200 research outputs found
Regularized Nonparametric Volterra Kernel Estimation
In this paper, the regularization approach introduced recently for
nonparametric estimation of linear systems is extended to the estimation of
nonlinear systems modelled as Volterra series. The kernels of order higher than
one, representing higher dimensional impulse responses in the series, are
considered to be realizations of multidimensional Gaussian processes. Based on
this, prior information about the structure of the Volterra kernel is
introduced via an appropriate penalization term in the least squares cost
function. It is shown that the proposed method is able to deliver accurate
estimates of the Volterra kernels even in the case of a small amount of data
points
Efficient Multidimensional Regularization for Volterra Series Estimation
This paper presents an efficient nonparametric time domain nonlinear system
identification method. It is shown how truncated Volterra series models can be
efficiently estimated without the need of long, transient-free measurements.
The method is a novel extension of the regularization methods that have been
developed for impulse response estimates of linear time invariant systems. To
avoid the excessive memory needs in case of long measurements or large number
of estimated parameters, a practical gradient-based estimation method is also
provided, leading to the same numerical results as the proposed Volterra
estimation method. Moreover, the transient effects in the simulated output are
removed by a special regularization method based on the novel ideas of
transient removal for Linear Time-Varying (LTV) systems. Combining the proposed
methodologies, the nonparametric Volterra models of the cascaded water tanks
benchmark are presented in this paper. The results for different scenarios
varying from a simple Finite Impulse Response (FIR) model to a 3rd degree
Volterra series with and without transient removal are compared and studied. It
is clear that the obtained models capture the system dynamics when tested on a
validation dataset, and their performance is comparable with the white-box
(physical) models
Kernel-based methods for Volterra series identification
Volterra series approximate a broad range of nonlinear systems. Their identification is challenging due to the curse of dimensionality: the number of model parameters grows exponentially with the complexity of the input-output response. This fact limits the applicability of such models and has stimulated recently much research on regularized solutions. Along this line, we propose two new strategies that use kernel-based methods. First, we introduce the multiplicative polynomial kernel (MPK). Compared to the standard polynomial kernel, the MPK is equipped with a richer set of hyperparameters, increasing flexibility in selecting the monomials that really influence the system output. Second, we introduce the smooth exponentially decaying multiplicative polynomial kernel (SEDMPK), that is a regularized version of MPK which requires less hyperparameters, allowing to handle also high-order Volterra series. Numerical results show the effectiveness of the two approaches. (C) 2021 Elsevier Ltd. All rights reserved
Statistical inference in mechanistic models: time warping for improved gradient matching
Inference in mechanistic models of non-linear differential equations is a challenging problem in current computational statistics. Due to the high computational costs of numerically solving the differential equations in every step of an iterative parameter adaptation scheme, approximate methods based on gradient matching have become popular. However, these methods critically depend on the smoothing scheme for function interpolation. The present article adapts an idea from manifold learning and demonstrates that a time warping approach aiming to homogenize intrinsic length scales can lead to a significant improvement in parameter estimation accuracy. We demonstrate the effectiveness of this scheme on noisy data from two dynamical systems with periodic limit cycle, a biopathway, and an application from soft-tissue mechanics. Our study also provides a comparative evaluation on a wide range of signal-to-noise ratios
Laplace deconvolution with noisy observations
In the present paper we consider Laplace deconvolution for discrete noisy
data observed on the interval whose length may increase with a sample size.
Although this problem arises in a variety of applications, to the best of our
knowledge, it has been given very little attention by the statistical
community. Our objective is to fill this gap and provide statistical treatment
of Laplace deconvolution problem with noisy discrete data. The main
contribution of the paper is explicit construction of an asymptotically
rate-optimal (in the minimax sense) Laplace deconvolution estimator which is
adaptive to the regularity of the unknown function. We show that the original
Laplace deconvolution problem can be reduced to nonparametric estimation of a
regression function and its derivatives on the interval of growing length T_n.
Whereas the forms of the estimators remain standard, the choices of the
parameters and the minimax convergence rates, which are expressed in terms of
T_n^2/n in this case, are affected by the asymptotic growth of the length of
the interval.
We derive an adaptive kernel estimator of the function of interest, and
establish its asymptotic minimaxity over a range of Sobolev classes. We
illustrate the theory by examples of construction of explicit expressions of
Laplace deconvolution estimators. A simulation study shows that, in addition to
providing asymptotic optimality as the number of observations turns to
infinity, the proposed estimator demonstrates good performance in finite sample
examples
Laplace deconvolution and its application to Dynamic Contrast Enhanced imaging
In the present paper we consider the problem of Laplace deconvolution with
noisy discrete observations. The study is motivated by Dynamic Contrast
Enhanced imaging using a bolus of contrast agent, a procedure which allows
considerable improvement in {evaluating} the quality of a vascular network and
its permeability and is widely used in medical assessment of brain flows or
cancerous tumors. Although the study is motivated by medical imaging
application, we obtain a solution of a general problem of Laplace deconvolution
based on noisy data which appears in many different contexts. We propose a new
method for Laplace deconvolution which is based on expansions of the
convolution kernel, the unknown function and the observed signal over Laguerre
functions basis. The expansion results in a small system of linear equations
with the matrix of the system being triangular and Toeplitz. The number of
the terms in the expansion of the estimator is controlled via complexity
penalty. The advantage of this methodology is that it leads to very fast
computations, does not require exact knowledge of the kernel and produces no
boundary effects due to extension at zero and cut-off at . The technique
leads to an estimator with the risk within a logarithmic factor of of the
oracle risk under no assumptions on the model and within a constant factor of
the oracle risk under mild assumptions. The methodology is illustrated by a
finite sample simulation study which includes an example of the kernel obtained
in the real life DCE experiments. Simulations confirm that the proposed
technique is fast, efficient, accurate, usable from a practical point of view
and competitive
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