摘要
文獻報導揉皺(Crumpling)噪音的分佈是個完美的SPL(Simple Power Law)，但分析數據的方法採用最小平方法(Least squares Method)，以殘差平方(Residual Sum of Squares)最小以及擬合參數的誤差大小，來當作判斷像不像SPL的依據；我們發現這個方法有統計上的缺失，會使得原本參數誤差很小的 SPL 的兩個分佈加起來後，變成「誤差更小」的SPL分佈。因此我們主張改採更嚴謹的統計方法:AIC(Akaike information criterion)，來判斷哪個分佈才是最佳的模型。
AIC證實單張用擰的方式所發出的噪音的確滿足SPL，但是如果改採揉皺方式,發現分佈變成Zipf-Mandelbrot distribution (ZMD)。我們利用不同層數的遮蔽率實驗證實並說明兩者差別的來源。
對於兩種不同材料的混合揉皺，我們發現剛開始是DPL (Double Power Law)，晚期會轉變為SPL。如果把（當單獨揉皺時）具備不同冪次的兩個薄膜比類成雙機制的複雜系統，我們的實驗結論支持一般人的認知和一篇PNAS論文的模型預測[1]：亦即加強系統內部的交互作用，可能導致統計行為的改變，並傾向演化成簡單的SPL。在揉皺的例子裡，隨著密度的增加，不同薄膜之間的交互作用顯然逐漸增強，晚期之所以呈現SPL，可以理解為典型的單張（「複合」）薄膜行為。

It has been reported that the distribution of crumpling noise obeys simple power law (SPL). But the analyses were baded on the Least Squares method which judges the goodness of fit by minimizing the residual sum of squares and error. We show that this method is flawed and may cause delusion when judging whether a set of data obeys SPL. By bridging the gap with mathematicians, we introduce a more rigorous statistical method: AIC (Akaike information criterion).
By use of AIC, we found the crumpling sound in fact obeys the Zipf-Mandelbrot distribution (ZMD), instead of SPL when the thin sheet is wringed. The delicacy of this difference is explained by a designed experiment and the attenuation effect.
To both test the robustness and SPL and simulate a system with more than one intrinsic mechanism, we also arranged for two different sheets to be crumpled together. By sorting the data according to different time order, we observed a subtle transition from Double power law (DPL) in the early phase to SPL when the crumpled ball is more compact. This result is consistent with the general belief and the prediction of a PNAS article [1] that enhanced interactions are capable of causing such a transition in statistical behavior.

中文摘要 1
英文摘要 2
第一章.緒論 3
第二章.實驗 6
2.1. 揉皺實驗 6
2.1.1 揉皺噪音實驗 6
2.1.2 遮蔽率測量實驗 10
2.1.3 擰紙實驗 11
2.2.分析揉皺實驗數據的演算法 : 12
2.3.初步實驗結果與Δβ問題的發現 : 15
第三章.AIC (Akaike information criterion) 19
3.1.簡介: 19
3.2.似然性(Likelihood): 21
3.3. 最大似然性(Maximum Likelihood) 22
3.4. 似然比檢驗 (Likelihood Ratio Test)[7] 26
第四章.實驗結果分析 27
4.1.單張揉皺 27
4.1.1 .擬合結果分析 27
4.1.2 Xmin的選擇與ZMD中γ的來源 32
4.1.3 擰材料實驗與材料之間摩擦的重要性評估 36
4.2.混合揉皺 38
五.結論和討論 41
附錄A: 理論模型建立 42
附錄B：Δβ問題的數學推導 45
參考資料. 48

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[2] William J. Reed and Kevin S. McKelvey“Power-law behaviour and parametric models for the size-distribution of forest fires”, Ecological Modelling.05/2002; DOI: 10.1016/S0304-3800(01)00483-5

[3] B. Gutenberg and C. F. Richter, “Frequency of earthquakes in California”, Bull. Seismol. Soc. Amer. 34, 185 (1944).

[4] M. E. J. Newman, “Power laws, Pareto distributions and Zipf's law”, Contemporary Physics 46, 323 (2005).

[5] P. Bak, C. Tang, and K. Wiesenfeld, “Self-organized criticality: an explanation of 1/f noise”, Phys. Rev. Lett. 59, 381 (1987); P. Bak, C. Tang, and K. Wiesenfeld, “Self- organized criticality”, Phys. Rev. A 38, 364 (1988); C. Tang and P. Bak, “Critical exponents and scaling relations for self-organized ritical phenomena”, Phys. Rev. Lett. 60, 2347 (1988).

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[8] More examples can be found in M. P. H. Stumpf and M. S. Porter, Science 335, 665 (2012).

[9] Paul A. Houle and James P. Sethna “Acoustic emission from crumpling paper”, Phys. Rev. E 54, 278 (1996).


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[13] J. deLeeuw, Introduction to Akaike (1973) Information Theory and an Extension of the Maximum Likelihood Principle, in Breakthroughs in Statistics, Volume I, edited by S. Kotz and N. L. Johenson (Springer, 1992), pp 599- 609.


[14] S. S. Wilks,“The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses”, The Annals of Mathematical Statistics 9, 60 (1938).