NLF: Nonlinear Forecasting for Dynamical Systems

A nonlinear equation is one in which a variable on the right side of the equation is raised to an exponent which is not equal to one (e.g., f (x) = x2). Nonlinear systems are generally thought of as systems whose motion is governed by mechanisms which behave as nonlinear equations. The sigmoidal-shaped oxygen-saturation curve of hemoglobin and the relation between vascular resistance and blood vessel diameter are only two of many examples of nonlinear systems in biology. Some nonlinear systems are capable of generating data sets of bewildering complexity. In fact, one special set of nonlinear systems, called "chaotic" systems, produces data sets so complicated that it is difficult to distinguish them from randomly generated data sets. Dynamical Systems, Inc., has developed a line of data analysis software for the study of chaotic systems. The company offers two packages that have been reviewed in previous issues of the Joumal. These two main packages include some of the most widely used methods for analyzing chaotic dynamics. The subject of this review, NLF: Nonlinear Forecasting for Dynamical Systems, is a stand-alone package which utilizes relatively novel analytic techniques. While chaotic systems might seem to be unpredictable by definition, in fact, parts of a chaotic system are predictable and the predictability varies for different parts of the system. The technique of nonlinear forecasting can be used to study how predictability varies within a chaotic system. Although the usefulness of this form of analysis is controversial, that debate is beyond the scope of this review. In order to run at all, the program requires a PC-compatible machine, 640K RAM, and VGA or EGA graphics. The company also strongly recommends a hard disk and a numeric co-processor. The computation-intensive nature of this program makes severe demands on even the nimblest of computers. In this reviewer's experience, a fairly fast computer should also be considered a necessity. The package itself contains a main analysis program, several helpful utilities, and a manual. The manual is thoroughly referenced and lucidly written. In addition to program instructions, an entire chapter of the manual is devoted to background mathematics of dynamical systems with special emphasis on chaotic systems and nonlinear forecasting. The main analysis program, NLF, is mouse-driven and contains on-screen and pull-down menus. After importing a specified data file, the program divides the data set into two parts, one which serves as the basis for creating a prediction and another which is used to test the prediction. At the outset, the user is allowed to tailor the analysis to the data set by specifying several quantities, including the number of state variables, the analysis variable, the embedding dimension, the embedding delay, and the algorithm. The program uses two algorithms for generating predictions, a fast, binary search tree algorithm and a slow "meat axe" approach. After the analysis is