Minimum Number of Probes for Brain Dynamics Observability

In this paper, we address the problem of placing sensor probes in the brain such that the system dynamics' are generically observable. The system dynamics whose states can encode for instance the fire-rating of the neurons or their ensemble following a neural-topological (structural) approach, and the sensors are assumed to be dedicated, i.e., can only measure a state at each time. Even though the mathematical description of brain dynamics is (yet) to be discovered, we build on its observed fractal characteristics and assume that the model of the brain activity satisfies fractional-order dynamics. Although the sensor placement explored in this paper is particularly considering the observability of brain dynamics, the proposed methodology applies to any fractional-order linear system. Thus, the main contribution of this paper is to show how to place the minimum number of dedicated sensors, i.e., sensors measuring only a state variable, to ensure generic observability in discrete-time fractional-order systems for a specified finite interval of time. Finally, an illustrative example of the main results is provided using electroencephalogram (EEG) data.Comment: arXiv admin note: text overlap with arXiv:1507.0720

Single shot parameter estimation via continuous quantum measurement

We present filtering equations for single shot parameter estimation using continuous quantum measurement. By embedding parameter estimation in the standard quantum filtering formalism, we derive the optimal Bayesian filter for cases when the parameter takes on a finite range of values. Leveraging recent convergence results [van Handel, arXiv:0709.2216 (2008)], we give a condition which determines the asymptotic convergence of the estimator. For cases when the parameter is continuous valued, we develop quantum particle filters as a practical computational method for quantum parameter estimation.Comment: 9 pages, 5 image

Degenerate Kalman filter error covariances and their convergence onto the unstable subspace

The characteristics of the model dynamics are critical in the performance of (ensemble) Kalman filters. In particular, as emphasized in the seminal work of Anna Trevisan and coauthors, the error covariance matrix is asymptotically supported by the unstable-neutral subspace only, i.e., it is spanned by the backward Lyapunov vectors with nonnegative exponents. This behavior is at the core of algorithms known as assimilation in the unstable subspace, although a formal proof was still missing. This paper provides the analytical proof of the convergence of the Kalman filter covariance matrix onto the unstable-neutral subspace when the dynamics and the observation operator are linear and when the dynamical model is error free, for any, possibly rank-deficient, initial error covariance matrix. The rate of convergence is provided as well. The derivation is based on an expression that explicitly relates the error covariances at an arbitrary time to the initial ones. It is also shown that if the unstable and neutral directions of the model are sufficiently observed and if the column space of the initial covariance matrix has a nonzero projection onto all of the forward Lyapunov vectors associated with the unstable and neutral directions of the dynamics, the covariance matrix of the Kalman filter collapses onto an asymptotic sequence which is independent of the initial covariances. Numerical results are also shown to illustrate and support the theoretical findings

A Framework for Phasor Measurement Placement in Hybrid State Estimation via Gauss-Newton

In this paper, we study the placement of Phasor Measurement Units (PMU) for enhancing hybrid state estimation via the traditional Gauss-Newton method, which uses measurements from both PMU devices and Supervisory Control and Data Acquisition (SCADA) systems. To compare the impact of PMU placements, we introduce a useful metric which accounts for three important requirements in power system state estimation: {\it convergence}, {\it observability} and {\it performance} (COP). Our COP metric can be used to evaluate the estimation performance and numerical stability of the state estimator, which is later used to optimize the PMU locations. In particular, we cast the optimal placement problem in a unified formulation as a semi-definite program (SDP) with integer variables and constraints that guarantee observability in case of measurements loss. Last but not least, we propose a relaxation scheme of the original integer-constrained SDP with randomization techniques, which closely approximates the optimum deployment. Simulations of the IEEE-30 and 118 systems corroborate our analysis, showing that the proposed scheme improves the convergence of the state estimator, while maintaining optimal asymptotic performance.Comment: accepted to IEEE Trans. on Power System

Efficiency and Sensitivity Analysis of Observation Networks for Atmospheric Inverse Modelling with Emissions

The controllability of advection-diffusion systems, subject to uncertain initial values and emission rates, is estimated, given sparse and error affected observations of prognostic state variables. In predictive geophysical model systems, like atmospheric chemistry simulations, different parameter families influence the temporal evolution of the system.This renders initial-value-only optimisation by traditional data assimilation methods as insufficient. In this paper, a quantitative assessment method on validation of measurement configurations to optimize initial values and emission rates, and how to balance them, is introduced. In this theoretical approach, Kalman filter and smoother and their ensemble based versions are combined with a singular value decomposition, to evaluate the potential improvement associated with specific observational network configurations. Further, with the same singular vector analysis for the efficiency of observations, their sensitivity to model control can be identified by determining the direction and strength of maximum perturbation in a finite-time interval.Comment: 30 pages, 10 figures, 5 table