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Papers by boumediene hamzi

Physica A: Statistical Mechanics and its Applications

To what extent can we forecast a time series without fitting to historical data? Can universal pa... more To what extent can we forecast a time series without fitting to historical data? Can universal patterns of probability help in this task? Deep relations between pattern Kolmogorov complexity and pattern probability have recently been used to make a priori probability predictions in a variety of systems in physics, biology and engineering. Here we study simplicity bias (SB)-an exponential upper bound decay in pattern probability with increasing complexity-in discretised time series extracted from the World Bank Open Data collection. We predict upper bounds on the probability of discretised series patterns, without fitting to trends in the data. Thus we perform a kind of 'forecasting without training data'. Additionally we make predictions about which of two discretised series is more likely with accuracy of ∼80%, much higher than a 50% baseline rate, just by using the complexity of each series. These results point to a promising perspective on practical time series forecasting and integration with machine learning methods.

Physica D: Nonlinear Phenomena

We consider the problem of learning Stochastic Differential Equations of the form dXt = f (Xt)dt ... more We consider the problem of learning Stochastic Differential Equations of the form dXt = f (Xt)dt + σ(Xt)dWt from one sample trajectory. This problem is more challenging than learning deterministic dynamical systems because one sample trajectory only provides indirect information on the unknown functions f , σ, and stochastic process dWt representing the drift, the diffusion, and the stochastic forcing terms, respectively. We propose a method that combines Computational Graph Completion [46] and data adapted kernels learned via a new variant of cross validation. Our approach can be decomposed as follows: (1) Represent the time-increment map Xt → X t+dt as a Computational Graph in which f , σ and dWt appear as unknown functions and random variables. (2) Complete the graph (approximate unknown functions and random variables) via Maximum a Posteriori Estimation (given the data) with Gaussian Process (GP) priors on the unknown functions. (3) Learn the covariance functions (kernels) of the GP priors from data with randomized cross-validation. Numerical experiments illustrate the efficacy, robustness, and scope of our method.

In this paper, we introduce the Controlled center dynamics for nonlinear discrete time systems wi... more In this paper, we introduce the Controlled center dynamics for nonlinear discrete time systems with uncontrollable linearization. This is a reduced order control system whose dimension is the number of uncontrollable modes and whose stabilizability properties determine the stabilizability properties of the full order system. After reducing the order of the system, the synthesis of a stabilizing controller is performed based on the reduced order control system. By changing the feedback, the stability properties of the controlled center dynamics will change, and thus the stability properties of the full order system will change too. Thus, choosing a feedback that stabilizes the controlled center dynamics allows stabilizing the full order system. This approach is a reduction technique for some classes of controlled differential equations.

The paper studies an extension to nonlinear systems of a recently proposed approach to the concep... more The paper studies an extension to nonlinear systems of a recently proposed approach to the concept of modal participation factors. First, a definition is given for local mode-in-state participation factors for smooth nonlinear autonomous systems. The definition is general, and, unlike in the more traditional approach, the resulting participation measures depend on the assumed uncertainty law governing the system initial condition. The work follows Hashlamoun, Hassouneh and Abed (2009) in taking a mathematical expectation (or set-theoretic average) of a modal contribution measure with respect to an assumed uncertain initial state. As in the linear case, it is found that a symmetry assumption on the distribution of the initial state results in a tractable calculation and an explicit and simple formula for mode-in-state participation factors.

We introduce a data-driven order reduction method for nonlinear control systems, drawing on recen... more We introduce a data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves linearly when lifted into a high (or infinite) dimensional feature space where balanced truncation may be carried out implicitly. This leads to a nonlinear reduction map which can be combined with a representation of the system belonging to a reproducing kernel Hilbert space to give a closed, reduced order dynamical system which captures the essential input-output characteristics of the original model. Empirical simulations illustrating the approach are also provided. I.

arXiv: Dynamical Systems, 2020

For certain dynamical systems it is possible to significantly simplify the study of stability by ... more For certain dynamical systems it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to a non-hyperbolic equilibrium point, and to obtain meaningful predictions of its behavior by analyzing a reduced dimensional problem. Since the manifold is usually not known, approximation methods are of great interest to obtain qualitative estimates. In this work, we use a data-based greedy kernel method to construct a suitable approximation of the manifold close to the equilibrium. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used to construct a surrogate model of the manifold. The method is tested on different examples which show promising performance and good accuracy.

arXiv: Dynamical Systems, 2015

Methods from learning theory are used in the state space of linear dynamical and control systems ... more Methods from learning theory are used in the state space of linear dynamical and control systems in order to estimate the system matrices. An application to stabilization via algebraic Riccati equations is included. The approach is illustrated via a series of numerical examples.

IFAC Proceedings Volumes, 1998

In this paper, we simplify and classify the equilibrium sets for nonlinear discrete time control ... more In this paper, we simplify and classify the equilibrium sets for nonlinear discrete time control systems which are not linearly controllable. Based on the normal forms of control systems, it is shown that this classification is intimately linked to a set of quadratic invariants.

Journal of Computational Dynamics, 2017

We introduce a data-based approach to estimating key quantities which arise in the study of nonli... more We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systemswith a reasonable expectation of success-once the nonlinear system has been mapped into a high or infinite dimensional feature space. In particular, we embed a nonlinear system in a reproducing kernel Hilbert space where linear theory can be used to develop computable, non-parametric estimators approximating controllability and observability energy functions for nonlinear systems. In all cases the relevant quantities are estimated from simulated or observed data. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system.

IFAC Proceedings Volumes, 2001

In this paper, we determine the normal forms for discrete-time control systems with a Naimark-Sa.... more In this paper, we determine the normal forms for discrete-time control systems with a Naimark-Sa.cker bifurcation. Based on the normal forms, we find stabilizability conditions and synthesize a quadratic stabilizing controller. Copyright© 2001 IFAC

IFAC Proceedings Volumes, 2005

IFAC Proceedings Volumes, 2006

Physica A: Statistical Mechanics and its Applications

To what extent can we forecast a time series without fitting to historical data? Can universal pa... more To what extent can we forecast a time series without fitting to historical data? Can universal patterns of probability help in this task? Deep relations between pattern Kolmogorov complexity and pattern probability have recently been used to make a priori probability predictions in a variety of systems in physics, biology and engineering. Here we study simplicity bias (SB)-an exponential upper bound decay in pattern probability with increasing complexity-in discretised time series extracted from the World Bank Open Data collection. We predict upper bounds on the probability of discretised series patterns, without fitting to trends in the data. Thus we perform a kind of 'forecasting without training data'. Additionally we make predictions about which of two discretised series is more likely with accuracy of ∼80%, much higher than a 50% baseline rate, just by using the complexity of each series. These results point to a promising perspective on practical time series forecasting and integration with machine learning methods.

Physica D: Nonlinear Phenomena

We consider the problem of learning Stochastic Differential Equations of the form dXt = f (Xt)dt ... more We consider the problem of learning Stochastic Differential Equations of the form dXt = f (Xt)dt + σ(Xt)dWt from one sample trajectory. This problem is more challenging than learning deterministic dynamical systems because one sample trajectory only provides indirect information on the unknown functions f , σ, and stochastic process dWt representing the drift, the diffusion, and the stochastic forcing terms, respectively. We propose a method that combines Computational Graph Completion [46] and data adapted kernels learned via a new variant of cross validation. Our approach can be decomposed as follows: (1) Represent the time-increment map Xt → X t+dt as a Computational Graph in which f , σ and dWt appear as unknown functions and random variables. (2) Complete the graph (approximate unknown functions and random variables) via Maximum a Posteriori Estimation (given the data) with Gaussian Process (GP) priors on the unknown functions. (3) Learn the covariance functions (kernels) of the GP priors from data with randomized cross-validation. Numerical experiments illustrate the efficacy, robustness, and scope of our method.

In this paper, we introduce the Controlled center dynamics for nonlinear discrete time systems wi... more In this paper, we introduce the Controlled center dynamics for nonlinear discrete time systems with uncontrollable linearization. This is a reduced order control system whose dimension is the number of uncontrollable modes and whose stabilizability properties determine the stabilizability properties of the full order system. After reducing the order of the system, the synthesis of a stabilizing controller is performed based on the reduced order control system. By changing the feedback, the stability properties of the controlled center dynamics will change, and thus the stability properties of the full order system will change too. Thus, choosing a feedback that stabilizes the controlled center dynamics allows stabilizing the full order system. This approach is a reduction technique for some classes of controlled differential equations.

The paper studies an extension to nonlinear systems of a recently proposed approach to the concep... more The paper studies an extension to nonlinear systems of a recently proposed approach to the concept of modal participation factors. First, a definition is given for local mode-in-state participation factors for smooth nonlinear autonomous systems. The definition is general, and, unlike in the more traditional approach, the resulting participation measures depend on the assumed uncertainty law governing the system initial condition. The work follows Hashlamoun, Hassouneh and Abed (2009) in taking a mathematical expectation (or set-theoretic average) of a modal contribution measure with respect to an assumed uncertain initial state. As in the linear case, it is found that a symmetry assumption on the distribution of the initial state results in a tractable calculation and an explicit and simple formula for mode-in-state participation factors.

We introduce a data-driven order reduction method for nonlinear control systems, drawing on recen... more We introduce a data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves linearly when lifted into a high (or infinite) dimensional feature space where balanced truncation may be carried out implicitly. This leads to a nonlinear reduction map which can be combined with a representation of the system belonging to a reproducing kernel Hilbert space to give a closed, reduced order dynamical system which captures the essential input-output characteristics of the original model. Empirical simulations illustrating the approach are also provided. I.

arXiv: Dynamical Systems, 2020

For certain dynamical systems it is possible to significantly simplify the study of stability by ... more For certain dynamical systems it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to a non-hyperbolic equilibrium point, and to obtain meaningful predictions of its behavior by analyzing a reduced dimensional problem. Since the manifold is usually not known, approximation methods are of great interest to obtain qualitative estimates. In this work, we use a data-based greedy kernel method to construct a suitable approximation of the manifold close to the equilibrium. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used to construct a surrogate model of the manifold. The method is tested on different examples which show promising performance and good accuracy.

arXiv: Dynamical Systems, 2015

Methods from learning theory are used in the state space of linear dynamical and control systems ... more Methods from learning theory are used in the state space of linear dynamical and control systems in order to estimate the system matrices. An application to stabilization via algebraic Riccati equations is included. The approach is illustrated via a series of numerical examples.

IFAC Proceedings Volumes, 1998

In this paper, we simplify and classify the equilibrium sets for nonlinear discrete time control ... more In this paper, we simplify and classify the equilibrium sets for nonlinear discrete time control systems which are not linearly controllable. Based on the normal forms of control systems, it is shown that this classification is intimately linked to a set of quadratic invariants.

Journal of Computational Dynamics, 2017

We introduce a data-based approach to estimating key quantities which arise in the study of nonli... more We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systemswith a reasonable expectation of success-once the nonlinear system has been mapped into a high or infinite dimensional feature space. In particular, we embed a nonlinear system in a reproducing kernel Hilbert space where linear theory can be used to develop computable, non-parametric estimators approximating controllability and observability energy functions for nonlinear systems. In all cases the relevant quantities are estimated from simulated or observed data. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system.

IFAC Proceedings Volumes, 2001

In this paper, we determine the normal forms for discrete-time control systems with a Naimark-Sa.... more In this paper, we determine the normal forms for discrete-time control systems with a Naimark-Sa.cker bifurcation. Based on the normal forms, we find stabilizability conditions and synthesize a quadratic stabilizing controller. Copyright© 2001 IFAC

IFAC Proceedings Volumes, 2005

IFAC Proceedings Volumes, 2006