On the covariance matrix of the stationary distribution of a noisy dynamical system

Abstract

In this paper, we analyze the relation between the stability of a noisy dynamical system based on linear approximation and the covariance matrix of its stationary distribution. We reformulate the theory of dynamical network biomarkers in terms of the covariance matrix and clarify the limiting behavior of the covariance matrix when a dynamical system approaches a bifurcation point. We also discuss the relation between the Jacobian matrix and principal component analysis. An application to a simple nonlinear network model is also demonstrated.

Identifying pre-disease signals before metabolic syndrome in mice by dynamical network biomarkers

“…We could not measure the recovery rate directly in this study because unrealizable repetitive measurements at short intervals from the same individual is generally required. On the other hand, the relative recovery rate can be estimated because the recovery rate is known to be approximately and inversely proportional to the largest eigenvalue of the sample covariance matrix of the state variables 19 . By using the 147 DNB genes, we estimated the relative recovery rate for each week, and observed that the estimated relative recovery rate was smallest at 5 weeks of age (Supplementary Fig.…”

“…The DNB theory has been applied to real data of many diseases, such as acute respiratory distress syndrome 5 , diabetes mellitus 12 , influenza 14 , cancer 13–15,17 , and Alzheimer’s disease 16 , as well as experimental data in cell biology, such as that of cell differentiation 18 . Moreover, researches on the improvement of the statistical methods 14,16 and refinement of the theory 19 are in progress.…”

Two Novel Methods for Extracting Synchronously Fluctuated Genes

“…Here, I briefly explain the reason why PCA can be used for extracting SFGs. When the stability of a living organism decreases, the stationary distribution of the state variables becomes wider if the assumptions used in the theory are applicable [11]. The expansion rate is largest in the direction with the largest vulnerability to disturbances.…”

“…The prediction that the decrease of the stability of a living organism may cause the increase of fluctuation levels and enhancement of synchrony or correlations for some gene expressions comes from theoretical analyses [1], [10], [11]. The theories are based on stationary, unbounded, linear, and monostable dynamical systems' limiting behavior.…”

“…The theories are based on stationary, unbounded, linear, and monostable dynamical systems' limiting behavior. Under these conditions, the variances of the state variables go to infinity [10], the correlation coefficients between the state variables go to either 1 or −1 if the dominant eigenvalue of the Jacobian matrix is real-valued [1], and the stationary distribution of the state variables expands along a two-dimensional plane determined by the dominant eigenvectors if the dominant eigenvalues of the Jacobian matrix are complexvalued [11]. On the other hand, real living organisms are nonstationary, bounded, nonlinear, and multistable, and state transitions can occur well before the limit state.…”