| Summary: | 碩士 === 國立清華大學 === 物理系 === 104 === Power laws have been reported in more than one-hundred instances of natural and man-made complex systems for its scale-free characteristics, which phenomenon sets off many researches, such as the frequency of occurrence of earthquake in geology, the stock market fluctuations in economics, population structure, and word count for Latin languages. In parallel to these empirical findings, Bak et al. in 1970 proposed the sandpile model and self-organized criticality, and Mandelbrot presented fractal interpretation. They all tried to construct a theoretical model in hope of explaining this popular phenomenon in a physicist-loving unified way.
In this thesis, we visit this problem from a different angle, namely, focusing on the statistical rigor to claim the power law. By adopting and proposing the Akaike information criterion (AIC), we examined several famous examples of power law and found their data are better described by double power laws (DPL). To lend more support to the legitimacy and powerfulness of AIC, we measured the acoustic emission of crumpling experiment in-depth. AIC was able to pick out DPL correctly for two different sheets crumpled separately. When the sheets are crumpled jointly, AIC indicates the statistics transits from favoring DPL to simple power law as the compaction increases. This is in support of the expectation that interactions can give rise to such a shift of statistical behavior.
In the second half of this thesis, armed with the AIC and accompanying experience on data analysis I shifted gear to study the Zipf’s law in Chinese, English and other Latin-based languages, also known as word count morphing. And after further generalization to music and graphs, we find that the morph of statistical distribution is common and depends on the specific method used for data analysis. Since the power law is beautifully simple in mathematics with its scale-free and self-similar feature, it is used as our primary subject to quantify different methods.
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