#### Reasoning about Uncertainties in Discrete-Time Dynamical Systems using Polynomial Forms.

Part of Advances in Neural Information Processing Systems 33 (NeurIPS 2020)

#### Authors

*Sriram Sankaranarayanan, Yi Chou, Eric Goubault, Sylvie Putot*

#### Abstract

<p>In this paper, we propose polynomial forms to represent distributions of state
variables over time for discrete-time stochastic dynamical systems. This
problem arises in a variety of applications in areas ranging from biology to
robotics. Our approach allows us to rigorously represent the probability
distribution of state variables over time, and provide guaranteed bounds on
the expectations, moments and probabilities of tail events involving the state
variables. First, we recall ideas from interval arithmetic, and use them to
rigorously represent the state variables at time t as a function of the
initial state variables and noise symbols that model the random
exogenous inputs encountered before time t. Next, we show how concentration
of measure inequalities can be employed to prove rigorous bounds on the tail
probabilities of these state variables. We demonstrate interesting
applications that demonstrate how our approach can be useful in some
situations to establish mathematically guaranteed bounds that are of a
different nature from those obtained through simulations with pseudo-random
numbers.</p>