Renormalization group approach to connect discrete- and continuous-time descriptions of Gaussian processes

Discretization of continuous stochastic processes is needed to numerically simulate them or to infer models from experimental time series. However, depending on the nature of the process, the same discretization scheme may perform very differently for the two tasks, if it is not accurate enough. Exact discretizations, which work equally well at any scale, are characterized by the property of invariance under coarse-graining. Motivated by this observation, we build an explicit renormalization group (RG) approach for Gaussian time series generated by autoregressive models. We show that the RG fixed points correspond to discretizations of linear SDEs, and only come in the form of first order Markov processes or non-Markovian ones. This fact provides an alternative explanation of why standard delay-vector embedding procedures fail in reconstructing partially observed noise-driven systems. We also suggest a possible effective Markovian discretization for the inference of partially observed underdamped equilibrium processes based on the exploitation of the Einstein relation.

Figure 1

Sketch of the RG transformation for a time series. The discretely observed coordinates $\left\{X_n\right\}$ play the role of spin variables on a lattice, with nearest neighbor and next-nearest neighbor coupling. The time step $\tau$ is the lattice spacing. Model parameters, $A^0$, are transformed into ${\tilde{A}}^0$ through coarse-graining, then rescaled to $A^1$ as we map $\tau \to \tau /2$.

Figure 2

Coarse-graining by decimation maps the ladder topology, corresponding to our AR(2) model, to a fully connected net. The new couplings are however not independent due to the nontrivial correlation properties of the random contributions.

Figure 3

Fixed-point manifolds of the RG map, projected on the plane of leading autoregressive coefficients ${\psi }_0,{\theta }_0$. The interior of the triangle is the basin of attraction of fixed point A. The equal sides are the basin of attraction of point B. The basins of points C and D are contained in the basis of the triangle (plus vertex on the top for D). The Euler AR(2) process has coordinates ($2,-1$) in this plane. Color-shaded trajectories represent the solution of recurrence relations from three sample initial conditions. Arrows show the direction of (discrete) moves and should not be interpreted as continuous flow lines.