Ensemble Observability of Linear Systems

Abstract

We address the observability problem for ensembles that are described by probability distributions. The problem is to reconstruct a probability distribution of the initial state from the time-evolution of the probability distribution of the output under a classical finite-dimensional linear system. We present two solutions to this problem, one based on formulating the problem as an inverse problem and the other one based on reconstructing all the moments of the distribution. The first approach leads us to a connection between the reconstruction problem and mathematical tomography problems. In the second approach we use the framework of tensor systems to describe the dynamics of the moments which leads to a more systems theoretic treatment of the reconstruction problem. Furthermore we show that both frameworks are inherently related. The appeal of having two dual viewpoints, the first being more geometric and the second one being more systems theoretic, is illuminated in several examples of theoretical or practical importance.

Identification of cell‐to‐cell heterogeneity through systems engineering approaches

“…5,30,39 However, the identifiability of an IBPM has not been examined as thoroughly as that of a deterministic model. Only available method for the IBPM is proposed by Zeng et al, 31,33 but their method is only applicable for a linear IBPM while the most of the models for intracellular processes are nonlinear. Therefore, their method is not implemented in this study.…”

“…30,31 Here, the identifiability of an IBPM studies the problem whether the PDFs of model parameters can be uniquely estimated from measurement distributions. [31][32][33] For an IBPM, the nonidentifiability of PDFs of some model parameters arises due to limited quality and quantity of measurements as well as the model structure. Consequently, the PDFs of only a small subset of model parameters are identifiable, and these subsets may not necessarily overlap with the parameter subset selected based on the priori knowledge on the system itself.…”

“…Consequently, the PDFs of only a small subset of model parameters are identifiable, and these subsets may not necessarily overlap with the parameter subset selected based on the priori knowledge on the system itself. 5,30,31 The above considerations form the motivation of this work. We propose a new methodology to estimate the PDFs of parameters from single-cell measurements by handling the identifiability issue without increasing the computational cost.…”

Structure theory for ensemble controllability, observability, and duality

“…The authors addressed the observability of the above ensemble system using the duality between controllability and observability of infinite-dimensional linear systems [18]. We also refer the reader to [19] for a related observability problem about estimating the probability distribution of the initial state. Specifically, the authors there considered a single time-invariant linear system: x(t) = Ax(t) + Bu(t) and y(t) = C x(t).…”

Dynamic Density Estimation in Heterogeneous Cell Populations

“…In contrast, we now consider noise-free measurements, as well as measurements that are subject to a multiplicative and log-normal distributed noise v v v × ∼ exp N (0 0 0, I I I · 10 −2 ) , as this is the main type of disturbance observed in biological experiments [17]. Ensemble observability: Following the theory developed in [18], ensemble observability denotes the property that the NDF can be reconstructed from the measured marginal distributions. Applying the conditions discovered in [18], the model (19) is ensemble observable.…”