To evaluate models of dynamical systems, researchers have traditionally used quantitative measures of short term prediction errors. However, for chaotic or stochastic systems, comparison of long term, qualitative behaviors may be more relevant. Let x = (x0. . . . , xn) be a sequence of real numbers generated by sampling a dynamical system or stochastic process and suppose y = (y0, . . . . yn) is another sequence, generated by a mathematical model of the process which generated x. In this paper we consider several ways of assigning a distance d(x, y) which measures the difference in long term behavior. © 1997 Elsevier Science B.V. All rights reserved.