andrew_gelman_stats andrew_gelman_stats-2010 andrew_gelman_stats-2010-346 knowledge-graph by maker-knowledge-mining
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Introduction: Mandelbrot on taxonomy (from 1955; the first publication about fractals that I know of): Searching for Mandelbrot on the blog led me to Akaike , who also recently passed away and also did interesting early work on self-similar stochastic processes. For example, this wonderful opening of his 1962 paper, “On a limiting process which asymptotically produces f^{-2} spectral density”: In the recent papers in which the results of the spectral analyses of roughnesses of runways or roadways are reported, the power spectral densities of approximately the form f^{-2} (f: frequency) are often treated. This fact directed the present author to the investigation of the limiting process which will provide the f^{-2} form under fairly general assumptions. In this paper a very simple model is given which explains a way how the f^{-2} form is obtained asymptotically. Our fundamental model is that the stochastic process, which might be considered to represent the roughness of the runway
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1 Mandelbrot on taxonomy (from 1955; the first publication about fractals that I know of): Searching for Mandelbrot on the blog led me to Akaike , who also recently passed away and also did interesting early work on self-similar stochastic processes. [sent-1, score-0.675]
2 This fact directed the present author to the investigation of the limiting process which will provide the f^{-2} form under fairly general assumptions. [sent-3, score-0.898]
3 In this paper a very simple model is given which explains a way how the f^{-2} form is obtained asymptotically. [sent-4, score-0.534]
4 Our fundamental model is that the stochastic process, which might be considered to represent the roughness of the runway, is obtained by alternative repetitions of roughening and smoothing. [sent-5, score-0.946]
5 We can easily get the limiting form of the spectrum for this model. [sent-6, score-0.543]
6 Further, by taking into account the physical meaning of roughening and smoothing we can formulate the conditions under which this general result assures that the f^{-2} form will eventually take place. [sent-7, score-1.041]
7 I’ve placed this in the Multilevel Modeling category because fractals are a form of multilevel model, although not always recognized as such: fractals are hi-tech physical science models, whereas multilevel modeling is associated with low-grade fields such as education and social science. [sent-10, score-1.495]
8 The connection is clear, though, when you consider Mandelbrot’s taxonomy model or the connection between Akaike’s dynamic model of roughness to complex models of student, teacher, and classroom effects. [sent-11, score-1.085]
9 Unfortunately, at the time I didn’t recognize the general importance of multilevel models, so all I could really do in the conversation was to express my awe and appreciation of his work. [sent-13, score-0.506]
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