cs249 lecture notes, Fall 2001 Week 14, Monday Quiz Thursday on matrix operations. Measuring fractals ------------------ With non-fractals, length converges on a limit as the resolution of the measurement increases. As scale gets smaller, resolution gets bigger. Define resolution = 1/scale. If you plot log(length) vs. log(resolution) you get a flat line. With (true, mathematical) fractals, the length increases constantly as the scale gets smaller. For example, in a Koch curve, the ruler with scale 1 yields length 3. At scale=1/3, the length is 4. Every time we cut the scale by 1/3, we discover a new level of complexity, and the length increases by 4/3. As a sequence, we could write s = (1/3)^k and u = (4/3)^k Better to write it in terms of resolution = 1/s 1/s = 3^k Taking the log, base 3, of both equations yields log 1/s = k and log u = k log (4/3) Combining yields: log u = log(4/3) log (1/s) Which means that on a log-log scale the length as a function of resolution is a straight line with slope log (4/3), which is 0.2619 Measuring real curves --------------------- Coastlines tend to look like fractals. Can we CHARACTERIZE their behavior using the notion of length and scale? Empirically, we can measure maps with increasing resolution. Plotting the results on a log-log scale provides a natural statistical test for fractal behavior. Eventually, of course, this behavior breaks down in the real world. But over a wide range of scales, real coastlines appear fractal. Furthermore, we can estimate the parameter d, in the relationship log u = d log (1/s) + b The intercept, b, has no physical meaning. It's the length when measured with a compass that has unit length. So it depends on the units you choose. But d doesn't depend on units, or even on the base of the logarithm. It's the FRACTAL DIMENSION, and it describes how fast the length grows as resolution increases. Raising both sides of the equation to a power, we get u = c (1/s)^d In other words, d is a measure of cragginess! 1) a circle has zero cragginess 2) the Koch curve is comparable to real coastlines 3) Britain gets 0.36, Maine is higher, California is lower Interesting comment in the reading about Spain and Portugal. How is this related to SOC? --------------------------- It's mostly not, except that we got some good info about log-log curves. In the one-dimensional sand model, the distribution of topple-sizes is Gaussian. In the two-dimensional model, the behavior is qualitatively different. There are many more small topples, and a small number of very large topples. There is a theory that suggests that the distribution of topple sizes is Pareto, which means that the number of topples of size t is proportional to (1/t)^a N(t) ~= (1/t)^a Taking the log of both sides yields log n = a log t + c So if we generate a histogram of topple-sizes, and plot it on a log-log scale, we should get something that looks like a straight line, and the slope estimates the coefficient a. Why is this an interesting model? --------------------------------- Empirically, we have seen that earthquakes and some other natural phenomena obey the same power law. Cellular automaton models suggest two, somewhat suprising, results 1) an explanatory model of power laws (good thing) 2) a dent in the notion that events have causes (bad thing?) Like chaos, SOC undermines what seemed to be an obvious truth about cause and effect. We would like to be able to predict earthquakes. How does the SOC model affect our approach to this problem?