Information Length Analysis of Linear Autonomous Stochastic Processes

Abstract

When studying the behaviour of complex dynamical systems, a statistical formulation can provide useful insights. In particular, information geometry is a promising tool for this purpose. In this paper, we investigate the information length for n-dimensional linear autonomous stochastic processes, providing a basic theoretical framework that can be applied to a large set of problems in engineering and physics. A specific application is made to a harmonically bound particle system with the natural oscillation frequency ω, subject to a damping γ and a Gaussian white-noise. We explore how the information length depends on ω and γ, elucidating the role of critical damping γ=2ω in information geometry. Furthermore, in the long time limit, we show that the information length reflects the linear geometry associated with the Gaussian statistics in a linear stochastic process.

Memory and Markov Blankets

“…In addition to thinking about the difference between joint and marginal densities, we can also think about divergences between the same densities at different times. Here, there is a point of contact with an important measure used to characterize itinerancy in non-equilibrium systems [ 26 , 27 , 28 ]. This approach borrows from information geometry [ 29 , 30 ], which applies methods from differential geometry to probability theory.…”

“…Itinerancy of this sort can be induced by giving the solenoidal flow a larger part to play. The (linear) association between solenoidal flow and the information length of the path to steady state has been convincingly demonstrated for stochastic systems that conform to a Laplace assumption [ 28 ]. Figure 4 reproduces this result, in addition to the other measures from Figure 3 , by varying θ .…”

Extreme Value Theory in Application to Delivery Delays

“…Delays are distributed randomly. Thus, the process dynamics are different from the Ornstein-Uhlenbeck process, as was considered in [ 56 ]. It is often called a jump-diffusion process or the Ornstein–Uhlenbeck processes driven by the Lévy process [ 57 , 58 ].…”

Causal Information Rate

“…To demonstrate how the methods Equations ( 3)-( 5) work, in this section, we investigate an analytically solvable, Kramers equation, governed by the following Langevin equations [31,42]:…”

“…We have recently proposed information-geometric theory as a powerful tool to understand non-equilibrium stochastic processes that often involve high temporal variabilities and large fluctuations [20][21][22][23][24][25][26][27][28][29][30][31][32], as often the case of rare, extreme events. This is based on the surprisal rate, r(x, t) = ∂ t s(x, t) = −∂ t ln p(x, t), where p(x, t) is a probability density function (PDF) of a random variable x at time t, and s(x, t) = − ln p(x, t) is a local entropy.…”