Extreme Value Statistics and the Lorenz Attractor
【摘要】：正 Extreme value statistics is anticipated to play an increasingly important role in systems theory research. There are two reasons behind this; one is that the number of actuators and sensors in systems envisioned today, e.g. micro-electromechanical arrays and nanotechnology, biomimetic control, molecular computing, etc. will contain hundreds of thousands of actuators and sensors, and the other is the increasing attention paid by systems theorists to natural and environmental systems in which one needs to forecast extremes in wind speed, temperature, rain fall etc. for safe and reliable design. Theory of extreme value statistics is based upon several rather stringent assumptions. An open question is whether physical models such as wind, rainfall etc. satisfy such assumptions. Since there does not appear to be any way to answer this question is a direct way, here we attempt to device an indirect, and somewhat anecdotal, way to test the hypothesis that one may make long term forecasts from medium term data series. We choose the Lorenz system as a simple model to test the predictive capability of extreme value statistics. This choice is motivated by the fact that the Lorenz model is often quoted as an example of a system that has the same qualitative characteristics as the weather patterns. Our main conclusion is that one must be a bit circumspect in using this methodology to make long term forecasts.