Dealing with Unknown Unknowns: Identification and Selection of Minimal Sensing for Fractional Dynamics with Unknown Inputs

TL;DR

This paper develops a new framework for analyzing fractional order complex networks with unknown external stimuli, enabling the estimation of system states and inputs with minimal measurements, validated on real EEG data.

Contribution

It introduces an alternating scheme for parameter and stimulus estimation, provides conditions for state and input recovery, and identifies minimal measurement subsets with guarantees.

Findings

Successful estimation of unknown stimuli in fractional networks.

Conditions established for exact state and input recovery.

Validated approach on real EEG data.

Abstract

This paper focuses on analysis and design of time-varying complex networks having fractional order dynamics. These systems are key in modeling the complex dynamical processes arising in several natural and man made systems. Notably, examples include neurophysiological signals such as electroencephalogram (EEG) that captures the variation in potential fields, and blood oxygenation level dependent (BOLD) signal, which serves as a proxy for neuronal activity. Notwithstanding, the complex networks originated by locally measuring EEG and BOLD are often treated as isolated networks and do not capture the dependency from external stimuli, e.g., originated in subcortical structures such as the thalamus and the brain stem. Therefore, we propose a paradigm-shift towards the analysis of such complex networks under unknown unknowns (i.e., excitations). Consequently, the main contributions of the present paper are threefold: (i) we present an alternating scheme that enables to determine the best estimate of the model parameters and unknown stimuli; (ii) we provide necessary and sufficient conditions to ensure that it is possible to retrieve the state and unknown stimuli; and (iii) upon these conditions we determine a small subset of variables that need to be measured to ensure that both state and input can be recovered, while establishing sub-optimality guarantees with respect to the smallest possible subset. Finally, we present several pedagogical examples of the main results using real data collected from an EEG wearable device.