Finding the likely behaviors of static continuous nonlinear systems

This paper describes two methods for predicting the likely behaviors of static continuous nonlinear systems with varying input values. The methods use a parameterized equation model and upper or lower bounds on the joint input density to bound the likelihood of a behavior, such as a state variable being inside a numeric range. Using a bound on the density instead of the density itself is desirable because the density's parameters and shape are not exactly known. The first method is limited to using lower density bounds. It finds rough bounds at first, and then refines them as more iterations of the method are allowed. The second method is a hit-or-miss version of sample-mean Monte Carlo. Unlike the first method, the second method can also handle upper density bounds, which are more useful than lower density bounds, but the generated probability bounds are only approximate. However, standard deviations on the bounds are given and become small as the sample size increases. In contrast to other researchers' methods, the two methods described here (1) find all the possible system behaviors, and tell how likely they are, (2) do not just approximate the distribution of possible outcomes without some measure of the error magnitude, (3) do not use discretized variable values, which limit the events one can find probability bounds for, (4) can handle density bounds, and (5) can handle such criteria as two state variables both being inside a numeric range.