Peak Time-Windowed Risk Estimation of Stochastic Processes

TL;DR

This work addresses the problem of upper-bounding extreme values of time-windowed risk for stochastic processes, using coherent risk measures such as the mean and ES. The authors lift the nonconvex risk-estimation task to a convex infinite-dimensional linear program in occupation measures with an augmented time variable, and then truncate via the Moment-SOS hierarchy to obtain computable semidefinite programs. They prove no-relaxation gaps and strong duality for the mean and ES cases, and establish convergence of the finite truncations under compactness and regularity. Numerical examples on nonlinear oscillators and a twist system illustrate the efficacy, providing rigorous, computable bounds that are relevant for applications like power-system safety and reliability.

Abstract

This paper develops a method to upper-bound extreme-values of time-windowed risks for stochastic processes. Examples of such risks include the maximum average or 90% quantile of the current along a transmission line in any 5-minute window. This work casts the time-windowed risk analysis problem as an infinite-dimensional linear program in occupation measures. In particular, we employ the coherent risk measures of the mean and the expected shortfall (conditional value at risk) to define the maximal time-windowed risk along trajectories. The infinite-dimensional linear program must then be truncated into finite-dimensional optimization problems, such as by using the moment-sum of squares hierarchy of semidefinite programs. The infinite-dimensional linear program will have the same optimal value as the original nonconvex risk estimation task under compactness and regularity assumptions, and the sequence of semidefinite programs will converge to the true value under additional properties of algebraic characterization. The scheme is demonstrated for risk analysis of example stochastic processes.

Figures (11)

• Figure 1: Comparison of instantaneous peak (square) and time-windowed average peak (star) of a signal $p(t)$ (black curve)
• Figure 2: Empirical Cumulative Density Function of the value of $p(t)$ in times $[1.25, 2.75]$
• Figure 3: Comparison of instantaneous peak (square) to time-windowed average peak (blue star) of a signal $p(t)$ from Figure \ref{['fig:demo_peak_inst_avg']} with stopping time $s \in \{h, \ t^*, \ T\}$. The magenta region highlights $p(t)$ in times $[s-h, s]$. The time-windowed average in this range is marked with the black star. The red dotted line depicts all time-windowed averages up to $t\leq s$.
• Figure 4: Trajectories of Twist system \ref{['eq:twist_dynamics']} starting from the gray-sphere initial set $X_0$.
• Figure 5: Evolution of time-windowed mean of $p(x)=x_3, h=1$ along trajectories in Figure \ref{['fig:twist_state']}.

Theorems & Definitions (27)

• Theorem 4.2
• proof
• Theorem 4.3
• proof
• Theorem 4.5
• proof
• Remark 1
• Remark 2
• Remark 3
• Theorem 5.2