arXiv subjects

Cristian R. Rojas

Publications and source records attributed to Cristian R. Rojas.

At least 73 records · Page 4Linked to original sources

arXiv2019

Consistency Analysis of the Simplified Refined Instrumental Variable Method for Continuous-time Systems

In this paper, we analyse the consistency of the Simplified Refined Instrumental Variable method for Continuous-time systems (SRIVC). It is well known that the intersample behaviour of the input signal influences the quality and accuracy of the results when estimating and simulating continuous-time models. Here, we present a comprehensive analysis on the consistency of the SRIVC estimator while taking into account the intersample behaviour of the input signal. The main result of the paper shows that, under some mild conditions, the SRIVC estimator is generically consistent. We also describe some conditions when consistency is not achieved, which is important from a practical standpoint. The theoretical results are supported by simulation examples.

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arXiv2018

Estimating Models with High-Order Noise Dynamics Using Semi-Parametric Weighted Null-Space Fitting

Standard system identification methods often provide inconsistent estimates with closed-loop data. With the prediction error method (PEM), this issue is solved by using a noise model that is flexible enough to capture the noise spectrum. However, a too flexible noise model (i.e., too many parameters) increases the model complexity, which can cause additional numerical problems for PEM. In this paper, we consider the weighted null-space fitting (WNSF) method. With this method, the system is first modeled using a non-parametric ARX model, which is then reduced to a parametric model of interest using weighted least squares. In the reduction step, a parametric noise model does not need to be estimated if it is not of interest. Because the flexibility of the noise model is increased with the sample size, this will still provide consistent estimates in closed loop and asymptotically efficient estimates in open loop. In this paper, we prove these results, and we derive the asymptotic covariance for the estimation error obtained in closed loop, which is optimal for an infinite-order noise model. For this purpose, we also derive a new technical result for geometric variance analysis, instrumental to our end. Finally, we perform a simulation study to illustrate the benefits of the method when the noise model cannot be parametrized by a low-order model.

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arXiv2018

An analysis of the SPARSEVA estimate for the finite sample data case

In this paper, we develop an upper bound for the SPARSEVA (SPARSe Estimation based on a VAlidation criterion) estimation error in a general scheme, i.e., when the cost function is strongly convex and the regularized norm is decomposable for a pair of subspaces. We show how this general bound can be applied to a sparse regression problem to obtain an upper bound for the traditional SPARSEVA problem. Numerical results are used to illustrate the effectiveness of the suggested bound.

math.ST↗

arXiv2018

Parametric Identification Using Weighted Null-Space Fitting

In identification of dynamical systems, the prediction error method using a quadratic cost function provides asymptotically efficient estimates under Gaussian noise and additional mild assumptions, but in general it requires solving a non-convex optimization problem. An alternative class of methods uses a non-parametric model as intermediate step to obtain the model of interest. Weighted null-space fitting (WNSF) belongs to this class. It is a weighted least-squares method consisting of three steps. In the first step, a high-order ARX model is estimated. In a second least-squares step, this high-order estimate is reduced to a parametric estimate. In the third step, weighted least squares is used to reduce the variance of the estimates. The method is flexible in parametrization and suitable for both open- and closed-loop data. In this paper, we show that WNSF provides estimates with the same asymptotic properties as PEM with a quadratic cost function when the model orders are chosen according to the true system. Also, simulation studies indicate that WNSF may be competitive with state-of-the-art methods.

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arXiv2018

An asymptotically optimal indirect approach to continuous-time system identification

The indirect approach to continuous-time system identification consists in estimating continuous-time models by first determining an appropriate discrete-time model. For a zero-order hold sampling mechanism, this approach usually leads to a transfer function estimate with relative degree 1, independent of the relative degree of the strictly proper real system. In this paper, a refinement of these methods is developed. Inspired by indirect PEM, we propose a method that enforces a fixed relative degree in the continuous-time transfer function estimate, and show that the resulting estimator is consistent and asymptotically efficient. Extensive numerical simulations are put forward to show the performance of this estimator when contrasted with other indirect and direct methods for continuous-time system identification.

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arXiv2017

Sparse Iterative Learning Control with Application to a Wafer Stage: Achieving Performance, Resource Efficiency, and Task Flexibility

Trial-varying disturbances are a key concern in Iterative Learning Control (ILC) and may lead to inefficient and expensive implementations and severe performance deterioration. The aim of this paper is to develop a general framework for optimization-based ILC that allows for enforcing additional structure, including sparsity. The proposed method enforces sparsity in a generalized setting through convex relaxations using $\ell_1$ norms. The proposed ILC framework is applied to the optimization of sampling sequences for resource efficient implementation, trial-varying disturbance attenuation, and basis function selection. The framework has a large potential in control applications such as mechatronics, as is confirmed through an application on a wafer stage.

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arXiv2017

Computing monotone policies for Markov decision processes: a nearly-isotonic penalty approach

This paper discusses algorithms for solving Markov decision processes (MDPs) that have monotone optimal policies. We propose a two-stage alternating convex optimization scheme that can accelerate the search for an optimal policy by exploiting the monotone property. The first stage is a linear program formulated in terms of the joint state-action probabilities. The second stage is a regularized problem formulated in terms of the conditional probabilities of actions given states. The regularization uses techniques from nearly-isotonic regression. While a variety of iterative method can be used in the first formulation of the problem, we show in numerical simulations that, in particular, the alternating method of multipliers (ADMM) can be significantly accelerated using the regularization step.

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arXiv2017

Asymptotically Efficient Identification of Known-Sensor Hidden Markov Models

We consider estimating the transition probability matrix of a finite-state finite-observation alphabet hidden Markov model with known observation probabilities. The main contribution is a two-step algorithm; a method of moments estimator (formulated as a convex optimization problem) followed by a single iteration of a Newton-Raphson maximum likelihood estimator. The two-fold contribution of this letter is, firstly, to theoretically show that the proposed estimator is consistent and asymptotically efficient, and secondly, to numerically show that the method is computationally less demanding than conventional methods - in particular for large data sets.

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arXiv2016

Successive Concave Sparsity Approximation for Compressed Sensing

In this paper, based on a successively accuracy-increasing approximation of the $\ell_0$ norm, we propose a new algorithm for recovery of sparse vectors from underdetermined measurements. The approximations are realized with a certain class of concave functions that aggressively induce sparsity and their closeness to the $\ell_0$ norm can be controlled. We prove that the series of the approximations asymptotically coincides with the $\ell_1$ and $\ell_0$ norms when the approximation accuracy changes from the worst fitting to the best fitting. When measurements are noise-free, an optimization scheme is proposed which leads to a number of weighted $\ell_1$ minimization programs, whereas, in the presence of noise, we propose two iterative thresholding methods that are computationally appealing. A convergence guarantee for the iterative thresholding method is provided, and, for a particular function in the class of the approximating functions, we derive the closed-form thresholding operator. We further present some theoretical analyses via the restricted isometry, null space, and spherical section properties. Our extensive numerical simulations indicate that the proposed algorithm closely follows the performance of the oracle estimator for a range of sparsity levels wider than those of the state-of-the-art algorithms.

cs.IT↗

arXiv2016

A Class of Nonconvex Penalties Preserving Overall Convexity in Optimization-Based Mean Filtering

$\ell_1$ mean filtering is a conventional, optimization-based method to estimate the positions of jumps in a piecewise constant signal perturbed by additive noise. In this method, the $\ell_1$ norm penalizes sparsity of the first-order derivative of the signal. Theoretical results, however, show that in some situations, which can occur frequently in practice, even when the jump amplitudes tend to $\infty$, the conventional method identifies false change points. This issue is referred to as stair-casing problem and restricts practical importance of $\ell_1$ mean filtering. In this paper, sparsity is penalized more tightly than the $\ell_1$ norm by exploiting a certain class of nonconvex functions, while the strict convexity of the consequent optimization problem is preserved. This results in a higher performance in detecting change points. To theoretically justify the performance improvements over $\ell_1$ mean filtering, deterministic and stochastic sufficient conditions for exact change point recovery are derived. In particular, theoretical results show that in the stair-casing problem, our approach might be able to exclude the false change points, while $\ell_1$ mean filtering may fail. A number of numerical simulations assist to show superiority of our method over $\ell_1$ mean filtering and another state-of-the-art algorithm that promotes sparsity tighter than the $\ell_1$ norm. Specifically, it is shown that our approach can consistently detect change points when the jump amplitudes become sufficiently large, while the two other competitors cannot.

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arXiv2016

Particle-based Gaussian process optimization for input design in nonlinear dynamical models

We propose a novel approach to input design for identification of nonlinear state space models. The optimal input sequence is obtained by maximizing a scalar cost function of the Fisher information matrix. Since the Fisher information matrix is unavailable in closed form, it is estimated using particle methods. In addition, we make use of Gaussian process optimization to find the optimal input and to mitigate the problem of a large computational cost incurred by the particle filter, as the method reduces the number of functional evaluations. Numerical examples are provided to illustrate the performance of the resulting algorithm.

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arXiv2015

Estimator Selection: End-Performance Metric Aspects

Recently, a framework for application-oriented optimal experiment design has been introduced. In this context, the distance of the estimated system from the true one is measured in terms of a particular end-performance metric. This treatment leads to superior unknown system estimates to classical experiment designs based on usual pointwise functional distances of the estimated system from the true one. The separation of the system estimator from the experiment design is done within this new framework by choosing and fixing the estimation method to either a maximum likelihood (ML) approach or a Bayesian estimator such as the minimum mean square error (MMSE). Since the MMSE estimator delivers a system estimate with lower mean square error (MSE) than the ML estimator for finite-length experiments, it is usually considered the best choice in practice in signal processing and control applications. Within the application-oriented framework a related meaningful question is: Are there end-performance metrics for which the ML estimator outperforms the MMSE when the experiment is finite-length? In this paper, we affirmatively answer this question based on a simple linear Gaussian regression example.

cs.IT↗

arXiv2015

Evaluation of Spectral Learning for the Identification of Hidden Markov Models

Hidden Markov models have successfully been applied as models of discrete time series in many fields. Often, when applied in practice, the parameters of these models have to be estimated. The currently predominating identification methods, such as maximum-likelihood estimation and especially expectation-maximization, are iterative and prone to have problems with local minima. A non-iterative method employing a spectral subspace-like approach has recently been proposed in the machine learning literature. This paper evaluates the performance of this algorithm, and compares it to the performance of the expectation-maximization algorithm, on a number of numerical examples. We find that the performance is mixed; it successfully identifies some systems with relatively few available observations, but fails completely for some systems even when a large amount of observations is available. An open question is how this discrepancy can be explained. We provide some indications that it could be related to how well-conditioned some system parameters are.

stat.ML↗

arXiv2015

Reweighted nuclear norm regularization: A SPARSEVA approach

The aim of this paper is to develop a method to estimate high order FIR and ARX models using least squares with re-weighted nuclear norm regularization. Typically, the choice of the tuning parameter in the reweighting scheme is computationally expensive, hence we propose the use of the SPARSEVA (SPARSe Estimation based on a VAlidation criterion) framework to overcome this problem. Furthermore, we suggest the use of the prediction error criterion (PEC) to select the tuning parameter in the SPARSEVA algorithm. Numerical examples demonstrate the veracity of this method which has close ties with the traditional technique of cross validation, but using much less computations.

math.OC↗

arXiv2015

Upper Bounds on the Error of Sparse Vector and Low-Rank Matrix Recovery

Suppose that a solution $\widetilde{\mathbf{x}}$ to an underdetermined linear system $\mathbf{b} = \mathbf{A} \mathbf{x}$ is given. $\widetilde{\mathbf{x}}$ is approximately sparse meaning that it has a few large components compared to other small entries. However, the total number of nonzero components of $\widetilde{\mathbf{x}}$ is large enough to violate any condition for the uniqueness of the sparsest solution. On the other hand, if only the dominant components are considered, then it will satisfy the uniqueness conditions. One intuitively expects that $\widetilde{\mathbf{x}}$ should not be far from the true sparse solution $\mathbf{x}_0$. We show that this intuition is the case by providing an upper bound on $\| \widetilde{\mathbf{x}} - \mathbf{x}_0\|$ which is a function of the magnitudes of small components of $\widetilde{\mathbf{x}}$ but independent from $\mathbf{x}_0$. This result is extended to the case that $\mathbf{b}$ is perturbed by noise. Additionally, we generalize the upper bounds to the low-rank matrix recovery problem.

cs.IT↗

arXiv2015

Approximate Regularization Paths for Nuclear Norm Minimization Using Singular Value Bounds -- With Implementation and Extended Appendix

The widely used nuclear norm heuristic for rank minimization problems introduces a regularization parameter which is difficult to tune. We have recently proposed a method to approximate the regularization path, i.e., the optimal solution as a function of the parameter, which requires solving the problem only for a sparse set of points. In this paper, we extend the algorithm to provide error bounds for the singular values of the approximation. We exemplify the algorithms on large scale benchmark examples in model order reduction. Here, the order of a dynamical system is reduced by means of constrained minimization of the nuclear norm of a Hankel matrix.

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arXiv2015

Bayesian Learning for Low-Rank matrix reconstruction

We develop latent variable models for Bayesian learning based low-rank matrix completion and reconstruction from linear measurements. For under-determined systems, the developed methods are shown to reconstruct low-rank matrices when neither the rank nor the noise power is known a-priori. We derive relations between the latent variable models and several low-rank promoting penalty functions. The relations justify the use of Kronecker structured covariance matrices in a Gaussian based prior. In the methods, we use evidence approximation and expectation-maximization to learn the model parameters. The performance of the methods is evaluated through extensive numerical simulations.

stat.ML↗

arXiv2015

Variance Analysis of Linear SIMO Models with Spatially Correlated Noise

Substantial improvement in accuracy of identified linear time-invariant single-input multi-output (SIMO) dynamical models is possible when the disturbances affecting the output measurements are spatially correlated. Using an orthogonal representation for the modules composing the SIMO structure, in this paper we show that the variance of a parameter estimate of a module is dependent on the model structure of the other modules, and the correlation structure of the disturbances. In addition, we quantify the variance-error for the parameter estimates for finite model orders, where the effect of noise correlation structure, model structure and signal spectra are visible. From these results, we derive the noise correlation structure under which the mentioned model parameterization gives the lowest variance, when one module is identified using less parameters than the other modules.

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