kalman filter and with concepts from
Koopman operator theory. More concretely, using diffusion maps, we construct
data-driven virtual state coordinates, which linearize the system model. Based
on these coordinates, we devise a data-driven framework for state estimation
using the Kalman filter. We demonstrate the strengths of our method with
respect to both parametric and non-parametric algorithms in three tracking
problems. In particular, applying the approach to actual recordings of
hippocampal neural activity in rodents directly yields a representation of the
position of the animals. We show that the proposed method outperforms competing
non-parametric algorithms in the examined stochastic problem formulations.
Additionally, we obtain results comparable to classical parametric algorithms,
which, in contrast to our method, are equipped with model knowledge.Comment: 15 pages, 12 figures, submitted to IEEE TS
Batch Nonlinear Continuous-Time Trajectory Estimation as Exactly Sparse Gaussian Process Regression
In this paper, we revisit batch state estimation through the lens of Gaussian
process (GP) regression. We consider continuous-discrete estimation problems
wherein a trajectory is viewed as a one-dimensional GP, with time as the
independent variable. Our continuous-time prior can be defined by any
nonlinear, time-varying stochastic differential equation driven by white noise;
this allows the possibility of smoothing our trajectory estimates using a
variety of vehicle dynamics models (e.g., `constant-velocity'). We show that
this class of prior results in an inverse kernel matrix (i.e., covariance
matrix between all pairs of measurement times) that is exactly sparse
(block-tridiagonal) and that this can be exploited to carry out GP regression
(and interpolation) very efficiently. When the prior is based on a linear,
time-varying stochastic differential equation and the measurement model is also
linear, this GP approach is equivalent to classical, discrete-time smoothing
(at the measurement times); when a nonlinearity is present, we iterate over the
whole trajectory to maximize accuracy. We test the approach experimentally on a
simultaneous trajectory estimation and mapping problem using a mobile robot
dataset.Comment: Submitted to Autonomous Robots on 20 November 2014, manuscript #
AURO-D-14-00185, 16 pages, 7 figure
Measurement-driven Quality Assessment of Nonlinear Systems by Exponential Replacement
We discuss the problem how to determine the quality of a nonlinear system
with respect to a measurement task. Due to amplification, filtering,
quantization and internal noise sources physical measurement equipment in
general exhibits a nonlinear and random input-to-output behaviour. This usually
makes it impossible to accurately describe the underlying statistical system
model. When the individual operations are all known and deterministic, one can
resort to approximations of the input-to-output function. The problem becomes
challenging when the processing chain is not exactly known or contains
nonlinear random effects. Then one has to approximate the output distribution
in an empirical way. Here we show that by measuring the first two sample
moments of an arbitrary set of output transformations in a calibrated setup,
the output distribution of the actual system can be approximated by an
equivalent exponential family distribution. This method has the property that
the resulting approximation of the statistical system model is guaranteed to be
pessimistic in an estimation theoretic sense. We show this by proving that an
equivalent exponential family distribution in general exhibits a lower Fisher
information measure than the original system model. With various examples and a
model matching step we demonstrate how this estimation theoretic aspect can be
exploited in practice in order to obtain a conservative measurement-driven
quality assessment method for nonlinear measurement systems.Comment: IEEE International Instrumentation and Measurement Technology
Conference (I2MTC), Taipei, Taiwan, 201
Asymptotic minimax risk of predictive density estimation for non-parametric regression
We consider the problem of estimating the predictive density of future
observations from a non-parametric regression model. The density estimators are
evaluated under Kullback--Leibler divergence and our focus is on establishing
the exact asymptotics of minimax risk in the case of Gaussian errors. We derive
the convergence rate and constant for minimax risk among Bayesian predictive
densities under Gaussian priors and we show that this minimax risk is
asymptotically equivalent to that among all density estimators.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ222 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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